# nLab prequantized Lagrangian correspondence

Classical Mechanics via Prequantum Correspondences

This entry is a sub-chapter of geometry of physics. See there for background

The following is effectively a derivation of, and an introduction to, classical mechanics by studying correspondences in what is called (as we will explain) the slice topos over the moduli stack of prequantum line bundles. One such correspondence in this slice topos is precisely a prequantized Lagrangian correspondence and the reader looking for just these should skip ahead to the section The classical action functional prequantizes Lagrangian correspondences. But for completeness and to introduce the technology used here, we start with introducing also more basic concepts, such as phase space etc.

## Surveys, textbooks and lecture notes

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Classical Mechanics via Prequantum Correspondences

## Model Layer

Here we discuss the notion of prequantized Lagrangian correspondences and how it serves to embed traditional classical mechanics in its formulation as Hamiltonian mechanics and Lagrangian mechanics into the general context of local prequantum field theory and its motivic quantization.

A traditional notion is that of a plain Lagrangian correspondence, which is a Lagrangian submanifold of the Cartesian product of a symplectic manifold and another one, with opposite symplectic structure. As discussed there, plain Lagrangian correspondences naturally generalize symplectomorphisms – hence transformations between phase spaces in physics – via correspondences of symplectic manifolds.

But in prequantum field theory proper, and in particular with an eye towards geometric quantization, one considers the prequantization of these symplectic manifolds by lifting them to prequantum circle bundles with principal connection. The notion of prequantized Lagrangian correspondence is the refinement of that of plain Lagrangian correspondence which does properly respect and reflect this prequantization information: a prequantized Lagrangian correspondence is a Lagrangian subspace as before, but now equipped with an explicit gauge transformation between the pullbacks of the two prequantum circle bundles to the correspondence space.

We discuss below how the concept of prequantized Lagrangian correspondences neatly induces and unifies ingredients and aspects of classical mechanics, notably the Hamiltonian mechanics of symplectic manifolds and the Lagrangian mechanics of action functionals associated to it via the Legendre transform. Specifically, below in The classical action prequantizes Hamiltonian correspondences we see that prequantized Lagrangian correspondences are diagrams which schematically express this data as follows:

$\array{ && {{space\,of} \atop {trajectories}} \\ & {}^{\mathllap{{initial}\atop {values}}}\swarrow && \searrow^{\mathrlap{{Hamiltonian} \atop {evolution}}} \\ phase\,space_{in} && \swArrow_{{action} \atop {functional}} && phase \,space_{out} \\ & {}_{\mathllap{{prequantum}\atop {bundle}_{in}}}\searrow && \swarrow_{\mathrlap{{prequantum} \atop {bundle}_{out}}} \\ && {{2\text{-}group} \atop {of\,phases}} } \,.$

This describes a diagram in what is called the slice topos of smooth sets over the moduli stack of prequantum circle bundles. Once formulated this way, there is an evident refinement to higher moduli stacks of prequantum n-bundles.

For $n = 2$ we show how the notion of prequantized Lagrangian correspondence is – still naturally in the context of local prequantum field theory – further refined from the context of symplectic manifolds to that of Poisson manifolds. Specifically, this is obtained by realizing that prequantized Lagrangian correspondences are really naturally to be regarded as correspondences-of-correspondences in a 2-category of correspondences, where now the new lower-order correspondences are instead boundary field theories for a 2d Chern-Simons theory (a non-perturbative Poisson sigma-model).

### Phase spaces and symplectic manifolds

Given a physical system, one says that its phase space is the space of its possible (“classical”) histories or trajectories. The first two of Newton's laws of motion say that trajectories of physical systems are (typically) determined by differential equations of second order, and therefore these spaces of trajectories are (typically) equivalent to initial value data of 0th and of 1st derivatives. In physics this data (or rather its linear dual) is referred to as the canonical coordinates and the canonical momenta, respectively, traditionally denoted by the symbols “$q$” and “$p$”. But being coordinates, these are actually far from being canonical in the mathematical sense; all that has invariant meaning is, locally, the surface element $\mathbf{d}p \wedge \mathbf{d}q$ spanned by a change of coordinates and momenta.

So far this says that a physical phase space is mathematically formalized by a sufficiently smooth manifold $X$ which is equipped with a closed and non-degenerate differential 2-form $\omega \in \Omega^2_{\mathrm{cl}}(X)$, hence by a symplectic manifold $(X,\omega)$.

The non-degeneracy of a symplectic form encodes the special property (as we will make explicit below) that (time) evolution of coordinates and momenta is uniquely induced by an action functional/Hamiltonian generating the evolution. This is however famously not the case for systems with gauge equivalences, hence such systems which have configurations that are nominally different but nevertheless physically equivalent. Presence of such gauge equivalences is not the exception, but the rule for physical systems, and therefore we want to include this case.

In the presence of gauge equivalences, the phase space form $\omega$ is still a closed differential 2-form, it just need not be non-degenerate anymore. While in such a case the pair $(X,\omega)$ could just be called a smooth manifold equipped with a closed differential 2-form}, it is traditional to call this a pre-symplectic manifold in order to amplify the indented use as a model for phase spaces. (Some authors demand that a pre-symplectic form be a closed form with constant rank, but here this technical condition will not be relevant and will not be considered.)

###### Example

The sigma-model describing the propagation of a particle on the real line $\mathbb{R}$ has as phase space the plane $\mathbb{R}^2 = T^\ast \mathbb{R}$ and as symplectic form its canonical volume form. Traditionally the two canonical coordinate functions on this phase space are denoted $q,p \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}$ (called the “canonical coordinate” and the “canonical momentum”, respectively), and in terms of these the symplectic form in this example is $\omega = \mathbf{d} q \wedge \mathbf{d} p$.

When dealing with spaces $X$ that are equipped with extra structure, such as $\omega \in \Omega^2_{\mathrm{cl}}(X)$, then it is useful to have a universal moduli space for these structures, and this will be central for our developments here. So we need a “smooth set$\mathbf{\Omega}^2_{\mathrm{cl}}$ of sorts, characterized by the property that there is a natural bijection between smooth closed differential 2-forms $\omega \in \Omega^2_{\mathrm{cl}}(X)$ and smooth maps $X \longrightarrow \mathbf{\Omega}^2_{\mathrm{cl}}$. Of course such a universal moduli spaces of closed 2-forms does not exist in the category of smooth manifolds. But it does exist canonically if we slightly generalize the notion of “smooth set” suitably.

