nLab Hamiltonian n-vector field

under construction

Idea

A Hamiltonian $n$-vector field is the $n$-dimensional analog of a Hamiltonian vector field as one passes from symplectic geometry to multisymplectic geometry/n-plectic geometry. Roughly, the transgression of a Hamiltonian $n$-vector field to mapping spaces out of an $(n-1)$-manifold yields an ordinary Hamiltonian vector field.

Definition

Definition

Let $X$ be a smooth manifold equipped with a degree $(n+1)$ differential form $\omega \in \Omega^{n+1}(X)$ for $n \in \mathbb{N}$, with the pair $(X,\omega)$ regarded as a multisymplectic manifold/n-plectic manifold.

Let moreover $H \;\colon\; X \longrightarrow \mathbb{R}$ be a smooth function, to be regarded as an extended Hamiltonian function, hence a de Donder-Weyl Hamiltonian.

Then a Hamiltonian $n$-vector field on $X$ is an $n$-multivector field $\mathbf{v} \in \Gamma(\wedge^{n} T X)$ satisfying the analog of Hamilton's equations, namely the differential equation of differential 1-forms on $X$

$$\iota_{\mathbf{v}} \omega = \mathbf{d} H \,.$$

Example

Let $\Sigma$ be a manifold to be regarded as a spacetime/worldvolume and let $E \to \Sigma$ be a vector bundle, to be regarded as a field bundle. Assume for simplicity of notation that both $\Sigma$ and $E$ are in fact Cartesian spaces (which is always true locally). Write $j^1 E$ for the corresponding jet bundle]], equipped with the canonical “kinetic” n-plectic form which in local coordinates reads

$$\omega \coloneqq \mathbf{d}_v\phi^a_{,i} \wedge \mathbf{d}_v q^a \wedge (\iota_{\partial_i} vol_\Sigma)$$

as discussed at multisymplectic geometry.

Then for $\phi^a$ and $\phi^a_{,i}$ smooth functions on $\Sigma$ and for vector fields

$$v_i \coloneqq \frac{\partial}{\partial \sigma^i} + \frac{\partial \phi^a}{\partial \sigma^i} + \frac{\partial \phi^a_{,j}}{\partial \sigma^j} \frac{\partial}{\partial \phi^a_{,i}}$$

the equation

$$\iota_{v_1 \cdots v_n} \omega = \pm \mathbf{d}H$$

is equivalent to the de Donder-Weyl equations

$$\frac{\partial \phi^a}{\partial \sigma^i} = \frac{\partial H}{\partial p^i_a} \;\;\;\; \frac{\partial p^i_a}{\partial \sigma^i} = \frac{\partial H}{\partial \phi^a} \,.$$

Proof

We have

$$\mathbf{d}_v \phi^a = \mathbf{d}\phi^a - \phi^a_{,i}\mathbf{d}\sigma^i$$

and

$$\mathbf{d}_v \phi^a_{,i} = \mathbf{d}\phi^a_{,i} - \phi^a_{,i j} \mathbf{d}\sigma^j \,.$$

The claimed equation comes from contracting in $\mathbf{d}\phi^a_{,i} \wedge \mathbf{d}\phi^a \wedge (\iota_{\partial_i}vol_\Sigma)$ all but the $i$th vector with the contracted volume form. The remaining contractions are then

$$\frac{\partial \phi^a}{\partial \sigma^i \partial \sigma^{[j}} \frac{\partial \phi_a}{\partial \sigma^{i]}} \mathbf{d}\sigma^j$$

and these cancel against the horizontal derivative contributions form above.

Interpretation in higher geometry

We discuss here how Hamiltonian $n$-vector fields are equivalently homorphisms from an $n$-dimensional surface into the Poisson bracket Lie n-algebra associated with the given n-plectic geometry.

Remark

By the discussion at n-plectic geometry, a (pre-)n-plectic manifold $(X,\omega)$ induces a Poisson bracket Lie n-algebra $\mathfrak{pois}(X,\omega)$, given as follows:

• in positive degree $k$ it has the degree $(n-k)$-differential forms on $X$

• in degree 0 it has pairs $(v,A)$ of Hamiltonian n-forms $A$ with their Hamiltonian vector fields $v$ (such that $\mathbf{d}A = \iota_v \omega$)

and

• whose n-ary Lie bracket for $n \geq 2$ is non-trivial only on $n$-tuples of degree-0 elements, where it is given by

  $$[(v_1, A_1), \cdots, (v_n,A_n)] = \pm \iota_{v_n}\cdots \iota_{v_1} \omega \;\;\; \in \Omega^1(X) \,,$$

• and whose unary Lie bracket is given by the de Rham differential, $\{-\} = \mathbf{d}$.

