From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 47

with the matrixM=[Mij]oftheform:

M=

∣∣

∣∣

∣∣

∣∣

∣∣

100 ... .... −ζΩ 0

−Ω 0 10 ... ... 0

0 −Ω 01 ... ... 0

... ... ... ... ... ...

000 ...−Ω 0 1

∣∣

∣∣

∣∣

∣∣

∣∣

, Ω 0 ≡ 1 −

β

M(Ω−μ). (1.226)

The partition function is then given by an integral over (complex or grassmann)
variables of gaussian type , which can be performed to get:

Z= lim→ 0 (detM)−ζ, (1.227)

where

detM=1+(−)M−^1 (−ζΩ 0 )(−Ω 0 )M−^1 =1−ζΩM 0. (1.228)

Hence

Z= lim→ 0

[

1 −ζ

(

β(Ω−μ)

M

)M]−ζ

=

[

1 −ζe−β(Ω−μ)

]−ζ

. (1.229)

1.3.3 Weyl quantization

In this section, we will give an introduction to the quantization `alaWeyl,the
interested reader can find a more exhaustive discussion in[ 14 ].
Weyl quantization is interesting from many points of view and has several
advantages. Firstly, it has the virtue of overcoming the problem mentioned in
the introduction about the fact that the CCR (1.2) betweenpandqimplies that
at least one of the two operators must be unbounded. Also, being founded on
geometric concepts, it can be generalizedto a generic phase space, i.e. a symplectic
manifold which is not necessarily a linear space. Then, The Weyl map (and its
inverse, the Wigner map) allow for a quantization on the space of functionsf(p, q)
over the entire phase space, and not on the space of functions on the configuration
space or on a suitable maximal Lagrangian subspace, as it is required by geometric
quantization. Finally, in this setting, it is transparent to discuss the limit→0,
which describes the quantumto classical transition.

1.3.3.1 The Weyl map

Let us start by considering a (real) vector spaceSendowed with a constant sym-
plectic structureω,sothatS≈R^2 nfor somen. We will denote withU(H)the
set of unitary operators on an abstract Hilbert spaceH.