nLab
Hamiltonian n-vector field

Contents

under construction

Contents

Idea

A Hamiltonian nn-vector field is the nn-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 nn-vector field to mapping spaces out of an (n1)(n-1)-manifold yields an ordinary Hamiltonian vector field.

Definition

Definition

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

Let moreover H:XH \;\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 nn-vector field on XX is an nn-multivector field vΓ( nTX)\mathbf{v} \in \Gamma(\wedge^{n} T X) satisfying the analog of Hamilton's equations, namely the differential equation of differential 1-forms on XX

ι vω=dH. \iota_{\mathbf{v}} \omega = \mathbf{d} H \,.
Example

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

ωd vϕ ,i ad vq a(ι ivol Σ) \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 ϕ a\phi^a and ϕ ,i a\phi^a_{,i} smooth functions on Σ\Sigma and for vector fields

v iσ i+ϕ aσ i+ϕ ,j aσ jϕ ,i a 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

ι v 1v nω=±dH \iota_{v_1 \cdots v_n} \omega = \pm \mathbf{d}H

is equivalent to the de Donder-Weyl equations

ϕ aσ i=Hp a ip a iσ i=Hϕ a. \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

d vϕ a=dϕ aϕ ,i adσ i \mathbf{d}_v \phi^a = \mathbf{d}\phi^a - \phi^a_{,i}\mathbf{d}\sigma^i

and

d vϕ ,i a=dϕ ,i aϕ ,ij adσ j. \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 dϕ ,i adϕ a(ι ivol Σ)\mathbf{d}\phi^a_{,i} \wedge \mathbf{d}\phi^a \wedge (\iota_{\partial_i}vol_\Sigma) all but the iith vector with the contracted volume form. The remaining contractions are then

ϕ aσ iσ [jϕ aσ i]dσ j \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 nn-vector fields are equivalently homorphisms from an nn-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,ω)(X,\omega) induces a Poisson bracket Lie n-algebra 𝔭𝔬𝔦𝔰(X,ω)\mathfrak{pois}(X,\omega), given as follows:

and

  • whose n-ary Lie bracket for n2n \geq 2 is non-trivial only on nn-tuples of degree-0 elements, where it is given by

    [(v 1,A 1),,(v n,A n)]=±ι v nι v 1ωΩ 1(X), [(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, {}=d\{-\} = \mathbf{d}.

This means that the nn-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,,v n}=±{H}. \{v_1, \cdots, v_n\} = \pm\{H\} \,.

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

ωdp a idq a(ι ivol Σ) \omega \coloneqq \mathbf{d}p^i_a \wedge \mathbf{d}q^a \wedge (\iota_{\partial_i} vol_\Sigma)

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

Proposition

The L-∞ algebra homomorphisms

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

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

on (j 1E) *(j^1 E)^\ast, such that the (v i)(v_i) satisfy the de Donder-Weyl equation? for Hamiltonian HH

ι v 1v nω=±dH. \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

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

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

k1k!{𝒥𝒥 kfactors}=0. \sum_k \frac{1}{k!}\{\underbrace{\mathcal{J}\wedge \cdots \wedge \mathcal{J}}_{k\; factors}\} = 0 \,.

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

𝒥=dσ i(v i,J i)+dσ i 1dσ i 2J i 1i 2++dσ 1dσ nH, \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 iv_i are vector fields on the extended phase space,

J i 1i kΩ nk((j 1E) *) J_{i_1 \cdots i_k} \in \Omega^{n-k}((j^1 E)^\ast)

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

ι v i 1ι v i kω=±dJ i 1i k \iota_{v_{i_1}} \cdots \iota_{v_{i_k}} \omega = \pm \mathbf{d} J_{i_1 \cdots i_k}

for all 1kn11 \leq k \leq n-1 and all {i j} j=1 k\{i_j\}_{j = 1}^k, and

ι v 1ι v nω=±dH. \iota_{v_{1}} \cdots \iota_{v_{n}} \omega = \pm \mathbf{d} H \,.

The condition on the projection means that

v i=σ i+ϕ aσ iϕ a+ϕ ,j aσ iϕ ,j a. 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 1i k±Hι i 1 i kvol Σ 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,ω)(X,\omega) by a prequantum n-bundle \nabla:

X B nU(1) conn ω F () Ω cl n \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 H /B nU(1) conn\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} of that cohesive (∞,1)-topos H\mathbf{H} of smooth ∞-groupoids, hence of the quantomorphism n-group)

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

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

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

X X B 2U(1) conn X X \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 𝔭𝔬𝔦𝔰(X,ω)\mathfrak{pois}(X,\omega) is a simplicial object which in simplicial degree nn has as elements the Maurer-Cartan elements of Ω (Σ n)𝔭𝔬𝔦𝔰(X,omgea)\Omega^\bullet(\Sigma_n)\otimes \mathfrak{pois}(X,\omgea), where Σ n=Δ n\Sigma_n = \Delta^n is the nn-dimensional simplex, regarded as a smooth manifold (with boundaries and corners).

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

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

This satisfying the Maurer-Cartan equation

dA+[A]+[AA]+[AAA]+=0 \mathbf{d}A + [A] + [A \wedge A] + [A\wedge A \wedge A] + \cdots = 0

then is equivalent to

[dσ 1v 1dσ 1v 1]+dσ 1dσ n =vol[H]=0 [\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,ω)(X,\omega) a pre-n-plectic manifold and 𝔭𝔬𝔦𝔰𝔰(X,ω)\mathfrak{poiss}(X,\omega) the corresponding Poisson bracket Lie n-algebra then L-∞ algebra homomorphisms of the form

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

are in bijection to tuples consisting of kk vector fields (v 1,,v k)(v_1, \cdots, v_k) on XX and differential forms {J i 1i l}\{J_{i_1 \cdots i_l}\} and a smooth function HH on XX such that

ι v 1 1v i lω=±dJ i 1i l \iota_{v_{1_1} \cdots v_{i_l}} \omega = \pm \mathbf{d} J_{i_1 \cdots i_l}
ι v 1 1v i nω=±dH \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,ω)(X,\omega) a pre-n-plectic manifold, (…)

let HC (X)H \in C^\infty(X) be a smooth function to be regarded as a de Donder-Weyl Hamiltonian and let (v 1,,v n)(v_1, \cdots, v_n) be a Hamiltonian nn-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,,v n),H):[n1]𝔭𝔬𝔦𝔰(X,ω). ((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,,v n),H)((v_1, \cdots, v_n), H) to a homomorphism of the form

((v 1,,v n,v 0),H,J,{K i}):[n1]𝔭𝔬𝔦𝔰(X,ω) ((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 0v_0 on XX such that

ι v 0dH= \iota_{v_0} \mathbf{d}H =
ι v 0ω=±dJ \iota_{v_0} \omega = \pm \mathbf{d}J
ι v 1v nω=±dH \iota_{v_1 \cdots v_n} \omega = \pm \mathbf{d}H
ι v 1v i1v i+1v nv 0ω=dK i \iota_{v_1 \cdots v_{i-1} v_{i+1} \cdots v_n v_0} \omega = \mathbf{d} K^i
  • de Donder-Weyl formalism?

References

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

Remark appears in remark 2.5.10 of

Last revised on October 2, 2013 at 02:19:42. See the history of this page for a list of all contributions to it.