A Short Course on Quantum Mechanics and Methods of Quantization 39- non-orthogonality condition
〈ξ|ξ′〉=e
∑
αξα∗ξ′α; (1.189)- resolution of identity
I=∫ (∏
αdξα∗dξα)
e∑
αξ∗αξα|ξ〉〈ξ|, (1.190)where the integration over Grassmann variables has to be of course suitably de-
fined. We have not time to explore this subject here and refer the interested reader
to the literature[30; 31].
As a final comment, we want to stress that the definition of generalized coherent
states can be extended to a more general groupG, including all nilpotent and
semi-simple Lie-groups, by looking at their UIRs: T :g→T(g),T(g)beingan
unitary operator on some Hilbert spaceH. As before, choosing a reference vector
|ψ 0 〉, we construct the setof coherent states asT(g)|ψ 0 〉, taking into account that
the fiducial vector might have a non-trivial isotropy groupG 0. This means that
coherent states are in this case labeled by points in the manifoldG/G 0 .Thisgives
a hint about the fact that coherent states are related to the geometry of symmetric
manifolds and the theory of co-adjoint orbits, a topic we cannot deal with here
(see[ 33 ]).
1.3.2 Feynman path integral
What were presented in the previous sections as well as the Weyl-Wigner ap-
proach, that will be considered in the next section, essentially rely on Dirac’s view
of quantization which starts from classical Poisson brackets involving functions of
positions and momenta satisfying Hamilton equations of motion. In his doctoral
thesis[ 16 ], Feynman presented a very different way to discuss a quantum mechan-
ical system starting from a classical one described in terms of a principle of least
action, and not necessarily by Hamilton equations of motion. Such an approach
was shown to be particularly suited to quantize systems with infinite number of
degrees of freedom (field theories), also in a relativistic context^21.
The aim of this section is to give an introduction to Feynman’s approach[ 17 ]to
quantization, by means of the now-called path-integral technique. The idea is not
to concentrate on the evolution of states and/or of operators, but to look directly
at probabilities. Thus one tries to construct the kernel of the evolution operator:
K(x, x′;t)=〈x, t|x′〉=〈x|e−ıtH
|x′〉, (1.191)(^21) See also sect. 3.8.1 of the third part of this volume.