nLab multisymplectic geometry

Idea

Traditional

Multisymplectic geometry is a generalization of symplectic geometry in the context of variational calculus and mechanical systems in which the symplectic form is generalized from a closed 2-form to a closed $n+1$-form, for $n \geq 1$ – the n-plectic form.

It is closely related to the de Donder-Weyl formalism of variational calculus. In the context of quantization it is meant to provide a refinement of geometric quantization which is well-adapted to $n$-dimensional quantum field theory. However, details of the multisymplectic quantization procedure remain under investigation.

From the $n$POV

We comment a bit on how to, presumably, think of multisymplectic geometry from the nPOV, in the context of higher geometric quantization. Readers may want to skip ahead to traditional technical discussion at Extended phase space.

Multisymplectic geometry is (or should be) to symplectic geometry as extended quantum field theory is to non-extended quantum field theory:

in the multisymplectic extended phase space of an $n$-dimensional field theory a state is not just a point, but an $n$-dimensional subspace.

See also n-plectic geometry.

Multisymplectic geometry is a generalization of symplectic geometry to cases where the symplectic 2-form is generalized to a higher degree differential form.

In as far as symplectic geometry encodes Hamiltonian mechanics, multisymplectic geometry may be regarded as resolving the symplectic geometry of the Hamiltonian mechanics of classical field theory: the kinematics of an $n$-dimensional field theory may be encoded in an degree $(n+1)$ symplectic form.

In this application to physics, multisymplectic geometry is also known as the covariant symplectic approach to field theory (e.g. section 2 here).

The idea is that under a suitable fiber integration multisymplectic geometry becomes ordinary symplectic form on the ordinary phase space of the theory, similar to, and in fact as a special case of, how for instance a line bundle on a loop space with a 2-form Chern class may arise by transgression from a bundle gerbe down on the original space, with a 3-form class.

By effectively undoing this implicit transgression in the ordinary Hamiltonian mechanics of classical field theory, multisymplectic geometry provides a general framework for a geometric, covariant formulation of classical field theory, where covariant formulation means that spacelike and timelike directions on a given space-time be treated on equal footing.

Extended phase spaces in covariant field theory

We discuss here the refinement in multisymplectic geometry of the covariant phase spaces of classical field theory/prequantum field theory from (pre-)symplectic manifolds of initial value data in a Cauchy surface to multisymplectic manifolds of local initial value data.

Recall that an ordinary phase space of a physical system is a symplectic manifold whose points correspond to the states of the system. The extended phase space of an $n$-dimensional quantum field theory is a multisymplectic space whose points correspond to pairs consisting of

• a point in the field theory’s parameter space – an “event”;

• a state of the theory “at that event”.

So extended phase spaces localizes the information about states : a point in here encodes not just the entire state of the system, but remembers explicitly what that state is like over any point in parameter space.

Covariant configuration bundle

Consider classical field theory over a parameter space $\Sigma$. From the point of view of FQFT $\Sigma$ will be one fixed cobordism on which we want to understand the (classical) field theory.

We assume that a field configuration on $\Sigma$ is a section $\phi : \Sigma \to E$ of some prescribed bundle $E \to \Sigma$: the field bundle.

Example

For instance an $n$-dimensional sigma-model quantum field theory is one whose field configurations on $\Sigma$ are given by maps

$$\phi : \Sigma \longrightarrow X$$

into some prescribed target space $X$. This is the case where $E = \Sigma \times X$ is a trivial bundle.

Remark

Beware of the standard source of confusion here when correlating this formalism with actual physics: the physical spacetime that we inhabit may be given either by $\Sigma$ or by $X$:

• in the description of the quantum mechanics of objects propagating in our physical spacetime, subject to forces exerted by fixed background gauge fields (such as electrons propagating in our particle accelerator, subject to the electromagnetic field in the accelerator tube), physical spacetime is identified with target space $X$, while $\Sigma$ is the worldvolume of the object that propagates through $X$. The field configurations on $\Sigma$ are really the maps $\Sigma \to X$ that determine how the object sits in spacetime.

• in quantum mechanics of fields on spacetime, such as the quantized electromagnetic field in a laser, it is $\Sigma$ which represents physical spacetime, and $X$ is some abstract space, for instance a smooth version of the classifying space $\mathcal{B}U(1)$, so that a field configuration $\Sigma \to X$ encodes a line bundle with connection that encodes a configuration of the electromagnetic field.

Definition

The configuration space of the system is the space of all field configurations, hence the space $\Gamma_\Sigma(E)$ of sections of the bundle $E$.

In the sigma-model example this is some incarnation of the mapping space $[\Sigma,X]$.