###### Definition

A smooth set or smooth 0-type $X$ is

1. an assignment to each $n \in \mathbb{N}$ of a set, to be written $X(\mathbb{R}^n)$ and to be called the set of smooth maps from $\mathbb{R}^n$ into $X$,

2. an assignment to each ordinary smooth function $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between Cartesian spaces of a function of sets $X(f) : X(\mathbb{R}^{n_2}) \to X(\mathbb{R}^{n_1})$, to be called the pullback of smooth functions into $X$ along $f$;

such that

1. this assignment respects composition of smooth functions;

2. this assignment respects the covering of Cartesian spaces by open disks: for every good open cover $\{\mathbb{R}^n \simeq U_i \hookrightarrow \mathbb{R}^n\}_i$, the set $X(\mathbb{R}^n)$ of smooth functions out of $\mathbb{R}^n$ into $X$ is in natural bijection with the set $\left\{ (\phi_i)_i \in \prod_i X(U_i) \;|\; \forall_{i,j}\; \phi_i|_{U_{i} \cap U_j} = \phi_j|_{U_{i} \cap U_j} \right\}$ of tuples of smooth functions out of the patches of the cover which agree on all intersections of two patches.

For more on this see at geometry of physics in the section Smooth sets.

While the formulation of this definition is designed to make transparent its geometric meaning, of course equivalently but more abstractly this says the following:

###### Definition

Write CartSp for the category of Cartesian spaces with smooth functions between them, and consider it as a site by equipping it with the coverage of good open covers. A smooth set or smooth 0-type is a sheaf on this site. The topos of smooth 0-types is the category of sheaves

$\mathrm{Smooth}0\mathrm{Type} \coloneqq \mathrm{Sh}(\mathrm{CartSp}) \,.$

In the following we will abbreviate the notation to

$\mathbf{H} \coloneqq \mathrm{Smooth}0\mathrm{Type} \,.$

The topos of prop. also has another site of definition.

###### Proposition

Write $SmoothMfd$ for the category of smooth manifolds regarded as a site with the standard Grothendieck topology of open covers. There is an equivalence of categories

$Sh(SmoothMfd) \simeq \mathrm{Smooth}0\mathrm{Type} \,.$
###### Proof

The canonical inclusion $CartSp\hookrightarrow SmoothMfd$ is readily seen to be a dense subsite. (This is just the statement that – by definition – every smooth manifold may be covered by Cartesian spaces.) The statement hence follows by the comparison lemma.

For the discussion of presymplectic manifolds, we need the following two examples.

###### Example

Every smooth manifold $X \in \mathrm{SmoothManifold}$ becomes a smooth 0-type by the assignment

$X \;\colon\; n \mapsto C^\infty(\mathbb{R}^n, X) \,.$

This construction extends to a full embedding of smooth manifolds into smooth sets

$SmoothManifold \hookrightarrow \mathbf{H}$
###### Proof

This follows via prop. by the Yoneda lemma.

###### Example

For $p \in \mathbb{N}$, write $\mathbf{\Omega}^p_{\mathrm{cl}}$ for the smooth set whose $n$-dimensional plots are smooth differential p-forms on $\mathbb{R}^n$:

$\mathbf{\Omega}^p_{\mathrm{cl}} \colon \mathbb{R}^n \mapsto \Omega^p_{\mathrm{cl}}(\mathbb{R}^n)$

and which sends a smooth function $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ to the pullback of differential forms along this map

$\array{ \mathbb{R}^{n_1} &\mapsto& \Omega^p_cl(\mathbb{R}^{n_1}) \\ \downarrow^{\mathrlap{f}} && \uparrow^{\mathrlap{f^\ast}} \\ \mathbb{R}^{n_2} &\mapsto& \Omega^p_cl(\mathbb{R}^{n_2}) } \,.$

For more on this example see at geometry of physics in the section Differential forms.

This solves the moduli problem for closed smooth differential forms:

###### Proposition

For $p \in \mathbb{N}$ and $X \in SmoothManifold \hookrightarrow Smooth0Type$, there is a natural bijection

$\mathbf{H}(X,\mathbf{\Omega}^p_{\mathrm{cl}}) \simeq \Omega^p_{\mathrm{cl}}(X)$

between morphisms (of smooth sets)

$X \longrightarrow \Omega^2_{cl}$

and smooth closed 2-forms

$\omega \in \Omega^2_{cl}(X)$

on $X$.

###### Proof

This follows via prop. by the Yoneda lemma.

So a presymplectic manifold $(X,\omega)$ is equivalently a map of smooth sets of the form

$\omega \;\colon\; X \longrightarrow \mathbf{\Omega}^2_{\mathrm{cl}} \,.$

### Canonical transformations and symplectomorphisms

An equivalence between two phase spaces, hence a re-expression of the “canonical coordinates” and “canonical momenta”, is called a canonical transformation in physics. Mathematically this is a symplectomorphism.

###### Definition

Given two symplectic manifolds $(X_1, \omega_1)$ and $(X_2, \omega_2)$ (which might be two copies of one single symplectic manifold), a symplectomorphism between them

$f \;\colon\; (X_1, \omega_1) \longrightarrow (X_2, \omega_2)$

is a diffeomorphism

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

of the underlying smooth manifolds, such that the pullback of the second symplectic form along $f$ equals the first,

$f^\ast \omega_2 = \omega_1 \,.$

The above formulation of pre-symplectic manifolds as maps into a moduli space of closed differential 2-forms yields the following formulation of symplectomorphisms, which is very simple in itself, but contains in it the seed of an important phenomenon:

###### Proposition

A symplectomorphism $f \colon (X_1, \omega_2) \longrightarrow (X_2, \omega_2)$ as above is, under the identification of prop. , equivalently a commuting diagram in $\mathbf{H}$ of the form

$\array{ X_1 && \stackrel{f}{\longrightarrow}&& X_2 \\ & {}_{\mathllap{\omega_1}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && \Omega^2_{cl} } \,.$
###### Proof

