under construction
On the formalization of the process of quantization – by abstract nonsense – from classical σ-model data to the corresponding quantum field theory .
The search is on for the abstract formalization of the process of quantization – the process that reads in a “classical field theory” – for instance presented in form of a gauge theory or in form of a σ-model background field data – and spits out the corresponding quantum field theory.
There is an open problem of mathematical (-physics) model building:
One that is to quantization as, say, symplectic geometry is to Hamiltonian mechanics?
The formalism of FQFT clearly suggests that the fundamental description of quantization is some natural operation on higher functors.
While for various aspects and facets of this question there are well-developed formalisms – such as geometric quantization or deformation quantization or BV-BRST formalism – a full answer is certainly still missing, not the least because the full formalization of the question itself has still to be established.
Considerable progress on this formulation of the question has been achieved with the formalization and proof of the cobordism hypothesis in On the Classification of Topological Field Theories by Jacob Lurie. This at least indicates what the result of any full quantization procedure should be in that it clarifies what exactly a TQFT FQFT is: a morphism from the (∞,n)-category of cobordisms $Z : Bord_n \to C$.
In On the Classification of Topological Field Theories Jacob Lurie indicates some first towards finding a similar formalization of “classical field theory” (in terms of his $(\infty,n)$-categories of “families”) and a systematic procedure for turning the classical theory into the quantum theory. These thoughts were further developed in the article Topological Quantum Field Theories from Compact Lie Groups . But for the moment, that, too, remains a bit sketchy.
For the purpose of the present entry this indication of a quantizaton proposal by Lurie et al. mainly serves as a reference for the idea itself that a formalization of something interesting is to be sought here, and of the kind of abstract nonsense answer one hopes to find. We will however discuss a somewhat different-looking approach. It may well be related to the Lurie-et al proposal in the end, but for the time being we shall not concentrate on that relation.
Rather, the approach for a formalization of the quantization procedure that shall be discussed at this entry here draws from a few different sources:
The observation that a classical background field that should serve as the input for a quantization of a $\sigma$-model that describes the dynamics of an object charged under this field is encoded by differential cocycles as as described at differential cohomology in an (∞,1)-topos.
The idea that by applying a pull-push quantization prescription a differential cocycle on a target space $X$ gives rise to a differential cocycle on a parameter space $\Sigma$, which may be thought of as one of the bordisms appearing in the FQFT-description of quantum field theory.
The pull-push operation here is akin to that in geometric ∞-function theory, where a quantum field theory is obtained from a σ-model target space object $X$ by homming extended cobordism cospans $\Sigma_in \to \Sigma \leftarrow \Sigma_{out}$ into the target object and then pull-pushing geometric function objects through the resulting spans of configuration space objects $[\Sigma_{in},X] \leftarrow [\Sigma,X] \to [\Sigma_{out},X]$.
The main result of David Ben-Zvi et. al.’s work on this approach is that they point out that as soon as the geometric function object one uses satisfies the two fundamental theorems of geometric infinity-function theory, a considerable amount of rich structure that has in parts been known by itself gets unified into one coherent elegant story: the nature of partition functions (i.e. traces), of centers, of Hochschild (co)homology, Deligne-Kontsevich-statements, etc. all are understood by means of a suitable geometric function theory as induced from the underlying geometry of configuration space objects $[\Sigma,X]$ as well as the loop space objects of $X$.
The resulting pull-push operation is an example or a generalization of what John Baez discusses under the term groupoidification.
The observation that a differential coccycle on a Lorentzian manifold $\Sigma$ gives rise to a local net of observables, as used in the formalizaton of QFT known as AQFT. (As described here).
So the procedure discussed here regards differential cocycles on target space as classicai field theories, regards their quantization as a way to obtain a differential cocycle on Lorentzian parameter space, and identifies this as a quantum field theory by associating a local net of observables to it.
These local nets, in turn, are akin to factorization algebras, which in the Euclidean (meaning non-Lorentzian setting) relate back to cobordism representations via the notion of topological chiral homology. However the – physically crucial – Lorentzian structure invoked here is not otherwise considered in these functorial axiomatization of quantum field theory.
The ambient context is the (∞,1)-topos $\mathbf{H} := Sh_{(\infty,1)}(CartSp)$ of Lie ∞-groupoids and the $(\infty,2)$-topos of smooth $(\infty,1)$-categories, which we model, respectively, as the left Bousfield localization of the model structure on functors $[CartSp^{op}, sSet_{Quillen}]$ and $[CartSp^{op}, sSet^+]$, respectively, where
CartSp is the site of Cartesian spaces,
$sSet_{Quillen}$ the standard model structure on simplicial sets;
and $sSet^+$ the model structure for Cartesian fibrations over the points, hence the simplicial model category for quaasi-categories.
For the definition of the path ∞-groupoid functor $\mathbf{\Pi}$ and the induced theory of differential cohomology in an (∞,1)-topos, we make use of the discussion at differential cohomology in an (∞,1)-topos -- survey.
When working with fibrant objects in the model, we will frequently use the constructions and notation from category of fibrant objects. Notably for $\mathbf{B}G$ a fibrant delooping object in the model we write $(\mathbf{B}G)^I$ for the path object we write $\mathbf{E}G$ for the pullback
and $\mathbf{E}G \to \mathbf{B}G$ for the remaining map, induced from $d_0 : (\mathbf{B}G)^I \to \mathbf{B}G$.