Remark

Beware that in low dimensions one often distinguishes between the space of configurations $\Sigma \to X$ and that of trajectories or histories $\Sigma \times \mathbb{R} \to X$. This comes from the case $\Sigma = *$ where for a particle propagating on $X$ the maps $[*,X] \simeq X$ are the possible configurations of the particle at a given parameter times, while maps $[* \times \mathbb{R}, X] = [\mathbb{R}, X]$ are the trajectories. But for the higher dimensional and extended field theories under discussion here, this distinction becomes a bit obsolete and trajectories become just a special case of configurations.

Remark

In the non-covariant approach one would try to consider the a cotangent bundle of the configuration space $\Gamma(E)$ as phase space . Contrary to that, in the covariant approach one considers the much smaller space $E$ instead. This is then called the covariant configuration space or covariant configuration bundle.

Definition

Write $J^1 E \to \Sigma$ for the first order jet bundle of the configuration space bundle $E \to \Sigma$. Its fiber over $s \in \Sigma$ are equivalence classses of germs of sections at $x$, where two germs are identified if their first derivatives coincide.

Covariant phase space

Definition

Given a vector bundle $E \to \Sigma$ over a smooth manifold of dimension $dim(\Sigma) = n+1$, the affine dual first jet bundle (or often just dual first jet bundle for short) $(J_1 E)^\ast \to \Sigma$ is the vector bundle whose fiber at $e \in E_s$ is the set of affine maps

$$J_e^1 E \longrightarrow \Lambda_s^{n+1} \Sigma$$

from the first jets at $e$ to the degree-$(n+1)$ differential forms at $s$ on $\Sigma$.

Definition

Given a spacetime/worldvolume $\Sigma$ and a field bundle $E \to \Sigma$, the extended covariant phase space is the multisymplectic manifold

• whose underlying manifold is the dual first jet bundle, def. , of the field bundle

  $$(J^1 E)^* \to E \,,$$

• equipped with the canonical degree-$(n+2)$ differential form

  $$\omega = d \alpha \,,$$

  where $\alpha$ is the canonical $(n+1)$-form

Given $\pi \colon E \to \Sigma$, with $\mathrm{dim} \Sigma =n+1$, the dual jet bundle $(J^1 E)^*$ is isomorphic to a particular vector sub-bundle of the $n+1$-form bundle $\Lambda^{n+1}T^{*}E$. To see this, first consider the following

Definition

Given a point $y \in E$, a tangent vector $v \in T_{y} E$ is said to be vertical if $d \pi(v) = 0$. Define

$$\Lambda^{n+1}_{1}T^{\ast}E \hookrightarrow \Lambda^{n+1} T^{\ast} E$$

to be the subbundle of the $n+1$-form bundle whose fiber at $y \in E$ consists of all $\beta \in \Lambda^{n+1} T^{*}_{y} E$ such that

$$\iota_{v_1}\iota_{v_2} \beta =0$$

for all vertical vectors $v_1,v_2 \in T_{y}E$. Sections of $\Lambda^{n+1}_{1}T^{*}E$ are called $n$-horizontal $\mathbf{n+1}$-forms.

In words, an $n$-horizontal $(n+1)$-form is one which has at most one “leg” not along $\Sigma$. This is made very explicit in the proof of the following proposition.

Proposition

Let $E \to \Sigma$ be a vector bundle over a smooth manifold $\Sigma$ of dimension $dim \Sigma = (n+1)$ and assume that $\Sigma$ is orientable, then there is an isomorphism

$$(J^1 E)^{\ast} \simeq \Lambda^{n+1}_{1} T^{\ast}E$$

of vector bundles over $\Sigma$ between the dual first jet bundle of $E$, def. , and the bundle of $n$-horizontal $(n+1)$-forms on $E$, def. .

Proof

It suffices to work locally with respect to a good open cover, so we reduce the statement to the special case of the sigma model i.e. the trivial bundle $E = \Sigma \times X$ over $\Sigma$. By the assumoption that $\Sigma$ admits an orientation we may pick a volume form $\vol \in \Gamma(\Lambda^{n+1}T^\ast \Sigma)$.

Let $q^1, \dots, q^{n+1}$ be local coordinates on $\Sigma$ and let $u^1, \dots , u^d$ be local coordinates on $X$. Then $\Lambda_1^{n+1} T^* E$ has a local basis of sections given by $(n+1)$-forms of two types: first, the wedge product of all $n+1$ cotangent vectors of type $\mathbf{d}q^i$:

$$\vol = \mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d}q^{n+1}$$

and second, wedge products of $n$ cotangent vectors of type $\mathbf{d}q^i$ and a single one of type $\mathbf{d}u^a$:

$$\mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d} q^{i-1} \wedge \mathbf{d}q^{i+1} \wedge \cdots \wedge \mathbf{d}q^{n+1} \wedge \mathbf{d}u^a \,.$$

