Quantum Mechanics for Mathematicians

then equation 46.11 can be written

∂E

∂t

(t,x) =∇A×B

The problem with these equations is that they are non-linear equations inA,
so the phase space of solutions is no longer a linear space, and different methods
are needed for quantization of the theory.

46.3 Gauss’s law and time-independent gauge transformations

Returning to theU(1) case, a problem with the temporal gauge is that Gauss’s
law (equation 46.4) is not necessarily satisfied. At the same time, the group
G 0 ⊂ Gof time-independent gauge transformation will act non-trivially on the
phase space of initial data of Maxwell’s equations, preserving the temporal gauge
conditionA 0 = 0 (see equation 46.6). We will see that the condition of invari-
ance under this group action can be used to impose Gauss’s law.
It is a standard fact from the theory of electromagnetism that

∇·E=ρ(x)

is the generalization of Gauss’s law to the case of a background electric charge
densityρ(x). A failure of Gauss’s law can thus be interpreted physically as
due to the inclusion of states with background electric charge, rather than just
electromagnetic fields in the vacuum.
There are two different ways to deal with this kind of problem:

• Before quantization, impose Gauss’s law as a condition on the phase space.


• After quantization, impose Gauss’s law as a condition on the states, defin-
  ing the physical state spaceHphys⊂Has the subspace of states satisfying

∇·Ê|ψ〉= 0

whereÊis the quantized electric field.

To understand what happens if one tries to implement one of these choices,
consider first a much simpler example, that of a non-relativistic particle in 3
dimensions, with a potential that does not depend on one configuration variable,
sayq 3. The system has a symmetry under translations in the 3 direction, and
the condition
P 3 |ψ〉= 0 (46.12)

on states will commute with time evolution since [P 3 ,H] = 0. In the Schr ̈odinger
representation, since

P 3 =−i

∂

∂q 3