This means that the $n$-extended Hamilton equation of def. reads in terms of the Poisson bracket Lie n-algebra equivalently thus (hgp 13, remark 2.5.10):

$$\{v_1, \cdots, v_n\} = \pm\{H\} \,.$$

Let now $\Sigma$ be a manifold to be regarded as a spacetime/worldvolume and let $E \to \Sigma$ be a vector bundle, to be regarded as a field bundle. Assume for simplicity of notation that both $\Sigma$ and $E$ are in fact Cartesian spaces (which is always true locally). Write $(j^1 E)^\ast$ for the coreresponding dual jet bundle, equipped with the canonical “kinetic” n-plectic form which in local coordinates reads

$$\omega \coloneqq \mathbf{d}p^i_a \wedge \mathbf{d}q^a \wedge (\iota_{\partial_i} vol_\Sigma)$$

as discussed at multisymplectic geometry. Write $\mathfrak{pois}(X,\omega)$ for the corresponding Poisson bracket Lie n-algebra.

Proposition

The L-∞ algebra homomorphisms

$$\mathbb{R}^n \longrightarrow \mathfrak{pois}((j^1 E)^\ast, \omega)$$

which induce the canonical map under the projection to vector fields on $\Sigma$ are equivalently tuples consisting of

• a smooth function $H$;

• $n$ vector fields $v_i$

on $(j^1 E)^\ast$, such that the $(v_i)$ satisfy the de Donder-Weyl equation? for Hamiltonian $H$

$$\iota_{v_1 \cdots v_n} \omega = \pm \mathbf{d}H \,.$$

Proof

By the discussion at Maurer-Cartan element, a homomorphism as indicated is equivalently an element

$$\mathcal{J} \in \wedge^\bullet(\mathbb{R}^n) \otimes \mathfrak{pois}(X,\omega)$$

of total degree 1, which satifies the Maurer-Cartan equation

$$\sum_k \frac{1}{k!}\{\underbrace{\mathcal{J}\wedge \cdots \wedge \mathcal{J}}_{k\; factors}\} = 0 \,.$$

If we suggestively write $\mathbf{d}\sigma^i$ for the generators of the Grassmann algebra $\wedge^\bullet(\mathbb{R}^n)$, then, by remark , $\mathcal{J}$ is of the form

$$\mathcal{J} = \mathbf{d}\sigma^i \otimes (v_i, J_i) + \mathbf{d}\sigma^{i_1} \wedge \mathbf{d}\sigma^{i_2} \otimes J_{i_1 i_2} + \cdots + \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^n \otimes H \,,$$

where the $v_i$ are vector fields on the extended phase space,

$$J_{i_1 \cdots i_k} \in \Omega^{n-k}((j^1 E)^\ast)$$

and $H$ is a smooth function. Again by remark , the Maurer-Cartan equation on this element is equivalent to

$$\iota_{v_{i_1}} \cdots \iota_{v_{i_k}} \omega = \pm \mathbf{d} J_{i_1 \cdots i_k}$$

for all $1 \leq k \leq n-1$ and all $\{i_j\}_{j = 1}^k$, and

$$\iota_{v_{1}} \cdots \iota_{v_{n}} \omega = \pm \mathbf{d} H \,.$$

The condition on the projection means that

$$v_i = \frac{\partial}{\partial\sigma^i} + \frac{\partial \phi^a}{\partial \sigma^i} \frac{\partial }{\partial \phi^a} + \frac{\partial \phi^a_{,j}}{\partial \sigma^i} \frac{\partial }{\partial \phi^a_{,j}} \,.$$

This way the last equation is the de Donder-Weyl equation?. But then

$$J_{i_1 \cdots i_k} \coloneqq \pm H \iota_{\partial_{i_1} \cdots \partial_{i_k}} vol_\Sigma$$

We can further interpret this in local prequantum field theory as follows:

Remark

By (hgp 13) the Poisson bracket Lie n-algebra is the Lie differentiation of the smooth automorphism n-group of any prequantization of $(X,\omega)$ by a prequantum n-bundle $\nabla$:

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

regarded as an object in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$ of that cohesive (∞,1)-topos $\mathbf{H}$ of smooth ∞-groupoids, hence of the quantomorphism n-group)

$$QuantMorph(X,\nabla) = \mathbf{Aut}_{\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}}(\nabla) \,.$$

In terms of this Lie integrated perspective, remark says that $H$-Hamitlonian $n$-vector fields are the infinitesimal n-morphisms in $\left(\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}\right)^{\Box_n}$ whose faces carry trivial labels.

