# nLab geometry of physics -- perturbative quantum field theory

Contents

These notes mean to give an expository but rigorous introduction to the basic concepts of relativistic perturbative quantum field theories, specifically those that arise as the perturbative quantization of Lagrangian field theories – such as quantum electrodynamics, quantum chromodynamics, and perturbative quantum gravity appearing in the standard model of particle physics.

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This is one chapter of geometry of physics.

Previous chapters: smooth sets, supergeometry.

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# Contents

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For broad introduction of the idea of the topic of perturbative quantum field theory see there and see

Here, first we consider classical field theory (or rather pre-quantum field theory), complete with BV-BRST formalism; then its deformation quantization via causal perturbation theory to perturbative quantum field theory. This mathematically rigorous (i.e. clear and precise) formulation of the traditional informal lore has come to be known as perturbative algebraic quantum field theory.

We aim to give a fully local discussion, where all structures arise on the “jet bundle over the field bundle” (introduced below) and “transgress” from there to the spaces of field histories over spacetime (discussed further below). This “Higher Prequantum Geometry” streamlines traditional constructions and serves the conceptualization in the theory. This is joint work with Igor Khavkine.

In full beauty these concepts are extremely general and powerful; but the aim here is to give a first precise idea of the subject, not a fully general account. Therefore we concentrate on the special case where spacetime is Minkowski spacetime (def. below), where the field bundle (def. below) is an ordinary trivial vector bundle (example below) and hence the Lagrangian density (def. below) is globally defined. Similarly, when considering gauge theory we consider just the special case that the gauge parameter-bundle is a trivial vector bundle and we concentrate on the case that the gauge symmetries are “closed irreducible” (def. below). But we aim to organize all concepts such that the structure of their generalization to curved spacetime and non-trivial field bundles is immediate.

This comparatively simple setup already subsumes what is considered in traditional texts on the subject; it captures the established perturbative BRST-BV quantization of gauge fields coupled to fermions on curved spacetimes – which is the state of the art. Further generalization, necessary for the discussion of global topological effects, such as instanton configurations of gauge fields, will be discussed elsewhere (see at homotopical algebraic quantum field theory).

Alongside the theory we develop the concrete examples of the real scalar field, the electromagnetic field and the Dirac field; eventually combining these to a disussion of quantum electrodynamics.

running examples

fieldfield bundleLagrangian densityequation of motion
real scalar fieldexpl. expl. expl.
Dirac fieldexpl. expl. expl.
electromagnetic fieldexpl. expl. expl.
Yang-Mills fieldexpl. ,
expl.
expl. expl.
B-fieldexpl. expl expl.

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fieldPoisson bracketcausal propagatorWightman propagatorFeynman propagator
real scalar fieldexpl. ,
expl.
prop. def. def.
Dirac fieldexpl. ,
expl.
prop. def. def.
electromagnetic fieldprop. prop.

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fieldgauge symmetrylocal BRST complexgauge fixing
electromagnetic fieldexpl. expl. expl.
Yang-Mills fieldexpl. expl.
B-fieldexpl. expl.

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interacting field theoryinteraction Lagrangian densityinteraction Wick algebra-element
phi^n theoryexp. expl.
quantum electrodynamicsexpl. expl.

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References

Pointers to the literature are given in each chapter, alongside the text. The following is a selection of these references.

The discussion of spinors in chapter 2. Spacetime follows Baez-Huerta 09.

For the jet bundle-formulation of variational calculus of Lagrangian field theory in 4. Field variations, and 5. Lagrangians we follow Anderson 89 and Olver 86; for 6. Symmetries augmented by Fiorenza-Rogers-Schreiber 13b.

The identification of polynomial observables with distributions in 7. Observables was observed by Paugam 12.

The discussion of the Peierls-Poisson bracket in 8. Phase space is based on Khavkine 14.

The derivation of wave front sets of propagators in 9. Propagators takes clues from Radzikowski 96 and uses results from Gelfand-Shilov 66.

For the general idea of BV-BRST formalism a good review is Henneaux 90.

The Lie algebroid-perspective on BRST complexes developed in chapter 10. Gauge symmetries, may be compared to Barnich 10.

For the local BV-BRST theory laid out in chapter 11. Reduced phase space we are following Barnich-Brandt-Henneaux 00.

For the BV-gauge fixing developed in 12. Gauge fixing we take clues from Fredenhagen-Rejzner 11a.

For the free quantum BV-operators in 13. Free quantum fields and the interacting quantum master equation in 15. Interacting quantum fields we are following Fredenhagen-Rejzner 11b, Rejzner 11, which in turn is taking clues from Hollands 07.

The discussion of quantization in 13. Quantization takes clues from Hawkins 04, Collini 16 and spells out the derivation of the Moyal star product from geometric quantization of symplectic groupoids due to Gracia-Bondia & Varilly 94.

The perspective on the Wick algebra in 14. Free quantum fields goes back to Dito 90 and was revived for pAQFT in Dütsch-Fredenhagen 00. The proof of the folklore result that the perturbative Hadamard vacuum state on the Wick algebra is indeed a state is cited from Dütsch 18.

The discussion of causal perturbation theory in 15. Interacting quantum fields follows the original Epstein-Glaser 73. The relevance here of the star product induced by the Feynman propagator was highlighted in Fredenhagen-Rejzner 12. The proof that the interacting field algebra of observables defined by Bogoliubov's formula is a causally local net in the sense of the Haag-Kastler axioms is that of Brunetti-Fredenhagen 00.

Our derivation of Feynman diagrammatics follows Keller 10, chapter IV, our derivation of the quantum master equation follows Rejzner 11, section 5.1.3, and our discussion of Ward identities is informed by Dütsch 18, chapter 4.

In chapter 16. Renormalization we take from Brunetti-Fredenhagen 00 the perspective of Epstein-Glaser renormalization via extension of distributions and from Brunetti-Dütsch-Fredenhagen 09 and Dütsch 10 the rigorous formulation of Gell-Mann Low renormalization group flow, UV-regularization, effective quantum field theory and Polchinski's flow equation.

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Acknowledgement

These notes profited greatly from discussions with Igor Khavkine and Michael Dütsch.

Thanks also to Marco Benini, Klaus Fredenhagen, Arnold Neumaier and Kasia Rejzner for helpful discussion.

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## Geometry

The geometry of physics is differential geometry. This is the flavor of geometry which is modeled on Cartesian spaces $\mathbb{R}^n$ with smooth functions between them. Here we briefly review the basics of differential geometry on Cartesian spaces.

In principle the only background assumed of the reader here is

1. usual naive set theory (e.g. Lawvere-Rosebrugh 03);

2. the concept of the continuum: the real line $\mathbb{R}$, the plane $\mathbb{R}^2$, etc.

3. the concepts of differentiation and integration of functions on such Cartesian spaces;

hence essentially the content of multi-variable differential calculus.

We now discuss:

As we uncover Lagrangian field theory further below, we discover ever more general concepts of “space” in differential geometry, such as smooth manifolds, diffeological spaces, infinitesimal neighbourhoods, supermanifolds, Lie algebroids and super Lie ∞-algebroids. We introduce these incrementally as we go along:

more general spaces in differential geometry introduced further below

higher differential geometry
differential geometrysmooth manifolds
(def. )
$\hookrightarrow$diffeological spaces
(def. )
$\hookrightarrow$smooth sets
(def. )
$\hookrightarrow$formal smooth sets
(def. )
$\hookrightarrow$super formal smooth sets
(def. )
$\hookrightarrow$super formal smooth ∞-groupoids
(not needed in fully perturbative QFT)
infinitesimal geometry,
Lie theory
infinitesimally thickened points
(def. )
superpoints
(def. )
Lie ∞-algebroids
(def. )
higher Lie theory
needed in QFT for:spacetime (def. )space of field histories
(def. )
Cauchy surface (def. ),
perturbation theory (def. )
Dirac field (expl. ), Pauli exclusion principleinfinitesimal gauge symmetry/BRST complex
(expl. )

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Abstract coordinate systems

What characterizes differential geometry is that it models geometry on the continuum, namely the real line $\mathbb{R}$, together with its Cartesian products $\mathbb{R}^n$, regarded with its canonical smooth structure (def. below). We may think of these Cartesian spaces $\mathbb{R}^n$ as the “abstract coordinate systems” and of the smooth functions between them as the “abstract coordinate transformations”.

We will eventually consider below much more general “smooth spaces$X$ than just the Cartesian spaces $\mathbb{R}^n$; but all of them are going to be understood by “laying out abstract coordinate systems” inside them, in the general sense of having smooth functions $f \colon \mathbb{R}^n \to X$ mapping a Cartesian space smoothly into them. All structure on generalized smooth spaces $X$ is thereby reduced to compatible systems of structures on just Cartesian spaces, one for each smooth “probe” $f\colon \mathbb{R}^n \to X$. This is called “functorial geometry”.

Notice that the popular concept of a smooth manifold (def./prop. below) is essentially that of a smooth space which locally looks just like a Cartesian space, in that there exist sufficiently many $f \colon \mathbb{R}^n \to X$ which are (open) isomorphisms onto their images. Historically it was a long process to arrive at the insight that it is wrong to fix such local coordinate identifications $f$, or to have any structure depend on such a choice. But it is useful to go one step further:

In functorial geometry we do not even focus attention on those $f \colon \mathbb{R}^n \to X$ that are isomorphisms onto their image, but consider all “probes” of $X$ by “abstract coordinate systems”. This makes differential geometry both simpler as well as more powerful. The analogous insight for algebraic geometry is due to Grothendieck 65; it was transported to differential geometry by Lawvere 67.

This allows to combine the best of two superficially disjoint worlds: On the one hand we may reduce all constructions and computations to coordinates, the way traditionally done in the physics literature; on the other hand we have full conceptual control over the coordinate-free generalized spaces analyzed thereby. What makes this work is that all coordinate-constructions are functorially considered over all abstract coordinate systems.

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###### Definition

(Cartesian spaces and smooth functions between them)

For $n \in \mathbb{N}$ we say that the set $\mathbb{R}^n$ of n-tuples of real numbers is a Cartesian space. This comes with the canonical coordinate functions

$x^k \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}$

which send an n-tuple of real numbers to the $k$th element in the tuple, for $k \in \{1, \cdots, n\}$.

For

$f \;\colon\; \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n'}$

any function between Cartesian spaces, we may ask whether its partial derivative along the $k$th coordinate exists, denoted

$\frac{\partial f}{\partial x^k} \;\colon\; \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n'} \,.$

If this exists, we may in turn ask that the partial derivative of the partial derivative exists

$\frac{\partial^2 f}{\partial x^{k_1} \partial x^{k_2}} \coloneqq \frac{\partial}{\partial x^{k_2}} \frac{\partial f}{\partial x^{k_1}}$

and so on.

A general higher partial derivative obtained this way is, if it exists, indexed by an n-tuple of natural numbers $\alpha \in \mathbb{N}^n$ and denoted

(1)$\partial_\alpha \;\coloneqq\; \frac{ \partial^{\vert \alpha \vert} f }{ \partial^{\alpha_1} x^1 \partial^{\alpha_2} x^2 \cdots \partial^{\alpha_n} x^n } \,,$

where ${\vert \alpha\vert} \coloneqq \underoverset{n}{i = 1}{\sum} \alpha_i$ is the total order of the partial derivative.

If all partial derivative to all orders $\alpha \in \mathbb{N}^n$ of a function $f \colon \mathbb{R}^n \to \mathbb{R}^{n'}$ exist, then $f$ is called a smooth function.

Of course the composition $g \circ f$ of two smooth functions is again a smooth function.

$\array{ && \mathbb{R}^{n_2} \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ \mathbb{R}^{n_1} && \underset{g \circ f}{\longrightarrow} && \mathbb{R}^{n_3} } \,.$

The inclined reader may notice that this means that Cartesian spaces with smooth functions between them constitute a category (“CartSp”); but the reader not so inclined may ignore this.

For the following it is useful to think of each Cartesian space as an abstract coordinate system. We will be dealing with various generalized smooth spaces (see the table below), but they will all be characterized by a prescription for how to smoothly map abstract coordinate systems into them.

###### Example

(coordinate functions are smooth functions)

Given a Cartesian space $\mathbb{R}^n$, then all its coordinate functions (def. )

$x^k \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}$

are smooth functions (def. ).

For

$f \colon \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$

any smooth function and $a \in \{1, 2, \cdots, n_2\}$ write

$f^a \coloneqq x^a \circ f \;\colon\; \mathbb{R}^{n_1} \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{x^a}{\longrightarrow} \mathbb{R}$

. for its composition with this coordinate function.

###### Example

(algebra of smooth functions on Cartesian spaces)

For each $n \in \mathbb{N}$, the set

$C^\infty(\mathbb{R}^n) \;\coloneqq\; Hom_{CartSp}(\mathbb{R}^n, \mathbb{R})$

of real number-valued smooth functions $f \colon \mathbb{R}^n \to \mathbb{R}$ on the $n$-dimensional Cartesian space (def. ) becomes a commutative associative algebra over the ring of real numbers by pointwise addition and multiplication in $\mathbb{R}$: for $f,g \in C^\infty(\mathbb{R}^n)$ and $x \in \mathbb{R}^n$

1. $(f + g)(x) \coloneqq f(x) + g(x)$

2. $(f \cdot g)(x) \coloneqq f(x) \cdot g(x)$.

The inclusion

$\mathbb{R} \overset{const}{\hookrightarrow} C^\infty(\mathbb{R}^n)$

is given by the constant functions.

We call this the real algebra of smooth functions on $\mathbb{R}^n$:

$C^\infty(\mathbb{R}^n) \;\in\; \mathbb{R} Alg \,.$

If

$f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$

is any smooth function (def. ) then pre-composition with $f$ (“pullback of functions”)

$\array{ C^\infty(\mathbb{R}^{n_2}) &\overset{f^\ast}{\longrightarrow}& C^\infty(\mathbb{R}^{n_1}) \\ g &\mapsto& f^\ast g \coloneqq g \circ f }$

is an algebra homomorphism. Moreover, this is clearly compatible with composition in that

$f_1^\ast(f_2^\ast g) = (f_2 \circ f_1)^\ast g \,.$

Stated more abstractly, this means that assigning algebras of smooth functions is a functor

$C^\infty(-) \;\colon\; CartSp \longrightarrow \mathbb{R} Alg^{op}$

from the category CartSp of Cartesian spaces and smooth functions between them (def. ), to the opposite of the category $\mathbb{R}$Alg of $\mathbb{R}$-algebras.

###### Definition

(local diffeomorphisms and open embeddings of Cartesian spaces)

A smooth function $f \colon \mathbb{R}^{n} \to \mathbb{R}^{n}$ from one Cartesian space to itself (def. ) is called a local diffeomorphism, denoted

$f \;\colon\; \mathbb{R}^{n} \overset{et}{\longrightarrow} \mathbb{R}^n$

if the determinant of the matrix of partial derivatives (the “Jacobian” of $f$) is everywhere non-vanishing

$det \left( \array{ \frac{\partial f^1}{\partial x^1}(x) &\cdots& \frac{\partial f^n}{\partial x^1}(x) \\ \vdots && \vdots \\ \frac{\partial f^1}{\partial x^n}(x) &\cdots& \frac{\partial f^n}{\partial x^n}(x) } \right) \;\neq\; 0 \phantom{AAAA} \text{for all} \, x \in \mathbb{R}^n \,.$

If the function $f$ is both a local diffeomorphism, as above, as well as an injective function then we call it an open embedding, denoted

$f \;\colon\; \mathbb{R}^n \overset{\phantom{A}et\phantom{A}}{\hookrightarrow} \mathbb{R}^n \,.$
###### Definition

(good open cover of Cartesian spaces)

For $\mathbb{R}^n$ a Cartesian space (def. ), a differentiably good open cover is

• $\left\{ \mathbb{R}^n \underoverset{et}{\phantom{AA}f_i\phantom{AA}}{\hookrightarrow} \mathbb{R}^n \right\}_{i \in I}$

of open embeddings (def. )

such that the images

$U_i \coloneqq im(f_i) \subset \mathbb{R}^n$

satisfy:

1. (open cover) every point of $\mathbb{R}^n$ is contained in at least one of the $U_i$;

2. (good) all finite intersections $U_{i_1} \cap \cdots \cap U_{i_k} \subset \mathbb{R}^n$ are either empty set or themselves images of open embeddings according to def. .

The inclined reader may notice that the concept of differentiably good open covers from def. is a coverage on the category CartSp of Cartesian spaces with smooth functions between them, making it a site, but the reader not so inclined may ignore this.

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fiber bundles

Given any context of objects and morphisms between them, such as the Cartesian spaces and smooth functions from def. it is of interest to fix one object $X$ and consider other objects parameterized over it. These are called bundles (def. ) below. For reference, we briefly discuss here the basic concepts related to bundles in the context of Cartesian spaces.

Of course the theory of bundles is mostly trivial over Cartesian spaces; it gains its main interest from its generalization to more general smooth manifolds (def./prop. below). It is still worthwhile for our development to first consider the relevant concepts in this simple case first.

For more exposition see at fiber bundles in physics.

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###### Definition

(bundles)

We say that a smooth function $E \overset{fb}{\to} X$ (def. ) is a bundle just to amplify that we think of it as exhibiting $E$ as being a “space over $X$”:

$\array{ E \\ \downarrow\mathrlap{fb} \\ X } \,.$

For $x \in X$ a point, we say that the fiber of this bundle over $x$ is the pre-image

(2)$E_x \coloneqq fb^{-1}(\{x\}) \subset E$

of the point $x$ under the smooth function. We think of $fb$ as exhibiting a “smoothly varying” set of fiber spaces over $X$.

Given two bundles $E_1 \overset{fb_1}{\to} X$ and $E_2 \overset{fb_2}{\to} X$ over $X$, a homomorphism of bundles between them is a smooth function $f \colon E_1 \to E_2$ (def. ) between their total spaces which respects the bundle projections, in that

$fb_2 \circ f = fb_1 \phantom{AAAA} \text{i.e.} \phantom{AAA} \array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}} \\ && X } \,.$

Hence a bundle homomorphism is a smooth function that sends fibers to fibers over the same point:

$f\left( (E_1)_x \right) \;\subset\; (E_2)_x \,.$

The inclined reader may notice that this defines a category of bundles over $X$, which is in fact just the slice category $CartSp_{/X}$; the reader not so inclined may ignore this.

###### Definition

(sections)

Given a bundle $E \overset{fb}{\to} X$ (def. ) a section is a smooth function $s \colon X \to E$ such that

$fb \circ s = id_X \phantom{AAAAA} \array{ && E \\ & {}^{\mathllap{s}}\nearrow & \downarrow\mathrlap{fb} \\ X &=& X } \,.$

This means that $s$ sends every point $x \in X$ to an element in the fiber over that point

$s(x) \in E_x \,.$

We write

$\Gamma_X(E) \coloneqq \left\{ \array{ && E \\ & {}^{\mathllap{s}}\nearrow & \downarrow^\mathrlap{fb} \\ X &=& X } \phantom{fb} \right\}$

for the set of sections of a bundle.

For $E_1 \overset{f_1}{\to} X$ and $E_2 \overset{f_2}{\to} X$ two bundles and for

$\array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}} \\ && X }$

a bundle homomorphism between them (def. ), then composition with $f$ sends sections to sections and hence yields a function denoted

$\array{ \Gamma_X(E_1) &\overset{f_\ast}{\longrightarrow}& \Gamma_X(E_2) \\ s &\mapsto& f \circ s } \,.$
###### Example

(trivial bundle)

For $X$ and $F$ Cartesian spaces, then the Cartesian product $X \times F$ equipped with the projection

$\array{ X \times F \\ \downarrow^\mathrlap{pr_1} \\ X }$

to $X$ is a bundle (def. ), called the trivial bundle with fiber $F$. This represents the constant smoothly varying set of fibers, constant on $F$

If $F = \ast$ is the point, then this is the identity bundle

$\array{ X \\ \downarrow\mathrlap{id} \\ X } \,.$

Given any bundle $E \overset{fb}{\to} X$, then a bundle homomorphism (def. ) from the identity bundle to $E \overset{fb}{\to} X$ is equivalently a section of $E \overset{fb}{\to} X$ (def. )

$\array{ X && \overset{s}{\longrightarrow} && E \\ & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{fb}} \\ && X }$
###### Definition

(fiber bundle)

A bundle $E \overset{fb}{\to} X$ (def. ) is called a fiber bundle with typical fiber $F$ if there exists a differentiably good open cover $\{U_i \hookrightarrow X\}_{i \in I}$ (def. ) such that the restriction of $fb$ to each $U_i$ is isomorphic to the trivial fiber bundle with fiber $F$ over $U_i$. Such diffeomorphisms $f_i \colon U_i \times F \overset{\simeq}{\to} E\vert_{U_i}$ are called local trivializations of the fiber bundle:

$\array{ U_i \times F &\underoverset{\simeq}{f_i}{\longrightarrow}& E\vert_{U_i} \\ & {}_{\mathllap{pr_1}}\searrow & \downarrow\mathrlap{fb\vert_{U_i}} \\ && U_i } \,.$
###### Definition

(vector bundle)

A vector bundle is a fiber bundle $E \overset{vb}{\to} X$ (def. ) with typical fiber a vector space $V$ such that there exists a local trivialization $\{U_i \times V \underoverset{\simeq}{f_i}{\to} E\vert_{U_i}\}_{i \in I}$ whose gluing functions

$U_i \cap U_j \times V \overset{f_i\vert_{U_i \cap U_j}}{\longrightarrow} E\vert_{U_i \cap U_j} \overset{f_j^{-1}\vert_{U_i \cap U_j}}{\longrightarrow} U_i \cap U_j \times V$

for all $i,j \in I$ are linear functions over each point $x \in U_i \cap U_j$.

A homomorphism of vector bundle is a bundle morphism $f$ (def. ) such that there exist local trivializations on both sides with respect to which $g$ is fiber-wise a linear map.

The inclined reader may notice that this makes vector bundles over $X$ a category (denoted $Vect_{/X}$); the reader not so inclined may ignore this.

###### Example

(module of sections of a vector bundle)

Given a vector bundle $E \overset{vb}{\to} X$ (def. ), then its set of sections $\Gamma_X(E)$ (def. ) becomes a real vector space by fiber-wise multiplication with real numbers. Moreover, it becomes a module over the algebra of smooth functions $C^\infty(X)$ (example ) by the same fiber-wise multiplication:

$\array{ C^\infty(X) \otimes_{\mathbb{R}} \Gamma_X(E) &\longrightarrow& \Gamma_X(E) \\ (f,s) &\mapsto& (x \mapsto f(x) \cdot s(x)) } \,.$

For $E_1 \overset{fb_1}{\to} X$ and $E_2 \overset{fb_2}{\to} X$ two vector bundles and

$\array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}} \\ && X }$

a vector bundle homomorphism (def. ) then the induced function on sections (def. )

$f_\ast \;\colon\; \Gamma_X(E_1) \longrightarrow \Gamma_X(E_2)$

is compatible with this action by smooth functions and hence constitutes a homomorphism of $C^\infty(X)$-modules.

The inclined reader may notice that this means that taking spaces of sections yields a functor

$\Gamma_X(-) \;\colon\; Vect_{/X} \longrightarrow C^\infty(X) Mod$

from the category of vector bundles over $X$ to that over modules over $C^\infty(X)$.

###### Example

(tangent vector fields and tangent bundle)

For $\mathbb{R}^n$ a Cartesian space (def. ) the trivial vector bundle (example , def. )

$\array{ T \mathbb{R}^n &\coloneqq& \mathbb{R}^n \times \mathbb{R}^n \\ \mathllap{tb}\downarrow && \downarrow\mathrlap{pr_1} \\ \mathbb{R}^n &=& \mathbb{R}^n }$

is called the tangent bundle of $\mathbb{R}^n$. With $(x^a)_{a = 1}^n$ the coordinate functions on $\mathbb{R}^n$ (def. ) we write $(\partial_a)_{a = 1}^n$ for the corresponding linear basis of $\mathbb{R}^n$ regarded as a vector space. Then a general section (def. )

$\array{ && T \mathbb{R}^n \\ & {}^{\mathllap{v}}\nearrow& \downarrow\mathrlap{tb} \\ \mathbb{R}^n &=& \mathbb{R}^n }$

of the tangent bundle has a unique expansion of the form

$v = v^a \partial_a$

where a sum over indices is understood (Einstein summation convention) and where the components $(v^a \in C^\infty(\mathbb{R}^n))_{a = 1}^n$ are smooth functions on $\mathbb{R}^n$ (def. ).