This is the naturality of the Yoneda lemma:

By prop. the identification of $\omega_i \in \Omega^2_{cl}(X_i)$ with $\omega_i \colon X \to \Omega^2_{cl}$ is via the natural equivalence

$Hom_{Smooth0Type}(X,\mathbf{\Omega}^2_{cl}) \stackrel{\simeq}{\longrightarrow} \Omega^2_{cl}(X) \,.$

This being natural means that for every morphism $X_1 \stackrel{f}{\longrightarrow} X_2$ there is a commuting diagram of the form

$\array{ Hom(X_2,\mathbf{\Omega}^2_{cl}) &\stackrel{\simeq}{\longrightarrow}& \Omega^2_{cl}(X_2) \\ \downarrow^{\mathrlap{(-) \circ f}} && \downarrow^{\mathrlap{f^\ast}} \\ Hom(X_1,\mathbf{\Omega}^2_{cl}) &\stackrel{\simeq}{\longrightarrow}& \Omega^2_{cl}(X_1) }$

where on the left we have the pre-composition operation on morphisms and on the right we have, by example , the pullback of differential forms.

Consider then the element

$(X_2 \stackrel{\omega_2}{\to} \mathbf{\Omega}^2_{cl}) \in Hom(X_2,\mathbf{\Omega}^2_{cl})$

in the top left set in this diagram. Sending it along the top and right maps yields the pullback of differential forms $f^\ast \omega_2 \in \Omega^2_{cl}(X_1)$. On the other hand, sending it along the left and bottom maps yields the differential form represented by the composite morphism $(X_1 \stackrel{f}{\to} X_2 \stackrel{\omega_1}{\to}\mathbf{\Omega}^2_{cl})$. Commutativity of the above naturality diagram means that these two elements of $\Omega^2_{cl}(X_1)$ coincide. This is the claim to be proven.

Situations like this are naturally interpreted in a slice topos:

###### Definition

For $A \in \mathbf{H}$ any smooth set, the slice topos $\mathbf{H}_{/A}$ is the category whose objects are objects $X \in \mathbf{H}$ equipped with maps $X \to A$, and whose morphisms are commuting diagrams in $\mathbf{H}$ of the form

$\array{ X &&\longrightarrow&& Y \\ & \searrow && \swarrow \\ && A }$
###### Definition

Write $\mathrm{SymplManifold}$ for the category with presymplectic smooth manifolds as objects and symplectomorphisms, def. , betwen them as morphisms

###### Proposition

The construction of prop. which sends a smooth symplectic manifold $(X,\omega)$ to the classifying morphism of smooth sets $(X \stackrel{\omega}{\longrightarrow}\mathbf{\Omega}^2_{cl})$ regarded as an object in the slice topos, def. constitutes a full and faithful functor

$SymplManifold \hookrightarrow Smooth0Type_{/\mathbf{\Omega}^2_{\mathrm{cl}}}$

of pre-symplectic manifolds with symplectomorphisms between them into the slice topos of smooth sets over the smooth moduli space of closed differential 2-forms.

By prop. .

### Trajectories and Lagrangian correspondences

A symplectomorphism clearly puts two symplectic manifolds “in relation” to each other. But it does so also in the formal sense of relations in mathematics. Recall:

###### Definition

For $X,Y \in$ Set two sets, a relation $R$ between elements of $X$ and elements of $Y$ is a subset of the Cartesian product set

$R \hookrightarrow X \times Y \,.$

More generally, for $X, Y \in \mathbf{H}$ two objects of a topos (such as the topos of smooth sets), then a relation $R$ between them is a subobject of their Cartesian product

$R \hookrightarrow X \times Y \,.$

In particular any function induces the relation$y$ is the image of $x$”:

###### Example

For $f \;\colon\; X \longrightarrow Y$ a function, its induced relation is the relation which is exhibited by the graph of $f$

$graph(f) \coloneqq \left\{ (x,y) \in X \times Y \;|\; f(x) = y \right\}$

canonically regarded as a subobject

$graph(f) \hookrightarrow X \times Y \,.$

Hence in the context of classical mechanics, in particular any symplectomorphism $f \;\colon\; (X_1, \omega_1) \longrightarrow (X_2, \omega_2)$ induces the relation

$graph(f) \hookrightarrow X_1 \times X_2 \,.$

Since we are going to think of $f$ as a kind of “physical process”, it is useful to think of the smooth set $graph(f)$ here as the space of trajectories of that process. To make this clearer, notice that we may equivalently rewrite every relation $R \hookrightarrow X \times Y$ as a diagram of the following form:

$\array{ && R \\ & {}^{\mathllap{i_X}}\swarrow && \searrow^{\mathrlap{i_Y}} \\ X && && Y } \;\; = \;\; \array{ && R \\ && \downarrow \\ && X \times Y \\ & {}^{\mathllap{p_X}}\swarrow && \searrow^{\mathrlap{p_Y}} \\ X && && Y }$

reflecting the fact that every element $(x \sim y) \in R$ defines an element $x = i_X(x \sim y) \in X$ and an element $y = i_Y(x \sim y) \in Y$.

Then if we think of $R = graph(f)$ we may read the relation as “there is a trajectory from an incoming configuration $x_1$ to an outgoing configuration $x_2$

$\array{ && graph(f) \\ & {}^{\mathllap{incoming}}\swarrow && \searrow^{\mathrlap{outgoing}} \\ X_1 && && X_2 } \,.$

Notice here that the defining property of a relation as a subset/subobject translates into the property of classical physics that there is at most one trajectory from some incoming configuration $x_1$ to some outgoing trajectory $x_2$ (for a fixed parameter time interval at least, we will formulate this precisely in the next section when we genuinely consider Hamiltonian correspondences).

In a more general context one could consider there to be several such trajectories, and even a whole smooth set of such trajectories between given incoming and outgoing configurations. Each such trajectory would “relate” $x_1$ to $x_2$, but each in a possible different way. We can also say that each trajectory makes $x_1$ correspond to $x_2$ in a different way, and that is the mathematical term usually used:

###### Defininition

For $X, Y \in \mathbf{H}$ two spaces, a correspondence between them is a diagram in $\mathbf{H}$ of the form

$\array{ && Z \\ & \swarrow && \searrow \\ X && && Y }$

with no further restrictions. Here $Z$ is also called the correspondence space.

