Fermionic Poisson bracket

Question

I'd like to understand the Poisson bracket for fermions in classical field theory defined on a cylinder (with coordinates $(t,x)$, $x$ being the compact direction) and propagating on $T^n$ with constant metric $G_{ij}$ (though $T^n$ probably isn't important to the discussion here).

The action is $$S = \frac{i}{4\pi}\int dt dx \,\,G_{ij} \left[ \psi^i(\partial_t + \partial_x)\psi^j + \bar{\psi}^i(\partial_t - \partial_x)\bar{\psi}^j\right] .$$

From this how can I give a general definition of the Poisson bracket? Some constraints on it should be the following:

1. It should be symmetric (since in the bosonic case it's anti-symmetric).

2. I should be able to recover the standard relation $$\{\psi^i(t,x), \psi^j(t,x') \}_{PB} = -2\pi i G^{ij}\, \delta(x-x')$$ from it.

One definition which seems to work is $$\{F,G\}_{PB} = -2\pi i \int dx\,\, G^{ij}\Big(\frac{\delta F }{\delta \psi^i } \frac{\delta G}{\delta \psi^j} + \frac{\delta F }{\delta \bar{\psi}^i } \frac{\delta G}{\delta \bar{\psi}^j}\Big).$$

However, if this is right, I'd like to see why on more general grounds.

Also one particular Poisson bracket I'm interested in computing is $\{(\partial_t - \partial_{x_1})\psi^i(x_1), \psi^j(x_2)\}$ which I get from the above definition to be $2\pi i \frac{\partial}{\partial x_1} \delta(x_1 - x_2)$. Is this sensible?

Edit: for reference, I'm looking at Appendix A in the following paper by Kapustin and Orlov:

http://arxiv.org/abs/hep-th/0010293

and trying to verify the Poisson Brackets in eqn. (48).

Answer 1

The super-Poisson bracket follows from a super-version of the Dirac-Bergmann or the Faddeev-Jackiw procedure. Diligent care must be taken to achieve consistent sign conventions when dealing with Grassmann-odd variables, see e.g. my Phys.SE answer here.

The singular Legendre transformation for fermions is also discussed in my Phys.SE answer here. In OP's case [which is the fermionic part of eq. (44) in Ref.1], the Lagrangian density reads

$$ {\cal L}~=~ \frac{i}{4\pi} G_{ij}\left[\psi^i \dot{\psi}^j + \bar{\psi}^i \dot{\bar{\psi}}^j\right]-{\cal H}, \tag{1}$$

where we have identified the Hamiltonian density

$$ {\cal H}~=~ \frac{i}{4\pi} G_{ij}\left[\psi^i \partial_x\psi^j - \bar{\psi}^i \partial_x\bar{\psi}^j\right].\tag{2}$$

The symplectic one-form potential can be transcribed from the kinetic term in (1):

$$ \vartheta(t) ~=~\frac{i}{4\pi}\int\! dx~G_{ij}\left[ \psi^i(x,t)~\mathrm{d}\psi^j(x,t) + \bar{\psi}^i(x,t) ~\mathrm{d}\bar{\psi}^j(x,t)\right], \tag{3}$$

where $\mathrm{d}$ denotes the exterior derivative in infinitely many dimensions. The symplectic two-form is then

$$\begin{align} \omega(t)~=~&\mathrm{d}\vartheta(t) ~=~\frac{i}{4\pi}\int\! dx~G_{ij}\left[ \mathrm{d}\psi^i(x,t)\wedge\mathrm{d}\psi^j(x,t) +\mathrm{d}\bar{\psi}^i(x,t) \wedge\mathrm{d}\bar{\psi}^j(x,t)\right]\cr ~=~&\frac{i}{4\pi}\int\! dx~dy~ G_{ij}~ \delta(x-y) \left[ \mathrm{d}\psi^i(x,t)\wedge\mathrm{d}\psi^j(x,t)+\mathrm{d}\bar{\psi}^i(x,t)\wedge\mathrm{d}\bar{\psi}^j(x,t)\right].\end{align}\tag{4} $$

The equal-time super-Poisson/Dirac bracket on fundamental fields is the inverse supermatrix of the supermatrix for the symplectic two-form (4):

$$ \{\psi^i(x,t), \psi^j(y,t)\}_{PB}~=~ \frac{2\pi}{i}(G^{-1})^{ij} ~\delta(x-y)~=~\{\bar{\psi}^i(x,t), \bar{\psi}^j(y,t)\}_{PB}, \tag{5}$$

and other fundamental super-Poisson brackets vanish. The equal-time super-Poisson bracket read

$$ \{F(t),G(t)\}_{PB} ~=~ \frac{2\pi}{i}\int \! dx (G^{-1})^{ij}\left[\frac{\delta_R F(t)}{\delta \psi^i(x,t)} \frac{\delta_L G(t)}{\delta \psi^j(x,t)}+\frac{\delta_R F(t)}{\delta \bar{\psi}^i(x,t)} \frac{\delta_L G(t)}{\delta \bar{\psi}^j(x,t)} \right]\tag{6} $$

for arbitrary equal-time functionals $F(t)$ and $G(t)$. Here the subscripts $L$ and $R$ refer to differentiation from left and right, respectively.

References:

1. A. Kapustin and D. Orlov, arXiv:hep-th/0010293.