Such a $v$ is also called a smooth tangent vector field on $\mathbb{R}^n$.

Each tangent vector field $v$ on $\mathbb{R}^n$ determines a partial derivative on smooth functions

$\array{ C^\infty(\mathbb{R}^n) &\overset{D_v}{\longrightarrow}& C^\infty(\mathbb{R}^n) \\ f &\mapsto& \mathrlap{ D_v f \coloneqq v^a \partial_a (f) \coloneqq \sum_a v^a \frac{\partial f}{\partial x^a} } } \,.$

By the product law of differentiation, this is a derivation on the algebra of smooth functions (example ) in that

1. it is an $\mathbb{R}$-linear map in that

$D_v( c_1 f_1 + c_2 f_2 ) = c_1 D_v f_1 + c_2 D_v f_2$
2. it satisfies the Leibniz rule

$D_v(f_1 \cdot f_2) = (D_v f_1) \cdot f_2 + f_1 \cdot (D_v f_2)$

for all $c_1, c_2 \in \mathbb{R}$ and all $f_1, f_2 \in C^\infty(\mathbb{R}^n)$.

Hence regarding tangent vector fields as partial derivatives constitutes a linear function

$D \;\colon\; \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n) \longrightarrow Der(C^\infty(\mathbb{R}^n))$

from the space of sections of the tangent bundle. In fact this is a homomorphism of $C^\infty(\mathbb{R}^n)$-modules (example ), in that for $f \in C^\infty(\mathbb{R}^n)$ and $v \in \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)$ we have

$D_{f v}(-) = f \cdot D_v(-) \,.$
###### Example

(vertical tangent bundle)

Let $E \overset{fb}{\to} \Sigma$ be a fiber bundle. Then its vertical tangent bundle $T_\Sigma E \overset{T fb}{\to} \Sigma$ is the fiber bundle (def. ) over $\Sigma$ whose fiber over a point is the tangent bundle (def. ) of the fiber of $E \overset{fb}{\to}\Sigma$ over that point:

$(T_\Sigma E)_x \coloneqq T(E_x) \,.$

If $E \simeq \Sigma \times F$ is a trivial fiber bundle with fiber $F$, then its vertical vector bundle is the trivial fiber bundle with fiber $T F$.

###### Definition

(dual vector bundle)

For $E \overset{vb}{\to} \Sigma$ a vector bundle (def. ), its dual vector bundle is the vector bundle whose fiber (2) over $x \in \Sigma$ is the dual vector space of the corresponding fiber of $E \to \Sigma$:

$(E^\ast)_x \;\coloneqq\; (E_x)^\ast \,.$

The defining pairing of dual vector spaces $(E_x)^\ast \otimes E_x \to \mathbb{R}$ applied pointwise induces a pairing on the modules of sections (def. ) of the original vector bundle and its dual with values in the smooth functions (def. ):

(3)$\array{ \Gamma_\Sigma(E) \otimes_{C^\infty(X)} \Gamma_\Sigma(E^\ast) &\longrightarrow& C^\infty(\Sigma) \\ (v,\alpha) &\mapsto& (v \cdot \alpha \colon x \mapsto \alpha_x(v_x) ) }$

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synthetic differential geometry

Below we encounter generalizations of ordinary differential geometry that include explicit “infinitesimals” in the guise of infinitesimally thickened points, as well as “super-graded infinitesimals”, in the guise of superpoints (necessary for the description of fermion fields such as the Dirac field). As we discuss below, these structures are naturally incorporated into differential geometry in just the same way as Grothendieck introduced them into algebraic geometry (in the guise of “formal schemes”), namely in terms of formally dual rings of functions with nilpotent ideals. That this also works well for differential geometry rests on the following three basic but important properties, which say that smooth functions behave “more algebraically” than their definition might superficially suggest:

###### Proposition

(the three magic algebraic properties of differential geometry)

1. embedding of Cartesian spaces into formal duals of R-algebras

For $X$ and $Y$ two Cartesian spaces, the smooth functions $f \colon X \longrightarrow Y$ between them (def. ) are in natural bijection with their induced algebra homomorphisms $C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y)$ (example ), so that one may equivalently handle Cartesian spaces entirely via their $\mathbb{R}$-algebras of smooth functions.

Stated more abstractly, this means equivalently that the functor $C^\infty(-)$ that sends a smooth manifold $X$ to its $\mathbb{R}$-algebra $C^\infty(X)$ of smooth functions (example ) is a fully faithful functor:

$C^\infty(-) \;\colon\; SmthMfd \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R} Alg^{op} \,.$
2. embedding of smooth vector bundles into formal duals of R-algebra modules

For $E_1 \overset{vb_1}{\to} X$ and $E_2 \overset{vb_2}{\to} X$ two vector bundle (def. ) there is then a natural bijection between vector bundle homomorphisms $f \colon E_1 \to E_2$ and the homomorphisms of modules $f_\ast \;\colon\; \Gamma_X(E_1) \to \Gamma_X(E_2)$ that these induces between the spaces of sections (example ).

More abstractly this means that the functor $\Gamma_X(-)$ is a fully faithful functor

$\Gamma_X(-) \;\colon\; VectBund_X \overset{\phantom{AAAA}}{\hookrightarrow} C^\infty(X) Mod$

Moreover, the modules over the $\mathbb{R}$-algebra $C^\infty(X)$ of smooth functions on $X$ which arise this way as sections of smooth vector bundles over a Cartesian space $X$ are precisely the finitely generated free modules over $C^\infty(X)$.

3. For $X$ a Cartesian space (example ), then any derivation $D \colon C^\infty(X) \to C^\infty(X)$ on the $\mathbb{R}$-algebra $C^\infty(X)$ of smooth functions (example ) is given by differentiation with respect to a uniquely defined smooth tangent vector field: The function that regards tangent vector fields with derivations from example

$\array{ \Gamma_X(T X) &\overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}& Der(C^\infty(X)) \\ v &\mapsto& D_v }$

is in fact an isomorphism.

(This follows directly from the Hadamard lemma.)

Actually all three statements in prop. hold not just for Cartesian spaces, but generally for smooth manifolds (def./prop. below; if only we generalize in the second statement from free modules to projective modules. However for our development here it is useful to first focus on just Cartesian spaces and then bootstrap the theory of smooth manifolds and much more from that, which we do below.

$\,$

$\,$

differential forms

We introduce and discuss differential forms on Cartesian spaces.

###### Definition

(differential 1-forms on Cartesian spaces and the cotangent bundle)

For $n \in \mathbb{N}$ a smooth differential 1-form $\omega$ on a Cartesian space $\mathbb{R}^n$ (def. ) is an n-tuple

$\left(\omega_i \in CartSp\left(\mathbb{R}^n,\mathbb{R}\right)\right)_{i = 1}^n$

of smooth functions (def. ), which we think of equivalently as the coefficients of a formal linear combination

$\omega = \omega_i d x^i$

on a set $\{d x^1, d x^2, \cdots, d x^n\}$ of cardinality $n$.

Here a sum over repeated indices is tacitly understood (Einstein summation convention).

Write

$\Omega^1(\mathbb{R}^k) \simeq CartSp(\mathbb{R}^k, \mathbb{R})^{\times k}\in Set$

for the set of smooth differential 1-forms on $\mathbb{R}^k$.

We may think of the expressions $(d x^a)_{a = 1}^n$ as a linear basis for the dual vector space $\mathbb{R}^n$. With this the differential 1-forms are equivalently the sections (def. ) of the trivial vector bundle (example , def. )

$\array{ T^\ast \mathbb{R}^n &\coloneqq& \mathbb{R}^n \times (\mathbb{R}^n)^\ast \\ \mathllap{cb}\downarrow && \downarrow\mathrlap{pr_1} \\ \mathbb{R}^n &=& \mathbb{R}^n }$

called the cotangent bundle of $\mathbb{R}^n$ (def. ):

$\Omega^1(\mathbb{R}^n) = \Gamma_{\mathbb{R}^n}(T^\ast \mathbb{R}^n) \,.$

This amplifies via example that $\Omega^1(\mathbb{R}^n)$ has the structure of a module over the algebra of smooth functions $C^\infty(\mathbb{R}^n)$, by the evident multiplication of differential 1-forms with smooth functions:

1. The set $\Omega^1(\mathbb{R}^k)$ of differential 1-forms in a Cartesian space (def. ) is naturally an abelian group with addition given by componentwise addition

\begin{aligned} \omega + \lambda & = \omega_i d x^i + \lambda_i d x^i \\ & = (\omega_i + \lambda_i) d x^i \end{aligned} \,,
2. The abelian group $\Omega^1(\mathbb{R}^k)$ is naturally equipped with the structure of a module over the algebra of smooth functions $C^\infty(\mathbb{R}^k)$ (example ), where the action $C^\infty(\mathbb{R}^k) \times\Omega^1(\mathbb{R}^k) \to \Omega^1(\mathbb{R}^k)$ is given by componentwise multiplication

$f \cdot \omega = ( f \cdot \omega_i) d x^i \,.$

Accordingly there is a canonical pairing between differential 1-forms and tangent vector fields (example )

(4)$\array{ \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n) \otimes_{\mathbb{R}} \Gamma_{\mathbb{R}^n}(T \ast \mathbb{R}^n) &\overset{\iota_{(-)}(-) }{\longrightarrow}& C^\infty(\mathbb{R}^n) \\ (v,\omega) &\mapsto& \mathrlap{ \iota_v \omega \coloneqq v^a \omega_a } }$

With differential 1-forms in hand, we may collect all the first-order partial derivatives of a smooth function into a single object: the exterior derivative or de Rham differential is the $\mathbb{R}$-linear function

(5)$\array{ C^\infty(\mathbb{R}^n) &\overset{d}{\longrightarrow}& \Omega^1(\mathbb{R}^n) \\ f &\mapsto& \mathrlap{ d f \coloneqq \frac{\partial f}{ \partial x^a} d x^a } } \,.$

Under the above pairing with tangent vector fields $v$ this yields the particular partial derivative along $v$:

$\iota_v d f = D_v f = v^a \frac{\partial f}{\partial x^a} \,.$

We think of $d x^i$ as a measure for infinitesimal displacements along the $x^i$-coordinate of a Cartesian space. If we have a measure of infintesimal displacement on some $\mathbb{R}^n$ and a smooth function $f \colon \mathbb{R}^{\tilde n} \to \mathbb{R}^n$, then this induces a measure for infinitesimal displacement on $\mathbb{R}^{\tilde n}$ by sending whatever happens there first with $f$ to $\mathbb{R}^n$ and then applying the given measure there. This is captured by the following definition:

###### Definition

(pullback of differential 1-forms)

For $\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k$ a smooth function, the pullback of differential 1-forms along $\phi$ is the function

$\phi^* \colon \Omega^1(\mathbb{R}^{k}) \to \Omega^1(\mathbb{R}^{\tilde k})$

between sets of differential 1-forms, def. , which is defined on basis-elements by

$\phi^* d x^i \;\coloneqq\; \frac{\partial \phi^i}{\partial \tilde x^j} d \tilde x^j$

and then extended linearly by

\begin{aligned} \phi^* \omega & = \phi^* \left( \omega_i d x^i \right) \\ & \coloneqq \left(\phi^* \omega\right)_i \frac{\partial \phi^i }{\partial \tilde x^j} d \tilde x^j \\ & = (\omega_i \circ \phi) \cdot \frac{\partial \phi^i }{\partial \tilde x^j} d \tilde x^j \end{aligned} \,.

This is compatible with identity morphisms and composition in that

(6)$(id_{\mathbb{R}^n})^\ast = id_{\Omega^1(\mathbb{R}^n)} \phantom{AAAA} (g \circ f)^\ast = f^\ast \circ g^\ast \,.$

Stated more abstractly, this just means that pullback of differential 1-forms makes the assignment of sets of differential 1-forms to Cartesian spaces a contravariant functor

$\Omega^1(-) \;\colon\; CartSp^{op} \longrightarrow Set \,.$

The following definition captures the idea that if $d x^i$ is a measure for displacement along the $x^i$-coordinate, and $d x^j$ a measure for displacement along the $x^j$ coordinate, then there should be a way to get a measure, to be called $d x^i \wedge d x^j$, for infinitesimal surfaces (squares) in the $x^i$-$x^j$-plane. And this should keep track of the orientation of these squares, with

$d x^j \wedge d x^i = - d x^i \wedge d x^j$

being the same infinitesimal measure with orientation reversed.

###### Definition

(exterior algebra of differential n-forms)

For $k,n \in \mathbb{N}$, the smooth differential forms on a Cartesian space $\mathbb{R}^k$ (def. ) is the exterior algebra

$\Omega^\bullet(\mathbb{R}^k) \coloneqq \wedge^\bullet_{C^\infty(\mathbb{R}^k)} \Omega^1(\mathbb{R}^k)$

over the algebra of smooth functions $C^\infty(\mathbb{R}^k)$ (example ) of the module $\Omega^1(\mathbb{R}^k)$ of smooth 1-forms.

We write $\Omega^n(\mathbb{R}^k)$ for the sub-module of degree $n$ and call its elements the differential n-forms.

Explicitly this means that a differential n-form $\omega \in \Omega^n(\mathbb{R}^k)$ on $\mathbb{R}^k$ is a formal linear combination over $C^\infty(\mathbb{R}^k)$ (example ) of basis elements of the form $d x^{i_1} \wedge \cdots \wedge d x^{i_n}$ for $i_1 \lt i_2 \lt \cdots \lt i_n$:

$\omega = \omega_{i_1, \cdots, i_n} d x^{i_1} \wedge \cdots \wedge d x^{i_n} \,.$

Now all the constructions for differential 1-forms above extent naturally to differential n-forms:

###### Definition

(exterior derivative or de Rham differential)

For $\mathbb{R}^n$ a Cartesian space (def. ) the de Rham differential $d \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)$ (5) uniquely extended as a derivation of degree +1 to the exterior algebra of differential forms (def. )

$d \;\colon\; \Omega^\bullet(\mathbb{R}^n) \longrightarrow \Omega^\bullet(\mathbb{R}^n)$

meaning that for $\omega_i \in \Omega^{k_i}(\mathbb{R})$ then

$d(\omega_1 \wedge \omega_2) \;=\; (d \omega_1) \wedge \omega_2 + \omega_1 \wedge d \omega_2 \,.$

In components this simply means that

\begin{aligned} d \omega & = d \left(\omega_{i_1 \cdots i_k} d x^{i_1} \wedge \cdots \wedge d x^{i_k}\right) \\ & = \frac{\partial \omega_{i_1 \cdots i_k}}{\partial x^{a}} d x^a \wedge d x^{i_1} \wedge \cdots \wedge d x^{i_k} \end{aligned} \,.

Since partial derivatives commute with each other, while differential 1-form anti-commute, this implies that $d$ is nilpotent

$d^2 = d \circ d = 0 \,.$

We say hence that differential forms form a cochain complex, the de Rham complex $(\Omega^\bullet(\mathbb{R}^n), d)$.

###### Definition

(contraction of differential n-forms with tangent vector fields)

The pairing $\iota_v \omega = \omega(v)$ of tangent vector fields $v$ with differential 1-forms $\omega$ (4) uniquely extends to the exterior algebra $\Omega^\bullet(\mathbb{R}^n)$ of differential forms (def. ) as a derivation of degree -1

$\iota_v \;\colon\; \Omega^{\bullet+1}(\mathbb{R}^n) \longrightarrow \Omega^\bullet(\mathbb{R}^n) \,.$

In particular for $\omega_1, \omega_2 \in \Omega^1(\mathbb{R}^n)$ two differential 1-forms, then

$\iota_{v} (\omega_1 \wedge \omega_2) \;=\; \omega_1(v) \omega_2 - \omega_2(v) \omega_1 \;\in\; \Omega^1(\mathbb{R}^n) \,.$
###### Proposition

(pullback of differential n-forms)

For $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a smooth function between Cartesian spaces (def. ) the operationf of pullback of differential 1-forms of def. extends as an $C^\infty(\mathbb{R}^k)$-algebra homomorphism to the exterior algebra of differential forms (def. ),

$f^\ast \;\colon\; \Omega^\bullet(\mathbb{R}^{n_2}) \longrightarrow \Omega^\bullet(\mathbb{R}^{n_1})$

given on basis elements by

$f^* \left( dx^{i_1} \wedge \cdots \wedge dx^{i_n} \right) = \left(f^* dx^{i_1} \wedge \cdots \wedge f^* dx^{i_n} \right) \,.$

This commutes with the de Rham differential $d$ on both sides (def. ) in that

$d \circ f^\ast = f^\ast \circ d \phantom{AAAAA} \array{ \Omega^\bullet(X) &\overset{f^\ast}{\longleftarrow}& \Omega^\bullet(Y) \\ \mathllap{d}\downarrow && \downarrow\mathrlap{d} \\ \Omega^\bullet(X) &\underset{f^\ast}{\longleftarrow}& \Omega^\bullet(Y) }$

hence that pullback of differential forms is a chain map of de Rham complexes.

This is still compatible with identity morphisms and composition in that

(7)$(id_{\mathbb{R}^n})^\ast = id_{\Omega^1(\mathbb{R}^n)} \phantom{AAAA} (g \circ f)^\ast = f^\ast \circ g^\ast \,.$

Stated more abstractly, this just means that pullback of differential n-forms makes the assignment of sets of differential n-forms to Cartesian spaces a contravariant functor

$\Omega^n(-) \;\colon\; CartSp^{op} \longrightarrow Set \,.$
###### Proposition

(Cartan's homotopy formula)

Let $X$ be a Cartesian space (def. ), and let $v \in \Gamma(T X)$ be a smooth tangent vector field (example ).

For $t \in \mathbb{R}$ write $\exp(t v) \colon X \overset{\simeq}{\to} X$ for the flow by diffeomorphisms along $v$ of parameter length $t$.

Then the derivative with respect to $t$ of the pullback of differential forms along $\exp(t v)$, hence the Lie derivative $\mathcal{L}_v \colon \Omega^\bullet(X) \to \Omega^\bullet(X)$, is given by the anticommutator of the contraction derivation $\iota_v$ (def. ) with the de Rham differential $d$ (def. ):

\begin{aligned} \mathcal{L}_v &\coloneqq \frac{d}{d t } \exp(t v)^\ast \omega \vert_{t = 0} \\ & = \iota_v d \omega + d \iota_v \omega \,. \end{aligned}

Finally we turn to the concept of integration of differential forms (def. below). First we need to say what it is that differential forms may be integrated over:

###### Definition

(smooth singular simplicial chains in Cartesian spaces)

For $n \in \mathbb{N}$, the standard n-simplex in the Cartesian space $\mathbb{R}^n$ (def. ) is the subset

$\Delta^n \;\coloneqq\; \left\{ (x^i)_{i = 1}^n \;\vert\; 0 \leq x^1 \leq \cdots \leq x^n \right\} \;\subset\; \mathbb{R}^n \,.$

More generally, a smooth singular n-simplex in a Cartesian space $\mathbb{R}^k$ is a smooth function (def. )

$\sigma \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}^k \,,$

to be thought of as a smooth extension of its restriction

$\sigma\vert_{\Delta^n} \;\colon\; \Delta^n \longrightarrow \mathbb{R}^k \,.$

(This is called a singular simplex because there is no condition that $\Sigma$ be an embedding in any way, in particular $\sigma$ may be a constant function.)

A singular chain in $\mathbb{R}^k$ of dimension $n$ is a formal linear combination of singular $n$-simplices in $\mathbb{R}^k$.

In particular, given a singular $n+1$-simplex $\sigma$, then its boundary is a singular chain of singular $n$-simplices $\partial \sigma$.

###### Definition

(fiber-integration of differential forms) over smooth singular chains in Cartesian spaces)

For $n \in \mathbb{N}$ and $\omega \in \Omega^n(\mathbb{R}^n)$ a differential n-form (def. ), which may be written as

$\omega = f d x^1 \wedge \cdots d x^n \,,$

then its integration over the standard n-simplex $\Delta^n \subset \mathbb{R}^n$ (def. ) is the ordinary integral (e.g. Riemann integral)

$\int_{\Delta^n} \omega \;\coloneqq\; \underset{0 \leq x^1 \leq \cdots \leq x^n \leq 1}{\int} f(x^1, \cdots, x^n) \, d x^1 \cdots d x^n \,.$

More generally, for

1. $\omega \in \Omega^n(\mathbb{R}^k)$ a differential n-forms;

2. $C = \underset{i}{\sum} c_i \sigma_i$ a singular $n$-chain (def. )

in any Cartesian space $\mathbb{R}^k$. Then the integration of $\omega$ over $x$ is the sum of the integrations, as above, of the pullback of differential forms (def. ) along all the singular n-simplices in the chain:

$\int_C \omega \;\coloneqq\; \underset{i}{\sum} c_i \int_{\Delta^n} (\sigma_i)^\ast \omega \,.$

Finally, for $U$ another Cartesian space, then fiber integration of differential forms along $U \times C \to U$ is the linear map

$\int_C \;\colon\; \Omega^{\bullet + dim(C)}(U \times C) \longrightarrow \Omega^\bullet(U)$

which on differential forms of the form $\omega_U \wedge \omega$ is given by

$\int_C \omega_U \wedge \omega \;\coloneqq\; (-1)^{\vert \omega_U\vert} \int_C \omega \,.$
###### Proposition

(Stokes theorem for fiber-integration of differential forms)

For $\Sigma$ a smooth singular simplicial chain (def. ) the operation of fiber-integration of differential forms along $U \times \Sigma \overset{pr_1}{\longrightarrow} U$ (def. ) is compatible with the exterior derivative $d_U$ on $U$ (def. ) in that

\begin{aligned} d \int_\Sigma \omega & = (-1)^{dim(\Sigma)} \int_\Sigma d_U \omega \\ & = (-1)^{dim(\Sigma)} \left( \int_\Sigma d \omega - \int_{\partial \Sigma} \omega \right) \end{aligned} \,,

where $d = d_U + d_\Sigma$ is the de Rham differential on $U \times \Sigma$ (def. ) and where the second equality is the Stokes theorem along $\Sigma$:

$\int_\Sigma d_\Sigma \omega = \int_{\partial \Sigma} \omega \,.$

$\,$

This concludes our review of the basics of (synthetic) differential geometry on which the following development of quantum field theory is based. In the next chapter we consider spacetime and spin.

## Spacetime

Relativistic field theory takes place on spacetime.

The concept of spacetime makes sense for every dimension $p+1$ with $p \in \mathbb{N}$. The observable universe has macroscopic dimension $3+1$, but quantum field theory generally makes sense also in lower and in higher dimensions. For instance quantum field theory in dimension 0+1 is the “worldline” theory of particles, also known as quantum mechanics; while quantum field theory in dimension $\gt p+1$ may be “KK-compactified” to an “effective” field theory in dimension $p+1$ which generally looks more complicated than its higher dimensional incarnation.

However, every realistic field theory, and also most of the non-realistic field theories of interest, contain spinor fields such as the Dirac field (example below) and the precise nature and behaviour of spinors does depend sensitively on spacetime dimension. In fact the theory of relativistic spinors is mathematically most natural in just the following four spacetime dimensions:

$p +1 = \phantom{AAAAA} \array{ 2+1,\; & 3+1,\; & \, & 5+1,\; &\, & \, & \, & \, 9+1 }$

In the literature one finds these four dimensions advertized for two superficially unrelated reasons:

1. in precisely these dimensions “twistors” exist (see there);

2. in precisely these dimensions “GS-superstrings” exist (see there).