An equivalence between two such correspondences is an equivalence $Z_1 \stackrel{\simeq}{\to}Z_2$ that gives a commuting diagram of the form

$\array{ && Z_1 \\ & {}^{}\swarrow && \searrow^{} \\ X &&\downarrow^{\mathrlap{\simeq}} && Y \\ & {}_{}\nwarrow && \nearrow_{} \\ && Z_2 }$

Correspondences between $X$ any $Y$ with such equivalences between them form a groupoid. (See at geometry of physics the section Essence of gauge theory: Groupoids and basic homotopy 1-type theory for more on this.) Hence we write

$Corr\left(\mathbf{H}\right)(X,Y) \in Grpd \,.$
###### Example

The correspondence induced by the graph of a function $f \colon X \to Y$ as in example is equivalent, in the sense of def. , to the correspondence

$\array{ && X \\ & {}^{\mathllap{id}}\swarrow && \searrow^{\mathrlap{f}} \\ X && && Y } \,.$

The equivalence

$\array{ && X \\ & {}^{\mathllap{id}}\swarrow && \searrow^{\mathrlap{f}} \\ X &&\downarrow^{\mathrlap{\simeq}} && Y \\ & {}_{\mathllap{i_X}}\nwarrow && \nearrow_{\mathrlap{i_Y}} \\ && graph(f) }$

is induced by

$X \stackrel{\simeq}{\longrightarrow} graph(f)$
$x \mapsto (x,f(x))$

Moreover, if we think of correspondences as modelling spaces of trajectories, then it is clear that their should be a notion of composition:

###### Definition

Given two consecutive correspondences, then their composite is the correspondence obtained by forming the fiber product of the two coincident morphisms:

$\left( \array{ && Y_1 &&&& Y_2 \\ & \swarrow && \searrow && \swarrow && \searrow \\ X_1 && && X_2 && && X_3 } \right) \;\;\;\; \mapsto \;\;\;\; \left( \array{ && Y_1 \circ_{X_2} Y_2 \\ & \swarrow && \searrow \\ X_1 && && X_3 } \right) \,.$
###### Remark

Heuristically, the composite space of trajectories $Y_1 \circ_{X_2} Y_2$ should consist precisely of those pairs of trajectories $( f, g ) \in Y_1 \times Y_2$ such that the endpoint of $f$ is the starting point of $g$. The space with this property is precisely the fiber product of $Y_1$ with $Y_2$ over $X_2$, denoted $Y_1 \underset{X_2}{\times} Y_2$ (also called the pullback of $Y_2 \longrightarrow X_2$ along $Y_1 \longrightarrow X_2$ and then abbreviated $(pb)$):

$\left( \array{ && Y_1 \circ_{X_2} Y_2 \\ & \swarrow && \searrow \\ X_1 && && X_3 } \right) \;\;\; = \;\;\; \left( \array{ && && Z_1 \underset{Y}{\times} Z_2 \\ && & \swarrow && \searrow \\ && Z_1 && (pb) && Z_2 \\ & \swarrow && \searrow && \swarrow && \searrow \\ X && && Y && && Z } \right) \,.$

Hence given a topos $\mathbf{H}$, correspondences between its objects form a category which composition the fiber product operation, where however the collection of morphisms between any two objects is not just a set, but is a groupoid (the groupoid of correspondences between two given objects and equivalences between them).

One says that correspondences form a (2,1)-category

$Corr(\mathbf{H}) \in (2,1)Cat \,.$

But for most purposes here, the reader unwilling to enter higher category theory can, to good approximation, pretend that correspondences form an ordinary category.

One reason for formalizing this notion of correspondences so much in the present context that it is useful now to apply it not just to the ambient topos $\mathbf{H}$ of smooth sets, but also to its slice topos $\mathbf{H}_{/\mathbf{\Omega}_{cl}^2}$ over the universal moduli space of closed differential 2-forms.

To see how this is useful in the present context, notice the following basic observation:

###### Definition

Given a symplectic manifold $(X,\omega)$, then a submanifold

$L \hookrightarrow X$

is called

###### Proposition

Let $f \colon X_1 \to X_2$ be a smooth function between smooth manifolds and let

$\array{ && graph(f) \\ && \downarrow \\ && X_1 \times X_2 \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X_1 && && X_2 }$

be the induced correspondence. If $\omega_1$ and $\omega_2$ are symplectic forms on $X_1$ and $X_2$, respectively, then $p_1^\ast \omega_1 - p_2^\ast \omega_2$ is a pre-symplectic form on $X_1 \times X_2$, and $f$ is a symplectomorphism precisely if $graph(f) \hookrightarrow X_1 \times X_2$ is a Lagrangian submanifold.

To capture this phenomenon, one traditionally sets:

###### Definition

For $(X_1,\omega_1)$ and $(X_2,\omega_2)$ two symplectic manifolds (not necessarily of the same dimension), an isotropic correspondence or Lagrangian correspondence between them is a correspondence of the underlying manifolds

$\array{ && R \\ && \downarrow \\ && X_1 \times X_2 \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X_1 && && X_2 }$

such that the correspondence space $R \hookrightarrow X_1 \times X_2$ is an isotropic submanifold or Lagrangian submanifold, respectively of the product symplectic manifold given by

$(X_1 \times X_2 , p_1^\ast \omega_1 - p_2^\ast \omega_2) \,.$
###### Proposition

Under the identification of prop. , isotropic correspondences as in def. are equivalent to diagrams of smooth sets of the form

$\array{ && R \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X_1 && \swArrow_= && X_2 \\ & {}_{\mathllap{\omega_1}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && \Omega^2_{cl} } \,.$

This in turn is equivalent to being a correspondence in the slice topos $\mathbf{H}_{/\Omega^2_{cl}}$, def. , under the identification of prop. .

