nLab off-shell Poisson bracket

Idea

Given a symplectic manifold $(X,\omega)$ and given a Hamiltonian function $H \colon X \longrightarrow \mathbb{R}$, there is a Poisson bracket on an algebra of functions on the smooth path space $[I,X]$ – the “space of histories” or “space of trajectories” – for $I = [0,1]$ the closed interval, which is such that its symplectic leaves are each a copy of $X$, but regarded as the space of initial conditions for evolution with respect to $H$ with a source term added.

The first statement was first observed for the Peierls bracket of local prequantum field theory in (Marolf 93, section II), but the construction there is not specific to the Peierls bracket. That the construction provides a foliation of trajectory space by symplectic leaves labeled by level sets of the Euler-Lagrange function was explicitly pointed out at (Brunetti-Fredenhagen-Ribeiro 12, top of p. 4). (Again for the Peierls bracket, but the statement holds more generally.) These references and that this means a symplectic foliation by source terms was highlighted out by (Khavkine 13).

On paths in a symplectic manifold

We describe here the off-shell Poisson bracket in the context of mechanics, hence for mechanical systems with finite-dimensional phase space. The basic idea is that sketched in (Marolf 93, section II), but we try to make it precise. Then we similarly look into the foliation by symplectic leaves as suggested by (Khavkine 13).

The trajectory space of a symplectic manifold

Let $(X,\omega)$ be a symplectic manifold. We write

$$\{-,-\} \;\colon\; C^\infty(X)\otimes C^\infty(X) \longrightarrow C^\infty(X)$$

for the Poisson bracket induced by the symplectic form $\omega$, hence by the Poisson bivector $\pi \coloneqq \omega^{-1}$.

For notational simplicity we will restrict attention to the special case that

$$X = \mathbb{R}^2 \simeq T^\ast \mathbb{R}$$

with canonical coordinates

$$q,p \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}$$

and symplectic form

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

The general case of the following discussion is a straightforward generalization of this, which is just notationally more inconvenient.

Write $I \coloneqq [0,1]$ for the standard interval regarded as a smooth manifold with boundary. The mapping space

$$P X \coloneqq [I, X]$$

canonically exists as a smooth space, but since $I$ is compact this structure canonically refines to that of a Fréchet manifold (see at manifold structure of mapping spaces). This implies that there is a good notion of tangent space $T P X$. The task now is to construct a certain Poisson bivector as a section $\pi \in \Gamma^{\wedge 2}(T P X)$.

Among the smooth functions on $P X$ are the evaluation maps

$$ev \;\colon\; P X \times I = [I,X] \times I \stackrel{}{\longrightarrow} X$$

whose components we denote, as usual, for $t \in I$ by

$$q(t) \coloneqq q \circ ev_t \;\colon\; P X \longrightarrow \mathbb{R}$$

and

$$p(t) \coloneqq p \circ ev_t \;\colon\; P X \longrightarrow \mathbb{R} \,.$$

Generally for $f \colon X \to \mathbb{R}$ any smooth function, we write $f(t) \coloneqq f \circ ev_t \in C^\infty(P X)$. This defines an embedding

$$C^\infty(X) \times I \hookrightarrow C^\infty(P X) \,.$$

Similarly we have

$$\dot q(t) \;\colon\; P X \longrightarrow \mathbb{R}$$

and

$$\dot q(t) \;\colon\; P X \longrightarrow \mathbb{R}$$

obtained by differentiation of $t \mapsto q(t)$ and $t \mapsto p(t)$.

Hamiltonian evolution

Let now

$$H \;\colon\; X \times I \longrightarrow \mathbb{R}$$

be a smooth function, to be regarded as a time-dependent Hamiltonian. This induces a time-dependent function on trajectory space, which we denote by the same symbol

$$H \;\colon\; P X \times I \stackrel{(ev,id)}{\longrightarrow} X \times X \stackrel{H}{\longrightarrow} \mathbb{R} \,.$$

Hence for $t \in I$ we write

$$H(t) \;\colon\; P X \times \{t\} \stackrel{(ev, id)}{\longrightarrow} X \times \{t\} \stackrel{H}{\longrightarrow} \mathbb{R}$$

for the function that assigns to a trajectors $(q(-),p(-)) \colon I \longrightarrow X$ its energy at (time) parameter value $t$.