If $y = (p,u) \in \Sigma \times X$, this basis gives an isomorphism

$$\Lambda^{n+1}_1 T^*_y E \;\; \simeq\;\; \left(\Lambda^{n+1} T^*_p \Sigma \right) \; \oplus \; \left(\Lambda^{n} T^*_p \Sigma \otimes T^*_u X \right) \,.$$

The volume form on $\Sigma$ also determines isomorphisms

$$\mathbb{R} \overset{\simeq}{\longrightarrow} \Lambda^{n+1} T^*_p \Sigma$$

$$c \mapsto c \, \vol_p$$

and

$$T_p \Sigma \overset{\simeq}{\longrightarrow} \Lambda^{n} T^*_p \Sigma$$

$$v \mapsto \iota_v \vol_p \,.$$

We thus have obtained an isomorphism

$$\Lambda^{n+1}_1 T^*_y E \;\; \cong \;\; \mathbb{R} \; \oplus \; T_p \Sigma \otimes T^*_u X \,.$$

On the other hand, the trivialization $E = \Sigma \times X$ gives an isomorphism of affine spaces

$$J^1_y E \; \; \cong \; \; T^*_p \Sigma \otimes T_u X$$

which has the side-effect of exhibiting on $J^1_y E$ the structure of a vector space. Since we’ve identified

$\Lambda^{n+1} T^*_p \Sigma$ with $\mathbb{R}$, an affine map from $J^1_y E$ to $\Lambda^{n+1} T^*_p \Sigma$ is just an element of $T_x \Sigma \otimes T^*_u X$ plus a constant. So, we obtain

$$(J^1_y E)^* \; \; \cong \; \; \mathbb{R} \; \oplus \; T_p \Sigma \otimes T^*_u X .$$

This gives a specific vector bundle isomorphism $(J^1 E)^* \cong \Lambda_1^{n+1} T^* E$, as desired.

Remark

In practice it is better to use the pulled back volume form $\pi^* \vol$ as a substitute for the coordinate-dependent $n+1$-form $\mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d}q^{n+1}$ on $E$. This gives another basis of sections of $\Lambda_1^{n+1} T^* E$, whose elements we write suggestively

$$dQ \coloneqq \pi^* \vol$$

and

$$dQ_i^a \coloneqq \left(\pi^* \iota_{\partial/\partial q^i} \vol\right) \wedge \mathbf{d}u^a \,.$$

Corresponding to this basis then there are local coordinates $P$ and $P^i_a$ on $\Lambda_1^{n+1} T^* E$, which combined with the coordinates $q^i$ and $u^a$ pulled back from $E$ give a local coordinate system on $\Lambda_1^{n+1} T^* E$.

In these coordinates the canonical $n+1$-form on $(J^1 E)^* \cong \Lambda_1^{n+1} T^* E$ is:

$$\alpha = P \wedge dQ + P^i_a \wedge dQ_i^a,$$

and the $n+2$ multisymplectic form is

$$\omega = \mathbf{d}P \wedge dQ + \mathbf{d}P^i_a \wedge dQ_i^a.$$

De Donder-Weyl-Hamilton field equations

We discuss the Euler-Lagrange equations of motion of a local field theory expressed in multisymplectic geoemtry via de Donder-Weyl formalism.

Definition

Given a field bundle $E \to \Sigma$ as above, a (first order) local Lagrangian is a smooth function

$$\mathbf{L} \;\colon\; J^1 E \longrightarrow \wedge^n T^\ast \Sigma$$

on the first jet bundle of $E$ with values in densities/volume forms. Equivalently this is a degree $(n,0)$-form on the jet bundle, in terms of variational bicomplex grading.

Definition

Given a local Lagrangian, def. , its local Legendre transform is the smooth function

$$\mathbb{F}\mathbf{L} \;\colon\; J^1 E \longrightarrow (J^1 E)^\ast$$

from first jets to the affine dual jet bundle, def. , which sends $\mathbf{L}$ to its first-order Taylor series.

This definition was proposed in (Forger-Romero 04, section 2.5).

Proposition

In terms of the local coordinates of remark the Legendre transform of def. is the function with coordinates

$$P^i_a = \frac{\partial \mathbf{L}}{\partial q^a_{, i}}$$

and

$$P = \mathbf{L} - \frac{\partial \mathbf{L}}{\partial q^a_{,i}}q^a_{,i} \,.$$

(Forger-Romero 04, section 2.5 (41)).

Remark

The second term in prop. is what is traditionally called the Legendre transform in multisymplectic geometry/de Donder-Weyl formalism. Def. may be regarded as explaining the conceptual role of this expression, in particular in view of the following proposition.