###### Proof

By prop. the commutativity of this diagram says precisely that on $R$ we have

$p_1^\ast \omega_1 = p_2^\ast \omega_2$

hence

$p_1^\ast \omega_1 - p_2^\ast \omega_2 = 0 \,.$

Therefore we have:

###### Proposition

For $(X_1, \omega_2)$ and $(X_2, \omega_2)$ two symplectic manifolds, there is a full embedding

$LagrangianCorrespondences\left(\left(X_1,\omega_1\right), \left(X_2, \omega_2\right)\right) \hookrightarrow Corr\left(\mathbf{H}_{/\mathbf{\Omega}^2_{cl}}\right)\left(\left(X_1,\omega_1\right), \left(X_2, \omega_2\right)\right)$

of the Lagrangian correspondences into the space of correspondences between the two manifolds as objects in the slice topos over the universal moduli space of closed differential 2-forms.

###### Example

The graph of a function $f \colon X_1\to X_2$ between symplectic manifold $(X_i, \omega_i)$ is a Lagrangian correspondence precisely if $f$ is a symplectomorphism.

###### Proof

Under the identification of

$\array{ && graph(f) \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X_1 && && X_2 }$

### Hamiltonian (time evolution) trajectories and Hamiltonian correspondences

An important class of symplectomorphisms are the following

###### Definition

Let $(X,\omega)$ be a symplectic manifold. The induced Poisson bracket $\{-,-\}$ takes a smooth function $H \in C^\infty(X)$ (the “Hamiltonian”) to the derivation $\{H,-\}$ on $C^\infty(X)$. This is equivalently a vector field $v_H \in \Gamma T X$, the corresponding Hamiltonian vector field.

A Hamiltonian symplectomorphisms from a symplectic manifold $(X,\omega)$ to itself, is a symplectomorphism $X \to X$ which is the flow of a Hamiltonian vector field for some parameter “time” $t \in \mathbb{R}$

$\exp( t \{H,-\}) \;\colon\; (X,\omega) \longrightarrow (X,\omega) \,.$

We call a Lagrangian correspondence, def. , induced from Hamiltonian symplectomorphisms a Hamiltonian correspondences.

###### Remark

Under the interpretation of correspondences as spaces of trajectories as in example , the smooth correspondence space of a Hamiltonian correspondence is naturally identified with the space of classical trajectories of the Hamiltonian dynamics of $H$

$Fields_{traj}^{class}(t) = graph\left( \exp(t\{H,-\}) \right)$

in that

1. every point in the space corresponds uniquely to a trajectory of parameter time length $t$ characterized as satisfying the equations of motion as given by Hamilton's equations for $H$;

2. the two projection maps to $X$ send a trajectory to its initial and to its final configuration, respectively.

###### Remark

Forming Hamiltonian correspondences consitutes a functor from 1-dimensional cobordisms with Riemannian structure to the category of correspondences in the slice topos:

$\exp((-)\{H,-\}) \;\colon\; Bord^{Riem}_1 \longrightarrow Corr_1(\mathbf{H}_{/\Omega^2})$

since for all (“time”) parameter valued $t_1, t_2 \in \mathbb{R}$ we have a composition (by fiber product) of correspondences exhibited by the following pasting diagram:

$\array{ &&&& graph\left(\exp\left(\left(t_1+t_2\right)\right) \left\{H,-\right\} \right) \\ && & \swarrow && \searrow \\ && graph\left(\exp\left(t_1\right)\left\{H,-\right\}\right) && (pb) && graph\left(\exp\left(t_2 \left\{H,-\right\}\right)\right) \\ & \swarrow && \searrow & & \swarrow && \searrow \\ X && && X && && X \\ & \searrow &&& \downarrow &&& \swarrow \\ &&&& \Omega^2_{cl} } \,.$

### The kinetic action and Planck’s constant

To naturally see why there would be any Hamiltonian associated to a (to some) symplectomorphism in the first place, we step back and consider local trivializations or local potentials for symplectic forms. Doing so turns out to give rise to what in physics is called the kinetic action, what in the context of geometric quantization is called prequantization and what in mathematics is called lifting to differential cohomology. All these concepts arise directly from the following simple consideration.

Given a pre-symplectic form $\omega \in \Omega^2_{\mathrm{cl}}(X)$, by the Poincaré lemma there is a good open cover $\{U_i \hookrightarrow X\}_i$ such that one can find smooth differential 1-forms $\theta_i \in \Omega^1(U_i)$ such that these are local trivializations/potentials for the symplectic form on each patch $U_i$ of the cover:

$\mathbf{d}\theta_i = \omega_{|U_i} \,.$

Physically such a 1-form is (up to a factor of 2) a choice of kinetic energy density called a kinetic Lagrangian $L_{\mathrm{kin}}$ (below in example we connect this statement to a maybe more familiar formla):

$\theta_i = 2 L_{\mathrm{kin}, i} \,.$
###### Example

Consider the phase space $(\mathbb{R}^2, \; \omega = \mathbf{d} q \wedge \mathbf{d} p)$ of example . Since $\mathbb{R}^2$ is a contractible topological space we consider the trivial covering ($\mathbb{R}^2$ covering itself) since this is already a good covering in this case. Then all the $\{g_{i j}\}$ are trivial and the data of a prequantization consists simply of a choise of 1-form $\theta \in \Omega^1(\mathbb{R}^2)$ such that

$\mathbf{d}\theta = \mathbf{d}q \wedge \mathbf{d}p \,.$

A standard such choice is

$\theta = - p \wedge \mathbf{d}q \,.$

Then given a trajectory $\gamma \colon [0,1] \longrightarrow X$ which satisfies Hamilton's equation for a standard kinetic energy term, then $(p \mathbf{d}q)(\dot\gamma)$ is this kinetic energy of the particle which traces out this trajectory.

Given a path $\gamma : [0,1] \to X$ in phase space, its kinetic action $S_{\mathrm{kin}}$ is supposed to be the integral of $L_{\mathrm{kin}}$ along this trajectory. In order to make sense of this in the generality where there is no globally defined $\theta$, there need to be functions $g_{i j} \in C^\infty(U_i \cap U_j, \mathbb{R})$ for each double intersection of patches of the cover, such that these the local $\theta$‘s differ on these double intersection only by the total derivative (de Rham differential $\mathbf{d}$ ) of these functions:

$\theta_j|_{U_j} - \theta_i|_{U_i} = \mathbf{d}g_{i j} \,.$

One then finds (from the theory of Cech cohomology) that if on triple intersections these functions satisfy

$g_{ij} + g_{j k} = g_{i k}$

then there is a well defined action functional

$S_{\mathrm{kin}}(\gamma) \in \mathbb{R}$

obtained by dividing $\gamma$ into small pieces that each map to a single patch $U_i$, integrating $\theta_i$ along this piece, and adding the contribution of $g_{i j}$ at the point where one switches from using $\theta_i$ to using $\theta_j$. Technically this is called the holonomy or parallel transport of the $(\mathbb{R},+)$-principal connection which is defined by the data $(\{\theta_i\}, \{g_{i j}\} )$.