However, both these explanations have a common origin in something simpler and deeper: Spacetime in these dimensions appears from the “Pauli matrices” with entries in the real normed division algebras. (In fact it goes deeper still, but this will not concern us here.)

This we explain now, and then we use this to obtain a slick handle on spinors in these dimensions, using simple linear algebra over the four real normed division algebras. At the end (in remark ) we give a dictionary that expresses these constructions in terms of the “two-component spinor notation” that is traditionally used in physics texts (remark below).

The relation between real spin representations and division algebras, is originally due to Kugo-Townsend 82, Sudbery 84 and others. We follow the streamlined discussion in Baez-Huerta 09 and Baez-Huerta 10.

A key extra structure that the spinors impose on the underlying Cartesian space of spacetime is its causal structure, which determines which points in spacetime (“events”) are in the future or the past of other points (def. below). This causal structure will turn out to tightly control the quantum field theory on spacetime in terms of the “causal additivity of the S-matrix” (prop. below) and the induced “causal locality” of the algebra of quantum observables (prop. below). To prepare the discussion of these constructions, we end this chapter with some basics on the causal structure of Minkowski spacetime.

$\,$

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Real division algebras

To amplify the following pattern and to fix our notation for algebra generators, recall these definitions:

###### Definition

(complex numbers)

The complex numbers $\mathbb{C}$ is the commutative algebra over the real numbers $\mathbb{R}$ which is generated from one generators $\{e_1\}$ subject to the relation

• $(e_1)^2 = -1$.
###### Definition

(quaternions)

The quaternions $\mathbb{H}$ is the associative algebra over the real numbers which is generated from three generators $\{e_1, e_2, e_3\}$ subject to the relations

1. for all $i$

$(e_i)^2 = -1$

2. for $(i,j,k)$ a cyclic permutation of $(1,2,3)$ then

1. $e_i e_j = e_k$

2. $e_j e_i = -e_k$

(graphics grabbed from Baez 02)

###### Definition

(octonions)

The octonions $\mathbb{O}$ is the nonassociative algebra over the real numbers which is generated from seven generators $\{e_1, \cdots, e_7\}$ subject to the relations

1. for all $i$

$(e_i)^2 = -1$

2. for $e_i \to e_j \to e_k$ an edge or circle in the diagram shown (a labeled version of the Fano plane) then

1. $e_i e_j = e_k$

2. $e_j e_i = -e_k$

and all relations obtained by cyclic permutation of the indices in these equations.

(graphics grabbed from Baez 02)

One defines the following operations on these real algebras:

###### Definition

(conjugation, real part, imaginary part and absolute value)

For $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$, let

$(-)^\ast \;\colon\; \mathbb{K} \longrightarrow \mathbb{K}$

be the antihomomorphism of real algebras

\begin{aligned} (r a)^\ast = r a^\ast &, \text{for}\;\; r \in \mathbb{R}, a \in \mathbb{K} \\ (a b)^\ast = b^\ast a^\ast &,\text{for}\;\; a,b \in \mathbb{K} \end{aligned}

given on the generators of def. , def. and def. by

$(e_i)^\ast = - e_i \,.$

This operation makes $\mathbb{K}$ into a star algebra. For the complex numbers $\mathbb{C}$ this is called complex conjugation, and in general we call it conjugation.

Let then

$Re \;\colon\; \mathbb{K} \longrightarrow \mathbb{R}$

be the function

$Re(a) \;\coloneqq\; \tfrac{1}{2}(a + a^\ast)$

(“real part”) and

$Im \;\colon\; \mathbb{K} \longrightarrow \mathbb{R}$

be the function

$Im(a) \;\coloneqq \; \tfrac{1}{2}(a - a^\ast)$

(“imaginary part”).

It follows that for all $a \in \mathbb{K}$ then the product of a with its conjugate is in the real center of $\mathbb{K}$

$a a^\ast = a^\ast a \;\in \mathbb{R} \hookrightarrow \mathbb{K}$

and we write the square root of this expression as

${\vert a\vert} \;\coloneqq\; \sqrt{a a^\ast}$

called the norm or absolute value function

${\vert -\vert} \;\colon\; \mathbb{K} \longrightarrow \mathbb{R} \,.$

This norm operation clearly satisfies the following properties (for all $a,b \in \mathbb{K}$)

1. $\vert a \vert \geq 0$;

2. ${\vert a \vert } = 0 \;\;\;\;\; \Leftrightarrow\;\;\;\;\;\; a = 0$;

3. ${\vert a b \vert } = {\vert a \vert} {\vert b \vert}$

and hence makes $\mathbb{K}$ a normed algebra.

Since $\mathbb{R}$ is a division algebra, these relations immediately imply that each $\mathbb{K}$ is a division algebra, in that

$a b = 0 \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; a = 0 \;\; \text{or} \;\; b = 0 \,.$

Hence the conjugation operation makes $\mathbb{K}$ a real normed division algebra.

###### Remark

(sequence of inclusions of real normed division algebras)

Suitably embedding the sets of generators in def. , def. and def. into each other yields sequences of real star-algebra inclusions

$\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O} \,.$

For example for the first two inclusions we may send each generator to the generator of the same name, and for the last inclusion me may choose

$\array{ 1 &\mapsto& 1 \\ e_1 &\mapsto & e_3 \\ e_2 &\mapsto& e_4 \\ e_3 &\mapsto& e_6 }$
###### Proposition

(Hurwitz theorem: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are the normed real division algebras)

The four algebras of real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, quaternions $\mathbb{H}$ and octonions $\mathbb{O}$ from def. , def. and def. respectively, which are real normed division algebras via def. , are, up to isomorphism, the only real normed division algebras that exist.

###### Remark

(Cayley-Dickson construction and sedenions)

While prop. says that the sequence from remark

$\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O}$

is maximal in the category of real normed non-associative division algebras, there is a pattern that does continue if one disregards the division algebra property. Namely each step in this sequence is given by a construction called forming the Cayley-Dickson double algebra. This continues to an unbounded sequence of real nonassociative star-algebras

$\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O} \hookrightarrow \mathbb{S} \hookrightarrow \cdots$

where the next algebra $\mathbb{S}$ is called the sedenions.

What actually matters for the following relation of the real normed division algebras to real spin representations is that they are also alternative algebras:

###### Definition

(alternative algebras)

Given any non-associative algebra $A$, then the trilinear map

$[-,-,-] \;-\; A \otimes A \otimes A \longrightarrow A$

given on any elements $a,b,c \in A$ by

$[a,b,c] \coloneqq (a b) c - a (b c)$

is called the associator (in analogy with the commutator $[a,b] \coloneqq a b - b a$ ).

If the associator is completely antisymmetric (in that for any permutation $\sigma$ of three elements then $[a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3]$ for $\vert \sigma \vert$ the signature of the permutation) then $A$ is called an alternative algebra.

If the characteristic of the ground field is different from 2, then alternativity is readily seen to be equivalent to the conditions that for all $a,b \in A$ then

$(a a)b = a (a b) \;\;\;\;\; \text{and} \;\;\;\;\; (a b) b = a (b b) \,.$

We record some basic properties of associators in alternative star-algebras that we need below:

###### Proposition

(properties of alternative star algebras)

Let $A$ be an alternative algebra (def. ) which is also a star algebra. Then (using def. ):

1. the associator vanishes when at least one argument is real

$[Re(a),b,c]$
2. the associator changes sign when one of its arguments is conjugated

$[a,b,c] = -[a^\ast,b,c] \,;$
3. the associator vanishes when one of its arguments is the conjugate of another

$[a,a^\ast, b] = 0 \,;$
4. the associator is purely imaginary

$Re([a,b,c]) = 0 \,.$
###### Proof

That the associator vanishes as soon as one argument is real is just the linearity of an algebra product over the ground ring.

Hence in fact

$[a,b,c] = [Im(a), Im(b), Im(c)] \,.$

This implies the second statement by linearity. And so follows the third statement by skew-symmetry:

$[a,a^\ast,b] = -[a,a,b] = 0 \,.$

The fourth statement finally follows by this computation:

\begin{aligned} \,[ a, b, c]^\ast & = -[c^\ast, b^\ast, a^\ast] \\ & = -[c,b,a] \\ & = -[a,b,c] \end{aligned} \,.

Here the first equation follows by inspection and using that $(a b)^\ast = b^\ast a^\ast$, the second follows from the first statement above, and the third is the anti-symmetry of the associator.

It is immediate to check that:

###### Proposition

($\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are real alternative algebras)

The real algebras of real numbers, complex numbers, def. ,quaternions def. and octonions def. are alternative algebras (def. ).

###### Proof

Since the real numbers, complex numbers and quaternions are associative algebras, their associator vanishes identically. It only remains to see that the associator of the octonions is skew-symmetric. By linearity it is sufficient to check this on generators. So let $e_i \to e_j \to e_k$ be a circle or a cyclic permutation of an edge in the Fano plane. Then by definition of the octonion multiplication we have

\begin{aligned} (e_i e_j) e_j &= e_k e_j \\ &= - e_j e_k \\ & = -e_i \\ & = e_i (e_j e_j) \end{aligned}

and similarly

\begin{aligned} (e_i e_i ) e_j &= - e_j \\ &= - e_k e_i \\ &= e_i e_k \\ &= e_i (e_i e_j) \end{aligned} \,.

The analog of the Hurwitz theorem (prop. ) is now this:

###### Proposition

($\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are precisely the alternative real division algebras)

The only division algebras over the real numbers which are also alternative algebras (def. ) are the real numbers themselves, the complex numbers, the quaternions and the octonions from prop. .

This is due to (Zorn 30).

For the following, the key point of alternative algebras is this equivalent characterization:

###### Proposition

(alternative algebra detected on subalgebras spanned by any two elements)

A nonassociative algebra is alternative, def. , precisely if the subalgebra? generated by any two elements is an associative algebra.

This is due to Emil Artin, see for instance (Schafer 95, p. 18).

Proposition is what allows to carry over a minimum of linear algebra also to the octonions such as to yield a representation of the Clifford algebra on $\mathbb{R}^{9,1}$. This happens in the proof of prop. below.

So we will be looking at a fragment of linear algebra over these four normed division algebras. To that end, fix the following notation and terminology:

###### Definition

(hermitian matrices with values in real normed division algebras)

Let $\mathbb{K}$ be one of the four real normed division algebras from prop. , hence equivalently one of the four real alternative division algebras from prop. .

Say that an $n \times n$ matrix with coefficients in $\mathbb{K}$

$A\in Mat_{n\times n}(\mathbb{K})$

is a hermitian matrix if the transpose matrix $(A^t)_{i j} \coloneqq A_{j i}$ equals the componentwise conjugated matrix (def. ):

$A^t = A^\ast \,.$

Hence with the notation

$(-)^\dagger \coloneqq ((-)^t)^\ast$

we have that $A$ is a hermitian matrix precisely if

$A = A^\dagger \,.$

We write $Mat_{2 \times 2}^{her}(\mathbb{K})$ for the real vector space of hermitian matrices.

###### Definition

(trace reversal)

Let $A \in Mat_{2 \times 2}^{her}(\mathbb{K})$ be a hermitian $2 \times 2$ matrix as in def. . Its trace reversal is the result of subtracting its trace times the identity matrix:

$\tilde A \;\coloneqq\; A - (tr A) 1_{n\times n} \,.$

$\,$

Minkowski spacetime in dimensions 3,4,6 and 10

We now discover Minkowski spacetime of dimension 3,4,6 and 10, in terms of the real normed division algebras $\mathbb{K}$ from prop. , equivalently the real alternative division algebras from prop. : this is prop./def. and def. below.

###### Proposition/Definition

(Minkowski spacetime as real vector space of hermitian matrices in real normed division algebras)

Let $\mathbb{K}$ be one of the four real normed division algebras from prop. , hence one of the four real alternative division algebras from prop. .

Then the real vector space of $2 \times 2$ hermitian matrices over $\mathbb{K}$ (def. ) equipped with the inner product $\eta$ whose quadratic form ${\vert -\vert^2_\eta}$ is the negative of the determinant operation on matrices is Minkowski spacetime:

(8)\begin{aligned} \mathbb{R}^{dim_{\mathbb{R}}(\mathbb{K})+1,1} & \coloneqq \left( \mathbb{R}^{dim_{\mathbb{R}(\mathbb{K})}+2} , {\vert -\vert^2_\eta} \right) & \coloneqq \left(Mat_{2 \times 2}^{her}(\mathbb{K}), -det \right) \end{aligned} \,.

hence

1. $\mathbb{R}^{2,1}$ for $\mathbb{K} = \mathbb{R}$;

2. $\mathbb{R}^{3,1}$ for $\mathbb{K} = \mathbb{C}$;

3. $\mathbb{R}^{5,1}$ for $\mathbb{K} = \mathbb{H}$;

4. $\mathbb{R}^{9,1}$ for $\mathbb{K} = \mathbb{O}$.

Here we think of the vector space on the left as $\mathbb{R}^{p,1}$ with

$p \coloneqq dim_{\mathbb{R}}(\mathbb{K})+1$

equipped with the canonical coordinates labeled $(x^\mu)_{\mu = 0}^p$.

As a linear map the identification is given by

$(x^0, x^1, \cdots, x^{d-1}) \;\mapsto\; \left( \array{ x^0 + x^1 & y \\ y^\ast & x^0 - x^1 } \right) \;\;\; \text{with}\; y \coloneqq x^2 1 + x^3 e_1 + x^4 e_2 + \cdots + x^{2 + dim_{\mathbb{R}(\mathbb{K})}} \,e_{dim_{\mathbb{R}}(\mathbb{K})-1} \,.$

This means that the quadratic form ${\vert - \vert^2_\eta}$ is given on an element $v = (v^\mu)_{\mu = 0}^p$ by

${\vert v \vert}^2_{\eta} \;=\; - (v^0)^2 + \underoverset{j = 1}{p}{\sum} (x^j)^2 \,.$

By the polarization identity the quadratic form ${\vert - \vert^2_\eta}$ induces a bilinear form

$\eta \;\colon\; \mathbb{R}^{p,1}\otimes \mathbb{R}^{p,1} \longrightarrow \mathbb{R}$

given by

\begin{aligned} \eta(v_1, v_2) & = \eta_{\mu \nu} v_1^\mu v_1^\nu \\ & \coloneqq - v_1^0 v_2^0 + \underoverset{j = 1}{p}{\sum} v_1^j v_2^j \end{aligned} \,.

This is called the Minkowski metric.

Finally, under the above identification the operation of trace reversal from def. corresponds to time reversal in that

$\widetilde{ \left( \array{ x^0 + x^1 & y \\ y^\ast & x^0 - x^1 } \right) } \;=\; \left( \array{ -x^0 + x^1 & y \\ y^\ast & -x^0 - x^1 } \right) \,.$
###### Proof

We need to check that under the given identification, the Minkowski norm-square is indeed given by minus the determinant on the corresponding hermitian matrices. This follows from the nature of the conjugation operation $(-)^\ast$ from def. :

\begin{aligned} - det \left( \array{ x^0 + x^1 & y \\ y^\ast & x^0 - x^1 } \right) & = -(x^0 + x^1)(x^0 - x^1) + y y^\ast \\ & = -(x^0)^2 + \underoverset{i = 1}{p}{\sum} (x^i)^2 \end{aligned} \,.
###### Remark

(physical units of length)

As the term “metric” suggests, in application to physics, the Minkowski metric $\eta$ in prop./def. is regarded as a measure of length: for $v \in \Gamma_x(T \mathbb{R}^{p,1})$ a tangent vector at a point $x$ in Minkowski spacetime, interpreted as a displacement from event $x$ to event $x + v$, then

1. if $\eta(v,v) \gt 0$ then

$\sqrt{\eta(v,v)} \in \mathbb{R}$

is interpreted as a measure for the spatial distance between $x$ and $x + v$;

2. if $\eta(v,v) \lt 0$ then

$\sqrt{-\eta(v,v)} \in \mathbb{R}$

is interpreted as a measure for the time distance between $x$ and $x + v$.

But for this to make physical sense, an operational prescription needs to be specified that tells the experimentor how the real number $\sqrt{\eta(v,v)}$ is to be translated into an physical distance between actual events in the observable universe.

Such an operational prescription is called a physical unit of length. For example “centimeter$cm$ is a physical unit of length, another one is “femtometer$fm$.

The combined information of a real number $\sqrt{\eta(v,v)} \in \mathbb{R}$ and a physical unit of length such as meter, jointly written

$\sqrt{\eta(v,v)} \, cm$

is a prescription for finding actual distance in the observable universe. Alternatively

$\sqrt{\eta(v,v)} \, fm$

is another prescription, that translates the same real number $\sqrt{\eta(v,v)}$ into another physical distance.

But of course they are related, since physical units form a torsor over the group $\mathbb{R}_{\gt 0}$ of non-negative real numbers, meaning that any two are related by a unique rescaling. For example

$fm = 10^{-13} cm \,,$

with $10^{-13} \in \mathbb{R}_{\gt 0}$.

This means that once any one prescription of turning real numbers into spacetime distances is specified, then any other such prescription is obtained from this by rescaling these real numbers. For example

\begin{aligned} \sqrt{\eta(v,v)} \, fm & = \left( 10^{-13} \sqrt{\eta(v,v)}\right) \,cm \\ & = \sqrt{ 10^{-26} \eta(v,v) } \, cm \end{aligned} \,.

The point to notice here is that, via the last line, we may think of this as rescaling the metric from $\eta$ to $10^{-30} \eta$.

In quantum field theory physical units of length are typically expressed in terms of a physical unit of “action”, called “Planck's constant$\hbar$, via the combination of units called the Compton wavelength

(9)$\ell_m = \frac{2\pi \hbar}{m c} \,.$

parameterized, in turn, by a physical unit of mass $m$. For the mass of the electron, the Compton wavelength is

$\ell_e = \frac{2\pi \hbar}{m_e c} \sim 386 \, fm \,.$

Another physical unit of length parameterized by a mass $m$ is the Schwarzschild radius $r_m \coloneqq 2 m G/c^2$, where $G$ is the gravitational constant. Solving the equation

$\array{ & \ell_m &=& r_m \\ \Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2 }$

for $m$ yields the Planck mass

$m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}} \,.$

The corresponding Compton wavelength $\ell_{m_{P}}$ is given by the Planck length $\ell_P$

$\ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} } \,.$
###### Definition

(Minkowski spacetime as a pseudo-Riemannian Cartesian space)

Prop./def. introduces Minkowski spacetime $\mathbb{R}^{p,1}$ for $p+1 \in \{3,4,6,10\}$ as a a vector space $\mathbb{R}^{p,1}$ equipped with a norm ${\vert - \vert_\eta}$. The genuine spacetime corresponding to this is this vector space regaded as a Cartesian space, i.e. with smooth functions (instead of just linear maps) to it and from it (def. ). This still carries one copy of $\mathbb{R}^{p,1}$ over each point $x \in \mathbb{R}^{p,1}$, as its tangent space (example )

$T_x \mathbb{R}^{p,1} \simeq \mathbb{R}^{p,1}$

and the Cartesian space $\mathbb{R}^{p,1}$ equipped with the Lorentzian inner product from prop./def. on each tangent space $T_x \mathbb{R}^{p,1}$ (a “pseudo-Riemannian Cartesian space”) is Minkowski spacetime as such.

We write

(10)$dvol_\Sigma \;\coloneqq\; d x^0 \wedge d x^1 \wedge \cdots \wedge d x^p \in \Omega^{p+1}(\mathbb{R}^{p,1})$

for the canonical volume form on Minkowski spacetime.

We use the Einstein summation convention: Expressions with repeated indices indicate summation over the range of indices.

For example a differential 1-form $\alpha \in \Omega^1(\mathbb{R}^{p,1})$ on Minkowski spacetime may be expanded as

$\alpha = \alpha_\mu d x^\mu \,.$

Moreover we use square brackets around indices to indicate skew-symmetrization. For example a differential 2-form $\beta \in \Omega^2(\mathbb{R}^{p,1})$ on Minkowski spacetime may be expanded as

\begin{aligned} \beta & = \beta_{\mu \nu} d x^\mu \wedge d x^\nu \\ & = \beta_{[\mu \nu]} d x^\mu \wedge d x^\nu \end{aligned}

$\,$

The identification of Minkowski spacetime (def. ) in the exceptional dimensions with the generalized Pauli matrices (prop./def. ) has some immediate useful implications:

###### Proposition

(Minkowski metric in terms of trace reversal)

In terms of the trace reversal operation $\widetilde{(-)}$ from def. , the determinant operation on hermitian matrices (def. ) has the following alternative expression

\begin{aligned} -det(A) & = A \tilde A \\ & = \tilde A A \end{aligned} \,.

and the Minkowski inner product from prop. has the alternative expression

\begin{aligned} \eta(A,B) & = \tfrac{1}{2}Re(tr(A \tilde B)) \\ & = \tfrac{1}{2} Re(tr(\tilde A B)) \end{aligned} \,.
###### Proposition

(special linear group $SL(2,\mathbb{K})$ acts by linear isometries on Minkowski spacetime )

For $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$ one of the four real normed division algebras (prop. ) the special linear group $SL(2,\mathbb{K})$ acts on Minkowski spacetime $\mathbb{R}^{p,1}$ in dimension $p+1 \in \{2+1, \,3+1, \, 5+1. \, 9+1\}$ (def. ) by linear isometries given under the identification with the Pauli matrices in prop./def. by conjugation:

$\array{ SL(2,\mathbb{K}) \times \mathbb{R}^{dim(\mathbb{K}+1,1)} & \simeq & SL(2, \mathbb{K}) \times Mat^{herm}_{2 \times 2}(\mathbb{K}) &\overset{}{\longrightarrow}& Mat^{herm}_{2 \times 2}(\mathbb{K}) & \simeq & \mathbb{R}^{dim(\mathbb{K}+1,1)} \\ && (G, A) &\mapsto& G \, A \, G^\dagger }$
###### Proof

For $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$ this is immediate from matrix calculus, but we spell it out now. While the argument does not directly apply to the case $\mathbb{K} = \mathbb{O}$ of the octonions, one can check that it still goes through, too.

First we need to see that the action is well defined. This follows from the associativity of matrix multiplication and the fact that forming conjugate transpose matrices is an antihomomorphism: $(G_1 G_2)^\dagger = G_2^\dagger G_1^\dagger$. In particular this implies that the action indeed sends hermitian matrices to hermitian matrices:

\begin{aligned} \left( G \, A \, G^\dagger \right)^\dagger & = \underset{= G}{\underbrace{\left( G^\dagger \right)}} \, \underset{= A}{\underbrace{A^\dagger}} \, G^\dagger \\ & = G \, A \, G^\dagger \end{aligned} \,.

By prop./def. such an action is an isometry precisely if it preserves the determinant. This follows from the multiplicative property of determinants: $det(A B) = det(A) det(B)$ and their compativility with conjugate transposition: $det(A^\dagger) = det(A^\ast)$, and finally by the assumption that $G \in SL(2,\mathbb{K})$ is an element of the special linear group, hence that its determinant is $1 \in \mathbb{K}$:

\begin{aligned} det\left( G \, A \, G^\dagger \right) & = \underset{ = 1}{\underbrace{det(G)}} \, det(A) \, \underset{= 1^\ast = 1}{\underbrace{det(G^\dagger)}} \\ & = det(A) \end{aligned} \,.

In fact the special linear groups of linear isometries in prop. are the spin groups (def. below) in these dimensions.

exceptional spinors and real normed division algebras

Lorentzian
spacetime
dimension
$\phantom{AA}$spin groupnormed division algebra$\,\,$ brane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\phantom{A}$ $\mathbb{R}$ the real numberssuper 1-brane in 3d
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\phantom{A}$ $\mathbb{C}$ the complex numberssuper 2-brane in 4d
$6 = 5+1$$Spin(5,1) \simeq$ SL(2,H)$\phantom{A}$ $\mathbb{H}$ the quaternionslittle string
$10 = 9+1$$Spin(9,1) {\simeq}$SL(2,O)$\phantom{A}$ $\mathbb{O}$ the octonionsheterotic/type II string

This we explain now.

