From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 211

3.5 FieldsversusParticles

We have assumed that the field operators act on a separable Hilbert spaceHwith-
out any special properties. In canonical quantization, we assumed that the space
of quantum states is given by functionals of the configuration space of square in-
tegrable classical fieldsMEq. (3.15). However, such a space has not a canonical
translation invariant Hilbert product. The reason being that in quantum mechan-
ics the equivalent space of functions of the configuration space is given by the space
of square integrable functionsL^2 with the standardL^2 -product with respect to the
Lebesgue measurednxofRn. However in infinite dimensional Hilbert spaces, the
equivalent Lebesgue measure is not well defined because the basic building blocks
of hypercubes of sizeLhave an infinite volume ifL>1 or zero volume ifL<1.
Thus, although all operators: the fieldsφ(f), the HamiltonianHˆand the momen-
tum operatorPˆare formally selfadjoint with respect to the naive generalization of
Lebesgue measureδφ, the definition of the quantum field theory requires a rigorous
definition of the Hilbert product and a redefinition of the physical observables.
The key ingredient is that the naive vacuum state Eq. (3.30) defines a good
measure in the space of functionals on space of classical fieldsM. Indeed, the
measure defined by

δμ(φ)=Ne−(φ,

√−∇ (^2) +m (^2) φ)
δφ=

∏

n∈Z^3

√

ωn

πe

−ωnφ^2 nδφn, (3.44)

is a well-defined probability measure. In Eq. (3.44),Ndenotes the normalization
factor which guarantees that the volume of the whole configuration space is unit.
According to Minlos’ theorem[ 12 ](see Appendix B) the Gaussian measureδμis
supported on the space of tempered distributionsS′(R^3 ) in the massive case and
on the space of generalized distributionsD′in the massless case.
The above definition requires a redefinition of all physical states and operators
by a similarity transformation

Ψ(φ)⇒e

(^12) (φ,√−∇ (^2) +m (^2) φ)
Ψ(φ); O⇒e
(^12) (φ,√−∇ (^2) +m (^2) φ)
Oe−
(^12) (φ,√−∇ (^2) +m (^2) φ)
.
The field operatorφ(f) remains unchanged whereas the canonical momentum
operator ˆπ(f) becomes
πˆ(f)=−i

∫

d^3 xf(x)

(

δ

δφ(x)−

√

−∇^2 +m^2 φ(x)

)

,

and both are selfadjoint with respect to the Hilbert product

(F,G)=

∫

S′(R^3 )

δμ(φ)F(φ)∗G(φ)

ofH=L^2 (S′(R^3 ),δμ).