Here for instance for $n = 2$ a 2-morphism in $\left(\mathbf{H}_{/\mathbf{B}^2 U(1)_{conn}}\right)^{\Box_2}$ looks, “viewed from the top”, like a diagram in $\mathbf{H}$ of the form

$$\array{ X && \stackrel{}{\longrightarrow} && X \\ & {}_{\mathllap{\nabla}}\searrow && \swarrow_{\mathrlap{\nabla}} \\ \downarrow && \mathbf{B}^2 U(1)_{conn} && \downarrow \\ & {}^{\mathllap{\nabla}}\nearrow && \nwarrow^{\mathrlap{\nabla}} \\ X && \underset{}{\longrightarrow} && X }$$

with a big 3-homotopy filling this pyramid.

The equation of remark is the Maurer-Cartan equation exhibiting an infinitesimal such situation.

More in detai: the Lie integration of $\mathfrak{pois}(X,\omega)$ is a simplicial object which in simplicial degree $n$ has as elements the Maurer-Cartan elements of $\Omega^\bullet(\Sigma_n)\otimes \mathfrak{pois}(X,\omgea)$, where $\Sigma_n = \Delta^n$ is the $n$-dimensional simplex, regarded as a smooth manifold (with boundaries and corners).

Write then $\mathbf{d}\sigma^i$ for the canonical basis of 1-forms on $\Sigma_n$, then consider an element in there is of the form

$$A = \mathbf{d}\sigma^i \otimes v_i + vol \otimes H \,.$$

This satisfying the Maurer-Cartan equation

$$\mathbf{d}A + [A] + [A \wedge A] + [A\wedge A \wedge A] + \cdots = 0$$

then is equivalent to

$$[\mathbf{d}\sigma^1 \otimes v_1 \wedge \cdots \wedge \mathbf{d}\sigma^1 \otimes v_1] + \underbrace{ \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^n }_{= vol} \otimes [H] = 0$$

which is again the de Donder-Weyl-Hamilton equation of motion.

Proposition

For $(X,\omega)$ a pre-n-plectic manifold and $\mathfrak{poiss}(X,\omega)$ the corresponding Poisson bracket Lie n-algebra then L-∞ algebra homomorphisms of the form

$$\mathbb{R}^k \longrightarrow \mathfrak{pois}(X,\omega)$$

are in bijection to tuples consisting of $k$ vector fields $(v_1, \cdots, v_k)$ on $X$ and differential forms $\{J_{i_1 \cdots i_l}\}$ and a smooth function $H$ on $X$ such that

$$\iota_{v_{1_1} \cdots v_{i_l}} \omega = \pm \mathbf{d} J_{i_1 \cdots i_l}$$

$$\iota_{v_{1_1} \cdots v_{i_n}} \omega = \pm \mathbf{d} H$$

holds.

The following is the higher/local analog of the symplectic Noether theorem.

For $(X,\omega)$ a pre-n-plectic manifold, (…)

let $H \in C^\infty(X)$ be a smooth function to be regarded as a de Donder-Weyl Hamiltonian and let $(v_1, \cdots, v_n)$ be a Hamiltonian $n$-vector field, hence a solution to the Hamilton-de Donder-Weyl equation of motion. By prop. this corresponds to an L-∞ algebra map of the form

$$((v_1, \cdots, v_n), H) \;\colon\; \mathbb{R}[n-1] \longrightarrow \mathfrak{pois}(X,\omega) \,.$$

Proposition

(higher symplectic Noether theorem)

The extension of $((v_1, \cdots, v_n), H)$ to a homomorphism of the form

$$((v_1, \cdots, v_n, v_0), H, J, \{K_i\}) \;\colon\; \mathbb{R}[n-1]\oplus \mathbb{R} \longrightarrow \mathfrak{pois}(X,\omega)$$

is equivalently the choice of a vector field $v_0$ on $X$ such that

$$\iota_{v_0} \mathbf{d}H =$$

$$\iota_{v_0} \omega = \pm \mathbf{d}J$$

$$\iota_{v_1 \cdots v_n} \omega = \pm \mathbf{d}H$$

$$\iota_{v_1 \cdots v_{i-1} v_{i+1} \cdots v_n v_0} \omega = \mathbf{d} K^i$$

Related concepts

• de Donder-Weyl formalism

References

The specific condition of def. appears as equation (4) in

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

Remark appears in remark 2.5.10 of

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)