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multisymplectic geometry

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Symplectic geometry

Physics

physics, mathematical physics, philosophy of physics

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theory (physics), model (physics)

experiment, measurement, computable physics

Contents

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+1n+1-form, for n1n \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 nn-dimensional quantum field theory. However, details of the multisymplectic quantization procedure remain under investigation.

From the nnPOV

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 spacehase 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 nn-dimensional field theory a state is not just a point, but an nn-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 nn-dimensional field theory may be encoded in an degree (n+1)(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 geomtry 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 nn-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 explcitly what that state is like over any point in parameter space.

Covariant configuration bundle

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

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

Example

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

ϕ:ΣX \phi : \Sigma \longrightarrow X

into some prescribed target space XX. This is the case where E=Σ×XE = \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 Σ\Sigma or by XX:

  • 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 identifid with target space XX, while Σ\Sigma is the worldvolume of the object that propagates through XX. The field configurations on Σ\Sigma are really the maps ΣX\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 Σ\Sigma which represents physical spacetime, and XX is some abstract space, for instance a smooth version of the classifying space U(1)\mathcal{B}U(1), so that a field configuration ΣX\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 Γ Σ(E)\Gamma_\Sigma(E) of sections of the bundle EE.

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

Remark

Beware that in low dimensions one often distinguishes between the space of configurations ΣX\Sigma \to X and that of trajectories or histories Σ×X\Sigma \times \mathbb{R} \to X. This comes from the case Σ=*\Sigma = * where for a particle propagating on XX the maps [*,X]X[*,X] \simeq X are the possible configurations of the particle at a given parameter times, while maps [*×,X]=[,X][* \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 Γ(E)\Gamma(E) as phase space . Contrary to that, in the covariant approach one considers the much smaller space EE instead. This is then called the covariant configuration space or covariant configuration bundle.

Definition

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

Covariant phase space

Definition

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

J e 1EΛ s n+1Σ J_e^1 E \longrightarrow \Lambda_s^{n+1} \Sigma

from the first jets at ee to the degree-(n+1)(n+1) differential forms at ss on Σ\Sigma.

Definition

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

  • whose underlying manifold is the dual first jet bundle, def. 3, of the field bundle

    (J 1E) *E, (J^1 E)^* \to E \,,
  • equipped with the canonical degree-(n+2)(n+2) differential form

    ω=dα, \omega = d \alpha \,,

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

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

Definition

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

Λ 1 n+1T *EΛ n+1T *E \Lambda^{n+1}_{1}T^{\ast}E \hookrightarrow \Lambda^{n+1} T^{\ast} E

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

ι v 1ι v 2β=0 \iota_{v_1}\iota_{v_2} \beta =0

for all vertical vectors v 1,v 2T yEv_1,v_2 \in T_{y}E. Sections of Λ 1 n+1T *E\Lambda^{n+1}_{1}T^{*}E are called nn-horizontal n+1\mathbf{n+1}-forms.

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

Proposition

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

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

of vector bundles over Σ\Sigma between the dual first jet bundle of EE, def. 3, and the bundle of nn-horizontal (n+1)(n+1)-forms on EE, def. 5.

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=Σ×XE = \Sigma \times X over Σ\Sigma. By the assumoption that Σ\Sigma admits an orientation we may pick a volume form volΓ(Λ n+1T *Σ)\vol \in \Gamma(\Lambda^{n+1}T^\ast \Sigma).

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

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

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

dq 1dq i1dq i+1dq n+1du 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)Σ×Xy = (p,u) \in \Sigma \times X, this basis gives an isomorphism

Λ 1 n+1T y *E(Λ n+1T p *Σ)(Λ nT p *ΣT u *X). \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 Σ\Sigma also determines isomorphisms

Λ n+1T p *Σ \mathbb{R} \overset{\simeq}{\longrightarrow} \Lambda^{n+1} T^*_p \Sigma
ccvol p c \mapsto c \, \vol_p

and

T pΣΛ nT p *Σ T_p \Sigma \overset{\simeq}{\longrightarrow} \Lambda^{n} T^*_p \Sigma
vι vvol p. v \mapsto \iota_v \vol_p \,.