$\,$

Lorentz group and spin group

###### Definition

(Lorentz group)

For $d \in \mathbb{N}$, write

$O(d-1,1) \hookrightarrow GL(\mathbb{R}^d)$

for the subgroup of the general linear group on those linear maps $A$ which preserve this bilinear form on Minkowski spacetime (def ), in that

$\eta(A(-),A(-)) = \eta(-,-) \,.$

This is the Lorentz group in dimension $d$.

The elements in the Lorentz group in the image of the special orthogonal group $SO(d-1) \hookrightarrow O(d-1,1)$ are rotations in space. The further elements in the special Lorentz group $SO(d-1,1)$, which mathematically are “hyperbolic rotations” in a space-time plane, are called boosts in physics.

One distinguishes the following further subgroups of the Lorentz group $O(d-1,1)$:

• $SO(d-1,1) \hookrightarrow O(d-1,1)$

is the subgroup of elements which have determinant +1 (as elements $SO(d-1,1)\hookrightarrow GL(d)$ of the general linear group);

• the proper orthochronous (or restricted) Lorentz group

$SO^+(d-1,1) \hookrightarrow SO(d-1,1)$

is the further subgroup of elements $A$ which preserve the time orientation of vectors $v$ in that $(v^0 \gt 0) \Rightarrow ((A v)^0 \gt 0)$.

###### Proposition

(connected component of Lorentz group)

As a smooth manifold, the Lorentz group $O(d-1,1)$ (def. ) has four connected components. The connected component of the identity is the proper orthochronous Lorentz group $SO^+(3,1)$ (def. ). The other three components are

1. $SO^+(d-1,1)\cdot P$

2. $SO^+(d-1,1)\cdot T$

3. $SO^+(d-1,1)\cdot P T$,

where, as matrices,

$P \coloneqq diag(1,-1,-1, \cdots, -1)$

is the operation of point reflection at the origin in space, where

$T \coloneqq diag(-1,1,1, \cdots, 1)$

is the operation of reflection in time and hence where

$P T = T P = diag(-1,-1, \cdots, -1)$

is point reflection in spacetime.

The following concept of the Clifford algebra (def. ) of Minkowski spacetime encodes the structure of the inner product space $\mathbb{R}^{d-1,1}$ in terms of algebraic operation (“geometric algebra”), such that the action of the Lorentz group becomes represented by a conjugation action (example below). In particular this means that every element of the proper orthochronous Lorentz group may be “split in half” to yield a double cover: the spin group (def. below).

###### Definition

(Clifford algebra)

For $d \in \mathbb{N}$, we write

$Cl(\mathbb{R}^{d-1,1})$

for the $\mathbb{Z}/2$-graded associative algebra over $\mathbb{R}$ which is generated from $d$ generators $\{\Gamma_0, \Gamma_1, \Gamma_2, \cdots, \Gamma_{d-1}\}$ in odd degree (“Clifford generators”), subject to the relation

(11)$\Gamma_{a} \Gamma_b + \Gamma_b \Gamma_a = - 2\eta_{a b}$

where $\eta$ is the inner product of Minkowski spacetime as in def. .

These relations say equivalently that

\begin{aligned} & \Gamma_0^2 = +1 \\ & \Gamma_i^2 = -1 \;\; \text{for}\; i \in \{1,\cdots, d-1\} \\ & \Gamma_a \Gamma_b = - \Gamma_b \Gamma_a \;\;\; \text{for}\; a \neq b \end{aligned} \,.

We write

$\Gamma_{a_1 \cdots a_p} \;\coloneqq\; \frac{1}{p!} \underset{{permutations \atop \sigma}}{\sum} (-1)^{\vert \sigma\vert } \Gamma_{a_{\sigma(1)}} \cdots \Gamma_{a_{\sigma(p)}}$

for the antisymmetrized product of $p$ Clifford generators. In particular, if all the $a_i$ are pairwise distinct, then this is simply the plain product of generators

$\Gamma_{a_1 \cdots a_n} = \Gamma_{a_1} \cdots \Gamma_{a_n} \;\;\; \text{if} \; \underset{i,j}{\forall} (a_i \neq a_j) \,.$

Finally, write

$\overline{(-)} \;\colon\; Cl(\mathbb{R}^{d-1,1}) \longrightarrow Cl(\mathbb{R}^{d-1,1})$

for the algebra anti-automorphism given by

$\overline{\Gamma_a} \coloneqq \Gamma_a$
$\overline{\Gamma_a \Gamma_b} \coloneqq \Gamma_b \Gamma_a \,.$
###### Remark

(vectors inside Clifford algebra)

By construction, the vector space of linear combinations of the generators in a Clifford algebra $Cl(\mathbb{R}^{d-1,1})$ (def. ) is canonically identified with Minkowski spacetime $\mathbb{R}^{d-1,1}$ (def. )

$\widehat{(-)} \;\colon\; \mathbb{R}^{d-1,1} \hookrightarrow Cl(\mathbb{R}^{d-1,1})$

via

$x_a \mapsto \Gamma_a \,,$

hence via

$v = v^a x_a \mapsto \hat v = v^a \Gamma_a \,,$

such that the defining quadratic form on $\mathbb{R}^{d-1,1}$ is identified with the anti-commutator in the Clifford algebra

$\eta(v_1,v_2) = -\tfrac{1}{2}( \hat v_1 \hat v_2 + \hat v_2 \hat v_1) \,,$

where on the right we are, in turn, identifying $\mathbb{R}$ with the linear span of the unit in $Cl(\mathbb{R}^{d-1,1})$.

The key point of the Clifford algebra (def. ) is that it realizes spacetime reflections, rotations and boosts via conjugation actions:

###### Example

(Clifford conjugation)

For $d \in \mathbb{N}$ and $\mathbb{R}^{d-1,1}$ the Minkowski spacetime of def. , let $v \in \mathbb{R}^{d-1,1}$ be any vector, regarded as an element $\hat v \in Cl(\mathbb{R}^{d-1,1})$ via remark .

Then

1. the conjugation action $\hat v \mapsto -\Gamma_a^{-1} \hat v \Gamma_a$ of a single Clifford generator $\Gamma_a$ on $\hat v$ sends $v$ to its

reflection at the hyperplane $x_a = 0$;

1. $\hat v \mapsto \exp(- \tfrac{\alpha}{2} \Gamma_{a b}) \hat v \exp(\tfrac{\alpha}{2} \Gamma_{a b})$

sends $v$ to the result of rotating it in the $(a,b)$-plane through an angle $\alpha$.

###### Proof

This is immediate by inspection:

For the first statement, observe that conjugating the Clifford generator $\Gamma_b$ with $\Gamma_a$ yields $\Gamma_b$ up to a sign, depending on whether $a = b$ or not:

$- \Gamma_a^{-1} \Gamma_b \Gamma_a = \left\{ \array{ -\Gamma_b & \vert \text{if}\, a = b \\ \Gamma_b & \vert \text{otherwise} } \right. \,.$

Therefore for $\hat v = v^b \Gamma_b$ then $\Gamma_a^{-1} \hat v \Gamma_a$ is the result of multiplying the $a$-component of $v$ by $-1$.

For the second statement, observe that

$-\tfrac{1}{2}[\Gamma_{a b}, \Gamma_c] = \Gamma_a \eta_{b c} - \Gamma_b \eta_{a c} \,.$

This is the canonical action of the Lorentzian special orthogonal Lie algebra $\mathfrak{so}(d-1,1)$. Hence

$\exp(-\tfrac{\alpha}{2} \Gamma_{ab}) \hat v \exp(\tfrac{\alpha}{2} \Gamma_{ab}) = \exp(\tfrac{1}{2}[\Gamma_{a b}, -])(\hat v)$

is the rotation action as claimed.

###### Remark

Since the reflections, rotations and boosts in example are given by conjugation actions, there is a crucial ambiguity in the Clifford elements that induce them:

1. the conjugation action by $\Gamma_a$ coincides precisely with the conjugation action by $-\Gamma_a$;

2. the conjugation action by $\exp(\tfrac{\alpha}{4} \Gamma_{a b})$ coincides precisely with the conjugation action by $-\exp(\tfrac{\alpha}{2}\Gamma_{a b})$.

###### Definition

(spin group)

For $d \in \mathbb{N}$, the spin group $Spin(d-1,1)$ is the group of even graded elements of the Clifford algebra $Cl(\mathbb{R}^{d-1,1})$ (def. ) which are unitary with respect to the conjugation operation $\overline{(-)}$ from def. :

$Spin(d-1,1) \;\coloneqq\; \left\{ A \in Cl(\mathbb{R}^{d-1,1})_{even} \;\vert\; \overline{A} A = 1 \right\} \,.$
###### Proposition

The function

$Spin(d-1,1) \longrightarrow GL(\mathbb{R}^{d-1,1})$

from the spin group (def. ) to the general linear group in $d$-dimensions given by sending $A \in Spin(d-1,1) \hookrightarrow Cl(\mathbb{R}^{d-1,1})$ to the conjugation action

$\overline{A}(-) A$

(via the identification of Minkowski spacetime as the subspace of the Clifford algebra containing the linear combinations of the generators, according to remark )

is

1. a group homomorphism onto the proper orthochronous Lorentz group (def. ):

$Spin(d-1,1) \longrightarrow SO^+(d-1,1)$
2. exhibiting a $\mathbb{Z}/2$-central extension.

###### Proof

That the function is a group homomorphism into the general linear group, hence that it acts by linear transformations on the generators follows by using that it clearly lands in automorphisms of the Clifford algebra.

That the function lands in the Lorentz group $O(d-1,1) \hookrightarrow GL(d)$ follows from remark :

\begin{aligned} \eta(\overline{A}v_1A , \overline{A} v_2 A) &= \tfrac{1}{2} \left( \left(\overline{A} \hat v_1 A\right) \left(\overline{A}\hat v_2 A\right) + \left(\overline{A} \hat v_2 A\right) \left(\overline{A} \hat v_1 A\right) \right) \\ & = \tfrac{1}{2} \left( \overline{A}(\hat v_1 \hat v_2 + \hat v_2 \hat v_1) A \right) \\ & = \overline{A} A \tfrac{1}{2}\left( \hat v_1 \hat v_2 + \hat v_2 \hat v_1\right) \\ & = \eta(v_1, v_2) \end{aligned} \,.

That it moreover lands in the proper Lorentz group $SO(d-1,1)$ follows from observing (example ) that every reflection is given by the conjugation action by a linear combination of generators, which are excluded from the group $Spin(d-1,1)$ (as that is defined to be in the even subalgebra).

To see that the homomorphism is surjective, use that all elements of $SO(d-1,1)$ are products of rotations in hyperplanes. If a hyperplane is spanned by the bivector $(\omega^{a b})$, then such a rotation is given, via example by the conjugation action by

$\exp(\tfrac{\alpha}{2} \omega^{a b}\Gamma_{a b})$

for some $\alpha$, hence is in the image.

That the kernel is $\mathbb{Z}/2$ is clear from the fact that the only even Clifford elements which commute with all vectors are the multiples $a \in \mathbb{R} \hookrightarrow Cl(\mathbb{R}^{d-1,1})$ of the identity. For these $\overline{a} = a$ and hence the condition $\overline{a} a = 1$ is equivalent to $a^2 = 1$. It is clear that these two elements $\{+1,-1\}$ are in the center of $Spin(d-1,1)$. This kernel reflects the ambiguity from remark .

$\,$

Spinors in dimensions 3, 4, 6 and 10

We now discuss how real spin representations (def. ) in spacetime dimensions 3,4, 6 and 10 are naturally induced from linear algebra over the four real alternative division algebras (prop. ).

###### Definition

(Clifford algebra via normed division algebra)

Let $\mathbb{K}$ be one of the four real normed division algebras from prop. , hence one of the four real alternative division algebras from prop. .

Define a real linear map

$\Gamma \;\colon\; \mathbb{R}^{dim_{\mathbb{R}}(\mathbb{K})+1,1} \longrightarrow End_{\mathbb{R}}(\mathbb{K}^4)$

from (the real vector space underlying) Minkowski spacetime to real linear maps on $\mathbb{K}^4$

$\Gamma(A) \left( \array{ \psi \\ \phi } \right) \;\coloneqq\; \left( \array{ - \tilde A \phi \\ A \psi } \right) \,.$

Here on the right we are using the isomorphism from prop. for identifying a spacetime vector with a $2 \times 2$-matrix, and we are using the trace reversal $\widetilde(-)$ from def. .

###### Remark

(Clifford multiplication via octonion-valued matrices)

Each operation of $\Gamma(A)$ in def. is clearly a linear map, even for $\mathbb{K}$ being the non-associative octonions. The only point to beware of is that for $\mathbb{K}$ the octonions, then the composition of two such linear maps is not in general given by the usual matrix product.

###### Proposition

(real spin representations via normed division algebras)

The map $\Gamma$ in def. gives a representation of the Clifford algebra $Cl(\mathbb{R}^{dim_{\mathbb{R}}(\mathbb{K})+1,1} )$ (this def.), i.e of

1. $Cl(\mathbb{R}^{2,1})$ for $\mathbb{K} = \mathbb{R}$;

2. $Cl(\mathbb{R}^{3,1})$ for $\mathbb{K} = \mathbb{C}$;

3. $Cl(\mathbb{R}^{5,1})$ for $\mathbb{K} = \mathbb{H}$;

4. $Cl(\mathbb{R}^{9,1})$ for $\mathbb{K} = \mathbb{O}$.

Hence this Clifford representation induces representations of the spin group $Spin(dim_{\mathbb{R}}(\mathbb{K})+1,1)$ on the real vector spaces

$S_{\pm } \coloneqq \mathbb{K}^2 \,.$

and hence on

$S \coloneqq S_+ \oplus S_- \,.$
###### Proof

We need to check that the Clifford relation

\begin{aligned} (\Gamma(A))^2 & = -\eta(A,A)1 \\ & = + det(A) \end{aligned}

is satisfied (where we used (11) and (8)). Now by definition, for any $(\phi,\psi) \in \mathbb{K}^4$ then

$(\Gamma(A))^2 \left( \array{ \phi \\ \psi } \right) \;=\; - \left( \array{ \tilde A(A \phi) \\ A(\tilde A \psi) } \right) \,,$

where on the right we have in each component ordinary matrix product expressions.

Now observe that both expressions on the right are sums of triple products that involve either one real factor or two factors that are conjugate to each other:

\begin{aligned} A (\tilde A \psi) & = \left( \array{ x_0 + x_1 & y \\ y^\ast & x_0 - x_1 } \right) \cdot \left( \array{ (-x_0 + x_1) \phi_1 + y \phi_2 \\ y^\ast \phi_1 - (x_0 + x_1)\phi_2 } \right) \\ & = \left( \array{ (-x_0^2 + x_1^2) \phi_1 + (x_0 + x_1)(y \phi_2) + y (y^\ast \phi_1) - y( (x_0 + x_1) \phi_2 ) \\ \cdots } \right) \end{aligned} \,.

Since the associators of triple products that involve a real factor and those involving both an element and its conjugate vanish by prop. (hence ultimately by Artin’s theorem, prop. ). In conclusion all associators involved vanish, so that we may rebracket to obtain

$(\Gamma(A))^2 \left( \array{ \phi \\ \psi } \right) \;=\; - \left( \array{ (\tilde A A) \phi \\ (A \tilde A) \psi } \right) \,.$

This implies the statement via the equality $-A \tilde A = -\tilde A A = det(A)$ (prop. ).

###### Proposition

(spinor bilinear pairings)

Let $\mathbb{K}$ be one of the four real normed division algebras and $S_\pm \simeq_{\mathbb{R}}\mathbb{K}^2$ the corresponding spin representation from prop. .

Then there are bilinear maps from two spinors (according to prop. ) to the real numbers

$\overline{(-)}(-) \;\colon\; S_+ \otimes S_-\longrightarrow \mathbb{R}$

as well as to $\mathbb{R}^{dim(\mathbb{K}+1,1)}$

$\overline{(-)}\Gamma (-) \;\colon\; S_\pm \otimes S_{\pm}\longrightarrow \mathbb{R}^{dim(\mathbb{K}+1,1)}$

given, respectively, by forming the real part (def. ) of the canonical $\mathbb{K}$-inner product

$\overline{(-)}(-) \colon S_+\otimes S_- \longrightarrow \mathbb{R}$
$(\psi,\phi)\mapsto \overline{\psi} \phi \coloneqq Re(\psi^\dagger \cdot \phi)$

and by forming the product of a column vector with a row vector to produce a matrix, possibly up to trace reversal (def. ) under the identification $\mathbb{R}^{dim(\mathbb{K})+1,1} \simeq Mat^{her}_{2 \times 2}(\mathbb{K})$ from prop. :

$S_+ \otimes S_+ \longrightarrow \mathbb{R}^{dim(\mathbb{K})+1,1}$
$(\psi , \phi) \mapsto \overline{\psi}\Gamma \phi \coloneqq \widetilde{\psi \phi^\dagger + \phi \psi^\dagger}$

and

$S_- \otimes S_- \longrightarrow \mathbb{R}^{dim(\mathbb{K}+1,1)}$
$(\psi , \phi) \mapsto {\psi \phi^\dagger + \phi \psi^\dagger}$

For $A \in Mat^{her}_{2 \times 2}(\mathbb{K})$ the $A$-component of this map is

$\eta(\overline{\psi}\Gamma \phi, A) = Re (\psi^\dagger (A\phi)) \,.$

These pairings have the following properties

1. both are $Spin(dim(\mathbb{K})+1,1)$-equivalent;

2. the pairing $\overline{(-)}\Gamma(-)$ is symmetric:

(12)$\overline{\psi_1} \,\Gamma\, \psi_2 = + \overline{\phi_2}\, \Gamma\, \psi_1 \phantom{AAAA} \text{for} \phantom{AA} \psi_1, \psi_2 \in S_+ \oplus S_-$
###### Remark

(two-component spinor notation)

In the physics/QFT literature the expressions for spin representations given by prop. are traditionally written in two-component spinor notation as follows:

• An element of $S_+$ is denoted $(\chi_a \in \mathbb{K})_{a = 1,2}$ and called a left handed spinor;

• an element of $S_-$ is denoted $(\xi^{\dagger \dot a})_{\dot a = 1,2}$ and called a right handed spinor;

• an element of $S = S_+ \oplus S_-$ is denoted

(13)$(\psi^\alpha) = \left( (\chi_a), (\xi^{\dagger \dot a}) \right)$

and called a Dirac spinor;

and the Clifford action of prop. corresponds to the generalized “Pauli matrices”:

• a hermitian matrix $A \in Mat^{her}_{2\times 2}(\mathbb{K})$ as in prop regarded as a linear map $S_- \to S_+$ via def. is denoted

$\left(x_\mu \sigma^\mu_{a \dot a}\right) \;\coloneqq\; \left( \array{ x_0 + x_1 & y \\ y^\ast & x_0 - x_1 } \right) \,;$
• the negative of the trace-reversal (def. ) of such a hermitian matrix, regarded as a linear map $S_+ \to S_-$, is denoted

$\left( x_\mu \widetilde \sigma^{\mu \dot a a} \right) \;\coloneqq\; - \left( \array{ -x_0 + x_1 & y \\ y^\ast & -x_0 - x_1 } \right) \,.$
• the corresponding Clifford generator $\Gamma(A) \;\colon\; S_+ \oplus S_- \to S_+ \oplus S_-$ (def. ) is denoted

$x_\mu (\gamma^\mu)_{\alpha \beta} \;\coloneqq\; \left( \array{ 0 & x_\mu \sigma^\mu_{a \dot b} \\ x_\mu \widetilde \sigma^{\mu \dot a b} } \right)$
• the bilinear spinor-to-vector pairing from prop. is written as the matrix multiplication

$\left( \overline{\psi} \, \gamma^\mu \, \phi\right) \;\coloneqq\; \overline{\psi}\,\Gamma \,\phi \,,$

where the Dirac conjugate $\overline{\psi}$ on the left is given on $(\psi_\alpha) = (\chi_a, \xi^{\dagger \dot c})$ by

(14)\begin{aligned} \overline{\psi} & \coloneqq \psi^\dagger \gamma^0 \\ & = ( \xi^a, \chi^\dagger_{\dot a} ) \end{aligned}

hence, with (13):

(15)\begin{aligned} \overline{\psi_1} \,\gamma^\mu\, \psi_2 & = \psi_1^\dagger \, \gamma^0 \gamma^\mu \, \psi_2 \\ & = (\xi_1)^a \, \sigma^\mu_{a \dot c}\, (\xi_2)^{\dagger \dot c} + (\chi_1)^\dagger_{\dot a} \, \widetilde \sigma^{\mu \dot a c} \, (\chi_2)_c \end{aligned}

Finally, it is common to abbreviate contractions with the Clifford algebra generators $(\gamma^\mu)$ by a slash, as in

$k\!\!\!/\, \;\coloneqq\; \gamma^\mu k_\mu$

or

(16)$i \partial\!\!\!/\, \;\coloneqq\; i \gamma^\mu \frac{\partial}{\partial x^\mu} \,.$

This is called the Feynman slash notation.

(e.g. Dermisek I-8, Dermisek I-9)

Below we spell out the example of the Lagrangian field theory of the Dirac field in detail (example ). For discussion of massive chiral spinor fields one also needs the following, here we just mention this for completeness:

###### Proposition

(chiral spinor mass pairing)

In dimension 2+1 and 3+1, there exists a non-trivial skew-symmetric pairing

$\epsilon \;\colon\; S \wedge S \longrightarrow \mathbb{R}$

which may be normalized such that in the two-component spinor basis of remark we have

(17)$\tilde \sigma^{\mu \dot a a} = \epsilon^{a b} \epsilon^{\dot a \dot b} \sigma^\mu_{b \dot b} \,.$
###### Proof

Take the non-vanishing components of $\epsilon$ to be

$\epsilon^{1 2} = \epsilon^{\dot 1 \dot 2} = \epsilon_{21} = \epsilon_{\dot 2 \dot 1} = 1$

and

$\epsilon^{2 1} = \epsilon^{\dot 2 \dot 1} = \epsilon_{1 2} = \epsilon_{\dot 1 \dot 2} = -1 \,.$

With this equation (17) is checked explicitly. It is clear that $\epsilon$ thus defined is skew symmetric as long as the component algebra is commutative, which is the case for $\mathbb{K}$ being $\mathbb{R}$ or $\mathbb{C}$.

$\,$

Causal structure

We need to consider the following concepts and constructions related to the causal structure of Minkowski spacetime $\Sigma$ (def. ).

###### Definition

(spacelike, timelike, lightlike directions; past and future)

Given two points $x,y \in \Sigma$ in Minkowski spacetime (def. ), write

$v \coloneqq y - x \in \mathbb{R}^{p,1}$

for their difference, using the vector space structure underlying Minkowski spacetime.

Recall the Minkowski inner product $\eta$ on $\mathbb{R}^{p,1}$, given by prop./def. . Then via remark we say that the difference vector $v$ is

1. spacelike if $\eta(v,v) \gt 0$,

2. timelike if $\eta(v,v) \lt 0$,

3. lightlike if $\eta(v,v) = 0$.

If $v$ is timelike or lightlike then we say that

1. $y$ is in the future of $x$ if $y^0 - x^0 \geq 0$;

2. $y$ is in the past of $x$ if $y^0 - x^0 \leq 0$.

###### Definition

(causal cones)

For $x \in \Sigma$ a point in spacetime (an event), we write

$V^+(x), V^-(x) \subset \Sigma$

for the subsets of events that are in the timelike future or in the timelike past of $x$, respectively (def. ) called the open future cone and open past cone, respectively, and

$\overline{V}^+(x), \overline{V}^-(x) \subset \Sigma$

for the subsets of events that are in the timelike or lightlike future or past, respectivel, called the closed future cone and closed past cone, respectively.

The union

$J(x) \coloneqq \overline{V}^+(x) \cup \overline{V}^-(x)$

of the closed future cone and past cone is called the full causal cone of the event $x$. Its boundary is the light cone.

