Chapter 36

Multi-particle Systems:

Momentum Space

Description

In chapter 9 we saw how to use symmetric or antisymmetric tensor products to
describe a fixed number of identical quantum systems (for instance, free parti-
cles). From very early on in the history of quantum mechanics, it became clear
that at least certain kinds of quantum particles, photons, required a formalism
that could describe arbitrary numbers of particles, as well as phenomena involv-
ing their creation and annihilation. This could be accomplished by thinking of
photons as quantized excitations of a classical electromagnetic field. In our
modern understanding of fundamental physics all elementary particles, not just
photons, are best described in this way, by quantum theories of fields. For free
particles the necessary theory can be understood as the quantum theory of the
harmonic oscillator, but with an infinite number of degrees of freedom, one for
each possible value of the momentum (or, Fourier transforming, each possible
value of the position). The symmetric (bosons) or antisymmetric (fermions)
nature of multi-particle quantum states is automatic in such a description as
quanta of oscillators.
Conventional textbooks on quantum field theory often begin with relativistic
systems, but we’ll start instead with the non-relativistic case. This is signifi-
cantly simpler, lacking the phenomenon of antiparticles that appears in the
relativistic case. It is also the case of relevance to condensed matter physics,
and applies equally well to bosonic or fermionic particles.
Quantum field theory is a large and complicated subject, suitable for a full-
year course at an advanced level. We’ll be giving only a very basic introduc-
tion, mostly just considering free fields, which correspond to systems of non-
interacting particles. Much of the complexity of the subject only appears when
one tries to construct quantum field theories of interacting particles.
For simplicity we’ll start with the case of a single spatial dimension. We’ll