We thus have obtained an isomorphism

Λ 1 n+1T y *ET pΣT u *X. \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=Σ×XE = \Sigma \times X gives an isomorphism of affine spaces

J y 1ET p *ΣT uX J^1_y E \; \; \cong \; \; T^*_p \Sigma \otimes T_u X

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

Λ n+1T p *Σ\Lambda^{n+1} T^*_p \Sigma with \mathbb{R}, an affine map from J y 1EJ^1_y E to Λ n+1T p *Σ\Lambda^{n+1} T^*_p \Sigma is just an element of T xΣT u *XT_x \Sigma \otimes T^*_u X plus a constant. So, we obtain

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

This gives a specific vector bundle isomorphism (J 1E) *Λ 1 n+1T *E(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 π *vol\pi^* \vol as a substitute for the coordinate-dependent n+1n+1-form dq 1dq n+1\mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d}q^{n+1} on EE. This gives another basis of sections of Λ 1 n+1T *E\Lambda_1^{n+1} T^* E, whose elements we write suggestively

dQπ *vol dQ \coloneqq \pi^* \vol

and

dQ i a(π *ι /q ivol)du a. 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 PP and P a iP^i_a on Λ 1 n+1T *E\Lambda_1^{n+1} T^* E, which combined with the coordinates q iq^i and u au^a pulled back from EE give a local coordinate system on Λ 1 n+1T *E\Lambda_1^{n+1} T^* E.

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

α=PdQ+P a idQ i a, \alpha = P \wedge dQ + P^i_a \wedge dQ_i^a,

and the n+2n+2 multisymplectic form is

ω=dPdQ+dP a idQ i a. \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 EΣE \to \Sigma as above, a (first order) local Lagrangian is a smooth function

L:J 1E nT *Σ \mathbf{L} \;\colon\; J^1 E \longrightarrow \wedge^n T^\ast \Sigma

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

Definition

Given a local Lagrangian, def. 6, its local Legendre transform is the smooth function

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

from first jets to the affine dual jet bundle, def. 3, which sends L\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 4 the Legendre transform of def. 7 is the function with coordinates

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

and

P=LLq ,i aq ,i a. 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. 2 is what is traditionally called the Legendre transform in multisymplectic geometry/de Donder-Weyl formalism. Def. 7 may be regarded as explaining the conceptual role of this expression, in particular in view of the following proposition.

Proposition

Given a local Lagrangian L\mathbf{L}, the pullback ω L\omega_{\mathbf{L}} of the canonical pre-n-plectic form ω\omega, def. 4, along the Legendre transform 𝔽L\mathbb{F}\mathbf{L}, def. 7, to the first jet bundle is the sum of the Euler-Lagrange equation EL LEL_{\mathbf{L}} and the canonical symplectic form d vθ L\mathbf{d}_v \theta_{\mathbf{L}} from covariant phase space formalism:

ω L 𝔽L *ω =EL L+d vθ L. \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. (ι v nι v 1)ω L=0(\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 Σ n1\Sigma_{n-1}, the transgression ω Σ Σ n1ω L\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 Σ=( d1;1,η)\Sigma = (\mathbb{R}^{d-1;1}, \eta) be Minkowski spacetime. Write the canonical coordinates

σ i:Σ. \sigma^i \;\colon\; \Sigma \longrightarrow \mathbb{R} \,.

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

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

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

{σ i},{ϕ a},{ϕ ,i a}:j 1(Σ×X)X. \{ \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(12g ijη abϕ ,i aϕ ,j a)dσ 1dσ d. 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

p a idσ 1dσ d u i aL =(g ijη abϕ ,j a)dσ 1dσ d \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 θ\theta in variational calculus, (see this remark at covariant phase space ) is

du aι i(ϕ ,i aL) =p a i(ι σ ivol)du a =p a idq i a, \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 4.

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 XX involves trajectories X\mathbb{R} \to X in XX, with parameter space Σ=\Sigma = \mathbb{R} the real line, thought of as the abstract “worldline” of the particle.

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

  • target space: XX, some manifoldspacetime;

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

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

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

Here a collection of vaues (q 0 i)(q^i_0) is a position of the particle and (v 0 i)(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 1EJ^1 E specified a parameter time and a corresponding position and velocity.

….