More generally for $S \subset \Sigma$ a subset of events we write

$\overline{V}^\pm(S) \;\coloneqq\; \underset{x \in S}{\cup} \overline{V}^{\pm}(x)$

for the union of the future/past closed cones of all events in the subset.

###### Definition

(compactly sourced causal support)

Consider a vector bundle $E \overset{}{\to} \Sigma$ (def. ) over Minkowski spacetime (def. ).

Write $\Gamma_{\Sigma}(E)$ for the spaces of smooth sections (def. ), and write

\begin{aligned} \Gamma_{cp}(E) & \,\text{compact support} \\ \Gamma_{\Sigma,\pm cp}(E) & \,\text{compactly sourced future/past support} \\ \Gamma_{\Sigma,scp}(E) & \,\text{spacelike compact support} \\ \Gamma_{\Sigma,(f/p)cp}(E) & \,\text{future/past compact support} \\ \Gamma_{\Sigma,tcp}(E) & \,\text{timelike compact support} \end{aligned}

for the subsets on those smooth sections whose support is

1. ($cp$) inside a compact subset,

2. ($\pm cp$) inside the closed future cone/closed past cone, respectively, of a compact subset,

3. ($scp$) inside the closed causal cone of a compact subset, which equivalently means that the intersection with every (spacelike) Cauchy surface is compact (Sanders 13, theorem 2.2),

4. ($fcp$) inside the past of a Cauchy surface (Sanders 13, def. 3.2),

5. ($pcp$) inside the future of a Cauchy surface (Sanders 13, def. 3.2),

6. ($tcp$) inside the future of one Cauchy surface and the past of another (Sanders 13, def. 3.2).

###### Definition

(causal order)

Consider the relation on the set $P(\Sigma)$ of subsets of spacetime which says a subset $S_1 \subset \Sigma$ is not prior to a subset $S_2 \subset \Sigma$, denoted $S_1 {\vee\!\!\!\wedge} S_2$, if $S_1$ does not intersect the causal past of $S_2$ (def. ), or equivalently that $S_2$ does not intersect the causal future of $S_1$:

\begin{aligned} S_1 {\vee\!\!\!\wedge} S_2 & \;\;\coloneqq\;\; S_1 \cap \overline{V}^-(S_2) = \emptyset \\ & \;\;\Leftrightarrow\;\; S_2 \cap \overline{V}^+(S_1) = \emptyset \end{aligned} \,.

(Beware that this is just a relation, not an ordering, since it is not relation.)

If $S_1 {\vee\!\!\!\wedge} S_2$ and $S_2 {\vee\!\!\!\wedge} S_1$ we say that the two subsets are spacelike separated and write

$S_1 {\gt\!\!\!\!\lt} S_2 \;\;\;\coloneqq\;\;\; S_1 {\vee\!\!\!\wedge} S_2 \;\text{and}\; S_2 {\vee\!\!\!\wedge} S_1 \,.$
###### Definition

(causal complement and causal closure of subset of spacetime)

For $S \subset X$ a subset of spacetime, its causal complement $S^\perp$ is the complement of the causal cone:

$S^\perp \;\coloneqq\; S \setminus J_X(S) \,.$

The causal complement $S^{\perp \perp}$ of the causal complement $S^\perp$ is called the causal closure. If

$S = S^{\perp \perp}$

then the subset $S$ is called a causally closed subset.

Given a spacetime $\Sigma$, we write

$CausClsdSubsets(\Sigma) \;\in\; Cat$

for the partially ordered set of causally closed subsets, partially ordered by inclusion $\mathcal{O}_1 \subset \mathcal{O}_2$.

###### Definition

For a causally closed subset $\mathcal{O} \subset \Sigma$ of spacetime (def. ) say that an adiabatic switching function or infrared cutoff function for $\mathcal{O}$ is a smooth function $g_{sw}$ of compact support (a bump function) whose restriction to some neighbourhood $U$ of $\mathcal{O}$ is the constant function with value $1$:

$Cutoffs(\mathcal{O}) \;\coloneqq\; \left\{ g_{sw} \in C^\infty_c(\Sigma) \;\vert\; \underset{ {U \supset \mathcal{O}} \atop { \text{neighbourhood} } }{\exists} \left( g_{sw}\vert_U = 1 \right) \right\} \,.$

Often we consider the vector space space $C^\infty(\Sigma)\langle g \rangle$ spanned by a formal variable $g$ (the coupling constant) under multiplication with smooth functions, and consider as adiabatic switching functions the corresponding images in this space,

$\array{ C_c^\infty(\Sigma) &\overset{\simeq}{\longrightarrow}& C_c^\infty(X)\langle g\rangle }$

which are thus bump functions constant over a neighbourhood $U$ of $\mathcal{O}$ not on 1 but on the formal parameter $g$:

$g_{sw}\vert_U = g \,$

In this sense we may think of the adiabatic switching as being the spacetime-depependent coupling “constant”.

The following lemma will be key in the derivation (proof of prop. below) of the causal locality of algebra of quantum observables in perturbative quantum field theory:

###### Lemma

(causal partition)

Let $\mathcal{O} \subset \Sigma$ be a causally closed subset (def. ) and let $f \in C^\infty_{cp}(\Sigma)$ be a compactly supported smooth function which vanishes on a neighbourhood $U \supset \mathcal{O}$, i.e. $f\vert_U = 0$.

Then there exists a causal partition of $f$ in that there exist compactly supported smooth functions $a,r \in C^\infty_{cp}(\Sigma)$ such that

1. they sum up to $f$:

$f = a + r$
2. their support satisfies the following causal ordering (def. )

$supp(a) {\vee\!\!\!\wedge} \mathcal{O} {\vee\!\!\!\wedge} supp(r) \,.$
###### Proof idea

By assumption $\mathcal{O}$ has a Cauchy surface. This may be extended to a Cauchy surface $\Sigma_p$ of $\Sigma$, such that this is one leaf of a foliation of $\Sigma$ by Cauchy surfaces, given by a diffeomorphism $\Sigma \simeq (-1,1) \times \Sigma_p$ with the original $\Sigma_p$ at zero. There exists then $\epsilon \in (0,1)$ such that the restriction of $supp(f)$ to the interval $(-\epsilon, \epsilon)$ is in the causal complement $\overline{\mathcal{O}}$ of the given region (def. ):

$supp(f) \cap (-\epsilon, \epsilon) \times \Sigma_p \;\subset\; \overline{\mathcal{O}} \,.$

Let then $\chi \colon \Sigma \to \mathbb{R}$ be any smooth function with

1. $\chi\vert_{(-1,0] \times \Sigma_p} = 1$

2. $\chi\vert_{(\epsilon,1) \times \Sigma_p} = 0$.

Then

$r \coloneqq \chi \cdot f \phantom{AAA} \text{and} \phantom{AAA} a \coloneqq (1-\chi) \cdot f$

are smooth functions as required.

$\,$

This concludes our discussion of spin and spacetime. In the next chapter we consider the concept of fields on spacetime.

## Fields

In this chapter we discuss these topics:

$\,$

A field history on a given spacetime $\Sigma$ (a history of spatial field configurations, see remark below) is a quantity assigned to each point of spacetime (each event), such that this assignment varies smoothly with spacetime points. For instance an electromagnetic field history (example below) is at each point of spacetime a collection of vectors that encode the direction in which a charged particle passing through that point would feel a force (the “Lorentz force”, see example below).

This is readily formalized (def. below): If $F$ denotes the smooth manifold of “values” that the given kind of field may take at any spacetime point, then a field history $\Phi$ is modeled as a smooth function from spacetime to this space of values:

$\Phi \;\colon\; \Sigma \longrightarrow F \,.$

It will be useful to unify spacetime and the space of field values (the field fiber) into a single manifold, the Cartesian product

$E \;\coloneqq\; \Sigma \times F$

and to think of this equipped with the projection map onto the first factor as a fiber bundle of spaces of field values over spacetime

$\array{ E &\coloneqq& \Sigma \times F \\ {}^{\mathllap{fb}}\downarrow & \swarrow_{\mathrlap{pr_1}} \\ \Sigma } \,.$

This is then called the field bundle, which specifies the kind of values that the given field species may take at any point of spacetime. Since the space $F$ of field values is the fiber of this fiber bundle (def. ), it is sometimes also called the field fiber. (See also at fiber bundles in physics.)

Given a field bundle $E \overset{fb}{\to}\Sigma$, then a field history is a section of that bundle (def. ). The discussion of field theory concerns the space of all possible field histories, hence the space of sections of the field bundle (example below). This is a very “large” generalized smooth space, called a diffeological space (def. below).

Or rather, in the presence of fermion fields such as the Dirac field (example below), the Pauli exclusion principle demands that the field bundle is a super-manifold, and that the fermionic space of field histories (example below) is a super-geometric generalized smooth space: a super smooth set (def. below).

This smooth structure on the space of field histories will be crucial when we discuss observables of a field theory below, because these are smooth functions on the space of field histories. In particular it is this smooth structure which allows to derive that linear observables of a free field theory are given by distributions (prop. ) below. Among these are the point evaluation observables (delta distributions) which are traditionally denoted by the same symbol as the field histories themselves.

Hence there are these aspects of the concept of “field” in physics, which are closely related, but crucially different:

$\,$

aspects of the concept of fields

aspecttermtypedescriptiondef.
field component$\phi^a$, $\phi^a_{,\mu}$$J^\infty_\Sigma(E) \to \mathbb{R}$coordinate function on jet bundle of field bundledef. , def.
field history$\Phi$, $\frac{\partial \Phi}{\partial x^\mu}$$\Sigma \to J^\infty_\Sigma(E)$jet prolongation of section of field bundledef. , def.
field observable$\mathbf{\Phi}^a(x)$, $\partial_{\mu} \mathbf{\Phi}^a(x),$$\Gamma_{\Sigma}(E) \to \mathbb{R}$derivatives of delta-functional on space of sectionsdef. , example
averaging of field observable$\alpha^\ast \mapsto \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)$$\Gamma_{\Sigma,cp}(E^\ast) \to Obs(E_{scp},\mathbf{L})$observable-valued distributiondef.
algebra of quantum observables$\left( Obs(E,\mathbf{L})_{\mu c},\, \star\right)$$\mathbb{C}Alg$non-commutative algebra structure on field observablesdef. , def.

$\,$

field bundles

###### Definition

(fields and field histories)

Given a spacetime $\Sigma$, then a type of fields on $\Sigma$ is a smooth fiber bundle (def. )

$\array{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }$

called the field bundle,

Given a type of fields on $\Sigma$ this way, then a field history of that type on $\Sigma$ is a term of that type, hence is a smooth section (def. ) of this bundle, namely a smooth function of the form

$\Phi \;\colon\; \Sigma \longrightarrow E$

such that composed with the projection map it is the identity function, i.e. such that

$fb \circ \Phi = id \phantom{AAAAAAA} \array{ && E \\ & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}} \\ \Sigma & = & \Sigma } \,.$

The set of such sections/field histories is to be denoted

(18)$\Gamma_\Sigma(E) \;\coloneqq\; \left\{ \array{ && E \\ & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}} \\ \Sigma &=& \Sigma } \phantom{fb} \right\}$
###### Remark

(field histories are histories of spatial field configurations)

Given a section $\Phi \in \Gamma_\Sigma(E)$ of the field bundle (def. ) and given a spacelike (def. ) submanifold $\Sigma_p \hookrightarrow \Sigma$ (def. ) of spacetime in codimension 1, then the restriction $\Phi\vert_{\Sigma_p}$ of $\Phi$ to $\Sigma_p$ may be thought of as a field configuration in space. As different spatial slices $\Sigma_p$ are chosen, one obtains such field configurations at different times. It is in this sense that the entirety of a section $\Phi \in \Gamma_\Sigma(E)$ is a history of field configurations, hence a field history (def ).

###### Remark

(possible field histories)

After we give the set $\Gamma_\Sigma(E)$ of field histories (18) differential geometric structure, below in example and example , we call it the space of field histories. This should be read as space of possible field histories; containing all field histories that qualify as being of the type specified by the field bundle $E$.

After we obtain equations of motion below in def. , these serve as the “laws of nature” that field histories should obey, and they define the subspace of those field histories that do solve the equations of motion; this will be denoted

$\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L}= 0} \overset{\phantom{AAA}}{\hookrightarrow} \Gamma_\Sigma(E)$

and called the on-shell space of field histories (41).

For the time being, not to get distracted from the basic idea of quantum field theory, we will focus on the following simple special case of field bundles:

###### Example

(trivial vector bundle as a field bundle)

In applications the field fiber $F = V$ is often a finite dimensional vector space. In this case the trivial field bundle with fiber $F$ is of course a trivial vector bundle (def. ).

Choosing any linear basis $(\phi^a)_{a = 1}^s$ of the field fiber, then over Minkowski spacetime (def. ) we have canonical coordinates on the total space of the field bundle

$( (x^\mu), ( \phi^a ) ) \,,$

where the index $\mu$ ranges from $0$ to $p$, while the index $a$ ranges from 1 to $s$.

If this trivial vector bundle is regarded as a field bundle according to def. , then a field history $\Phi$ is equivalently an $s$-tuple of real-valued smooth functions $\Phi^a \colon \Sigma \to \mathbb{R}$ on spacetime:

$\Phi = ( \Phi^a )_{a = 1}^s \,.$
###### Example

(field bundle for real scalar field)

If $\Sigma$ is a spacetime and if the field fiber

$F \coloneqq \mathbb{R}$

is simply the real line, then the corresponding trivial field bundle (def. )

$\array{ \Sigma \times \mathbb{R} \\ {}^{\mathllap{pr_1}}\downarrow \\ \Sigma }$

is the trivial real line bundle (a special case of example ) and the corresponding field type (def. ) is called the real scalar field on $\Sigma$. A configuration of this field is simply a smooth function on $\Sigma$ with values in the real numbers:

(19)$\Gamma_\Sigma(\Sigma \times \mathbb{R}) \;\simeq\; C^\infty(\Sigma) \,.$
###### Example

(field bundle for electromagnetic field)

On Minkowski spacetime $\Sigma$ (def. ), let the field bundle (def. ) be given by the cotangent bundle

$E \coloneqq T^\ast \Sigma \,.$

This is a trivial vector bundle (example ) with canonical field coordinates $(a_\mu)$.

A section of this bundle, hence a field history, is a differential 1-form

$A \in \Gamma_\Sigma(T^\ast \Sigma) = \Omega^1(\Sigma)$

on spacetime (def. ). Interpreted as a field history of the electromagnetic field on $\Sigma$, this is often called the vector potential. Then the de Rham differential (def. ) of the vector potential is a differential 2-form

$F \coloneqq d A$

known as the Faraday tensor. In the canonical coordinate basis 2-forms this may be expanded as

(20)$F = \underoverset{i = 1}{p}{\sum} E_i d x^0 \wedge d x^i + \underset{1 \leq i \lt j \leq p}{\sum} B_{i j} d x^i \wedge d x^j \,.$

Here $(E_i)_{i = 1}^p$ are called the components of the electric field, while $(B_{i j})$ are called the components of the magnetic field.

###### Example

(field bundle for Yang-Mills field over Minkowski spacetime)

Let $\mathfrak{g}$ be a Lie algebra of finite dimension with linear basis $(t_\alpha)$, in terms of which the Lie bracket is given by

(21)$[t_\alpha, t_\beta] \;=\; \gamma^\gamma{}_{\alpha \beta} t_\gamma \,.$

Over Minkowski spacetime $\Sigma$ (def. ), consider then the field bundle which is the cotangent bundle tensored with the Lie algebra $\mathfrak{g}$

$E \coloneqq T^\ast \Sigma \otimes \mathfrak{g} \,.$

This is the trivial vector bundle (example ) with induced field coordinates

$( a_\mu^\alpha ) \,.$

A section of this bundle is a Lie algebra-valued differential 1-form

$A \in \Gamma_\Sigma(T^\ast \Sigma \otimes \mathfrak{g}) = \Omega^1(\Sigma, \mathfrak{g}) \,.$

with components

$A^\ast(a_\mu^\alpha) = A^\alpha_\mu \,.$

This is called a field history for Yang-Mills gauge theory (at least if $\mathfrak{g}$ is a semisimple Lie algebra, see example below).

For $\mathfrak{g} = \mathbb{R}$ is the line Lie algebra, this reduces to the case of the electromagnetic field (example ).

For $\mathfrak{g} = \mathfrak{su}(3)$ this is a field history for the gauge field of the strong nuclear force in quantum chromodynamics.

For readers familiar with the concepts of principal bundles and connections on a bundle we include the following example which generalizes the Yang-Mills field over Minkowski spacetime from example to the situation over general spacetimes.

###### Example

(general Yang-Mills field in fixed topological sector)

Let $\Sigma$ be any spacetime manifold and let $G$ be a compact Lie group with Lie algebra denoted $\mathfrak{g}$. Let $P \overset{is}{\to} \Sigma$ be a $G$-principal bundle and $\nabla_0$ a chosen connection on it, to be called the background $G$-Yang-Mills field.

Then the field bundle (def. ) for $G$-Yang-Mills theory in the topological sector $P$ is the tensor product of vector bundles

$E \coloneqq \left(P \times^{ad}_G \mathfrak{g}\right) \otimes_\Sigma \left( T^\ast \Sigma \right)$

of the adjoint bundle of $P$ and the cotangent bundle of $\Sigma$.

With the choice of $\nabla_0$, every (other) connection $\nabla$ on $P$ uniquely decomposes as

$\nabla = \nabla_0 + A \,,$

where

$A \in \Gamma_\Sigma(E)$

is a section of the above field bundle, hence a Yang-Mills field history.

The electromagnetic field (def. ) and the Yang-Mills field (def. , def. ) with differential 1-forms as field histories are the basic examples of gauge fields (we consider this in more detail below in Gauge symmetries). There are also higher gauge fields with differential n-forms as field histories:

###### Example

(field bundle for B-field)

On Minkowski spacetime $\Sigma$ (def. ), let the field bundle (def. ) be given by the skew-symmetrized tensor product of vector bundles of the cotangent bundle with itself

$E \coloneqq \wedge^2_\Sigma T^\ast \Sigma \,.$

This is a trivial vector bundle (example ) with canonical field coordinates $(b_{\mu \nu})$ subject to

$b_{\mu \nu} \;=\; - b_{\nu \mu} \,.$

A section of this bundle, hence a field history, is a differential 2-form (def. )

$B \in \Gamma_\Sigma(\wedge^2_\Sigma T^\ast \Sigma) = \Omega^2(\Sigma)$

on spacetime.

$\,$

space of field histories

Given any field bundle, we will eventually need to regard the set of all field histories $\Gamma_\Sigma(E)$ as a “smooth set” itself, a smooth space of sections, to which constructions of differential geometry apply (such as for the discussion of observables and states below ). Notably we need to be talking about differential forms on $\Gamma_\Sigma(E)$.

However, a space of sections $\Gamma_\Sigma(E)$ does not in general carry the structure of a smooth manifold; and it carries the correct smooth structure of an infinite dimensional manifold only if $\Sigma$ is a compact space (see at manifold structure of mapping spaces). Even if it does carry infinite dimensional manifold structure, inspection shows that this is more structure than actually needed for the discussion of field theory. Namely it turns out below that all we need to know is what counts as a smooth family of sections/field histories, hence which functions of sets

$\Phi_{(-)} \;\colon\; \mathbb{R}^n \longrightarrow \Gamma_\Sigma(E)$

from any Cartesian space $\mathbb{R}^n$ (def. ) into $\Gamma_\Sigma(E)$ count as smooth functions, subject to some basic consistency condition on this choice.

This structure on $\Gamma_\Sigma(E)$ is called the structure of a diffeological space:

###### Definition

(diffeological space)

A diffeological space $X$ is

1. a set $X_s \in$ Set;

2. for each $n \in \mathbb{N}$ a choice of subset

$X(\mathbb{R}^n) \subset Hom_{Set}(\mathbb{R}^n_s, X_s) = \left\{ \mathbb{R}^n_s \to X_s \right\}$

of the set of functions from the underlying set $\mathbb{R}^n_s$ of $\mathbb{R}^n$ to $X_s$, to be called the smooth functions or plots from $\mathbb{R}^n$ to $X$;

3. for each smooth function $f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$ between Cartesian spaces (def. ) a choice of function

$f^\ast \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1})$

to be thought of as the precomposition operation

$\left( \mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X \right) \;\overset{f^\ast}{\mapsto}\; \left( \mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{\Phi}{\to} X \right)$

such that

1. (constant functions are smooth)

$X(\mathbb{R}^0) = X_s \,,$
1. If $id_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \to \mathbb{R}^n$ is the identity function on $\mathbb{R}^n$, then $\left(id_{\mathbb{R}^n}\right)^\ast \;\colon\; X(\mathbb{R}^n) \to X(\mathbb{R}^n)$ is the identity function on the set of plots $X(\mathbb{R}^n)$;

2. If $\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{g}{\to} \mathbb{R}^{n_3}$ are two composable smooth functions between Cartesian spaces (def. ), then pullback of plots along them consecutively equals the pullback along the composition:

$f^\ast \circ g^\ast = (g \circ f)^\ast$

i.e.

$\array{ && X(\mathbb{R}^{n_2}) \\ & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}} \\ X(\mathbb{R}^{n_1}) && \underset{ (g \circ f)^\ast }{\longleftarrow} && X(\mathbb{R}^{n_3}) }$
2. (gluing)

If $\{ U_i \overset{f_i}{\to} \mathbb{R}^n\}_{i \in I}$ is a differentiably good open cover of a Cartesian space (def. ) then the function which restricts $\mathbb{R}^n$-plots of $X$ to a set of $U_i$-plots

$X(\mathbb{R}^n) \overset{( (f_i)^\ast )_{i \in I} }{\hookrightarrow} \underset{i \in I}{\prod} X(U_i)$

is a bijection onto the set of those tuples $(\Phi_i \in X(U_i))_{i \in I}$ of plots, which are “matching families” in that they agree on intersections:

$\phi_i\vert_{U_i \cap U_j} = \phi_j \vert_{U_i \cap U_j} \phantom{AAAAAA} \array{ && U_i \cap U_j \\ & \swarrow && \searrow \\ U_i && && U_j \\ & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}} \\ && X }$

Finally, given $X_1$ and $X_2$ two diffeological spaces, then a smooth function between them

$f \;\colon\; X_1 \longrightarrow X_2$

is

• a function of the underlying sets

$f_s \;\colon\; (X_1)_s \longrightarrow (X_2)_s$

such that

• for $\Phi \in X(\mathbb{R}^n)$ a plot of $X_1$, then the composition $f_s \circ \Phi_s$ is a plot $f_\ast(\Phi)$ of $X_2$:

$\array{ && \mathbb{R}^n \\ & {}^{\mathllap{\Phi}}\swarrow && \searrow^{\mathrlap{f_\ast(\Phi)}} \\ X_1 && \underset{f}{\longrightarrow} && X_2 } \,.$

(Stated more abstractly, this says simply that diffeological spaces are the concrete sheaves on the site of Cartesian spaces from def. .)

For more background on diffeological spaces see also geometry of physics – smooth sets.

###### Example

(Cartesian spaces are diffeological spaces)

Let $X$ be a Cartesian space (def. ) Then it becomes a diffeological space (def. ) by declaring its plots $\Phi \in X(\mathbb{R}^n)$ to the ordinary smooth functions $\Phi \colon \mathbb{R}^n \to X$.

Under this identification, a function $f \;\colon\; (X_1)_s \to (X_2)_s$ between the underlying sets of two Cartesian spaces is a smooth function in the ordinary sense precisely if it is a smooth function in the sense of diffeological spaces.

Stated more abstractly, this statement is an example of the Yoneda embedding over a subcanonical site.

More generally, the same construction makes every smooth manifold a smooth set.

