From Classical Mechanics to Quantum Field Theory

26 From Classical Mechanics to Quantum Field Theory. A Tutorial

the vector fieldXA:H→TH via x−→(x, Ax), then we find:

∇fA=XA, (1.114)

XfA=J(XA). (1.115)

1.3 Methods of Quantization

In the following, we will give an introduction to some quantization techniques
which are widely used in the framework of Quantum Field Theory[5; 21]: coherent
states, Feynman path integral and finally the Weyl-Wigner map. To do so, we will
need some basic knowledge of the Heisenberg-Weyl algebra and group, that we
will recall in the next subsection.

1.3.0.1 The Heisenberg-Weyl Group

We describe the algebra Heisenberg-Weyl algebra and group in detail just for one
degree of freedom, since the generalization to an arbitrary number is obvious.
Starting from the canonical degrees of freedomq,p, we can construct the so-
called creation/annihilation operators:

a†=

qˆ√−ıpˆ

2 

,a=

qˆ√+ıpˆ

2 

(1.116)

which, in virtue of the CCR (1.2), satisfy:

[a, a†]=I,[a,I]=[a†,I]=0. (1.117)

We say that {ˆq,p,ˆI} or equivalently {a, a†,I} generate a Lie algebra, the
Heisenberg-Weyl algebraw 1.
Introducing the (imaginary) elements:
e 1 =ıp/ˆ

√

,e 2 =ıq/ˆ

√

,e 3 =ıI, (1.118)

we can equivalently say that the Heisenberg-Weyl algebra w 1 is the real 3-
dimensional Lie algebra generated by the set{e 1 ,e 2 ,e 3 }such that:

[e 1 ,e 2 ]=e 3 ,[e 1 ,e 3 ]=[e 2 ,e 3 ]=0. (1.119)

A generic elementx∈w 1 can therefore be written in one of the following form:

x =x 1 e 1 +x 2 e 2 +se 3 x 1 ,x 2 ,s∈R (1.120)

=ısI+ı(Ppˆ−Qqˆ)/ s, P, Q∈R (1.121)

=ısI+(αa†−α∗a) s∈R,α∈C, (1.122)

where the relationship between the coefficients of the three expressions can be
easily derived by the reader.