Define then the Euler-Lagrange density induced by $H$ to be the functions

$$EL(t) \;\colon\; P X \longrightarrow \mathbb{R}^2$$

with components

$$EL(t) = \left( \array{ \dot q(t) - \frac{\partial H}{\partial p}(t) \\ \dot p(t) + \frac{\partial H}{\partial p}(t) } \right) \,.$$

The trajectories $\gamma \colon I \to X$ on which $EL(t)$ vanishes for all $t \in I$ are equivalently those

• for which the tangent vector $\dot \gamma \in T_{\gamma}X$ is a Hamiltonian vector for $H$;

• which satisfy Hamilton's equations of motion for $H$.

Since the differential equations $EL = 0$ have a unique solution for given initial data $(q(0), p(0))$, the evaluation map

$$\left\{ \gamma \in P X | \forall_{t \in I}\; EL_\gamma(t) = 0 \right\} \stackrel{\gamma \mapsto \gamma(0)}{\longrightarrow} X$$

is an equivalence (an isomorphism of smooth spaces).

The off-shell Poisson bracket

Write

$$Poly(P X) \hookrightarrow C^\infty(P X)$$

for the subalgebra? of smooth functions on path space which are

• polynomials

• of integrals over $I$

• of the smooth functions in the image of $C^\infty(X) \times I \hookrightarrow C^\infty(P X)$

• and all their derivatives along $I$.

Define a bilinear function

$$\{-,-\} \;\colon\; Poly(P X) \otimes Poly(P X) \longrightarrow Poly(P X)$$

as the unique function which is a derivation in both arguments and moreover is a solution to the differential equations

$$\frac{\partial}{\partial t_2} \left\{f(t_1), q(t_2)\right\} = \left\{ f(t_1), \frac{\partial H}{\partial p}(t_2) \right\}$$

$$\frac{\partial}{\partial t_2} \left\{f(t_1), p(t_2)\right\} = - \left\{ f(t_1), \frac{\partial H}{\partial q}(t_2) \right\}$$

subject to the initial conditions

$$\{f(t), q(t)\} = \{f,q\}$$

$$\{f(t), p(t)\} = \{f,p\}$$

for all $t \in I$, where on the right we have the original Poisson bracket on $X$.

This bracket directly inherits skew-symmetry and the Jacobi identity from the Poisson bracket of $(X, \omega)$, hence equips the vector space $Poly(P X)$ with the structure of a Lie bracket. Since it is by construction also a derivation of $Poly(P X)$ as an associative algebra, we have that

$$\left( Poly\left(P X\right), \; \left\{ -,- \right\} \right) \;\;\; \in P_1 Alg$$

is a Poisson algebra. This is the “off-shell Poisson algebra” on the space of trajectories in $(X,\omega)$.

The symplectic leaves

Observe that by construction of the off-shell Poisson bracket, specifically by the differential equations defining it, the Euler-Lagrange function $EL$ generate a Poisson ideal.

For instance

$$\left( \array{ \frac{\partial}{\partial t_2} \left\{f(t_1), q(t_2)\right\} &=& \left\{ f(t_1), \frac{\partial H}{\partial p}(t_2) \right\} \\ \frac{\partial}{\partial t_2} \left\{f(t_1), p(t_2)\right\} &=& - \left\{ f(t_1), \frac{\partial H}{\partial q}(t_2) \right\} } \right) \;\;\; \Leftrightarrow \;\;\; \left( \left\{ f(t_1), \; EL(t) \right\} = 0 \right) \,.$$

Moreover, since $\{EL(t) = 0\}$ are equations of motion the Poisson reduction defined by this Poisson idea is the subspace of those trajectories which are solutions of Hamilton's equations, hence the “on-shell trajectories”.

