Quantum Mechanics for Mathematicians

This state space is two complex dimensional, with an arbitrary state

f(η) =c 1 1 +c 2 η

withcj complex numbers. The inner product on this space is given by the
fermionic integral

(f 1 (η),f 2 (η)) =

∫

f 1 ∗(η)f 2 (η)dη

with
f∗(ξ) =c 1 1 +c 2 η

With respect to this inner product, one has

(1,1) = (η,η) = 0, (1,η) = (η,1) = 1

This inner product is indefinite and can take on negative values, since

(1−η, 1 −η) =− 2

Having such negative-norm states ruins any standard interpretation of this
as a physical system, since this negative number is supposed to the probability of
finding the system in this state. Such quantum systems are called “ghosts”, and
do have applications in the description of various quantum systems, but only
when a mechanism exists for the negative-norm states to cancel or otherwise be
removed from the physical state space of the theory.

31.3 Spinors and the Bargmann-Fock construc-

tion

While the fermionic analog of the Schr ̈odinger construction does not give a uni-
tary representation of the spin group, it turns out that the fermionic analog of
the Bargmann-Fock construction does, on the fermionic oscillator state space
discussed in chapter 27. This will work for the case of a positive definite sym-
metric bilinear form (·,·). Note though that this will only work for fermionic
phase spacesRnwithneven, since a complex structure on the phase space is
needed.
The corresponding pseudo-classical system will be the classical fermionic
oscillator studied in section 30.3.2. Recall that this uses a choice of complex
structureJon the fermionic phase spaceR^2 d, with the standard choiceJ=J 0
coming from the relations

θj=

1

√

2

(ξ 2 j− 1 −iξ 2 j), θj=

1

√

2

(ξ 2 j− 1 +iξ 2 j) (31.2)

forj= 1,...,dbetween real and complex coordinates. Here (·,·) is positive-
definite, and theξjare coordinates with respect to an orthonormal basis, so we
have the standard relation{ξj,ξk}+=δjkand theθj,θjsatisfy

{θj,θk}+={θj,θk}+= 0, {θj,θk}+=δjk