From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 203

where the functional derivativeδφδ is given by

δF

δφ

=

δF

δφ

−∂μ

[

δF

δ(∂μφ)

]

.

The Poisson brackets of fundamental fields are

{φ(x 1 ),φ(x 2 )}={π(x 1 ),π(x 2 )}=0,

{φ(x 1 ),π(x 2 )}=δ^3 (x 1 −x 2 ),

because of the basic rules of functional derivation

δφ(x 1 )

δφ(x 2 )

=δ^3 (x 1 −x 2 );

δπ(x 1 )

δπ(x 2 )

=δ^3 (x 1 −x 2 ).

The appearance of delta functions in the Poisson structure of the fields reflects
the fact that a mathematically sound analysis of field theory requires the use of
distributions. This will be even more necessary for the quantum fields. Thus, it is
convenient to consider smeared field functionals. To a classical field test function
f which might be more regular thatL^2 (R^3 ) fields (e.g. f∈S(R^3 ) for massive
fields, orf∈D(R^3 ) for massless fields) we can associate a smeared field defined
by the image of the linear functional

φ(f)=

∫

d^3 xf(x)φ(x); π(f)=

∫

d^3 xf(x)π(x)

inL^2 (R^3 ).
The Poisson structure can be expressed in terms of smeared fieldsφ(f)as

{φ(f 1 ),φ(f 2 )}={π(f 1 ),π(f 2 )}=0,

{φ(f 1 ),π(f 2 )}=(f 1 ,f 2 )

where (·,·) denotes the canonical Hilbert product ofL^2 (R^3 ).
By choosing an orthonormal Hilbert basis of test functionsfninL^2 (R^3 )we
can get a discrete representation of the Poisson structure,

{φn,φm}={φn,φm}=0; {φn,πm}=δmn

whereφn=φ(fn)andπn=π(fn).
In that representation, the Hamiltonian operator Eq. (3.17) becomes

H=

1

2

∑∞

n=0

πn^2 −

1

2

∑∞

n,m=0

Δmnφmφn+

1

2

m^2

∑∞

n=0

φn^2 , (3.18)

where

Δmn=(fm,Δfn)=(fm,∇^2 fn). (3.19)