Significance of symplectic form in classical field theory

Question

I'm trying to understand the significance of construction presented to me in field theory class. Let me first briefly describe it and then ask questions.

Given two solutions $\phi_1$, $\phi_2$ of the scalar wave equation $( \Box + m^2 ) \phi_i =0, $ $i=1,2$ one can define a conserved current, given by

$$ j[\phi_1, \phi_2] = \phi_1 \nabla \phi _2 - \phi_2 \nabla \phi_1, \tag{1} $$ $$ \nabla \cdot j =0 . \tag{2} $$

This allows one to constuct a symplectic form one the space of solutions. One chooses a Cauchy surface $\Sigma$ with future directed unit normal vector $N$ and defines

$$ \{ \phi_1 , \phi_2 \} = \int _{\Sigma} N \cdot j[\phi_1, \phi_2] d^3 x. \tag{3} $$

Furthermore, one can show that for any solution $\phi$ one can choose a function $\rho$ such that following representation holds:

$$ \hat{\phi}(k) = (2 \pi)^{3/2} \hat{D}(k) \hat{\rho}(k), \tag{4} $$

where hat denotes the Fourier transform and $D$ is Pauli-Jordan distribution, which satisfies

$$ \hat D (k) = \frac{i}{2 \pi} \mathrm{sgn} (k) \delta (k^2 -m^2).\tag{5} $$

Furthermore this representation is unique up to addition of a function with Fourier transform vanishing on the mass shell, or putting in a different way

$$ \phi _{\rho_1}=\phi_{\rho_2} \iff \exists \chi : \rho_1-\rho_2=(\Box + m^2) \chi. \tag{6}$$

One then constructs a quotient space, dividing space of all $\rho$ by space of all $(\Box +m^2) \chi$. On this space the symplectic form $ \sigma (\rho_1, \rho_2)=\{ \phi_{\rho_1}, \phi_{\rho_2} \} $ is well-defined and non-degenerate. It can also be rewritten as

$$ \sigma (\rho_1, \rho_2) = \int \rho_1(x) D(x-y) \rho_2(y) d^4 x d^4 y.\tag{7} $$

First question: are these symplectic forms ($\sigma(\cdot, \cdot)$ and $\{ \cdot, \cdot \}$) somehow related to Poisson bracket on phase space in Hamiltonian mechanics? I would expect something like that to be true, but for that one would need to somehow interpret $\rho$ as a function on some infinite-dimensional phase space. I am wondering if this can be done. And second, but closely related question: what is the intrepretation of these $\rho$ functions? Our lecturer told us that they should be thought of as degrees of freedom of the field but again, I don't quite see it. Some intuition here would be nice.

Answer 1

1. The first part of OP's construction is directly related to the covariant Hamiltonian formalism for a real scalar field with Lagrangian density $$ {\cal L} ~=~ \frac{1}{2}\partial_{\alpha} \phi ~\partial^{\alpha} \phi -{\cal V}(\phi), \tag{CW4} $$ see e.g. Ref. [CW] and this Phys.SE post. See also the Wronskian method in this Phys.SE post. [In this answer we use the $(+,-,-,-)$ Minkowski signature convention and set Planck's constant $\hbar=1$ to one.] OP's eqs. (1)-(3) correspond in Ref. [CW] to the symplectic 2-form current $$ J^{\alpha}(x) ~=~ \delta \phi_{\rm cl}(x) \wedge \partial^{\alpha} \delta\phi_{\rm cl} (x); \tag{CW14} $$ which is conserved $$ \partial_{\alpha} J^{\alpha}(x)~\approx~0 ;\tag{CW15} $$ and the symplectic 2-form $$ \omega ~=~\int_{\Sigma} \!\mathrm{d}\Sigma_{\alpha} ~J^{\alpha} \tag{CW16}$$ on the space of classical solutions, respectively. (Note that Ref. [CW] denotes the infinite-dimensional exterior derivative with a $\delta$ rather than a $\mathrm{d}$.) If we pick the standard initial time surface $\Sigma=\{x^0=0\}$, we get back to the standard symplectic 2-form $$ \omega ~=~\int_{\Sigma} \delta \phi_{\rm cl} \wedge \delta \dot{\phi}_{\rm cl}. \tag{CW17}$$

2. In the second part of OP's construction, we specialize to a quadratic potential $$ {\cal V}(\phi) ~=~\frac{1}{2}m^2\phi^2, \tag{A}$$ i.e. a free field.

OP's last eq. (7) corresponds to the standard non-equal-time commutator $$[\phi(x),\phi (y)]~=~ i\Delta(x\!-\!y) , \tag{IZ3-55} $$ where $$\begin{align} \Delta&(x\!-\!y) \cr &= \frac{1}{i} \int \! \frac{d^4k}{(2\pi)^3} \delta(k^2\!-\!m^2) ~{\rm sgn}(k^0)~ e^{-ik\cdot (x-y)},\end{align} \tag{IZ3-56}$$ see e.g. Ref. [IZ]. To compare with OP's eq. (7), smear the commutator (IZ3-55) with two test functions $\rho_1$ and $\rho_2$. Differentiation wrt. to time $y^0$ yields $$\begin{align} [\phi(x),\pi (y)] ~=~&[\phi(x),\dot{\phi} (y)]\cr ~=~& i\cos(\omega_{\bf k} (x^0\!-\!y^0))~ \delta^3({\bf x}\!-\!{\bf y}), \cr \omega_{\bf k}~:=~&\sqrt{{\bf k}^2+m^2}.\end{align} \tag{B} $$ Eqs. (IZ3-55), (IZ3-56) and (B) imply the standard equal-time CCR, $$\begin{align} [\phi(t, {\bf x}),\phi (t, {\bf y})] ~=~&0, \cr [\phi(t, {\bf x}),\pi (t, {\bf y})] ~=~&i\delta^3({\bf x}\!-\!{\bf y}), \cr [\pi(t, {\bf x}),\pi (t, {\bf y})]~=~&0 . \end{align} \tag{IZ3-3} $$ The CCR (IZ3-3) in turn is related to the standard canonical Poisson bracket $$\begin{align} \{\phi(t, {\bf x}),\phi (t, {\bf y})\}_{PB} ~=~&0, \cr \{\phi(t, {\bf x}),\pi (t, {\bf y})\}_{PB}~=~&\delta^3({\bf x}\!-\!{\bf y}), \cr \{\pi(t, {\bf x}),\pi (t, {\bf y})\}_{PB} ~=~&0 \end{align} \tag{C} $$ via the correspondence principle between quantum mechanics and classical mechanics, cf. e.g. this Phys.SE post.

References:

• [CW] C. Crnkovic & E. Witten, Covariant description of canonical formalism in geometrical theories. Published in Three hundred years of gravitation (Eds. S. W. Hawking and W. Israel), (1987) 676.

• [IZ] C. Itzykson & J.B. Zuber, QFT, 1985, p.117-118.