Quantum Mechanics for Mathematicians

then ∫

Fdξj=

∂

∂ξj

F=

∂

∂ξj

∂

∂ξj

G= 0

using the fact that repeated derivatives give zero.

30.2 Pseudo-classical mechanics and the fermionic Poisson bracket

The basic structure of Hamiltonian classical mechanics depends on an even
dimensional phase spaceM=R^2 dwith a Poisson bracket{·,·}on functions on
this space. Time evolution of a functionfon phase space is determined by

d

dt

f={f,h}

for some Hamiltonian functionh. This says that taking the derivative of any
function in the direction of the velocity vector of a classical trajectory is the
linear map
f→{f,h}

on functions. As we saw in chapter 14, since this linear map is a derivative, the
Poisson bracket will have the derivation property, satisfying the Leibniz rule

{f 1 ,f 2 f 3 }=f 2 {f 1 ,f 3 }+{f 1 ,f 2 }f 3

for arbitrary functionsf 1 ,f 2 ,f 3 on phase space. Using the Leibniz rule and
antisymmetry, Poisson brackets can be calculated for any polynomials, just
from knowing the Poisson bracket on generatorsqj,pj (or, equivalently, the
antisymmetric bilinear form Ω(·,·)), which we chose to be

{qj,qk}={pj,pk}= 0, {qj,pk}=−{pk,qj}=δjk

Notice that we have a symmetric multiplication on generators, while the Poisson
bracket is antisymmetric.
To get pseudo-classical mechanics, we think of the Grassmann algebra Λ∗(Rn)
as our algebra of classical observables, an algebra we can think of as functions
on a “fermionic” phase spaceV=Rn(note that in the fermionic case, the phase
space does not need to be even dimensional). We want to find an appropriate
notion of fermionic Poisson bracket operation on this algebra, and it turns out
that this can be done. While the standard Poisson bracket is an antisymmetric
bilinear form Ω(·,·) on linear functions, the fermionic Poisson bracket will be
based on a choice of symmetric bilinear form on linear functions, equivalently,
a notion of inner product (·,·).
Denoting the fermionic Poisson bracket by{·,·}+, for a multiplication anti-
commutative on generators one has to adjust signs in the Leibniz rule, and the