From Classical Mechanics to Quantum Field Theory

234 From Classical Mechanics to Quantum Field Theory. A Tutorial

functions is the topologicaldualof the space of distributions where the measure is
supported.
LetS(Rn) be the space of fast decreasing smoothC∞(Rn) functions andCa
positive, symmetric, bounded operator inS(Rn). The Minlos theorem states that
there is a unique Borelian Gaussian measure withCcovariance in the space of
tempered distributionsS′(Rn), which is the topological dual ofS(Rn). The same
holds for the space of smooth functions of compact supportD(Rn) and its dual
D′(Rn), the space of generalized distributions.
In the first case the generating function

GC(f)=〈ei(f,g)〉C=

∫

S′(Rn,C)

dμC(g)ei(f,g)=e−

(^12) (f,Cf)
,
is defined only for test functionsf∈S(Rn), whereas in the second case it is only
defined for functions of compact supportD(Rn).
There are two special Gaussian measures which arise in quantum field theory.
One is defined by the covariance operator
C 0 =(−∇^2 +m^2 )−
(^12)
inR^3 that corresponds to the measure defined by the ground state of a free bosonic
field theory. The second one is defined by the covariance operators
CE=(−Δ+m^2 )−^1
inR^4 , which corresponds to the measure given by the Euclidean functional in-
tegral of a free bosonic theory. In physical termsCEis known as the Euclidean
propagator of a scalar field.

3.12 Appendix3.Peierlsbrackets

There is an alternative canonical approach to classical dynamics developed by R.
Peierls which provides a relativistic covariant description of classical field theory.
The standard approach uses the Poisson structure based on equal time com-
mutators Eq. (3.6) Eq. (3.18). However, in relativistic theories the simultaneity of
space-like separated points is not a relativistic invariant notion. For such a reason
R. Peierls introduced an equivalent dynamical approach which explicitly preserves
relativistic covariance.
The phase space in classical mechanicsT∗Mcontains all Cauchy data (x, p)∈
T∗Mand a canonical symplectic structure

ω 0 =

∑n

i=1

dxi∧dpi