However, requiring this condition on triple overlaps as an equation between $\mathbb{R}$-valued functions makes the local patch structure trivial: if this is possible then one can in fact already find a single $\theta \in \Omega^1(X)$ and functions $h_i \in C^\infty(U_i, \mathbb{R})$ such that $\theta_i = \theta|_{U_i} + \mathbf{d}h_i$. This has the superficially pleasant effect that the action is simply the integral against this globally defined 1-form, $S_{\mathrm{kin}} = \int_{[0,1]} \gamma^\ast L_{\mathrm{kin}}$, but it also means that the pre-symplectic form $\omega$ is exact, which is not the case in many important examples. (In more abstract terms what this is saying is that every $(\mathbb{R},+)$-principal bundle over a manifolds is trivializable.)

On the other hand, what really matters in prequantum physics is not the action functional $S_{\mathrm{kin}} \in \mathbb{R}$ itself, but the exponentiated action

$\exp\left( \tfrac{i}{\hbar} S \right) \in \mathbb{R}/(2\pi \hbar)\cdot\mathbb{Z} \,,$

which takes values in the quotient of the additive group of real numbers by integral multiples of Planck's constant $2\pi \hbar$.

In more detail, consider the canonical inclusion

$\mathbb{Z} \hookrightarrow \mathbb{R}$

of the integers as an addiditve subgroup of the real numbers. Strictly speaking what appears in physics is the real line on which a unit is chosen as part of the identification of mathematical formalism with physical reality, one should really consider all possible additive group homomorphisms $\mathbb{Z}\to \mathbb{R}$. These are parameterized by

$h \in (\mathbb{R}- \{0\}) \hookrightarrow \mathbb{R}$
$(-)\cdot h \;\colon\; \mathbb{Z} \longrightarrow \mathbb{R}$

and this “physical unit$h$ is what is called Planck’s constant.

In particular the induced circle group is identified as the quotient of $\mathbb{R}$ by $h \mathbb{Z}$, in this sense

$U(1) \simeq \mathbb{R}/h \mathbb{Z}$

and under this identification its quotient map is expressed in terms of the exponential function $\exp \colon z \mapsto \sum_{k = 0}^\infty \frac{z^k}{k!} \in \mathbb{C}$ as

$\exp(2 \pi \tfrac{i}{h}(-)) = \exp(\tfrac{i}{\hbar} (-)) \;\colon\; \mathbb{R} \longrightarrow U(1) \,,$

where

$\hbar \coloneqq h/2\pi \,.$

The resulting short exact sequence is the real exponential exact sequence

$0 \to \mathbb{Z} \longrightarrow \mathbb{R} \stackrel{\exp(\tfrac{i}{\hbar}(-))}{\longrightarrow} U(1) \to 0 \,.$

This is the source of the ubiquity of the expression $\exp(\tfrac{i}{\hbar} (-))$ in quantum physics, say in the path integral, where the exponentiated action functional appears as $\exp(\tfrac{i}{\hbar} S)$.

### Pre-Quantization and Differential cohomology

By the above discussion, for the exponentiated kinetic action functional to be well defined, one only needs that the equation $g_{i j} + g_{j k} = g_{i k}$ on triple intersection holds modulo addition of an integral multiple of Planck's constant $h = 2\pi \hbar$.

If this is the case, then one says that the data $(\{\theta_i\}, \{g_{i j}\})$ defines equivalently

on $X$, with curvature the given symplectic 2-form $\omega$.

Such data is called a pre-quantization of the symplectic manifold $(X,\omega)$. Since it is the exponentiated action functional $\exp(\frac{i}{\hbar} S)$ that enters the quantization of the given mechanical system (for instance as the integrand of a path integral), the prequantization of a symplectic manifold is indeed precisely the data necessary before quantization.

Therefore, in the spirit of the above discussion of pre-symplectic structures, we would like to refine the smooth moduli space of closed differential 2-forms to a moduli space of prequantized differential 2-forms.

Again this does naturally exist if only we allow for a good notion of “space”. An additional phenomenon to be taken care of now is that while pre-symplectic forms are either equal or not, their pre-quantizations can be different and yet be equivalent:

because there is still a remaining freedom to change this data without changing the exponentiated action along a closed path: we say that a choice of functions $h_i \in C^\infty(U_i, \mathbb{R}/(2\pi\hbar)\mathbb{Z})$ defines an equivalence between $(\{\theta_i\}, \{g_{i j}\})$ and $(\{\tilde \theta_i\}, \{\tilde g_{i j}\})$ if $\tilde \theta_i - \theta_i = \mathbf{d}h_i$ and $\tilde g_{i j} - g_{i j} = h_j - h_i$.

This means that the space of prequantizations of $(X,\omega)$ is similar to an orbifold: it has points which are connected by gauge equivalences: there is a groupoid of pre-quantum structures on a manifold $X$.