Proposition

Given a local Lagrangian $\mathbf{L}$, the pullback $\omega_{\mathbf{L}}$ of the canonical pre-n-plectic form $\omega$, def. , along the Legendre transform $\mathbb{F}\mathbf{L}$, def. , to the first jet bundle is the sum of the Euler-Lagrange equation $EL_{\mathbf{L}}$ and the canonical symplectic form $\mathbf{d}_v \theta_{\mathbf{L}}$ from covariant phase space formalism:

$$\begin{aligned} \omega_{\mathbf{L}} & \coloneqq \mathbb{F}\mathbf{L}^\ast \omega \\ & = EL_{\mathbf{L}} + \mathbf{d}_v \theta_{\mathbf{L}} \end{aligned} \,.$$

It follows that

1. $(\iota_{v_n} \cdots \iota_{v_1}) \omega_{\mathbf{L}} = 0$ is the Euler-Lagrange equation of motion in de Donder-Weyl-Hamilton-form;

2. for any Cauchy surface $\Sigma_{n-1}$, the transgression $\omega_\Sigma \coloneqq \int_{\Sigma_{n-1}}\omega_{\mathbf{L}}$ is the canonical pre-symplectic form on phase space (as discussed there).

This statement is essentially the content of (Forger-Romero 04, equation (54) and theorem 1). In the above form in terms of variational bicomplex notions this statement has been amplified by Igor Khavkine.

Examples

Free field theory

We write out the multisymplectic geometry corresponding to a free field theory.

Let $\Sigma = (\mathbb{R}^{d-1;1}, \eta)$ be Minkowski spacetime. Write the canonical coordinates

$$\sigma^i \;\colon\; \Sigma \longrightarrow \mathbb{R} \,.$$

Let $(X,g)$ be a Riemannian manifold. For simplicity of notation we assume that $X \simeq \mathbb{R}^k$ is a vector space, too. Write its canonical coordinates as

$$\phi^a \;\colon\; X \longrightarrow \mathbb{R} \,.$$

Let $X \times \Sigma \to \Sigma$ be the field bundle. Its first jet bundle then has canonical coordinates

$$\{ \sigma^i \}, \{\phi^a\}, \{\phi^a_{,i}\} \;\colon\; j_\infty^1(\Sigma \times X) \longrightarrow X \,.$$

Definition

The local Lagrangian for free field theory with this field bundle is

$$L \coloneqq \left( \frac{1}{2} g^{i j} \eta_{a b} \phi^a_{,i} \phi^a_{,j} \right) \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d \,.$$

The canonical momenta are

$$\begin{aligned} p_a^i \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d & \coloneqq \frac{\partial}{\partial u^a_i} L \\ & = \left( g^{i j} \eta_{a b} \phi^a_{,j} \right) \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d \end{aligned}$$

So the boundary term $\theta$ in variational calculus, (see this remark at covariant phase space ) is

$$\begin{aligned} \mathbf{d}u^a \wedge \iota_{\partial_i} \left( \frac{\partial}{\partial \phi^a_{,i}}L \right) & = p^i_a \wedge (\iota_{\partial_{\sigma^i}} vol) \wedge \mathbf{d}u^a \\ & = p^i_a \wedge dq_i^a \,, \end{aligned}$$

where in the last line we adopted the notation of remark .

This shows that the canonical multisymplectic form is the “covariant symplectic potential current density” which is induced by the free field Lagrangian.

See also (Forger Romero 04, section 3.2).

Bosonic particle propagating on a manifold

scratch

Ordinary point particle mechanics on a manifold $X$ involves trajectories $\mathbb{R} \to X$ in $X$, with parameter space $\Sigma = \mathbb{R}$ the real line, thought of as the abstract “worldline” of the particle.

• parameter space: $\Sigma = \mathbb{R}$, the worldline;

• target space: $X$, some manifold – spacetime;

• configuration bundle: $(E \to \mathbb{R}) = (\mathbb{R}\times X \to \mathbb{R})$;

• jet bundle: $J^1 E = \mathbb{R} \times T X$ .

for $U \subset E = \mathbb{R} \times X$ a local patch with coordinate functions $\{t, q^i\}$, there are canonically induced coordinates on $J^1 E$ written $\{t,q^i, v^i\}$.

Here a collection of vaues $(q^i_0)$ is a position of the particle and $(v^i_0)$ is a velocity of the particle. Notice that in this covariant approach these are not positions and velocities “at a given time”. Rather, a point in $J^1 E$ specified a parameter time and a corresponding position and velocity.

….

Let $U \to X$ be a local patch of $X$ with canonical coordinates $\{x^i\}$.

The canonical 2-form on the extended phase space in this case is traditionally locally written as

$$\omega|_U = \mathbf{d} \alpha|_U = \mathbf{d}( p_i \wedge \mathbf{d} q^i + H \wedge d t ) \,.$$

… blah-blah-blah…

Electromagnetism

• parameter space: $\Sigma$ is spacetime;

A field configuration of the electromagnetic field is a line bundle with connection on $\Sigma$. If we assume the corresponding bundle to be trivial, then this is just a 1-form on $\Sigma$. So in this simplified case we can take

• configuration bundle: $T^* \Sigma$ is the cotangent bundle of $\Sigma$.