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

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

ω| U=dα| U=d(p idq i+Hdt). \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: Σ\Sigma is spacetime;

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

Bosonic string propagating on a manifold

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

  • target space. XX spacetime;

  • covariant configuration bundle E=Σ×XE = \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 Σ\Sigma is the cylinder ×S 1\mathbb{R}\times S^1 and XX is dd-dimensional Minkowski spacetime, 1,d1\mathbb{R}^{1,d-1}. A solution of the classical bosonic string is then a map ϕ:ΣX\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 ϕ\phi by equipping ×S 1\mathbb{R} \times S^{1} with its standard Minkowski metric and then solving the 1+11+1 dimensional field theory specified by the Lagrangian density

=12g ijη abϕ aq iϕ bq j. \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 iq^i (i=0,1)(i = 0,1) are standard coordinates on ×S 1\mathbb{R} \times S^1 and g=diag(1,1)g=\mathrm{diag}(1,-1) is the Minkowski metric on ×S 1\mathbb{R} \times S^1, while ϕ a\phi^a are the coordinates of the map ϕ\phi in 1,d1\mathbb{R}^{1,d-1} and η=diag(1,1,,1)\eta = \mathrm{diag}(1,-1,\cdots,-1) is the Minkowski metric on 1,d1\mathbb{R}^{1,d-1}. The corresponding Euler-Lagrange equation is just a version of the wave equation:

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

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

=12g ijη abu i au j b, \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 i au^a_{i}.

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

=p a iu i a=12η abg ijp a ip b j, \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 i au^a_{i} are defined implicitly by p a i=/u i ap_a^{i}=\partial \mathcal{L} / \partial u^{a}_{i}, and p a ip_a^{i} are coordinates on the fiber T u *XT qΣT^{*}_{u}X \otimes T_{q}\Sigma. Note that \mathcal{H} differs from the standard (non-covariant) Hamiltonian density for a field theory:

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

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

π a iq i=u a| u=ϕ,p=π \frac{\partial \pi^{i}_{a}}{\partial q^i} = - \left.\frac{\partial \mathcal{H}}{\partial u^{a}} \right \vert_{u=\phi,p=\pi}
ϕ aq i=p a i| u=ϕ,p=π. \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 1E) *(J^1 E)^*, and this space is equipped with a canonical 2-form α\alpha whose exterior derivative ω=dα\omega = d \alpha is a multisymplectic 3-form. Using the isomorphism

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

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

α=edq 0dq 1+(p a 0du adq 1p a 1du adq 0). \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 1E) *(J^1 E)^* is thus

ω=dedq 0dq 1+(dp a 0du adq 1dp a 1du adq 0). \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 ee may be considered as canonically conjugate to the area form dq 0dq 1dq^{0} \wedge dq^{1}.

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

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

Note that SS is constructed from ϕ\phi, π\pi and from the constraint e+=0e + \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 SS, a tangent bivector v=v 0v 1v=v_{0} \wedge v_{1} can be defined as

v 0=q 0+ϕ aq 0u a+π a iq 0p a i 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=q 1+ϕ aq 1u a+π a iq 1p 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 SS is generated by solutions to Hamilton’s equations if and only if

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

Hamiltonian nn-dimensional flow

Relation to nn-symplectic manifolds

There is also the notion of

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 nn-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)(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)(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, (n1)(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 (n1)(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 (n1)(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

Much of the material in the section on covariant field theoryhase space) is based on this.

Other discussions include

The relation to covariant phase space methods is discussed in

A discussion of Hamiltonian n-forms as observables is in

Other texts include

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

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 nn-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] M. Henneaux, C. 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 is given in

Higher order moment maps are considered in

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

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

  • Luca Vitagliano, L-infinity Algebras From Multicontact Geometry (arXiv.1311.2751)

Relation to covariant phase space formalism

The relation of multisymplectic formalism to covariant phase space and variational bicomplex methods is discussed in

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

On quantization of multisymplectic geometry

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

  • I.V. Kanatchikov, Geometric (pre)quantization in the polysymplectic approach to field theory , (arXiv:hep-th/0112263).

  • I.V. Kanatchikov, DeDonder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory , (arXiv:hep-th/9810165)

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:

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.

Revised on November 13, 2013 02:20:42 by Urs Schreiber (82.169.114.243)