In just the same way then that above we found a smooth moduli space $\mathbf{\Omega}^2_{cl}$ of closed differential 2-forms, one can find a smooth groupoid (for more on this see at geometry of physics the section Smooth homotopy types ), which we denote

$\mathbf{B}U(1)_{\mathrm{conn}} \in \mathbf{H}$
###### Proposition

The smooth groupoid $\mathbf{B}U(1)_{\mathrm{conn}}$ is characterized as follows,

1. For $X$ a smooth manifold, maps

$\nabla \colon X \longrightarrow \mathbf{B}U(1)_{conn}$

are equivalent to the above prequantum data $(\{\theta_i\}, \{g_{i j}\})$ on $X$;

2. for $\nabla_1, \nabla_2 \colon X \longrightarrow \mathbf{B}U(1)_{conn}$ two such maps, homotopies

$\array{ & \nearrow \searrow^{\mathrlap{\nabla_1}} \\ X & \Downarrow & \mathbf{B}U(1)_{conn} \\ & \searrow \nearrow_{\mathrlap{\nabla_2}} }$

between these are equivalent to the above gauge transformations $(\{h_i\})$ between this data

$(\theta_2)_i - (\theta_1)_i = \mathbf{d} \tfrac{\hbar}{i} log (h_i) \,.$
###### Proposition

There is a universal curvature map, a morphism of smooth groupoids

$F \;\colon\; \mathbf{B}U(1)_{\mathrm{conn}} \longrightarrow \mathbf{\Omega}^2_{\mathrm{cl}}$

which is such that for $\nabla \colon X \longrightarrow \mathbf{B}U(1)_{conn}$ a $U(1)$-principal connection, the composite

$F_\nabla \;\colon\; X \stackrel{\nabla}{\longrightarrow} \mathbf{B}U(1)_{conn} \stackrel{F_{(-)}}{\longrightarrow} \mathbf{\Omega}^2_{cl}$

is its curvature 2-form.

Hence this is the map that sends $(\{\theta_i\}, \{g_{i j}\})$ to $\omega$ with $\omega|_{U_i} = \mathbf{d}\theta_i$.

Therefore:

###### Definition

A prequantization of a symplectic manifold $(X,\omega)$ is – if it exists – a choice of circle group-principal connection $\nabla$ on $X$ whose curvature 2-form is the given symplectic form

$F_\nabla = \omega \,.$

In terms of the classifying morphism of differential forms as in prop. this reads as follows.

###### Proposition

Given a presymplectic manifold $(X,\omega)$, regarded equivalently as an object $(X \stackrel{\omega}{\longrightarrow} \mathbf{\Omega}^2_{cl}) \in \mathbf{H}_{/\mathbf{\Omega}^2_{cl}}$ by prop. , then a prequantization of $(X,\omega)$, def. , is equivalently a choice of lift $\nabla$ in

$\array{ X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}U(1)_{conn} \\ & {}_{\mathllap{\omega}}\searrow & \downarrow^{\mathrlap{F_{(-)}}} \\ && \mathbf{\Omega}^2_{cl} } \,.$

Phrased this way, there is an evident concept of prequantization of Lagrangian correspondences:

###### Definition

Given prequantized symplectic manifolds $(X_i,\nabla_i)$ as in prop. , and given a Lagrangian correspondence as in prop. , then a prequantization of this correspondence is a lift of the whole diagram through the universal curvature map of prop. :

$\array{ && Z \\ & \swarrow && \searrow \\ X_1 && && X_2 \\ & {}_{\mathllap{\omega_1}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && \mathbf{\Omega}^2_{cl} } \;\;\;\; \mapsto \;\;\;\; \array{ && Z \\ & \swarrow && \searrow \\ X_1 && \swArrow_{\simeq} && X_2 \\ & {}_{\mathllap{\nabla_1}}\searrow && \swarrow_{\mathrlap{\nabla_2}} \\ && \mathbf{B}U(1)_{conn} \\ && \downarrow^{\mathrlap{F}} \\ && \mathbf{\Omega}^2_{cl} } \,.$
###### Remark

This means in words that a prequantized Lagrangian correspondence is a prequantization of the in- and out-going symplectic manifolds together with a choice of equivalence/gauge transformation between the two prequantum circle bundles pulled back to the correspondences space.

###### Remark

By duality in the smmetric monoidal (2,1)-category of correspondences, a prequantized Lagragian correspondence is equivalently a diagram of the form

$\array{ && Y \\ & \swarrow && \searrow \\ \ast && \swArrow && X_1 \times X_2 \\ & \searrow && \swarrow_{\mathrlap{\nabla_2- \nabla}} \\ \\ && \mathbf{B}U(1)_{conn} }$

hence a trivialization of the product of one prequantum bundle with the negative (the inverse under tensor product) of the other, on the correspondence space.

### Hamiltonian flows, the Legendre transform and the Hamilton-Jacobi action

###### Proposition

Consider the phase space $(\mathbb{R}^2, \; \omega = \mathbf{d} q \wedge \mathbf{d} p)$ of example equipped with its canonical prequantization by $\theta = p \, \mathbf{d}q$ from example ,

Then smooth 1-parameter flows of this data via prequantized correspondences, def. ,

$t \;\;\;\; \mapsto \;\;\;\; \array{ X && \stackrel{f_t}{\longrightarrow} && X \\ & {}_{\mathllap{\theta}} \searrow & \swArrow_{ F_t } & \swarrow_{\mathrlap{\theta}} \\ && \mathbf{B}U(1)_{conn} }$

are in bijection with smooth functions $H \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$.

This bijection works by regarding $H$ as a Hamiltonian, def. , and assigning the flow $f_t = \exp(t \{H,-\})$ of its Hamiltonian vector field

$t \;\; \mapsto \;\; \array{ X && \stackrel{\exp(t \{H,-\})}{\longrightarrow} && X \\ & {}_{\mathllap{\theta}} \searrow & \swArrow_{\exp( \tfrac{i}{\hbar} S_t )} & \swarrow_{\mathrlap{\theta}} \\ && \mathbf{B}U(1)_{conn} } \,,$

where the prequantization is given by

• $S_t \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}$ is the Hamilton-Jacobi action of the classical trajectories induced by $H$,

• which is the integral $S_t = \int_{0}^t L \, d t$ of the Lagrangian $L \,d t$ induced by $H$,

• which is the Legendre transform of the Hamiltonian

$L \coloneqq p \frac{\partial H}{\partial p} - H \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R} \,.$
###### Proof

By prop. the prequantum filler of the diagram is given by a function $F_t =\exp(\tfrac{i}{\hbar} S_t)$ satisfying

$f_t^\ast \theta - \theta = -\mathbf{d}S_t \,.$

By standard Lie theory a smooth such 1-parameter flow is fixed by its derivative by $t$. For the above equation this yields

$\mathcal{L}_v \theta = -\mathbf{d}L$

where

1. $v \in \Gamma(T X)$ is the vector field of the flow $t\mapsto f_t$;

2. $\mathcal{L}_v$ is the Lie derivative along $v$;

3. $L \coloneqq \frac{\partial S}{\partial t}$.

