From Classical Mechanics to Quantum Field Theory

202 From Classical Mechanics to Quantum Field Theory. A Tutorial

The quantization is usually formulated in the Hamiltonian formalism. Thus,
it is necessary to start from the classical canonical formalism. LetMbe the
configuration space of square integrable classical fields at any fixed time (e.g.
t=0),

M=

{

φ(x)=φ(x,0);‖φ‖^2 =

∫

d^3 x|φ(x,0)|^2 <∞

}

. (3.15)

The Legendre transformation maps the tangent spaceTMinto the cotangent
spaceT∗M, fixing the value of the canonical momentum

π=

δL

δφ ̇

=φ, ̇

from the Lagrangian

L=

∫

d^3 x

(

1

2

φ ̇^2 −^1

2

(∇φ)^2 −V(φ)

)

,

whereφ ̇=∂tφ. The corresponding Hamiltonian is given by

H=

∫

d^3 x

(

1

2

π^2 +

1

2

(∇φ)^2 +V(φ)

)

. (3.16)

In the case of a free massive theory with massm,V(φ)=^12 m^2 φ^2 and the Hamil-
tonian reads

H=

1

2

(

‖π‖^2 +‖∇φ‖^2 +m^2 ‖φ‖^2

)

, (3.17)

where we have used theL^2 (R^3 ) norm introduced in Eq. (3.15).
In this case the classical vacuum solution is uniqueφ= 0. However, in the
massless casem= 0 the vacuum is degenerated, because any constant configura-
tionφ= cte is a solution with finite energy, although such configurations in the
massive case have infinite energy.
The symplectic structure ofT∗M

ω=

∫

d^3 xδπ∧δφ

induces a Poisson structure in the space of functionals ofT∗M.Giventwolocal
functionalsF(φ, π),G(φ, π) of the canonical variables of the form

F(φ, π)=

∫

d^3 xF(φ, π), G(φ, π)=

∫

d^3 xG(φ, π).

Their Poisson bracket is defined by

{F,G}≡

∫

d^3 x

[

δF

δφ

δG

δπ

−

δF

δπ

δG

δφ

]

,