Bosonic string propagating on a manifold

• parameter space: $\Sigma$ some 2-dimensional manifold – the worldsheet – for instance $\Sigma = S^1 \times \mathbb{R}$ models a closed string propagating without interaction.

• target space. $X$ spacetime;

• covariant configuration bundle $E = \Sigma \times X$.

We will work out the covariant Hamiltonian formalism (also known as the de Donder-Weyl formalism) for this example in detail. We follow here the exposition found in Hélein 02.

For simplicity we will only consider the case where $\Sigma$ is the cylinder $\mathbb{R}\times S^1$ and $X$ is $d$-dimensional Minkowski spacetime, $\mathbb{R}^{1,d-1}$. A solution of the classical bosonic string is then a map $\phi : \Sigma \to X$ which is a critical point of the area subject to certain boundary conditions.

Equivalently, by exploiting symmetries in the variational problem, one can describe solutions $\phi$ by equipping $\mathbb{R} \times S^{1}$ with its standard Minkowski metric and then solving the $1+1$ dimensional field theory specified by the Lagrangian density

$$\mathcal{L}=\frac{1}{2} g^{ij}\eta_{ab} \frac{\partial \phi^{a}}{\partial q^i}\frac{\partial \phi^{b}}{\partial q^j}.$$

Here $q^i$ $(i = 0,1)$ are standard coordinates on $\mathbb{R} \times S^1$ and $g=\mathrm{diag}(1,-1)$ is the Minkowski metric on $\mathbb{R} \times S^1$, while $\phi^a$ are the coordinates of the map $\phi$ in $\mathbb{R}^{1,d-1}$ and $\eta = \mathrm{diag}(1,-1,\cdots,-1)$ is the Minkowski metric on $\mathbb{R}^{1,d-1}$. The corresponding Euler-Lagrange equation is just a version of the wave equation:

$$g^{ij}\partial_{i} \partial_{j} \phi^a =0.$$

The space $E=\Sigma \times X$ can be thought of as a trivial bundle over $\Sigma$, and the graph of a function $\phi : \Sigma \to X$ is a smooth section of $E$. We write the coordinates of a point $(q,u)\in E$ as $\left(q^i,u^a \right)$. Let $J^1 E \to E$ be the first jet bundle of $E$. We may regard $J^1 E$ as a vector bundle whose fiber over $(q,u)\in E$ is $T^*_q \Sigma \otimes T_u X$.
The Lagrangian density for the string can be defined as a smooth function on $J^1 E$:

$$\mathcal{L}=\frac{1}{2} g^{ij}\eta_{ab}u^{a}_{i}u^{b}_{j},$$

which depends in this example only on the fiber coordinates $u^a_{i}$.

From the Lagrangian $\mathcal{L} : J^{1}E \to \mathbb{R}$, the de Donder-Weyl Hamiltonian $\mathcal{H} : T \Sigma \otimes T^*X \to \mathbb{R}$ can be constructed via a Legendre transform. It is given as follows:

$$\mathcal{H}= p^{i}_{a}u^{a}_{i}- \mathcal{L} =\frac{1}{2} \eta^{ab}g_{ij}p_{a}^{i}p_{b}^{j},$$

where $u^a_{i}$ are defined implicitly by $p_a^{i}=\partial \mathcal{L} / \partial u^{a}_{i}$, and $p_a^{i}$ are coordinates on the fiber $T^{*}_{u}X \otimes T_{q}\Sigma$. Note that $\mathcal{H}$ differs from the standard (non-covariant) Hamiltonian density for a field theory:

$$p^{0}_{a}u^{a}_{0} - \mathcal{L}.$$

Let $\phi$ be a section of $E$ and let $\pi$ be a smooth section of $T \Sigma \otimes T^*X$ restricted to $\phi(\Sigma)$ with fiber coordinates $\pi_{a}^{i}$. It is then straightforward to show that $\phi$ is a solution of the Euler-Lagrange equations if and only if $\phi$ and $\pi$ satisfy the following system of equations:

$$\frac{\partial \pi^{i}_{a}}{\partial q^i} = - \left.\frac{\partial \mathcal{H}}{\partial u^{a}} \right \vert_{u=\phi,p=\pi}$$

$$\frac{\partial \phi^{a}}{\partial q^i} = \left.\frac{\partial \mathcal{H}}{\partial p^i_{a}} \right \vert_{u=\phi,p=\pi}.$$

This system of equations is a generalization of Hamilton’s equations for the point particle.