###### Example

(diffeological space of field histories)

Let $E \overset{fb}{\to} \Sigma$ be a smooth field bundle (def. ). Then the set $\Gamma_\Sigma(E)$ of field histories/sections (def. ) becomes a diffeological space (def. )

(22)$\Gamma_\Sigma(E) \in DiffeologicalSpaces$

by declaring that a smooth family $\Phi_{(-)}$ of field histories, parameterized over any Cartesian space $U$ is a smooth function out of the Cartesian product manifold of $\Sigma$ with $U$

$\array{ U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E \\ (u,x) &\mapsto& \Phi_u(x) }$

such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.

$\array{ && E \\ & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}} \\ U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma } \,.$

The following example is included only for readers who wonder how infinite-dimensional manifolds fit in. Since we will never actually use infinite-dimensional manifold-structure, this example is may be ignored.

###### Example

(Fréchet manifolds are diffeological spaces)

Consider the particular type of infinite-dimensional manifolds called Fréchet manifolds. Since ordinary smooth manifolds $U$ are an example, for $X$ a Fréchet manifold there is a concept of smooth functions $U \to X$. Hence we may give $X$ the structure of a diffeological space (def. ) by declaring the plots over $U$ to be these smooth functions $U \to X$, with the evident postcomposition action.

It turns out that then that for $X$ and $Y$ two Fréchet manifolds, there is a natural bijection between the smooth functions $X \to Y$ between them regarded as Fréchet manifolds and [regarded as . Hence it does not matter which of the two perspectives we take (unless of course a more general than a enters the picture, at which point the second definition generalizes, whereas the first does not).]

Stated more abstractly, this means that Fréchet manifolds form a full subcategory of that of diffeological spaces (this prop.):

$FrechetManifolds \hookrightarrow DiffeologicalSpaces \,.$

If $\Sigma$ is a compact smooth manifold and $E \simeq \Sigma \times F \to \Sigma$ is a trivial fiber bundle with fiber $F$ a smooth manifold, then the set of sections $\Gamma_\Sigma(E)$ carries a standard structure of a Fréchet manifold (see at manifold structure of mapping spaces). Under the above inclusion of Fréchet manifolds into diffeological spaces, this smooth structure agrees with that from example (see this prop.)

Once the step from smooth manifolds to diffeological spaces (def. ) is made, characterizing the smooth structure of the space entirely by how we may probe it by mapping smooth Cartesian spaces into it, it becomes clear that the underlying set $X_s$ of a diffeological space $X$ is not actually crucial to support the concept: The space is already entirely defined structurally by the system of smooth plots it has, and the underlying set $X_s$ is recovered from these as the set of plots from the point $\mathbb{R}^0$.

This is crucial for field theory: the spaces of field histories of fermionic fields (def. below) such as the Dirac field (example below) do not have underlying sets of points the way diffeological spaces have. Informally, the reason is that a point is a bosonic object, while and the nature of fermionic fields is the opposite of bosonic.

But we may just as well drop the mentioning of the underlying set $X_s$ in the definition of generalized smooth spaces. By simply stripping this requirement off of def. we obtain the following more general and more useful definition (still “bosonic”, though, the supergeometric version is def. below):

###### Definition

(smooth set)

A smooth set $X$ is

1. for each $n \in \mathbb{N}$ a choice of set

$X(\mathbb{R}^n) \in Set$

to be called the set of smooth functions or plots from $\mathbb{R}^n$ to $X$;

2. for each smooth function $f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$ between Cartesian spaces a choice of function

$f^\ast \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1})$

to be thought of as the precomposition operation

$\left( \mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X \right) \;\overset{f^\ast}{\mapsto}\; \left( \mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{\Phi}{\to} X \right)$

such that

1. If $id_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \to \mathbb{R}^n$ is the identity function on $\mathbb{R}^n$, then $\left(id_{\mathbb{R}^n}\right)^\ast \;\colon\; X(\mathbb{R}^n) \to X(\mathbb{R}^n)$ is the identity function on the set of plots $X(\mathbb{R}^n)$.

2. If $\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{g}{\to} \mathbb{R}^{n_3}$ are two composable smooth functions between Cartesian spaces, then consecutive pullback of plots along them equals the pullback along the composition:

$f^\ast \circ g^\ast = (g \circ f)^\ast$

i.e.

$\array{ && X(\mathbb{R}^{n_2}) \\ & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}} \\ X(\mathbb{R}^{n_1}) && \underset{ (g \circ f)^\ast }{\longleftarrow} && X(\mathbb{R}^{n_3}) }$
1. (gluing)

If $\{ U_i \overset{f_i}{\to} \mathbb{R}^n\}_{i \in I}$ is a differentiably good open cover of a Cartesian space (def. ) then the function which restricts $\mathbb{R}^n$-plots of $X$ to a set of $U_i$-plots

$X(\mathbb{R}^n) \overset{( (f_i)^\ast )_{i \in I} }{\hookrightarrow} \underset{i \in I}{\prod} X(U_i)$

is a bijection onto the set of those tuples $(\Phi_i \in X(U_i))_{i \in I}$ of plots, which are “matching families” in that they agree on intersections:

$\phi_i\vert_{U_i \cap U_j} = \phi_j \vert_{U_i \cap U_j} \phantom{AAAA} \text{i.e.} \phantom{AAAA} \array{ && U_i \cap U_j \\ & \swarrow && \searrow \\ U_i && && U_j \\ & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}} \\ && X }$

Finally, given $X_1$ and $X_2$ two smooth sets, then a smooth function between them

$f \;\colon\; X_1 \longrightarrow X_2$

is

• for each $n \in \mathbb{N}$ a function

$f_\ast(\mathbb{R}^n) \;\colon\; X_1(\mathbb{R}^n) \longrightarrow X_2(\mathbb{R}^n)$

such that

• for each smooth function $g \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between Cartesian spaces we have

$g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2}) = f_\ast(\mathbb{R}^{n_1}) \circ g^\ast_1 \phantom{AAAAA} \text{i.e.} \phantom{AAAAA} \text{i.e.} \phantom{AAAAA} \array{ X_1(\mathbb{R}^{n_2}) &\overset{f_\ast(\mathbb{R}^{n_2})}{\longrightarrow}& X_2(\mathbb{R}^{n_2}) \\ \mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2} \\ X_1(\mathbb{R}^{n_1}) &\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}& X_2(\mathbb{R}^{n_1}) }$

Stated more abstractly, this simply says that smooth sets are the sheaves on the site of Cartesian spaces from def. .

Basing differential geometry on smooth sets is an instance of the general approach to geometry called functorial geometry or topos theory. For more background on this see at geometry of physics – smooth sets.

First we verify that the concept of smooth sets is a consistent generalization:

###### Example

(diffeological spaces are smooth sets)

Every diffeological space $X$ (def. ) is a smooth set (def. ) simply by forgetting its underlying set of points and remembering only its sets of plot.

In particular therefore each Cartesian space $\mathbb{R}^n$ is canonically a smooth set by example .

Moreover, given any two diffeological spaces, then the morphisms $f \colon X \to Y$ between them, regarded as diffeological spaces, are the same as the morphisms as smooth sets.

Stated more abstractly, this means that we have full subcategory inclusions

$CartesianSpaces \overset{\phantom{AAA}}{\hookrightarrow} DiffeologicalSpaces \overset{\phantom{AAA}}{\hookrightarrow} SmoothSets \,.$

Recall, for the next proposition , that in the definition of a smooth set $X$ the sets $X(\mathbb{R}^n)$ are abstract sets which are to be thought of as would-be smooth functions “$\mathbb{R}^n \to X$”. Inside def. this only makes sense in quotation marks, since inside that definition the smooth set $X$ is only being defined, so that inside that definition there is not yet an actual concept of smooth functions of the form “$\mathbb{R}^n \to X$”.

But now that the definition of smooth sets and of morphisms between them has been stated, and seeing that Cartesian space $\mathbb{R}^n$ are examples of smooth sets, by example , there is now an actual concept of smooth functions $\mathbb{R}^n \to X$, namely as smooth sets. For the concept of smooth sets to be consistent, it ought to be true that this a posteriori concept of smooth functions from Cartesian spaces to smooth sets coincides wth the a priori concept, hence that we “may remove the quotation marks” in the above. The following proposition says that this is indeed the case:

###### Proposition

(plots of a smooth set really are the smooth functions into the smooth set)

Let $X$ be a smooth set (def. ). For $n \in \mathbb{R}$, there is a natural function

$Hom_{SmoothSet}(\mathbb{R}^n , X) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} X(\mathbb{R}^n)$

from the set of homomorphisms of smooth sets from $\mathbb{R}^n$ (regarded as a smooth set via example ) to $X$, to the set of plots of $X$ over $\mathbb{R}^n$, given by evaluating on the identity plot $id_{\mathbb{R}^n}$.

This function is a bijection.

This says that the plots of $X$, which initially bootstrap $X$ into being as declaring the would-be smooth functions into $X$, end up being the actual smooth functions into $X$.

###### Proof

This elementary but profound fact is called the Yoneda lemma, here in its incarnation over the site of Cartesian spaces (def. ).

A key class of examples of smooth sets (def. ) that are not diffeological spaces (def. ) are universal smooth moduli spaces of differential forms:

###### Example

(universal smooth moduli spaces of differential forms)

For $k \in \mathbb{N}$ there is a smooth set (def. )

$\mathbf{\Omega}^k \;\in\; SmoothSet$

defined as follows:

1. for $n \in \mathbb{N}$ the set of plots from $\mathbb{R}^n$ to $\mathbf{\Omega}^k$ is the set of smooth differential k-forms on $\mathbb{R}^n$ (def. )

$\mathbf{\Omega}^k(\mathbb{R}^n) \;\coloneqq\; \Omega^k(\mathbb{R}^n)$
2. for $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a smooth function (def. ) the operation of pullback of plots along $f$ is just the pullback of differential forms $f^\ast$ from prop.

$\array{ \mathbb{R}^{n_1} && \Omega^k(\mathbb{R}^{n_1}) \\ \downarrow^{\mathrlap{f}} && \uparrow^{\mathrlap{f^\ast}} \\ \mathbb{R}^{n_2} && \Omega^k(\mathbb{R}^{n_2}) }$

That this is functorial is just the standard fact (7) from prop. .

For $k = 1$ the smooth set $\mathbf{\Omega}^0$ actually is a diffeological space, in fact under the identification of example this is just the real line:

$\mathbf{\Omega}^0 \simeq \mathbb{R}^1 \,.$

But for $k \geq 1$ we have that the set of plots on $\mathbb{R}^0 = \ast$ is a singleton

$\mathbf{\Omega}^{k \geq 1}(\mathbb{R}^0) \simeq \{0\}$

consisting just of the zero differential form. The only diffeological space with this property is $\mathbb{R}^0 = \ast$ itself. But $\mathbf{\Omega}^{k \geq 1}$ is far from being that trivial: even though its would-be underlying set is a single point, for all $n \geq k$ it admits an infinite set of plots. Therefore the smooth sets $\mathbf{\Omega}^k$ for $k \geq$ are not diffeological spaces.

That the smooth set $\mathbf{\Omega}^k$ indeed deserves to be addressed as the universal moduli space of differential k-forms follows from prop. : The universal moduli space of $k$-forms ought to carry a universal differential $k$-forms $\omega_{univ} \in \Omega^k(\mathbf{\Omega}^k)$ such that every differential $k$-form $\omega$ on any $\mathbb{R}^n$ arises as the pullback of differential forms of this universal one along some modulating morphism $f_\omega \colon X \to \mathbf{\Omega}^k$:

$\array{ \{\omega\} &\overset{(f_\omega)^\ast}{\longleftarrow}& \{\omega_{univ}\} \\ \\ X &\underset{f_\omega}{\longrightarrow}& \mathbf{\Omega}^k }$

But with prop. this is precisely what the definition of the plots of $\mathbf{\Omega}^k$ says.

Similarly, all the usual operations on differential form now have their universal archetype on the universal moduli spaces of differential forms

In particular, for $k \in \mathbb{N}$ there is a canonical morphism of smooth sets of the form

$\mathbf{\Omega}^k \overset{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^{k+1}$

defined over $\mathbb{R}^n$ by the ordinary de Rham differential (def. )

(23)$\Omega^k(\mathbb{R}^n) \overset{d}{\longrightarrow} \Omega^{k+1}(\mathbb{R}^n) \,.$

That this satisfies the compatibility with precomposition of plots

$\array{ \mathbb{R}^{n_1} && \Omega^k(\mathbb{R}^{n_1}) &\overset{d}{\longrightarrow}& \Omega^{k+1}(\mathbb{R}^{n_1}) \\ {}^{\mathllap{f}}\downarrow && \uparrow^{\mathrlap{f^\ast}} && \uparrow^{\mathrlap{f^\ast}} \\ \mathbb{R}^{n_2} && \Omega^k(\mathbb{R}^{n_2}) &\underset{d}{\longrightarrow}& \Omega^k( \mathbb{R}^{n_2} ) }$

is just the compatibility of pullback of differential forms with the de Rham differential of from prop. .

The upshot is that we now have a good definition of differential forms on any diffeological space and more generally on any smooth set:

###### Definition

(differential forms on smooth sets)

Let $X$ be a diffeological space (def. ) or more generally a smooth set (def. ) then a differential k-form $\omega$ on $X$ is equivalently a morphism of smooth sets

$X \longrightarrow \mathbf{\Omega}^k$

from $X$ to the universal smooth moduli space of differential froms from example .

Concretely, by unwinding the definitions of $\mathbf{\Omega}^k$ and of morphisms of smooth sets, this means that such a differential form is:

• for each $n \in \mathbb{N}$ and each plot $\mathbb{R}^n \overset{\Phi}{\to} X$ an ordinary differential form

$\Phi^\ast(\omega) \in \Omega^\bullet(\mathbb{R}^n)$

such that

• for each smooth function $f \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between Cartesian spaces the ordinary pullback of differential forms along $f$ is compatible with these choices, in that for every plot $\mathbb{R}^{n_2} \overset{\Phi}{\to} X$ we have

$f^\ast\left(\Phi^\ast(\omega)\right) = ( f^\ast \Phi )^\ast(\omega)$

i.e.

$\array{ \mathbb{R}^{n_1} && \overset{f}{\longrightarrow} && \mathbb{R}^{n_2} \\ & {}_{\mathllap{f^\ast \Phi}}\searrow && \swarrow_{\mathrlap{\Phi}} \\ && X } \phantom{AAAA} \array{ \Omega^\bullet( \mathbb{R}^{n_1} ) && \overset{f^\ast}{\longleftarrow} && \Omega^\bullet(\mathbb{R}^{n_2}) \\ & {}_{\mathllap{(f^\ast \Phi)^\ast}}\nwarrow && \nearrow_{\mathrlap{\Phi^\ast}} \\ && \Omega^\bullet(X) } \,.$

We write $\Omega^\bullet(X)$ for the set of differential forms on the smooth set $X$ defined this way.

Moreover, given a differential k-form

$X \overset{\omega}{\longrightarrow} \mathbf{\Omega}^k$

on a smooth set $X$ this way, then its de Rham differential $d \omega \in \Omega^{k+1}(X)$ is given by the composite of morphisms of smooth sets with the universal de Rham differential from (23):

(24)$d \omega \;\colon\; X \overset{\omega}{\longrightarrow} \mathbf{\Omega}^k \overset{d}{\longrightarrow} \mathbf{\Omega}^{k+1} \,.$

Explicitly this means simply that for $\Phi \colon U \to X$ a plot, then

$\Phi^\ast (d\omega) \;=\; d\left( \Phi^\ast \omega\right) \;\in\; \Omega^{k+1}(U) \,.$

The usual operations on ordinary differential forms directly generalize plot-wise to differential forms on diffeological spaces and more generally on smooth sets:

###### Definition

(exterior differential and exterior product on smooth sets)

Let $X$ be a diffeological space (def. ) or more generally a smooth set (def. ). Then

1. For $\omega \in \Omega^n(X)$ a differential form on $X$ (def. ) its exterior differential

$d \omega \in \Omega^{n+1}(X)$

is defined on any plot $\mathbb{R}^n \overset{\Phi}{\to} X$ as the ordinary exterior differential of the pullback of $\omega$ along that plot:

$\Phi^\ast(d \omega) \coloneqq d \Phi^\ast(\omega) \,.$
2. For $\omega_1 \in \Omega^{n_1}$ and $\omega_2 \in \Omega^{n_2}(X)$ two differential forms on $X$ (def. ) then their exterior product

$\omega_1 \wedge \omega_2 \;\in\; \Omega^{n_1 + n_2}(X)$

is the differential form defined on any plot $\mathbb{R}^n \overset{\Phi}{\to} X$ as the ordinary exterior product of the pullback of th differential forms $\omega_1$ and $\omega_2$ to this plot:

$\Phi^\ast(\omega_1 \wedge \omega_2) \;\coloneqq\; \Phi^\ast(\omega_1) \wedge \Phi^\ast(\omega_2) \,.$

$\,$

Infinitesimal geometry

It is crucial in field theory that we consider field histories not only over all of spacetime, but also restricted to submanifolds of spacetime. Or rather, what is actually of interest are the restrictions of the field histories to the infinitesimal neighbourhoods (example below) of these submanifolds. This appears notably in the construction of phase spaces below. Moreover, fermion fields such as the Dirac field (example below) take values in graded infinitesimal spaces, called super spaces (discussed below). Therefore “infinitesimal geometry”, sometimes called formal geometry (as in “formal scheme”) or synthetic differential geometry or synthetic differential supergeometry, is a central aspect of field theory.

In order to mathematically grasp what infinitesimal neighbourhoods are, we appeal to the first magic algebraic property of differential geometry from prop. , which says that we may recognize smooth manifolds $X$ dually in terms of their commutative algebras $C^\infty(X)$ of smooth functions on them

$C^\infty(-) \;\colon\; SmoothManifolds \overset{\phantom{AAA}}{\hookrightarrow} (\mathbb{R} Algebras)^{op} \,.$

But since there are of course more algebras $A \in \mathbb{R}Algebras$ than arise this way from smooth manifolds, we may turn this around and try to regard any algebra $A$ as defining a would-be space, which would have $A$ as its algebra of functions.

For example an infinitesimally thickened point should be a space which is “so small” that every smooth function $f$ on it which vanishes at the origin takes values so tiny that some finite power of them is not just even more tiny, but actually vanishes:

###### Definition

(infinitesimally thickened Cartesian space)

$\mathbb{D} \coloneqq Spec(A)$

is represented by a commutative algebra $A \in \mathbb{R}Alg$ which as a real vector space is a direct sum

$A \simeq_{\mathbb{R}} \langle 1 \rangle \oplus V$

of the 1-dimensional space $\langle 1 \rangle = \mathbb{R}$ of multiples of 1 with a finite dimensional vector space $V$ that is a nilpotent ideal in that for each element $a \in V$ there exists a natural number $n \in \mathbb{N}$ such that

$a^{n+1} = 0 \,.$

More generally, an infinitesimally thickened Cartesian space

$\mathbb{R}^n \times \mathbb{D} \;\coloneqq\; \mathbb{R}^n \times Spec(A)$

is represented by a commutative algebra

$C^\infty(\mathbb{R}^n) \otimes A \;\in\; \mathbb{R} Alg$

which is the tensor product of algebras of the algebra of smooth functions $C^\infty(\mathbb{R}^n)$ on an actual Cartesian space of some dimension $n$ (example ), with an algebra of functions $A \simeq_{\mathbb{R}} \langle 1\rangle \oplus V$ of an infinitesimally thickened point, as above.

We say that a smooth function between two infinitesimally thickened Cartesian spaces

$\mathbb{R}^{n_1} \times Spec(A_1) \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \times Spec(A_2)$

is by definition dually an $\mathbb{R}$-algebra homomorphism of the form

$C^\infty(\mathbb{R}^{n_1}) \otimes A_1 \overset{f^\ast}{\longleftarrow} C^\infty(\mathbb{R}^{n_2}) \otimes A_2 \,.$
###### Example

(infinitesimal neighbourhoods in the real line )

Consider the quotient algebra of the formal power series algebra $\mathbb{R}[ [\epsilon] ]$ in a single parameter $\epsilon$ by the ideal generated by $\epsilon^2$:

$(\mathbb{R}[ [\epsilon] ])/(\epsilon^2) \;\simeq_{\mathbb{R}}\; \mathbb{R} \oplus \epsilon \mathbb{R} \,.$

(This is sometimes called the algebra of dual numbers, for no good reason.) The underlying real vector space of this algebra is, as show, the direct sum of the multiples of 1 with the multiples of $\epsilon$. A general element in this algebra is of the form

$a + b \epsilon \in (\mathbb{R}[\epsilon])/(\epsilon^2)$

where $a,b \in \mathbb{R}$ are real numbers. The product in this algebra is given by “multiplying out” as usual, and discarding all terms proportional to $\epsilon^2$:

$\left( a_1 + b_1 \epsilon \right) \cdot \left( a_2 + b_2 \epsilon \right) \;=\; a_1 a_2 + ( a_1 b_2 + b_1 a_2 ) \epsilon \,.$

We may think of an element $a + b \epsilon$ as the truncation to first order of a Taylor series at the origin of a smooth function on the real line

$f \;\colon\; \mathbb{R} \to \mathbb{R}$

where $a = f(0)$ is the value of the function at the origin, and where $b = \frac{\partial f}{\partial x}(0)$ is its first derivative at the origin.

Therefore this algebra behaves like the algebra of smooth function on an infinitesimal neighbourhood $\mathbb{D}^1$ of $0 \in \mathbb{R}$ which is so tiny that its elements $\epsilon \in \mathbb{D}^1 \hookrightarrow \mathbb{R}$ become, upon squaring them, not just tinier, but actually zero:

$\epsilon^2 = 0 \,.$

This intuitive picture is now made precise by the concept of infinitesimally thickened points def. , if we simply set

$\mathbb{D}^1 \;\coloneqq\; Spec\left( \mathbb{R}[ [\epsilon] ]/(\epsilon^2) \right)$

and observe that there is the inclusion of infinitesimally thickened Cartesian spaces

$\mathbb{D}^1 \overset{\phantom{AA}i\phantom{AA} }{\hookrightarrow} \mathbb{R}^1$

which is dually given by the algebra homomorphism

$\array{ \mathbb{R} \oplus \epsilon \mathbb{R} &\overset{i^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^1) \\ f(0) + \frac{\partial f}{\partial x}(0) &\longleftarrow& \{f\} }$

which sends a smooth function to its value $f(0)$ at zero plus $\epsilon$ times its derivative at zero. Observe that this is indeed a homomorphism of algebras due to the product law of differentiation, which says that

\begin{aligned} i^\ast(f \cdot g) & = (f \cdot g)(0) + \frac{\partial f \cdot g}{\partial x}(0) \epsilon \\ & = f(0) \cdot g(0) + \left( \frac{\partial f}{\partial x}(0) \cdot g(0) + f(0) \cdot \frac{\partial g}{\partial x}(0) \right) \epsilon \\ & = \left( f(0) + \frac{\partial f}{\partial x}(0) \epsilon \right) \cdot \left( g(0) + \frac{\partial g}{\partial x}(0) \epsilon \right) \end{aligned}

Hence we see that restricting a smooth function to the infinitesimal neighbourhood of a point is equivalent to restricting attention to its Taylor series to the given order at that point:

$\array{ \mathbb{D}^1 &\overset{i}{\hookrightarrow}& \mathbb{R}^1 \\ & {}_{\mathllap{(\epsilon \mapsto f(0) + \frac{\partial f}{\partial x}(0) \epsilon) }}\searrow & \downarrow_{\mathrlap{f}} \\ && \mathbb{R}^1 }$

Similarly for each $k \in \mathbb{N}$ the algebra

$(\mathbb{R}[ [ \epsilon ] ])/(\epsilon^{k+1})$

may be thought of as the algebra of Taylor series at the origin of $\mathbb{R}$ of smooth functions $\mathbb{R} \to \mathbb{R}$, where all terms of order higher than $k$ are discarded. The corresponding infinitesimally thickened point is often denoted

$\mathbb{D}^1(k) \;\coloneqq\; Spec\left( \left(\mathbb{R}[ [\epsilon] ]\right)/(\epsilon^{k+1}) \right) \,.$

This is now the subobject of the real line

$\mathbb{D}^1(k) \overset{\phantom{AAA}}{\hookrightarrow} \mathbb{R}^1$

on those elements $\epsilon$ such that $\epsilon^{k+1} = 0$.

