master basic computational techniques, and one that to this day resists math-
ematician’s attempts to prove that many examples of such theories have even
the basic expected properties. In the quantum theory of charged particles inter-
acting with an electromagnetic field (see chapter 45), when the electromagnetic
field is treated classically one still has a Hamiltonian quadratic in the field oper-
ators for the particles. But if the electromagnetic field is treated as a quantum
system, it acquires its own field operators, and the Hamiltonian is no longer
quadratic in the fields but instead gives an interacting quantum field theory.
Even if one restricts attention to the quantum fields describing one kind
of particle, there may be interactions between particles that add terms to the
Hamiltonian that will be higher order than quadratic. For instance, if there is an
interaction between such particles described by an interaction energyv(y−x),
this can be described by adding the following quartic term to the Hamiltonian
1
2∫∞
−∞∫∞
−∞Ψ̂†(x)Ψ̂†(y)v(y−x)Ψ(̂y)Ψ(̂x)dxdyThe study of “many-body” quantum systems with interactions of this kind is a
major topic in condensed matter physics.
Digression(The Lagrangian density and the path integral). While we have
worked purely in the Hamiltonian formalism, another approach would have been
to start with an action for this system and use Lagrangian methods. An action
that will give the Schr ̈odinger equation as an Euler-Lagrange equation is
S=
∫∞
−∞∫∞
−∞(
iψ∂
∂tψ−h)
dxdt=
∫∞
−∞∫∞
−∞(
iψ∂
∂tψ+ψ1
2 m∂^2
∂x^2ψ)
dxdt=
∫∞
−∞∫∞
−∞(
iψ∂
∂tψ−1
2 m|
∂
∂xψ|^2)
dxdtwhere the last form comes by using integration by parts to get an alternate form
ofhas mentioned in section 37.2. In the Lagrangian approach to field theory,
the action is an integral over space and time of a Lagrangian density, which in
this case is
L(x,t) =iψ∂
∂t
ψ−1
2 m|
∂
∂x
ψ|^2Defining a canonical conjugate momentum forψas∂L
∂ψ ̇gives as momentumvariableiψ. This justifies the Poisson bracket relation
{Ψ(x),iΨ(x′)}=δ(x−x′)but, as expected for a case where the equation of motion is first-order in time,
the canonical momentum coordinateiΨ(x)is not independent of the coordinate
Ψ(x). The spaceH 1 of wavefunctions is already a phase space rather than