Subatomic Physics

204 Additive Conservation Laws
Equation (7.16) is an example of a conservation law. We have stated in the
introduction that each conservation law is related to a corresponding symmetry
principle. What is the symmetry principle that gives rise to the conservation of the
electric charge? To answer this question, we repeat the arguments of Section 7.1
specifically for electric charge conservation. While reading the following derivation,
it is a good idea to follow the more general steps in Section 7.1 in parallel. Assume
that ψ describes a state with charge q and that it satisfies a Schrödinger equation,
Eq. (7.1):
i� dψ
= Hψ. (7.19)
dt
If Q is the charge operator, we know from Eqs. (7.5) and (7.6) that 〈Q〉 is conserved
if H and Q commute. ψ then can also be chosen to be an eigenfunction of Q,
Qψ = qψ, (7.20)
and the eigenvalue q is also conserved. What symmetry guarantees that H and
Q commute? The answer to this question was given by Weyl (5) who considered a
transformation of the type of Eq. (7.12):
ψ ′ = e iɛQ ψ (7.21)
where ɛ is an arbitrary real parameter and Q the charge operator. The transformation
is called a “global” gauge transformation, (6) since it is independent of space and
time coordinates. Gauge invariance means that ψ ′ satisfies the same Schrödinger
equation as does ψ:
or
i� dψ′
dt
= Hψ′
i� d
dt (eiɛQ ψ)=He iɛQ ψ.
Multiplying from the left with exp(−iɛQ), noting that Q is a time-independent and
Hermitian operator, and comparing with Eq. (7.19) give
e −iɛQ He iɛQ = H. (7.22)
Since ɛ is an arbitrary parameter, it can be taken to be so small that ɛQ ≪ 1.
Expanding the exponential yields
(1 − iɛQ)H(1 + iɛQ) =H
5H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York, 1950, pp. 100,
214.
6The word “gauge” stems from a translation of Hermann Weyl’s first introduction of the subject
in 1919 as a scale invariance; H. Weyl, Ann. Physik 59, 101 (1919). The idea lay dormant for
about forty years because Weyl’s use of it was shown to be incorrect.