From Classical Mechanics to Quantum Field Theory

236 From Classical Mechanics to Quantum Field Theory. A Tutorial

In the case of the one-dimensional harmonic oscillator, the Poisson bracket of
position operator is

{x(t 1 ),x(t 2 )}=

m

ω

sinω(t 2 −t 1 ).

The construction of Peierls brackets depends on the LagrangianL 0 of the
theory. In this sense it is not as universal as the Poisson brackets induced by the
symplectic structure ofT∗M.
The generalization to field theory is straightforward and the result for a free
scalar theory is the following. In the case of the scalar field, a general solution of
the field equations

(∂μ∂μ+m^2 )φ(x)=0

can be obtained via its Fourier transform

(−k^2 +m^2 )φ ̃(k)=0,

whose general solution can be written asφ ̃(p)=2πˆα(k)δ(k^2 −m^2 ), where ˆα(k)
is an arbitrary function ofkμ. The solution in position space obtained by inverse
Fourier transform is

φ(x)=

∫

d^4 k

(2π)^4

(2π)δ(k^2 −m^2 )θ(k^0 )

[

α(k)e−ik·x+α(k)∗eik·x

]

=

∫

d^3 k

(2π)^3

1

2 ωk

[

α(k)e−iωkt+k·x+α(k)∗eiωkt−k·x

]

The corresponding Peierls bracket is given by

{φ(x),φ(y)}=Δ(x−y),

where Δ(x−y) is the causal propagator 3.49. At equal timesx 0 =y 0 the Peierls
bracket reduce to the Poisson bracket, and vanishes.
In this framework the covariant quantization rule is a generalization of the
canonical one: to replacethe Peierls brackets by operator commutators.

Bibliography.................................

[1]L.Alvarez-Gaum ́e and M. A. V ́azquez-Mozo,An Invitation to Quantum Field Theory,
Springer (2011)
[2]M.Asorey,Conformal Invariance in Quantum Field Theory and Statistical Mechanics,
Forts. Phys. 40 (1992) 92
[3]M.Asorey,A.IbortandG.Marmo,Global theory of quantum boundary conditions
and topology change,Int.J.Mod.Phys.A20, (2005) 1001
[4] T. Banks,Modern Quantum Field Theory, Cambridge U. Press (2008)
[5]J.D.BjorkenandS.D.Drell,Relativistic Quantum Fields, McGraw-Hill (1965)