As remarked above, the initial value map canonically identifies this on-shell trajectory space with the original phase space manifold $X$. Moreover, by the very construction of the off-shell Poisson bracket as being the original Poisson bracket at equal times, hence in particular at time $t = 0$, it follows that restricted to the zero locus $EL = 0$ the off-shell Poisson bracket becomes symplectic.

All this clearly remains true with the function $EL$ replaced by the function $EL - J$, for $J \in C^\infty(I)$ any function of the (time) parameter (since $\{J,-\} = 0$). Any such choice of $J$ hence defines a symplectic subspace

$$\left\{ \gamma \in P X \;|\; \forall_{t \in I}\; EL_\gamma(t) = J \right\}$$

of the off-shell Poisson structure on trajectory space. Hence $\left(O X, \left\{-,-\right\}\right)$ has a foliation by symplectic leaves with the leaf space being the smooth space $C^\infty(I)$ of smooth functions on the interval.

Notice that changing $EL \mapsto EL - J$ corresponds changing the time-dependent Hamiltonian $H$ as

$$H \mapsto H - J q \,.$$

Such a term linear in the canonical coordinates (the fields) is a source term. (The action functionals with such source terms added serve as integrands of generating functions for correlators in statistical mechanics and in quantum mechanics.)

Boundary field theory interpretation

Hence in conclusion we find the following statement:

The trajectory space (history space) of a mechanical system carries a natural Poisson structure whose symplectic leaves are the subspaces of those trajectories which satisfy the equations of motion with a fixed source term and hence whose symplectic leaf space is the space of possible sources.

Notice what becomes of this statement as we consider the the 2d Chern-Simons theory induced by the off-shell Poisson bracket (the non-perturbative Poisson sigma-model) whose moduli stack of fields is the symplectic groupoid $SG\left(P X, \left\{-,-\right\}\right)$ induced by the Poisson structure.

By the discussion at motivic quantization in the section The Poisson manifold at the boundary of the 2d Chern-Simons theory, the Poisson space $\left(P X, \left\{-,-\right\}\right)$ defines a boundary field theory (in the sense of local prequantum field theory) for this 2d Chern-Simons theory, exhibited by a boundary correspondence of the form

$$\array{ && P X \\ & \swarrow && \searrow \\ \ast && \swArrow_{\xi} && SG\left(P X, \left\{-,-\right\}\right) \\ & \searrow && \swarrow_{\mathrlap{\chi}} \\ && \mathbf{B}^2 U(1) \\ && \downarrow^{\mathrlap{\rho}} \\ && KU Mod } \,.$$

Notice that the symplectic groupoid is a version of the symplectic leaf space of the given Poisson manifold (its 0-truncation is exactly the leaf space). Hence in the case of the off-shell Poisson bracket, the symplectic groupoid is the space of sources of a mechanical system. At the same time it is the moduli space of fields of the 2d Chern-Simons theory of which the mechanical system is the boundary field theory. Hence the fields of the bulk field theory are identified with the sources of the boundary field theory. Hence conceptually the above boundary correspondence diagram is of the following form

$$\array{ && Fields \\ & \swarrow && \searrow \\ \ast && \swArrow_{} && Sources \\ & \searrow && \swarrow_{\mathrlap{}} \\ && Phases } \,.$$

Such a relation

bulk field theory	boundary field theory
field	source

between bulk fields and boundary sources is the characteristic feature of what is called the holographic principle in its realization as the AdS-CFT correspondence.

References

The off-shell extension of the Peierls bracket is observed in section II of

• Don Marolf, Poisson Brackets on the Space of Histories Annals of Physics Volume 236, Issue 2, December 1994, Pages 374-391 (arXiv:hep-th/9308141)

The observation that the off-shell bracket has a symplectic foliation by the level sets of the Euler-Lagrange functions appears on the top of p. 4 of

• Romeo Brunetti, Klaus Fredenhagen, Pedro Ribeiro, Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields (arXiv:1209.2148)

All this and the interpretation of the resulting symplectic foliation as a foliation by source terms has been highlighted by

• Igor Khavkine, personal communication