As explained above, the covariant phase space for the bosonic string is the dual jet bundle $(J^1 E)^*$, and this space is equipped with a canonical 2-form $\alpha$ whose exterior derivative $\omega = d \alpha$ is a multisymplectic 3-form. Using the isomorphism

$$(J^1 E)^* \cong T \Sigma \otimes T^*X \times \mathbb{R} ,$$

a point in $(J^1 E)^{*}$ gets coordinates $(q^i,u^a,p^{i}_{a},e)$. In terms of these coordinates,

$$\alpha= e dq^{0} \wedge dq^{1} + \left(p_{a}^{0} du^{a} \wedge dq^{1} - p_{a}^{1} du^{a} \wedge dq^{0} \right) .$$

The multisymplectic structure on $(J^1 E)^*$ is thus

$$\omega = de \wedge dq^0 \wedge dq^{1} + \left(dp_a^0 \wedge du^a \wedge dq^{1} - dp_a^1 \wedge du^a \wedge dq^{0} \right) .$$

So, the variable $e$ may be considered as canonically conjugate to the area form $dq^{0} \wedge dq^{1}$.

As before, let $\phi$ be a section of $E$ and let $\pi$ be a smooth section of $T \Sigma \otimes T^*X$ restricted to $\phi(\Sigma)$. Consider the submanifold $S \subset (J^1 E)^*$ with coordinates:

$$(q^i,\phi^{a}(q^j),\pi_{a}^{i}(q^j),-\mathcal{H}).$$

Note that $S$ is constructed from $\phi$, $\pi$ and from the constraint $e + \mathcal{H}=0$. This constraint is analogous to the one that is used in finding constant energy solutions in the extended phase space approach to classical mechanics. At each point in $S$, a tangent bivector $v=v_{0} \wedge v_{1}$ can be defined as

$$v_{0} =\frac{\partial}{\partial q^{0}} + \frac{\partial \phi^a}{\partial q^{0}}\frac{\partial}{\partial u^a} + \frac{\partial \pi_{a}^{i}}{\partial q^{0}} \frac{\partial}{\partial p_{a}^{i}}$$

$$v_{1} = \frac{\partial}{\partial q^{1}} + \frac{\partial \phi^a}{\partial q^{1}} \frac{\partial}{\partial u^a} + \frac{\partial \pi_{a}^{i}}{\partial q^{1}} \frac{\partial}{\partial p_{a}^{i}}.$$

Explicit computation reveals that the submanifold $S$ is generated by solutions to Hamilton’s equations if and only if

$$\omega(v_{0},v_{1},\cdot)=0.$$

Hamiltonian $n$-dimensional flow

• Hamiltonian n-vector field

Relation to $n$-symplectic manifolds

There is also the notion of

• n-symplectic manifolds.

Which is different, but related…

….blah-blah….

Survey of developments in the field

There is in this sense a covariant form of the Legendre transformation which associates to every field variable as many conjugated momenta – the multimomenta – as there are space-time dimensions. These, together with the field variables, those of $n$-dimensional space-time, and an extra variable, the energy variable, span the multiphase space [1]. For a recent exposition on the differential geometry of this construction, see [2]. Multiphase space, together with a closed, nondegenerate differential $(n+1)$-form, the multisymplectic form, is an example of a multisymplectic manifold [3].

Among the achievements of the multisymplectic approach is a geometric formulation of the relation of infinitesimal symmetries and covariantly conserved quantities (Noether's theorem), see [4] for a recent review, and [5,6] for a clarification of the improvement techniques (“Belinfante-Rosenfeld formula”) of the energy-momentum tensor in classical field theory.

Multisymplectic geometry also provides convenient sets of variational integrators for the numerical study of partial differential equations [7].

Since in multisymplectic geometry, the symplectic 2-form of classical Hamiltonian mechanics is replaced by a closed differential form of higher tensor degree, multivector fields and differential forms have their natural appearance. (See [8] for an interpretation of multivector fields as describing solutions to field equations infinitesimally.) Multivector fields form a graded Lie algebra? with the Schouten bracket (see [9] for a review and unified viewpoint). Using the multisymplectic $(n+1)$-form, one can construct a new bracket for the differential forms, the Poisson forms [10], generalizing a well-known formula for the Poisson brackets related to a symplectic 2-form.

A remarkable fact is that in order to establish a Jacobi identity, the multisymplectic form has to have a potential, a condition that is not seen in symplectic geometry. Further, the admissible differential forms, the Poisson forms, are subject to the rather strong restrictions on their dependence on the multimomentum variables [11]. In particular, $(n-1)$-forms in this context can be shown to arise exactly from the covariantly conserved currents of Noether symmetries [11], which allows their pairing with spacelike hypersurfaces to yield conserved charges in a geometric way.

Not much is known about the interpretation of Poisson forms of form degree between zero and n-1. However, as $(n-1)$-forms describe vector fields and hence collections of lines [2, 10], and as (certain) functions describe n-vector fields and hence collections of bundle sections [8], it seems natural to speculate that the intermediate forms may be useful for the branes of string theory.

