37.1 Quantum field operators
The multi-particle formalism developed in chapter 36 works well to describe
states of multiple free particles, but does so purely in terms of states with well-
defined momenta, with no information at all about their position. Instead of
starting witht= 0 momentum eigenstates|p〉and corresponding Schr ̈odinger
solutionsA(p) as continuous basis elements of the single-particle spaceH 1 , the
position operatorQeigenstates|x〉could be used. The solution that is such
an eigenstate att= 0 will be denoted Ψ(x), the conjugate solution will be
writtenΨ(x). The Ψ(x),Ψ(x) can be thought of as the complex coordinates of
an oscillator for each value ofx.
The corresponding quantum state space would naively be a Fock space for
an infinite number of degrees of freedom, with an occupation number for each
value ofx. This could be made well-defined by introducing a spatial cutoff and
discretizing space, so thatxonly takes on a finite number of values. However,
such states in the occupation number basis would not be free particle energy
eigenstates. While a state with a well-defined momentum evolves as a state
with the same momentum, a state with well-defined position at some time does
not evolve into states with well-defined positions (its wavefunction immediately
spreads out).
One does however want to be able to discuss states with well-defined posi-
tions, in order to describe amplitudes for particle propagation, and to introduce
local interactions between particles. One approach is to try and define opera-
tors corresponding to creation or annihilation of a particle at a fixed position,
by taking a Fourier transform of the annihilation and creation operators for
momentum eigenstates. Quantum fields could be defined as
Ψ(̂x) =√^1
2 π∫∞
−∞eipxa(p)dp (37.1)and its adjoint
Ψ̂†(x) =√^1
2 π∫∞
−∞e−ipxa†(p)dpNote that, just likea(p) anda†(p), these are not self-adjoint operators, and
thus not themselves observables, but physical observables can be constructed
by taking simple (typically quadratic) combinations of them. As explained in
section 36.5, for multi-particle systems the state spaceHis complicated to
describe and work with, it is the operators that behave simply. These will
generally be built out of either the field operatorsΨ(̂x),Ψ̂†(x), or annihilation
and creation operatorsa(p),a†(p), with the Fourier transform relating the two
possibilities.
In the continuum formalism the annihilation and creation operators satisfy
the distributional equation
[a(p),a†(p′)] =δ(p−p′)