The following example shows that infinitesimal thickening is invisible for ordinary spaces when mapping out of these. In contrast example further below shows that the morphisms into an ordinary space out of an infinitesimal space are interesting: these are tangent vectors and their higher order infinitesimal analogs.

###### Example

(infinitesimal line $\mathbb{D}^1$ has unique global point)

For $\mathbb{R}^n$ any ordinary Cartesian space (def. ) and $D^1(k) \hookrightarrow \mathbb{R}^1$ the order-$k$ infinitesimal neighbourhood of the origin in the real line from example , there is exactly only one possible morphism of infinitesimally thickened Cartesian spaces from $\mathbb{R}^n$ to $\mathbb{D}^1(k)$:

$\array{ \mathbb{R}^n && \overset{\exists !}{\longrightarrow} &6 \mathbb{D}^1(k) \\ & {}_{\mathllap{\exists !}}\searrow && \nearrow_{\mathrlap{\exists !}} \\ && \mathbb{R}^0 = \ast } \,.$
###### Proof

By definition such a morphism is dually an algebra homomorphism

$C^\infty(\mathbb{R}^n) \overset{f^\ast}{\longleftarrow} \left( \mathbb{R}[ [\epsilon] ])/(\epsilon^{k+1} \right) \simeq_{\mathbb{R}} \mathbb{R} \oplus \mathcal{O}(\epsilon)$

from the higher order “algebra of dual numbers” to the algebra of smooth functions (example ).

Now this being an $\mathbb{R}$-algebra homomorphism, its action on the multiples $c \in \mathbb{R}$ of the identity is fixed:

$f^\ast(1) = 1 \,.$

All the remaining elements are proportional to $\epsilon$, and hence are nilpotent. However, by the homomorphism property of an algebra homomorphism it follows that it must send nilpotent elements $\epsilon$ to nilpotent elements $f(\epsilon)$, because

\begin{aligned} \left(f^\ast(\epsilon)\right)^{k+1} & = f^\ast\left( \epsilon^{k+1}\right) \\ & = f^\ast(0) \\ & = 0 \end{aligned}

But the only nilpotent element in $C^\infty(\mathbb{R}^n)$ is the zero element, and hence it follows that

$f^\ast(\epsilon) = 0 \,.$

Thus $f^\ast$ as above is uniquely fixed.

###### Example

(synthetic tangent vector fields)

Let $\mathbb{R}^n$ be a Cartesian space (def. ), regarded as an infinitesimally thickened Cartesian space (def. ) and consider $\mathbb{D}^1 \coloneqq Spec( (\mathbb{R}[ [\epsilon] ])/(\epsilon^2) )$ the first order infinitesimal line from example .

Then homomorphisms of infinitesimally thickened Cartesian spaces of the form

$\array{ \mathbb{R}^n \times \mathbb{D}^1 && \overset{\tilde v}{\longrightarrow} && \mathbb{R}^n \\ & {}_{\mathllap{pr_1}}\searrow && \swarrow_{\mathrlap{id}} \\ && \mathbb{R}^n }$

hence smoothly $X$-parameterized collections of morphisms

$\tilde v_x \;\colon\; \mathbb{D}^1 \longrightarrow \mathbb{R}^n$

which send the unique base point $\Re(\mathbb{D}^1) = \ast$ (example ) to $x \in \mathbb{R}^n$, are in natural bijection with tangent vector fields $v \in \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)$ (example ).

###### Proof

By definition, the morphisms in question are dually $\mathbb{R}$-algebra homomorphisms of the form

$(C^\infty(\mathbb{R}^n) \oplus \epsilon C^\infty(\mathbb{R}^n)) \longleftarrow C^\infty(\mathbb{R}^n)$

which are the identity modulo $\epsilon$. Such a morphism has to take any function $f \in C^\infty(\mathbb{R}^n)$ to

$f + (\partial f) \epsilon$

for some smooth function $(\partial f) \in C^\infty(\mathbb{R}^n)$. The condition that this assignment makes an algebra homomorphism is equivalent to the statement that for all $f_1,f_2 \in C^\infty(\mathbb{R}^n)$ we have

$(f_1 f_2 + (\partial (f_1 f_2))\epsilon ) \;=\; (f_1 + (\partial f_1) \epsilon) \cdot (f_2 + (\partial f_2) \epsilon) \,.$

Multiplying this out and using that $\epsilon^2 = 0$, this is equivalent to

$\partial(f_1 f_2) = (\partial f_1) f_2 + f_1 (\partial f_2) \,.$

This in turn means equivalently that $\partial\colon C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)$ is a derivation.

With this the statement follows with the third magic algebraic property of smooth functions (prop. ): derivations of smooth functions are vector fields.

We need to consider infinitesimally thickened spaces more general than the thickenings of just Cartesian spaces in def. . But just as Cartesian spaces (def. ) serve as the local test geometries to induce the general concept of diffeological spaces and smooth sets (def. ), so using infinitesimally thickened Cartesian spaces as test geometries immediately induces the corresponding generalization of smooth sets with infinitesimals:

###### Definition

(formal smooth set)

A formal smooth set $X$ is

1. for each infinitesimally thickened Cartesian space $\mathbb{R}^n \times Spec(A)$ (def. ) a set

$X(\mathbb{R}^n \times Spec(A)) \in Set$

to be called the set of smooth functions or plots from $\mathbb{R}^n \times Spec(A)$ to $X$;

2. for each smooth function $f \;\colon\; \mathbb{R}^{n_1} \times Spec(A_1) \longrightarrow \mathbb{R}^{n_2} \times Spec(A_2)$ between infinitesimally thickened Cartesian spaces a choice of function

$f^\ast \;\colon\; X(\mathbb{R}^{n_2} \times Spec(A_2)) \longrightarrow X(\mathbb{R}^{n_1} \times Spec(A_1))$

to be thought of as the precomposition operation

$\left( \mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X \right) \;\overset{f^\ast}{\mapsto}\; \left( \mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{\Phi}{\to} X \right)$

such that

1. If $id_{\mathbb{R}^n \times Spec(A)} \;\colon\; \mathbb{R}^n \times Spec(A) \to \mathbb{R}^n \times Spec(A)$ is the identity function on $\mathbb{R}^n \times Spec(A)$, then $\left(id_{\mathbb{R}^n \times Spec(A)}\right)^\ast \;\colon\; X(\mathbb{R}^n \times Spec(A)) \to X(\mathbb{R}^n \times Spec(A))$ is the identity function on the set of plots $X(\mathbb{R}^n \times Spec(A))$;

2. If $\mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{g}{\to} \mathbb{R}^{n_3} \times Spec(A_3)$ are two composable smooth functions between infinitesimally thickened Cartesian spaces, then pullback of plots along them consecutively equals the pullback along the composition:

$f^\ast \circ g^\ast = (g \circ f)^\ast$

i.e.

$\array{ && X(\mathbb{R}^{n_2} \times Spec(A_2)) \\ & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}} \\ X(\mathbb{R}^{n_1} \times Spec(A_1)) && \underset{ (g \circ f)^\ast }{\longleftarrow} && X(\mathbb{R}^{n_3} \times Spec(A_3)) }$
1. (gluing)

If $\{ U_i \times Spec(A) \overset{f_i \times id_{Spec(A)}}{\to} \mathbb{R}^n \times Spec(A)\}_{i \in I}$ is such that

$\{ U_i \overset{f_i }{\to} \mathbb{R}^n \}_{i \in I}$

in a differentiably good open cover (def. ) then the function which restricts $\mathbb{R}^n \times Spec(A)$-plots of $X$ to a set of $U_i \times Spec(A)$-plots

$X(\mathbb{R}^n \times Spec(A)) \overset{( (f_i)^\ast )_{i \in I} }{\hookrightarrow} \underset{i \in I}{\prod} X(U_i \times Spec(A))$

is a bijection onto the set of those tuples $(\Phi_i \in X(U_i))_{i \in I}$ of plots, which are “matching families” in that they agree on intersections:

$\phi_i\vert_{((U_i \cap U_j) \times Spec(A)} = \phi_j \vert_{(U_i \cap U_j)\times Spec(A)}$

i.e.

$\array{ && (U_i \cap U_j) \times Spec(A) \\ & \swarrow && \searrow \\ U_i\times Spec(A) && && U_j \times Spec(A) \\ & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}} \\ && X }$

Finally, given $X_1$ and $X_2$ two formal smooth sets, then a smooth function between them

$f \;\colon\; X_1 \longrightarrow X_2$

is

• for each infinitesimally thickened Cartesian space $\mathbb{R}^n \times Spec(A)$ (def. ) a function

$f_\ast(\mathbb{R}^n \times Spec(A)) \;\colon\; X_1(\mathbb{R}^n \times Spec(A)) \longrightarrow X_2(\mathbb{R}^n \times Spec(A))$

such that

• for each smooth function $g \colon \mathbb{R}^{n_1} \times Spec(A_1) \to \mathbb{R}^{n_2} \times Spec(A_2)$ between infinitesimally thickened Cartesian spaces we have

$g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2} \times Spec(A_2)) = f_\ast(\mathbb{R}^{n_1} \times Spec(A_1)) \circ g^\ast_1$

i.e.

$\array{ X_1(\mathbb{R}^{n_2} \times Spec(A_2)) &\overset{f_\ast(\mathbb{R}^{n_2}\times Spec(A_2) )}{\longrightarrow}& X_2(\mathbb{R}^{n_2} \times Spec(A_2)) \\ \mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2} \\ X_1(\mathbb{R}^{n_1} \times Spec(A_1)) &\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}& X_2(\mathbb{R}^{n_1} \times Spec(A_1)) }$

(Dubuc 79)

Basing infinitesimal geometry on formal smooth sets is an instance of the general approach to geometry called functorial geometry or topos theory. For more background on this see at geometry of physics – manifolds and orbifolds.

We have the evident generalization of example to smooth geometry with infinitesimals:

###### Example

(infinitesimally thickened Cartesian spaces are formal smooth sets)

For $X$ an infinitesimally thickened Cartesian space (def. ), it becomes a formal smooth set according to def. by taking its plots out of some $\mathbb{R}^n \times \mathbb{D}$ to be the homomorphism of infinitesimally thickened Cartesian spaces:

$X(\mathbb{R}^n \times \mathbb{D}) \;\coloneqq\; Hom_{FormalCartSp}( \mathbb{R}^n \times \mathbb{D}, X ) \,.$

(Stated more abstractly, this is an instance of the Yoneda embedding over a subcanonical site.)

###### Example

(smooth sets are formal smooth sets)

Let $X$ be a smooth set (def. ). Then $X$ becomes a formal smooth set (def. ) by declaring the set of plots $X(\mathbb{R}^n \times \mathbb{D})$ over an infinitesimally thickened Cartesian space (def. ) to be equivalence classes of pairs

$\mathbb{R}^n \times \mathbb{D} \longrightarrow \mathbb{R}^{k} \,, \phantom{AA} \mathbb{R}^k \longrightarrow X$

of a morphism of infinitesimally thickened Cartesian spaces and of a plot of $X$, as shown, subject to the equivalence relation which identifies two such pairs if there exists a smooth function $f \colon \mathbb{R}^k \to \mathbb{R}^{k'}$ such that

$\array{ && \mathbb{R}^n \times \mathbb{D} \\ & \swarrow && \searrow \\ \mathbb{R}^k && \overset{f}{\longrightarrow} && \mathbb{R}^{k'} \\ \mathbb{R}^k && \underset{f}{\longrightarrow} && \mathbb{R}^{k'} \\ & \searrow && \swarrow \\ && X }$

Stated more abstractly this says that $X$ as a formal smooth set is the left Kan extension (see this example) of $X$ as a smooth set along the functor that includes Cartesian spaces (def. ) into infinitesimally thickened Cartesian spaces (def. ).

###### Definition

(reduction and infinitesimal shape)

For $\mathbb{R}^n \times \mathbb{D}$ an infinitesimally thickened Cartesian space (def. ) we say that the underlying ordinary Cartesian space $\mathbb{R}^n$ (def. ) is its reduction

$\Re\left( \mathbb{R}^n \times \mathbb{D} \right) \;\coloneqq\; \mathbb{R}^n \,.$

There is the canonical inclusion morphism

$\Re\left( \mathbb{R}^n \times \mathbb{D} \right) = \mathbb{R}^n \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R}^n \times \mathbb{D}$

which dually corresponds to the homomorphism of commutative algebras

$C^\infty(\mathbb{R}^n) \longleftarrow C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}} A$

which is the identity on all smooth functions $f \in C^\infty(\mathbb{R}^n)$ and is zero on all elements $a \in V \subset A$ in the nilpotent ideal of $A$ (as in example ).

Given any formal smooth set $X$, we say that its infinitesimal shape or de Rham shape (also: de Rham stack) is the formal smooth set $\Im X$ (def. ) defined to have as plots the reductions of the plots of $X$, according to the above:

$(\Im X)( U ) \;\coloneqq\: X(\Re(U)) \,.$

There is a canonical morphism of formal smooth set

$\eta_X \;\colon\; X \longrightarrow \Im X$

which takes a plot

$U = \mathbb{R}^n \times \mathbb{D} \overset{f}{\longrightarrow} X$

to the composition

$\mathbb{R}^n \hookrightarrow \mathbb{R}^n \times \mathbb{D} \overset{f}{\hookrightarrow} X$

regarded as a plot of $\Im X$.

###### Example

(mapping space out of an infinitesimally thickened Cartesian space)

Let $X$ be an infinitesimally thickened Cartesian space (def. ) and let $Y$ be a formal smooth set (def. ). Then the mapping space

$[X,Y] \;\in\; FormalSmoothSet$

of smooth functions from $X$ to $Y$ is the formal smooth set whose $U$-plots are the morphisms of formal smooth sets from the Cartesian product of infinitesimally thickened Cartesian spaces $U \times X$ to $Y$, hence the $U \times X$-plots of $Y$:

$[X,Y](U) \;\coloneqq\; Y(U \times X) \,.$
###### Example

(synthetic tangent bundle)

Let $X \coloneqq \mathbb{R}^n$ be a Cartesian space (def. ) regarded as an infinitesimally thickened Cartesian space () and thus regarded as a formal smooth set (def. ) by example . Consider the infinitesimal line

$\mathbb{D}^1 \hookrightarrow \mathbb{R}^1$

from example . Then the mapping space $[\mathbb{D}^1, X]$ (example ) is the total space of the tangent bundle $T X$ (example ). Moreover, under restriction along the reduction $\ast \longrightarrow \mathbb{D}^1$, this is the full tangent bundle projection, in that there is a natural isomorphism of formal smooth sets of the form

$\array{ T X &\simeq& [\mathbb{D}^1, X] \\ {}^{\mathllap{tb}}\downarrow && \downarrow^{\mathrlap{ [ \ast \to \mathbb{D}^1, X ] }} \\ X &\simeq& [\ast, X] }$

In particular this implies immediately that smooth sections (def. ) of the tangent bundle

$\array{ && [\mathbb{D}^1, X] & \simeq T X \\ & {}^{\mathllap{v}}\nearrow & \downarrow \\ X &=& X }$

are equivalently morphisms of the form

$\array{ && X \\ & {}^{\mathllap{\tilde v}}\nearrow & \downarrow^{\mathrlap{id}} \\ X \times \mathbb{D}^1 &\underset{pr_1}{\longrightarrow}& X }$

which we had already identified with tangent vector fields (def. ) in example .

###### Proof

This follows by an analogous argument as in example , using the Hadamard lemma.

While in infinitesimally thickened Cartesian spaces (def. ) only infinitesimals to any finite order may exist, in formal smooth sets (def. ) we may find infinitesimals to any arbitrary finite order:

###### Example

(infinitesimal neighbourhood)

Let $X$ be a formal smooth sets (def. ) $Y \hookrightarrow X$ a sub-formal smooth set. Then the infinitesimal neighbourhood to arbitrary infinitesimal order of $Y$ in $X$ is the formal smooth set $N_X Y$ whose plots are those plots of $X$

$\mathbb{R}^n \times Spec(A) \overset{f}{\longrightarrow} X$

such that their reduction (def. )

$\mathbb{R}^n \hookrightarrow \mathbb{R}^n \times Spec(A) \overset{f}{\longrightarrow} X$

factors through a plot of $Y$.

This allows to grasp the restriction of field histories to the infinitesimal neighbourhood of a submanifold of spacetime, which will be crucial for the discussion of phase spaces below.

###### Definition

(field histories on infinitesimal neighbourhood of submanifold of spacetime)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle (def. ) and let $S \hookrightarrow \Sigma$ be a submanifold of spacetime.

We write $N_\Sigma(S) \hookrightarrow \Sigma$ for its infinitesimal neighbourhood in $\Sigma$ (def. ).

Then the set of field histories restricted to $S$, to be denoted

(25)$\Gamma_{S}(E) \coloneqq \Gamma_{N_\Sigma(S)}( E\vert_{N_\Sigma S} ) \in \mathbf{H}$

is the set of section of $E$ restricted to the infinitesimal neighbourhood $N_\Sigma(S)$ (example ).

$\,$

We close the discussion of infinitesimal differential geometry by explaining how we may recover the concept of smooth manifolds inside the more general formal smooth sets (def./prop. below). The key point is that the presence of infinitesimals in the theory allows an intrinsic definition of local diffeomorphisms/formally étale morphism (def. and example below). It is noteworthy that the only role this concept plays in the development of field theory below is that smooth manifolds admit triangulations by smooth singular simplices (def. ) so that the concept of fiber integration of differential forms is well defined over manifolds.

###### Definition

(local diffeomorphism of formal smooth sets)

Let $X,Y$ be formal smooth sets (def. ). Then a morphism between them is called a local diffeomorphism or formally étale morphism, denoted

$f \;\colon\; X \overset{et}{\longrightarrow} Y \,,$

if $f$ if for each infinitesimally thickened Cartesian space (def. ) $\mathbb{R}^n \times \mathbb{D}$ we have a natural identification between the $\mathbb{R}^n \times \mathbb{D}$-plots of $X$ with those $\mathbb{R}^n n\times \mathbb{D}$-plots of $Y$ whose reduction (def. ) comes from an $\mathbb{R}^n$-plot of $X$, hence if we have a natural fiber product of sets of plots

$X(\mathbb{R}^n \times \mathbb{D}) \;\simeq\; Y(\mathbb{R}^n \times \mathbb{D}) \underset{Y(\mathbb{R}^n)}{\times^f} X(\mathbb{R}^n)$

i. e.

$\array{ && X(\mathbb{R}^n \times \mathbb{D}) \\ & \swarrow && \searrow \\ Y(\mathbb{R}^n \times \mathbb{D}) && \text{(pb)} && X(\mathbb{R}^n) \\ & \searrow && \swarrow \\ && Y(\mathbb{R}^n ) }$

for all infinitesimally thickened Cartesian spaces $\mathbb{R}^n \times \mathbb{D}$.

Stated more abstractly, this means that the naturality square of the unit of the infinitesimal shape $\Im$ (def. ) is a pullback square

$\array{ X &\overset{\eta_X}{\longrightarrow}& \Im X \\ {}^{\mathllap{f}}\downarrow &\text{(pb)}& \downarrow^{\mathrlap{\Im f}} \\ Y &\underset{\eta_Y}{\longrightarrow}& \Im Y }$
###### Example

(local diffeomorphism between Cartesian spaces from the general definition)

For $X,Y \in CartSp$ two ordinary Cartesian spaces (def. ), regarded as formal smooth sets by example then a morphism $f \colon X \to Y$ between them is a local diffeomorphism in the general sense of def. precisely if it is so in the ordinary sense of def. .

###### Definition/Proposition

(smooth manifolds)

A smooth manifold $X$ of dimension $n \in \mathbb{N}$ is

such that

1. there exists an indexed set $\{ \mathbb{R}^n \overset{\phi_i}{\to} X\}_{i \in I}$ of morphisms of formal smooth sets (def. ) from Cartesian spaces $\mathbb{R}^n$ (def. ) (regarded as formal smooth sets via example , example and example ) into $X$, (regarded as a formal smooth set via example and example ) such that

1. every point $x \in X_s$ is in the image of at least one of the $\phi_i$;

2. every $\phi_i$ is a local diffeomorphism according to def. ;

2. the final topology induced by the set of plots of $X$ makes $X_s$ a paracompact Hausdorff space.

For more on smooth manifolds from the perspective of formal smooth sets see at geometry of physics – manifolds and orbifolds.

$\,$

fermion fields and supergeometry

Field theories of interest crucially involve fermionic fields (def. below), such as the Dirac field (example below), which are subject to the “Pauli exclusion principle”, a key reason for the stability of matter. Mathematically this principle means that these fields have field bundles whose field fiber is not an ordinary manifold, but an odd-graded supermanifold (more on this in remark and remark below).

This “supergeometry” is an immediate generalization of the infinitesimal geometry above, where now the infinitesimal elements in the algebra of functions may be equipped with a grading: one speaks of superalgebra.

The “super”-terminology for something as down-to-earth as the mathematical principle behind the stability of matter may seem unfortunate. For better or worse, this terminology has become standard since the middle of the 20th century. But the concept that today is called supercommutative superalgebra was in fact first considered by Grassmann 1844 who got it right (“Ausdehnungslehre”) but apparently was too far ahead of his time and remained unappreciated.

Beware that considering supergeometry does not necessarily involve considering “supersymmetry”. Supergeometry is the geometry of fermion fields (def. below), and that all matter fields in the observable universe are fermionic has been experimentally established since the Stern-Gerlach experiment in 1922. Supersymmetry, on the other hand, is a hypothetical extension of spacetime-symmetry within the context of supergeometry. Here we do not discuss supersymmetry; for details see instead at geometry of physics – supersymmetry.

###### Definition

(supercommutative superalgebra)

A real $\mathbb{Z}/2$-graded algebra or superalgebra is an associative algebra $A$ over the real numbers together with a direct sum decomposition of its underlying real vector space

$A \simeq_{\mathbb{R}} A_{even} \oplus A_{odd} \,,$

such that the product in the algebra respects the multiplication in the cyclic group of order 2 $\mathbb{Z}/2 = \{even, odd\}$:

$\left. \array{ A_{even} \cdot A_{even} \\ A_{odd} \cdot A_{odd} } \right\} \subset A_{even} \phantom{AAAA} \left. \array{ A_{odd} \cdot A_{even} \\ A_{even} \cdot A_{odd} } \right\} \subset A_{odd} \,.$

This is called a supercommutative superalgebra if for all elements $a_1, a_2 \in A$ which are of homogeneous degree ${\vert a_i \vert} \in \mathbb{Z}/2 = \{even, odd\}$ in that

$a_i \in A_{{\vert a_i\vert}} \subset A$

we have

$a_1 \cdot a_2 = (-1)^{{\vert a_1 \vert \vert a_2 \vert}} a_2 \cdot a_1 \,.$
$f \;\colon\; A \longrightarrow A'$

is a homomorphism of associative algebras over the real numbers such that the $\mathbb{Z}/2$-grading is respected in that

$f(A_{even}) \subset A'_{even} \phantom{AAAAA} f(A_{odd}) \subset A'_{odd} \,.$

For more details on superalgebra see at geometry of physics – superalgebra.

###### Example

(basic examples of supercommutative superalgebras)

Basic examples of supercommutative superalgebras (def. ) include the following:

1. Every commutative algebra $A$ becomes a supercommutative superalgebra by declaring it to be all in even degree: $A = A_{even}$.

2. For $V$ a finite dimensional real vector space, then the Grassmann algebra $A \coloneqq \wedge^\bullet_{\mathbb{R}} V^\ast$ is a supercommutative superalgebra with $A_{even} \coloneqq \wedge^{even} V^\ast$ and $A_{odd} \coloneqq \wedge^{odd} V^\ast$.

