Chapter 37

Multi-particle Systems and

Field Quantization

The multi-particle formalism developed in chapter 36 is based on the idea of
taking as dual phase space the space of solutions to the free particle Schr ̈odinger
equation, then quantizing using the Bargmann-Fock method. Continuous basis
elementsA(p),A(p) are momentum operator eigenstates which after quantiza-
tion become creation and annihilation operatorsa†(p),a(p). These act on the
multi-particle state space by adding or subtracting a free particle of momentum
p.
Instead of the solutionsA(p), which are momentum space delta-functions
localized atpatt= 0, one could use solutions Ψ(x) that are position space
delta-functions localized atxatt= 0. Quantization using these basis elements
ofH 1 (and conjugate basis elements ofH 1 ) will give operatorsΨ̂†(x) andΨ(̂x),
which are the quantum field operators. These are related to the operators
a†(p),a(p) by the Fourier transform.
The operatorsΨ(̂x) andΨ̂†(x) can be given a physical interpretation as act-
ing on states by subtracting or adding a particle localized atx. Such states with
localized position have the same problem of non-normalizability as momentum
eigenstates. In addition, unlike states with fixed momentum, they are not sta-
ble energy eigenstates so will immediately evolve into something non-localized
(just as in the single-particle case discussed in section 12.5). Quantum fields are
however very useful in the study of theories of interacting particles, since the
interactions in such theories are typically local, taking place at a pointxand
describable by adding terms to the Hamiltonian operator involving multiplying
field operators at the same pointx. The difficulties involved in properly defining
products of these operators and calculating their dynamical effects will keep us
from starting the study of such interacting theories.