The Hamiltonian, infinite dimensional formulation of classical field theory requires the choice of a spacelike hypersurface (“Cauchy surface”) [12] which manifestly breaks the general covariance of the theory at hand. For $(n-1)$-forms, the above mentioned new bracket reduces to the Peierls-deWitt bracket after integration over the spacelike hypersurface [13]. With the choice of a hypersurface, a constraint analysis [14] à la Dirac [15,16] can be performed [17]. Again, the necessary breaking of general covariance raises the need for an alternative formulation of all this [18]; first attempts have been made to carry out a Marsden-Weinstein reduction [19] for multisymplectic manifolds with symmetries [20]. However, not very much is known about how to quantize a multisymplectic geometry, see [21] for an approach using a path integral.

This discussion so far concerns field theories of first order, i.e. where the Lagrangian depends on the fields and their first derivatives. Higher order theories can be reduced to first order ones for the price of introducing auxiliary fields. A direct treatment would involve higher order jet bundles [22]. A definition of the covariant Legendre transform and the multiphase space has been given for this case [3].

References

On classical multisymplectic geometry

A comprehensive source on covariant field theory with plenty of further references is

• Mark J. Gotay, James Isenberg, Jerrold Marsden, Richard Montgomery, Momentum maps and classical relativistic fields. Part I: covariant field theory (arXiv:physics/9801019)

Much of the material in the section on covariant field theory is based on this.

Further discussions:

• Frédéric Hélein, Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory [arXiv:math-ph/0212036]

• Narciso Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, SIGMA 5 (2009) 100 [doi;10.3842/SIGMA.2009.100, arXiv:math-ph/0506022]

The relation to covariant phase space methods is discussed in

• Frédéric Hélein, Multisymplectic formalism and the covariant phase, in Variational Problems in Differential Geometry, Cambridge University Press (2011) 94-126 [doi:10.1017/CBO9780511863219.007, arXiv:1106.2086]

A discussion of Hamiltonian $n$-forms as observables is in

• Frédéric Hélein, The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables, Adv. Theor. Math. Phys. 8 (2004) 735-777 [doi:10.4310/ATMP.2004.v8.n4.a4, arXiv:math-ph/0401047]

Other texts include

• Carlo Rovelli, Covariant hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space $\mathcal{G}$ (arXiv:gr-qc/0207043)

Much of the above survey of recent developments and of the following list of references is reproduced from the web-page

• Cornelius Mertzlufft-Paufler, Multisymplectic geometry

References mentioned above are

• [1] J. Kijowski, W. Szczyrba, A Canonical Structure For Classical Field Theories . Commun. Math. Phys. 46 (1976) 183.

• [2] M. J. Gotay, J. Isenberg, J. E. Marsden: Momentum maps and classical relativistic fields. I: Covariant field theory arXiv:physics/9801019v2.

• [3] M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. I: Covariant Hamiltonian formalism In M. Francaviglia (ed.), Mechanics, analysis and geometry: 200 years after Lagrange Amsterdam etc.: North-Holland (1991), 203–235.

• [4] M. de Leon, D. Martin de Diego, A. Santamaria-Merino, Symmetries in Classical Field Theory arXiv:math-ph/0404013.

• [5] M. J. Gotay, J. E. Marsden: Stress-energy-momentum tensors and the Belinfante–Rosenfeld formula Contemp. Math. vol. 132, AMS, Providence, 1992, 367–392.

• [6] M. Forger, H. Römer, Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem Ann. Phys. (N.Y.) 309 (2004) 306–389. arXiv:hep-th/0307199.

• [7] A. Lew, J. E. Marsden, M. Ortiz, M. West, An overview of variational integrators In L. P. Franca (ed.), Finite Element Methods: 70’s and Beyond. Barcelona (2003).

• [8] C. Paufler, H. Römer, The Geometry of Hamiltonian $n$-vector fields in Multisymplectic Field Theory J. Geom. Phys. 44, No.1(2002), 52–69. arXiv:math-ph/0102008.

• [9] Y. Kosmann-Schwarzbach: Derived brackets. Lett. Math. Phys. 69 (2004) 61–87 arXiv:math.DG/0312524.

• [10] M. Forger, C. Paufler, H. Römer: The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory Rev. Math. Phys. 15 (2003) 705 arXiv:math-ph/0202043.

• [10] M. Forger, C. Paufler, H. Römer, Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory arXiv:math-ph/0407057.

• [11] M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. II: Space + time decomposition Differ. Geom. Appl. 1(4) (1991), 375–390.

• [12] M. O. Salles, Campos Hamiltonianos e Colchete de Poisson na Teoria Geométrica dos Campos , PhD thesis, IME-USP, June 2004.

