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
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 $$ \omega ~=~\int_{\Sigma} \delta \phi_{\rm cl} \wedge \delta \dot{\phi}_{\rm cl}. \tag{CW17}$$
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 $$ \Delta(x\!-\!y) ~=~ \frac{1}{i} \int \! \frac{d^4k}{(2\pi)^3} \delta(k^2\!-\!m^2) ~{\rm sgn}(k^0)~ e^{-ik\cdot (x-y)}, \tag{IZ3-56}$$ 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 $$ [\phi(x),\pi (y)]~=~[\phi(x),\dot{\phi} (y)]~=~ i\cos(\omega_{\bf k} (x^0\!-\!y^0))~ \delta^3({\bf x}\!-\!{\bf y}), \qquad \omega_{\bf k}~:=~\sqrt{{\bf k}^2+m^2}. \tag{B} $$ Eqs. (IZ3-55), (IZ3-56) and (B) imply the standard equal-time CCR, $$ [\phi(t, {\bf x}),\phi (t, {\bf y})]~=~0, \quad [\phi(t, {\bf x}),\pi (t, {\bf y})]~=~i\delta^3({\bf x}\!-\!{\bf y}), \quad [\pi(t, {\bf x}),\pi (t, {\bf y})]~=~0 . \quad \tag{IZ3-3} $$ The CCR (IZ3-3) in turn is related to the standard canonical Poisson bracket $$ \{\phi(t, {\bf x}),\phi (t, {\bf y})\}_{PB}~=~0, \quad \{\phi(t, {\bf x}),\pi (t, {\bf y})\}_{PB}~=~\delta^3({\bf x}\!-\!{\bf y}), \quad \{\pi(t, {\bf x}),\pi (t, {\bf y})\}_{PB}~=~0 \quad \tag{C} $$ 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.
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