More explicitly, if $V = \mathbb{R}^s$ is a Cartesian space with canonical dual coordinates $(\theta^i)_{i = 1}^s$ then the Grassmann algebra $\wedge^\bullet (\mathbb{R}^s)^\ast$ is the real algebra which is generated from the $\theta^i$ regarded in odd degree and hence subject to the relation

$\theta^i \cdot \theta^j = - \theta^j \cdot \theta^i \,.$

In particular this implies that all the $\theta^i$ are infinitesimal (def. ):

$\theta^i \cdot \theta^i = 0 \,.$
3. For $A_1$ and $A_2$ two supercommutative superalgebras, there is their tensor product supercommutative superalgebra $A_1 \otimes_{\mathbb{R}} A_2$. For example for $X$ a smooth manifold with ordinary algebra of smooth functions $C^\infty(X)$ regarded as a supercommutative superalgebra by the first example above, the tensor product with a Grassmann algebra (second example above) is the supercommutative superalgebta

$C^\infty(X) \otimes_{\mathbb{R}} \wedge^\bullet((\mathbb{R}^s)\ast)$

whose elements may uniquely be expanded in the form

$f + f_i \theta^i + f_{i j} \theta^i \theta^j + f_{i j k} \theta^i \theta^j \theta^k + \cdots + f_{i_1 \cdots i_s} \theta^{i_1} \cdots \theta^{i_s} \,,$

where the $f_{i_1 \cdots i_k} \in C^\infty(X)$ are smooth functions on $X$ which are skew-symmetric in their indices.

The following is the direct super-algebraic analog of the definition of infinitesimally thickened Cartesian spaces (def. ):

###### Definition

(super Cartesian space)

A superpoint $Spec(A)$ is represented by a super-commutative superalgebra $A$ (def. ) which as a $\mathbb{Z}/2$-graded vector space (super vector space) is a direct sum

$A \simeq_{\mathbb{R}} \langle 1 \rangle \oplus V$

of the 1-dimensional even vector space $\langle 1 \rangle = \mathbb{R}$ of multiples of 1, with a finite dimensional super vector space $V$ that is a nilpotent ideal in $A$ in that for each element $a \in V$ there exists a natural number $n \in \mathbb{N}$ such that

$a^{n+1} = 0 \,.$

More generally, a super Cartesian space $\mathbb{R}^n \times Spec(A)$ is represented by a super-commutative algebra $C^\infty(\mathbb{R}^n) \otimes A \in \mathbb{R} Alg$ which is the tensor product of algebras of the algebra of smooth functions $C^\infty(\mathbb{R}^n)$ on an actual Cartesian space of some dimension $n$, with an algebra of functions $A \simeq_{\mathbb{R}} \langle 1\rangle \oplus V$ of a superpoint (example ).

Specifically, for $s \in \mathbb{N}$, there is the superpoint

(26)$\mathbb{R}^{0 \vert s} \;\coloneqq\; Spec\left( \wedge^\bullet (\mathbb{R}^s)^\ast \right)$

whose algebra of functions is by definition the Grassmann algebra on $s$ generators $(\theta^i)_{i = 1}^s$ in odd degree (example ).

We write

\begin{aligned} \mathbb{R}^{b\vert s} & \coloneqq \mathbb{R}^b \times \mathbb{R}^{0 \vert s} \\ & = \mathbb{R}^b \times Spec( \wedge^\bullet(\mathbb{R}^s)^\ast ) \\ & = Spec\left( C^\infty(\mathbb{R}^b) \otimes_{\mathbb{R}} \wedge^\bullet (\mathbb{R}^s)^\ast \right) \end{aligned}

for the corresponding super Cartesian spaces whose algebra of functions is as in example .

We say that a smooth function between two super Cartesian spaces

$\mathbb{R}^{n_1} \times Spec(A_1) \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \times Spec(A_2)$

is by definition dually a homomorphism of supercommutative superalgebras (def. ) of the form

$C^\infty(\mathbb{R}^{n_1}) \otimes A_1 \overset{f^\ast}{\longleftarrow} C^\infty(\mathbb{R}^{n_2}) \otimes A_2 \,.$
###### Example

(superpoint induced by a finite dimensional vector space)

Let $V$ be a finite dimensional real vector space. With $V^\ast$ denoting its dual vector space write $\wedge^\bullet V^\ast$ for the Grassmann algebra that it generates. This being a supercommutative algebra, it defines a superpoint (def. ).

We denote this superpoint by

$V_{odd} \simeq \mathbb{R}^{0 \vert dim(V)} \,.$

All the differential geometry over Cartesian space that we discussed above generalizes immediately to super Cartesian spaces (def. ) if we stricly adhere to the super sign rule which says that whenever two odd-graded elements swap places, a minus sign is picked up. In particular we have the following generalization of the de Rham complex on Cartesian spaces discussed above.

###### Definition

(super differential forms on super Cartesian spaces)

For $\mathbb{R}^{b\vert s}$ a super Cartesian space (def. ), hence the formal dual of the supercommutative superalgebra of the form

$C^\infty(\mathbb{R}^{b\vert s}) \;=\; C^\infty(\mathbb{R}^b) \otimes_{\mathbb{R}} \wedge^\bullet \mathbb{R}^s$

with canonical even-graded coordinate functions $(x^i)_{i = 1^b}$ and odd-graded coordinate functions $(\theta^j)_{j = 1}^s$.

Then the de Rham complex of super differential forms on $\mathbb{R}^{b\vert s}$ is, in super-generalization of def. , the $\mathbb{Z} \times (\mathbb{Z}/2)$-graded commutative algebra

$\Omega^\bullet(\mathbb{R}^{b|s}) \;\coloneqq\; C^\infty(\mathbb{R}^{b|s}) \otimes_{\mathbb{R}} \wedge^\bullet \langle d x^1, \cdots, d x^b, \; d \theta^1, \cdots, d\theta^s \rangle$

which is generated over $C^\infty(\mathbb{R}^{b\vert s})$ from new generators

$\underset{ \text{deg} = (1,even) }{\underbrace{ d x^i }} \phantom{AAAAA} \underset{ \text{deg} = (1,odd) }{ \underbrace{ d \theta^j } }$

whose differential is defined in degree-0 by

$d f \;\coloneqq\; \frac{\partial f}{\partial x^i} d x^i + \frac{\partial f}{\partial \theta^j} d \theta^j$

and extended from there as a bigraded derivation of bi-degree $(1,even)$, in super-generalization of def. .

Accordingly, the operation of contraction with tangent vector fields (def. ) has bi-degree $(-1,\sigma)$ if the tangent vector has super-degree $\sigma$:

generatorbi-degree
$\phantom{AA} x^a$(0,even)
$\phantom{AA} \theta^\alpha$(0,odd)
$\phantom{AA} dx^a$(1,even)
$\phantom{AA} d\theta^\alpha$(1,odd)
derivationbi-degree
$\phantom{AA} d$(1,even)
$\phantom{AA}\iota_{\partial x^a}$(-1, even)
$\phantom{AA}\iota_{\partial \theta^\alpha}$(-1,odd)

This means that if $\alpha \in \Omega^\bullet(\mathbb{R}^{b\vert s})$ is in bidegree $(n_\alpha, \sigma_\alpha)$, and $\beta \in \Omega^\bullet(\mathbb{R}^{b \vert \sigma})$ is in bidegree $(n_\beta, \sigma_\beta)$, then

$\alpha \wedge \beta \; = \; (- 1)^{n_\alpha n_\beta + \sigma_\alpha \sigma_\beta} \; \beta \wedge \alpha \,.$

Hence there are two contributions to the sign picked up when exchanging two super-differential forms in the wedge product:

1. there is a “cohomological sign” which for commuting an $n_1$-forms past an $n_2$-form is $(-1)^{n_1 n_2}$;

2. in addition there is a “super-grading” sign which for commuting a $\sigma_1$-graded coordinate function past a $\sigma_2$-graded coordinate function (possibly under the de Rham differential) is $(-1)^{\sigma_1 \sigma_2}$.

For example:

$x^{a_1} (dx^{a_2}) \;=\; + (dx^{a_2}) x^{a_1}$
$\theta^\alpha (dx^a) \;=\; + (dx^a) \theta^\alpha$
$\theta^{\alpha_1} (d\theta^{\alpha_2}) \;=\; - (d\theta^{\alpha_2}) \theta^{\alpha_1}$
$dx^{a_1} \wedge d x^{a_2} \;=\; - d x^{a_2} \wedge d x^{a_1}$
$dx^a \wedge d \theta^{\alpha} \;=\; - d\theta^{\alpha} \wedge d x^a$
$d\theta^{\alpha_1} \wedge d \theta^{\alpha_2} \;=\; + d\theta^{\alpha_2} \wedge d \theta^{\alpha_1}$

Beware that there is also another sign rule for super differential forms used in the literature. See at signs in supergeometry for further discussion.

$\,$

It is clear now by direct analogy with the definition of formal smooth sets (def. ) what the corresponding supergeometric generalization is. For definiteness we spell it out yet once more:

###### Definition

(super smooth set)

A super smooth set $X$ is

1. for each super Cartesian space $\mathbb{R}^n \times Spec(A)$ (def. ) a set

$X(\mathbb{R}^n \times Spec(A)) \in Set$

to be called the set of smooth functions or plots from $\mathbb{R}^n \times Spec(A)$ to $X$;

2. for each smooth function $f \;\colon\; \mathbb{R}^{n_1} \times Spec(A_1) \longrightarrow \mathbb{R}^{n_2} \times Spec(A_2)$ between super Cartesian spaces a choice of function

$f^\ast \;\colon\; X(\mathbb{R}^{n_2} \times Spec(A_2)) \longrightarrow X(\mathbb{R}^{n_1} \times Spec(A_1))$

to be thought of as the precomposition operation

$\left( \mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X \right) \;\overset{f^\ast}{\mapsto}\; \left( \mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{\Phi}{\to} X \right)$

such that

1. If $id_{\mathbb{R}^n \times Spec(A)} \;\colon\; \mathbb{R}^n \times Spec(A) \to \mathbb{R}^n \times Spec(A)$ is the identity function on $\mathbb{R}^n \times Spec(A)$, then $\left(id_{\mathbb{R}^n \times Spec(A)}\right)^\ast \;\colon\; X(\mathbb{R}^n \times Spec(A)) \to X(\mathbb{R}^n \times Spec(A))$ is the identity function on the set of plots $X(\mathbb{R}^n \times Spec(A))$.

2. If $\mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{g}{\to} \mathbb{R}^{n_3} \times Spec(A_3)$ are two composable smooth functions between infinitesimally thickened Cartesian spaces, then pullback of plots along them consecutively equals the pullback along the composition:

$f^\ast \circ g^\ast = (g \circ f)^\ast$

i.e.

$\array{ && X(\mathbb{R}^{n_2} \times Spec(A_2)) \\ & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}} \\ X(\mathbb{R}^{n_1} \times Spec(A_1)) && \underset{ (g \circ f)^\ast }{\longleftarrow} && X(\mathbb{R}^{n_3} \times Spec(A_3)) }$
1. (gluing)

If $\{ U_i \times Spec(A) \overset{f_i \times id_{Spec(A)}}{\to} \mathbb{R}^n \times Spec(A)\}_{i \in I}$ is such that

$\{ U_i \overset{f_i }{\to} \mathbb{R}^n \}_{i \in I}$

is a differentiably good open cover (def. ) then the function which restricts $\mathbb{R}^n \times Spec(A)$-plots of $X$ to a set of $U_i \times Spec(A)$-plots

$X(\mathbb{R}^n \times Spec(A)) \overset{( (f_i)^\ast )_{i \in I} }{\hookrightarrow} \underset{i \in I}{\prod} X(U_i \times Spec(A))$

is a bijection onto the set of those tuples $(\Phi_i \in X(U_i))_{i \in I}$ of plots, which are “matching families” in that they agree on intersections:

$\phi_i\vert_{((U_i \cap U_j) \times Spec(A)} = \phi_j \vert_{(U_i \cap U_j)\times Spec(A)}$

i.e.

$\array{ && (U_i \cap U_j) \times Spec(A) \\ & \swarrow && \searrow \\ U_i\times Spec(A) && && U_j \times Spec(A) \\ & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}} \\ && X }$

Finally, given $X_1$ and $X_2$ two super formal smooth sets, then a smooth function between them

$f \;\colon\; X_1 \longrightarrow X_2$

is

• for each super Cartesian space $\mathbb{R}^n \times Spec(A)$ a function

$f_\ast(\mathbb{R}^n \times Spec(A)) \;\colon\; X_1(\mathbb{R}^n \times Spec(A)) \longrightarrow X_2(\mathbb{R}^n \times Spec(A))$

such that

• for each smooth function $g \colon \mathbb{R}^{n_1} \times Spec(A_1) \to \mathbb{R}^{n_2} \times Spec(A_2)$ between super Cartesian spaces we have

$g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2} \times Spec(A_2)) = f_\ast(\mathbb{R}^{n_1} \times Spec(A_1)) \circ g^\ast_1$

i.e.

$\array{ X_1(\mathbb{R}^{n_2} \times Spec(A_2)) &\overset{f_\ast(\mathbb{R}^{n_2}\times Spec(A_2) )}{\longrightarrow}& X_2(\mathbb{R}^{n_2} \times Spec(A_2)) \\ \mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2} \\ X_1(\mathbb{R}^{n_1} \times Spec(A_1)) &\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}& X_2(\mathbb{R}^{n_1} \times Spec(A_1)) }$

Basing supergeometry on super formal smooth sets is an instance of the general approach to geometry called functorial geometry or topos theory. For more background on this see at geometry of physics – supergeometry.

In direct generalization of example we have:

###### Example

(super Cartesian spaces are super smooth sets)

Let $X$ be a super Cartesian space (def. ) Then it becomes a super smooth set (def. ) by declaring its plots $\Phi \in X(\mathbb{R}^n \times \mathbb{D})$ to the algebra homomorphisms $C^\infty(\mathbb{R}^n \times \mathbb{D}) \leftarrow C^\infty(\mathbb{R}^{b\vert s})$.

Under this identification, morphisms between super Cartesian spaces are in natural bijection with their morphisms regarded as super smooth sets.

Stated more abstractly, this statement is an example of the Yoneda embedding over a subcanonical site.

Similarly, in direct generalization of prop. we have:

###### Proposition

(plots of a super smooth set really are the smooth functions into the smooth smooth set)

Let $X$ be a super smooth set (def. ). For $\mathbb{R}^n \times \mathbb{D}$ any super Cartesian space (def. ) there is a natural function

$Hom_{SmoothSet}(\mathbb{R}^n , X) \overset{\simeq}{\longrightarrow} X(\mathbb{R}^n)$

from the set of homomorphisms of super smooth sets from $\mathbb{R}^n \times \mathbb{D}$ (regarded as a super smooth set via example ) to $X$, to the set of plots of $X$ over $\mathbb{R}^n \times \mathbb{D}$, given by evaluating on the identity plot $id_{\mathbb{R}^n \times \mathbb{D}}$.

This function is a bijection.

This says that the plots of $X$, which initially bootstrap $X$ into being as declaring the would-be smooth functions into $X$, end up being the actual smooth functions into $X$.

###### Proof

This is the statement of the Yoneda lemma over the site of super Cartesian spaces.

We do not need to consider here supermanifolds more general than the super Cartesian spaces (def. ). But for those readers familiar with the concept we include the following direct analog of the characterization of smooth manifolds according to def./prop. :

###### Definition/Proposition

(supermanifolds)

A supermanifold $X$ of dimension super-dimension $(b,s) \in \mathbb{N} \times \mathbb{N}$ is

such that

1. there exists an indexed set $\{ \mathbb{R}^{b\vert s} \overset{\phi_i}{\to} X\}_{i \in I}$ of morphisms of super smooth sets (def. ) from super Cartesian spaces $\mathbb{R}^{b\vert s}$ (def. ) (regarded as super smooth sets via example into $X$, such that

1. for every plot $\mathbb{R}^n \times \mathbb{D} \to X$ there is a differentiably good open cover (def. ) restricted to which the plot factors through the $\mathbb{R}^{b\vert s}_i$;

2. every $\phi_i$ is a local diffeomorphism according to def. , now with respect not just to infinitesimally thickened points, but with respect to superpoints;

2. the bosonic part of $X$ is a smooth manifold according to def./prop. .

Finally we have the evident generalization of the smooth moduli space $\mathbf{\Omega}^\bullet$ of differential forms from example to supergeometry

###### Example

(universal smooth moduli spaces of super differential forms)

For $n \in \mathbf{M}$ write

$\mathbf{\Omega}^n \;\in\; SuperSmoothSet$

for the super smooth set (def. ) whose set of plots on a super Cartesian space $U \in SuperCartSp$ (def. ) is the set of super differential forms (def. ) of cohomolgical degree $n$

$\mathbf{\Omega}^n(U) \;\coloneqq\; \Omega^n(U)$

and whose maps of plots is given by pullback of super differential forms.

The de Rham differential on super differential forms applied plot-wise yields a morpism of super smooth sets

(27)$d \;\colon\; \mathbf{\Omega}^n \longrightarrow \mathbf{\Omega}^{n+1} \,.$

As before in def. we then define for any super smooth set $X \in SuperSmoothSet$ its set of differential $n$-forms to be

$\Omega^n(X) \;\coloneqq\; Hom_{SuperSmoothSet}(X,\mathbf{\Omega}^n)$

and we define the de Rham differential on these to be given by postcomposition with (27).

$\,$

###### Definition

(bosonic fields and fermionic fields)

For $\Sigma$ a spacetime, such as Minkowski spacetime (def. ) if a fiber bundle $E \overset{fb}{\longrightarrow} \Sigma$ with total space a super Cartesian space (def. ) (or more generally a supermanifold, def./prop. ) is regarded as a super-field bundle (def. ), then

In components, if $E = \Sigma \times F$ is a trivial bundle with fiber a super Cartesian space (def. ) with even-graded coordinates $(\phi^a)$ and odd-graded coordinates $(\psi^A)$, then the $\phi^a$ are called the bosonic field coordinates, and the $\psi^A$ are called the fermionic field coordinates.

What is crucial for the discussion of field theory is the following immediate supergeometric analog of the smooth structure on the space of field histories from example :

###### Example

(supergeometric space of field histories)

Let $E \overset{fb}{\to} \Sigma$ be a super-field bundle (def. , def. ).

Then the space of sections, hence the space of field histories, is the super formal smooth set (def. )

$\Gamma_\Sigma(E) \in SuperSmoothSet$

whose plots $\Phi_{(-)}$ for a given Cartesian space $\mathbb{R}^n$ and superpoint $\mathbb{D}$ (def. ) with the Cartesian products $U \coloneqq \mathbb{R}^n \times \mathbb{D}$ and $U \times \Sigma$ regarded as super smooth sets according to example are defined to be the morphisms of super smooth set (def. )

$\array{ U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E }$

which make the following diagram commute:

$\array{ && E \\ & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}} \\ U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma } \,.$

Explicitly, if $\Sigma$ is a Minkowski spacetime (def. ) and $E = \Sigma \times F$ a trivial field bundle with field fiber a super vector space (example , example ) this means dually that a plot $\Phi_{(-)}$ of the super smooth set of field histories is a homomorphism of supercommutative superalgebras (def. )

$\array{ C^\infty(U \times \Sigma) &\overset{\left(\Phi_{(-)}(-)\right)^\ast}{\longleftarrow}& C^\infty(E) }$

which make the following diagram commute:

$\array{ && C^\infty(E) \\ & {}^{\mathllap{\left( \Phi_{(-)}(-) \right)^\ast }}\nearrow & \uparrow^{\mathrlap{fb^\ast}} \\ C^\infty(U \times \Sigma) &\underset{pr_2^\ast}{\longleftarrow}& C^\infty(\Sigma) } \,.$

We will focus on discussing the supergeometric space of field histories (example ) of the Dirac field (def. below). This we consider below in example ; but first we discuss now some relevant basics of general supergeometry.

Example is really a special case of a general relative mapping space-construction as in example . This immediately generalizes also to the supergeometric context.

###### Definition

(super-mapping space out of a super Cartesian space)

Let $X$ be a super Cartesian space (def. ) and let $Y$ be a super smooth set (def. ). Then the mapping space

$[X,Y] \;\in\; SuperSmoothSet$

of super smooth functions from $X$ to $Y$ is the super formal smooth set whose $U$-plots are the morphisms of super smooth set from the Cartesian product of super Cartesian space $U \times X$ to $Y$, hence the $U \times X$-plots of $Y$:

$[X,Y](U) \;\coloneqq\; Y(U \times X) \,.$

In direct generalization of the synthetic tangent bundle construction (example ) to supergeometry we have

###### Definition

(odd tangent bundle)

Let $X$ be a super smooth set (def. ) and $\mathbb{R}^{0 \vert 1}$ the superpoint (26) then the supergeometry-mapping space

$\array{ T_{odd} X & \coloneqq& [\mathbb{R}^{0\vert 1}, X] \\ {}^{\mathllap{tb_{odd}}}\downarrow && \downarrow^{\mathrlap{ [ \ast \to \mathbb{R}^{0 \vert 1}, X ] }} \\ X & = & X }$

is called the odd tangent bundle of $X$.

###### Example

(mapping space of superpoints)

Let $V$ be a finite dimensional real vector space and consider its corresponding superpoint $V_{odd}$ from exampe . Then the mapping space (def. ) out of the superpoint $\mathbb{R}^{0\vert 1}$ (def. ) into $V_{odd}$ is the Cartesian product $V_{odd} \times V$

$[\mathbb{R}^{0\vert 1}, V_{odd}] \;\simeq\; V_{odd} \times V \,.$

By def. this says that $V_{odd} \times V$ is the “odd tangent bundle” of $V_{odd}$.

###### Proof

Let $U$ be any super Cartesian space. Then by definition we have the following sequence of natural bijections of sets of plots

\begin{aligned} \left[ \mathbb{R}^{0\vert 1}, V_{odd} \right](U) & = Hom_{SuperSmoothSet}( \mathbb{R}^{0\vert 1} \times U, V_{odd} ) \\ & \simeq Hom_{\mathbb{R}sAlg}( \wedge^\bullet(V^\ast)\,,\, C^\infty(U)[\theta]/(\theta^2) ) \\ & \simeq Hom_{\mathbb{R}Vect}( V^\ast \,,\, (C^\infty(U)_{odd} \oplus C^\infty(U)_{even}\langle \theta\rangle ) \\ & \simeq Hom_{\mathbb{R}Vect}( V^\ast\,,\, C^\infty(U)_{odd} ) \,\times\, Hom_{\mathbb{R}Vect}( V^\ast, C^\infty(U)_{even} ) \\ & \simeq V_{odd}(U) \times V(U) \\ & \simeq (V_{odd} \times V)(U) \end{aligned}

Here in the third line we used that the Grassmann algebra $\wedge^\bullet V^\ast$ is free on its generators in $V^\ast$, meaning that a homomorphism of supercommutative superalgebras out of the Grassmann algebra is uniquely fixed by the underlying degree-preserving linear function on these generators. Since in a Grassmann algebra all the generators are in odd degree, this is equivalently a linear map from $V^\ast$ to the odd-graded real vector space underlying $C^\infty(U)[\theta](\theta^2)$, which is the direct sum $C^\infty(U)_{odd} \oplus C^\infty(U)_{even}\langle \theta \rangle$.

Then in the fourth line we used that finite direct sums are Cartesian products, so that linear maps into a direct sum are pairs of linear maps into the direct summands.

That all these bijections are natural means that they are compatible with morphisms $U \to U'$ and therefore this says that $[\mathbb{R}^{0\vert 1}, V_{odd}]$ and $V_{odd} \times V$ are the same as seen by super-smooth plots, hence that they are isomorphic as super smooth sets.

With this supergeometry in hand we finally turn to defining the Dirac field species:

###### Example

(field bundle for Dirac field)

For $\Sigma$ being Minkowski spacetime (def. ), of dimension $2+1$, $3+1$, $5+1$ or $9+1$, let $S$ be the spin representation from prop. , whose underlying real vector space is