Quantum Mechanics for Mathematicians

For the first part of the theorem, the map

L→μL

is a vector space isomorphism of the space of antisymmetric matrices and
Λ^2 (Rn). To show that it is a Lie algebra isomorphism, one can use an analogous
argument to that of the proof of 16.2. Here one considers the action

ξ→{μL,ξ}+

ofμL∈so(n,R) on an arbitrary

ξ=

∑

j

cjξj

and uses the super-Jacobi identity relating the fermionic Poisson brackets of
μL,μL′,ξ.

30.3 Examples of pseudo-classical mechanics

In pseudo-classical mechanics, the dynamics will be determined by choosing a
Hamiltonianhin Λ∗(Rn). Observables will be other functionsF ∈Λ∗(Rn),
and they will satisfy the analog of Hamilton’s equations

d

dt

F={F,h}+

We’ll consider two of the simplest possible examples.

30.3.1 The pseudo-classical spin degree of freedom

Using pseudo-classical mechanics, a “classical” analog can be found for some-
thing that is quintessentially quantum: the degree of freedom that appears in
the qubit or spin^12 system that first appeared in chapter 3. TakingV=R^3 with
the standard inner product as fermionic phase space, we have three generators
ξ 1 ,ξ 2 ,ξ 3 ∈V∗satisfying the relations

{ξj,ξk}+=δjk

and an 8 dimensional space of functions with basis

1 , ξ 1 , ξ 2 , ξ 3 , ξ 1 ξ 2 , ξ 1 ξ 3 , ξ 2 ξ 3 , ξ 1 ξ 2 ξ 3

If we want the Hamiltonian function to be non-trivial and of even degree, it
will have to be a linear combination

h=B 12 ξ 1 ξ 2 +B 13 ξ 1 ξ 3 +B 23 ξ 2 ξ 3