By Cartan's magic formula this equation is equivalent to

$\iota_v \omega = -\mathbf{d}L - \mathbf{d} \iota_v \theta \,.$

This is the symplectic form of Hamilton's equations for $v$ and says that

$H \coloneqq - L - \iota_v \theta$

is a Hamiltonian that makes $v$ a Hamiltonian vector field. The correction term is

\begin{aligned} \iota_v \theta &= \iota_v ( p \, \mathbf{d}q ) \\ & = p \partial_v q \\ \end{aligned} \,.

But since $v$ is Hamiltonian, this is given by one component of Hamilton's equations $\iota_v (\mathbf{d}p \wedge \mathbf{d}q) = \mathbf{d}H$ saying that $\partial_v q = \frac{\partial H}{\partial p}$.

Hence in summary the flow is Hamiltonian and the pre-quntum filler is the choice of Hamiltonian $H$ specified by

$\frac{\partial S}{\partial t} = L = p \frac{\partial H}{\partial p} - H \,.$
###### Remark

In particular, this induces a functor

$\exp(\tfrac{i}{\hbar} S) \;\colon\; Bord_1^{Riem} \longrightarrow Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}}) \,.$

In summary, prop. and remark say that a prequantized Lagrangian correspondence is conceptually of the following form

$\array{ && {{space\,of} \atop {trajectories}} \\ & {}^{\mathllap{{initial}\atop {values}}}\swarrow && \searrow^{\mathrlap{{Hamiltonian} \atop {evolution}}} \\ phase\,space_{in} && \swArrow_{{action} \atop {functional}} && phase \,space_{out} \\ & {}_{\mathllap{{prequantum}\atop {bundle}_{in}}}\searrow && \swarrow_{\mathrlap{{prequantum} \atop {bundle}_{out}}} \\ && {{2-group} \atop {of\,phases}} } \,.$
###### Remark

The proof of prop. recovers, from general abstract input, precisely all the ingredients known in physics as canonical transformations.

The proposition says that the slice topos $\mathbf{H}_{/\mathbf{B}U(1)_{conn}}$ unifies classical mechanics in its two incarnations as Hamiltonian mechanics and as Lagrangian mechanics, where the relation between the two via the Legendre transform is exhibited by the homotopies that fill diagrams in the slice topos over $\mathbf{B}U(1)_{conn}$.

Hamiltonian$\leftarrow$ Legendre transform $\rightarrow$Lagrangian
Lagrangian correspondenceprequantizationprequantized Lagrangian correspondence

### Heisenberg group and Poisson bracket

Above we have interpreted maps $f \colon X \to Y$ as correspondences between $X$ and $Y$ by taking the correspondence space to be the graph of $f$. There is also another natural way to regard maps as correspondences: we may simply take $X$ as the correspondence space, take the left map out of it to be the identity and the right map to be $f$ itself:

$\left( X \stackrel{f}{\longrightarrow} Y \right) \;\; \mapsto \;\; \left( \array{ && X \\ & {}^{\mathllap{id}}\swarrow && \searrow^{\mathrlap{f}} \\ X && && Y } \right) \,.$

Consider now those correspondences which are equivalences (isomorphisms) in the category of correspondences $Corr_1(\mathbf{H})$. If we forget the smooth structure on everything and consider just correspondences of the underlying sets, hence $Corr_1(Set)$, then it is easy to see that under the cardinality map correspondences are given by matrices with cardinality entries and composition of correspondence by fiber product induces matrix multiplication.

Therefore for a correspondence to be an equivalence-transformation it has to be of the form above, induced by a direct map, which in addition is an equivalence $f \colon X \stackrel{\simeq}{\longrightarrow} Y$.

###### Proposition

Let $(X,\omega)$ be a symplectic manifold and choose any prequantization $(L,\nabla)$, thought of, via remark , as an object in the slice (2,1)-topos, $\nabla \in \mathbf{H}_{/\mathbf{B}U(1)_{conn}}$. Then

See (hgp 13)

For some reason, the quantomorphism group which is the Lie integration of the Poisson bracket is less famous than the Heisenberg group that sits inside it:

###### Remark

Suppose that $(X,\omega)$ itself has the structure of a group (for instance if $(X,\omega)$ is a symplectic vector space such as $(\mathbb{R}^{2n}, \sum_i p_i \mathbf{d}q^i)$ ), then the subgroup of the quantomorphism group whose underlying diffeomorphisms are given by the action of $X$ is the Heisenberg group of $X$.

### Hamiltonian actions and moment maps

For $G$ a Lie group, a Hamiltonian action of $G$ on $(X,\omega)$ is equivalently an action by prequantized Lagrangian correspondences, hence a group homomorphism

$G \longrightarrow \mathbf{Aut}_\nabla(Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}})) \,.$

The Lie differentiation of this is the corresponding moment map.

See (hgp 13)

## Semantic Layer

We now discuss the above constructions more abstractly in cohesive topos theory.

under construction

given $V$ then the delooping $\mathbf{B} \mathbf{Aut}(V)$ of its automorphism group $\mathbf{Aut}(V)$ is the 1-image factorization of the name $\ast \stackrel{\vdash V}{\longrightarrow} Type$ of $V$

$\ast \stackrel{}{\longrightarrow} \mathbf{B}\mathbf{Aut}(V) \hookrightarrow Type \,.$

for the slice over $\mathbf{B}U(1)_{conn}$ this needs to be subjected to differential concretification

(…)

## Syntactic Layer

We now discuss the above constructions yet more abstractly in homotopy type theory.

(…)

prequantization is lift through dependent sum along the universal curvature map of prop.

$\underset{F_{(-)}}{\sum} \;\colon\; \mathbf{H}_{/\mathbf{B}U(1)_{conn}} \longrightarrow \mathbf{H}_{/\mathbf{\Omega}^2_{cl}}$

(…)

## References

As far as it is not covered by traditional material, the above discussion is taken from

For more and related references see there and see at motivic quantization.

Last revised on November 14, 2014 at 08:04:00. See the history of this page for a list of all contributions to it.