• [13] M. J. Gotay, J. M. Nester, Generalized constraint algorithm and special presymplectic manifolds In G. E. Kaiser, J. E. Marsden, Geometric methods in mathematical physics, Proc. NSF-CBMS Conf., Lowell/Mass. 1979, Berlin: Springer-Verlag, Lect. Notes Math. 775 (1980) 78–80.

• [14] P. A. M. Dirac, Lectures on Quantum Mechanic Belfer Graduate School of Science, Yeshiva University, N.Y., 1964.

• [15] Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems Princeton University Press, 1992.

• [16] M. J. Gotay, J. Isenberg, J. E. Marsden, R. Montgomery, Momentum Maps and Classical Relativistic Fields II: Canonical Analysis of Field Theories (2004) arXiv:math-ph/0411032.

• [17] N. P. Landsman, Against the Wheeler-DeWitt equation Class. Quan. Grav. 12 (1995) L119-L124. arXiv:gr-qc/9510033.

• [18] J. E. Marsden, A. Weinstein Reduction of symplectic manifolds with symmetry Rept. Math. Phys. 5 (1974) 121–130.

• [19] F. Munteanu, A. M. Rey, M. Salgado, The Günther’s formalism in classical field theory: momentum map and reduction J. Math. Phys. 45, No. 5 (2004) 1730–1750.

• [20] D. Bashkirov, G. Sardanashvily, Covariant Hamiltonian Field Theory. Path Integral Quantization arXiv:hep-th/0402057.

• [21] D. J. Saunders, The Geometry of Jet Bundles Lond. Math. Soc. Lect. Notes Ser. 142, Cambr. Univ. Pr., Cambridge, 1989.

A higher categorial interpretation of 2-plectic geometry and connection to $L_\infty$-algebras is given in

• John Baez, Chris Rogers, Categorified symplectic geometry and the string Lie 2-algebra. arXiv:0901.4721.

• John Baez, Alexander E. Hoffnung, Chris Rogers, Categorified symplectic geometry and the classical string (arXiv:0808.0246)

• Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068); $L_\infty$-algebras from multisymplectic geometry, Lett. Math. Phys., 100(1):29–50 (2012); 2-plectic geometry, Courant algebroids, and categorified prequantization, J. Symplectic Geom. 11(1):53–91 (2013)

• Antonio Michele Miti, Marco Zambon, Observables on multisymplectic manifolds and higher Courant algebroids, arXiv:2209.05836

Higher order moment maps are considered in

• Thomas Bruun Madsen, Andrew Swann, Closed forms and multi-moment maps (arXiv:1110.6541)

• Martin Callies, Yaël Frégier, Chris Rogers, Marco Zambon, Homotopy moment maps, Advances in Mathematics 303 (2016) 954–1043 (arXiv:1304.2051)

A higher differential geometry-generalization of contact geometry in line with multisymplectic geometry/n-plectic geometry is discussed in

• Luca Vitagliano, $L_\infty$ algebras from multicontact geometry, Diff. Geom. Appl. 39 (2015) 147-165 arXiv:1311.2751 doi

Relation to covariant phase space formalism

On the relation of multisymplectic formalism to covariant phase space and variational bicomplex methods:

• Michael Forger, Sandro Romero, Covariant Poisson brackets in geometric field theory, Commun. Math. Phys. 256 (2005) 375-410 [arXiv:math-ph/0408008, doi:10.1007/s00220-005-1287-8]

• Narciso Román-Roy, Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories, SIGMA 5 (2009), 100, 25 pages (arXiv:math-ph/0506022)

On multisymplectic boundary field theory:

• José Zapata, Multisymplectic effective General Boundary Field Theory (2013) [pdf, pdf]

On quantization of multisymplectic geometry

The following articles discuss the quantization procedure for multisymplectic geometry, generalizing geometric quantization of symplectic geometry.

• Igor V. Kanatchikov, De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory, Rept. Math. Phys. 43 (1999) 157-170 [arXiv:hep-th/9810165, doi:10.1016/S0034-4877(99)80024-X]

• Igor V. KanatchikovGeometric (pre)quantization in the polysymplectic approach to field theory, in Differential Geometry and Its Applications, Proc. Conf., Opava (August 2001) [arXiv:hep-th/0112263, pdf]

Kanatchikov’s “algebra of observables” is what he calls a “higher-order Gerstenhaber algebra”. (The “bracket” in this structure fails to satisfy Leibniz’s rule as a derivation of the product.) The relationship between it and the Lie superalgebra of observables constructed by Forger, Paufler, and Roemer is discussed in this paper:

• Michael Forger, C. Paufler, and H. Roemer, The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory , (arXiv:math-ph/0202043)

and (Forger-Romero 04) above.

Kanatchikov’s formalism was used by S.P. Hrabak to propose a multisymplectic refinement of BV-BRST formalism. See there for more details.