We describe the general theory for the simple example of the charged particle.
The background field for the charged particle that we want to consider is the electromagnetic field. The data involved is
the target space $X$ – a smooth manifold;
the structure group or gauge group $G = U(1)$;
a choice of representation
taken to be the canonical representation on $V = \mathbb{C}$;
the background field given by a
a $U(1)$-principal bundle $P \to X$ classified by a cocycle $g : X \to \mathbf{B}U(1)$ in $\mathbf{H}$ which in the model is given by an anafunctor $X \stackrel{\simeq}{\leftarrow} Y \to \mathbf{B}U(1)$;
a connection $\nabla$ on this bundle, which in the model is given by a diagram
and whose field strength is given by the composite
Let $\Sigma = \mathbb{R}$ be the parameter space, the worldline, regarded as a Lorentzian manifold and write $\mathbf{\Pi}(\Sigma)$ for the Lorentzian path category.
To define the kinetic action, we first form the $\rho$-associated background field
and then pull this back to the extended configuration space $X \times \Sigma$.
The morphisms in the product category $\mathbf{\Pi}(X) \times \mathbf{\Pi}(\Sigma)$ are paths $\gamma_X : [0,1] \to X$ in $X$ on whose base we have a (pseudo)Riemannian metric, which is the pullback of the metric $\mu_\Sigma$ on $\Sigma$ along $\gamma_\Sigma : [0,1] \to \Sigma$. We can consider the kinetic action to be a differential cocycle
which sends the path $\gamma_X : [0,1] \to X$ of parameter length $\gamma_\Sigma : [0,1] \to \Sigma$ to
Notice that $\mathbf{\Pi}(X)\times \mathbf{\Pi}(\Sigma) \simeq \mathbf{\Pi}(X \times \Sigma)$.
The total action is differential cocycle
We want to consider the diagram
and use it to obtain a differential cocycle on $\Sigma$, by forming something like a lax pullback (“comma object”) of the point inclusion
where the left vertical morphism picks the ground field $\mathbb{C}$, along this cocycle. For the underlying cocycle this is obtained as the ordinary pullback of $\mathbf{E}Vect$ defined as the ordinary pullback
where $I$ is the interval category, $Vect^I$ the functor category and $k : * \to Vect$ picks the ground field vector space. Via the remaining map $d_0 : Vect^I \to Vect$ this maps to $Vect$ and then further to $Vect \coprod_{\mathbf{B}U(1)} \mathbf{E}\mathbf{B}U(1)$.
The pullback of
along our differential cocycle, i. e. the pullback of the top part of the diagram
is over $X \times \Sigma$ the total space of the pullback of the underlying vector bundle $E$ on $X$ to $X \times \Sigma$, and over $\mathbf{\Pi}(X \times \Sigma)$ is a groupoid $E_{\mathbf{\Pi}}$
To see what $E \to E_{\mathbf{\Pi}}$ is like, first notice that we have a fibration sequence
with the bottom right square a lax pullback and everything else homotopy pullbacks. Here $V$ is the vector space that $\rho : \mathbf{B}U(1) \to Vect$ is a representation on (which for the electromagnetic field will be $\mathbb{C}$ itself).
The groupoid $E_{\mathbf{\Pi}}$ has the following description:
its objects are triples $(x,\sigma, v)$, where $v \in E_{x}$ is a vector in the fiber of $E$ over $x$;
its morphisms are triples $(x \stackrel{\gamma}{\to} y, \sigma \to \sigma', v)$ that go from $(x,\sigma,v)$ to $(y, \sigma', \exp(\int_0^1 |...|) \rho(\nabla(\gamma)))(v)$, i.e. from a vector in the fiber over the source to the corresponding vector in the fiber over the target, obtained by evaluating the action functional on the path.
To obtain from this a cocycle on $\Sigma$, we proceed as follows: we regard an interval $\sigma := [\sigma_{in}, \sigma_{out}] \in \mathbf{\Pi}(\Sigma)_1$ as a cospan
where in the top row we regard these subsets of $\Sigma$ as discrete smooth sub-categories, and in the bottom row form the path $\infty$-groupoids.
Then we take sections of $\array{E &\to& E_{\mathbf{\Pi}}\\ \downarrow && \downarrow \\ \Sigma &\to& \mathbf{\Pi}(\Sigma)}$ to produce a span of sections
of smooth $\infty$-groupoids.
Consider an $\infty$-groupoid
over the left foot. Under groupoid cardinality, if $\Psi$ is tame, this corresponds to a collection of rational numbers over vectors in fibers of $E$. Under “degrupoidification” we may think of this as specifying a section $|\Psi| \in \Gamma(E)$. The above span is supposed to give us the propagation of this state along $\sigma$.
To determine this, consider the special case where $\Psi$ is a “delta-section”, $* \mapsto (x,\sigma_{in}, v)$ supported on a single vector $v$ in a single fiber $E_x$ over $x$. Then its pull-push through this span yields the $\infty$-groupoid over $E$, which over $(y,\sigma_{out},w)$ consists of the set of paths $\gamma : x \to y$ such that $w = \exp(\int...)\rho(\gamma)(v)$, i.e. such that $w$ is the vector obtained from applying the action to $v$ along this path.
If everything were suitably finite, we could take cardinalities of the result and obtain the familiar path integral (sum)