48
2. Electromagnetism as a Gauge Theory
invariance under a change of V by a constant can be extended to a local invariance (which is a much more restrictive condition to satisfy). Hence there
is a beginning of a suggestion that one might almost ‘derive’ the complete
Maxwell equations, which unify electricity and magnetism, from the requirement that the theory be expressed in terms of potentials in such a way as
to be invariant under local (gauge) transformations on those potentials. Certainly special relativity must play a role too: this also links electricity and
magnetism, via the magnetic effects of charges as seen by an observer moving
relative to them. If a 4-vector potential A
μ is postulated, and it is then demanded that the theory involve it only in a way which is insensitive to local
changes of the form (2.15), one is led naturally to the idea that the physical fields enter only via the quantity F
μν , which is invariant under (2.15).
From this, one might conjecture the field equation on grounds of Lorentz
covariance.
It goes without saying that this is certainly not a ‘proof’ or ‘derivation’ of
the Maxwell equations. Nevertheless, the idea that dynamics (in this case, the
complete interconnection of electric and magnetic effects) may be intimately
related to a local invariance requirement (in this case, electromagnetic gauge
invariance) turns out to be a fruitful one. As indicated in section 2.1, it is
generally the case that, when a certain global invariance is generalized to a
local one, the existence of a new ‘compensating’ field is entailed, interacting in
a specified way. The first example of dynamical theory ‘derived’ from a local
invariance requirement seems to be the theory of Yang and Mills (1954) (see
also Shaw 1955). Their work was extended by Utiyama (1956), who developed
a general formalism for such compensating fields. As we have said, these types
of dynamical theories, based on local invariance principles, are called gauge
theories.
It is a remarkable fact that the interactions in the Standard Model of particle physics are of precisely this type. We have briefly discussed the Maxwell
equations in this light, and we will continue with (quantum) electrodynamics in the following two sections. The two other fundamental interactions
– the strong interaction between quarks and the weak interaction between
quarks and leptons – also seem to be described by gauge theories (of essentially the Yang–Mills type), as we shall see in detail in the second volume of
this book. A fourth example, but one which we shall not pursue in this book,
is that of general relativity (the theory of gravitational interactions). Utiyama
(1956) showed that this theory could be arrived at by generalizing the global
(space–time independent) coordinate transformations of special relativity to
local ones; as with electromagnetism, the more restrictive local invariance
requirements entailed the existence of a new field – the gravitational one –
with an (almost) prescribed form of interaction. Unfortunately, despite this
‘gauge’ property, no consistent quantum field theory of general relativity is
known.
In order to proceed further, we must now discuss how such (gauge) ideas
are incorporated into quantum mechanics.
2. Electromagnetism as a Gauge Theory
invariance under a change of V by a constant can be extended to a local invariance (which is a much more restrictive condition to satisfy). Hence there
is a beginning of a suggestion that one might almost ‘derive’ the complete
Maxwell equations, which unify electricity and magnetism, from the requirement that the theory be expressed in terms of potentials in such a way as
to be invariant under local (gauge) transformations on those potentials. Certainly special relativity must play a role too: this also links electricity and
magnetism, via the magnetic effects of charges as seen by an observer moving
relative to them. If a 4-vector potential A
μ is postulated, and it is then demanded that the theory involve it only in a way which is insensitive to local
changes of the form (2.15), one is led naturally to the idea that the physical fields enter only via the quantity F
μν , which is invariant under (2.15).
From this, one might conjecture the field equation on grounds of Lorentz
covariance.
It goes without saying that this is certainly not a ‘proof’ or ‘derivation’ of
the Maxwell equations. Nevertheless, the idea that dynamics (in this case, the
complete interconnection of electric and magnetic effects) may be intimately
related to a local invariance requirement (in this case, electromagnetic gauge
invariance) turns out to be a fruitful one. As indicated in section 2.1, it is
generally the case that, when a certain global invariance is generalized to a
local one, the existence of a new ‘compensating’ field is entailed, interacting in
a specified way. The first example of dynamical theory ‘derived’ from a local
invariance requirement seems to be the theory of Yang and Mills (1954) (see
also Shaw 1955). Their work was extended by Utiyama (1956), who developed
a general formalism for such compensating fields. As we have said, these types
of dynamical theories, based on local invariance principles, are called gauge
theories.
It is a remarkable fact that the interactions in the Standard Model of particle physics are of precisely this type. We have briefly discussed the Maxwell
equations in this light, and we will continue with (quantum) electrodynamics in the following two sections. The two other fundamental interactions
– the strong interaction between quarks and the weak interaction between
quarks and leptons – also seem to be described by gauge theories (of essentially the Yang–Mills type), as we shall see in detail in the second volume of
this book. A fourth example, but one which we shall not pursue in this book,
is that of general relativity (the theory of gravitational interactions). Utiyama
(1956) showed that this theory could be arrived at by generalizing the global
(space–time independent) coordinate transformations of special relativity to
local ones; as with electromagnetism, the more restrictive local invariance
requirements entailed the existence of a new field – the gravitational one –
with an (almost) prescribed form of interaction. Unfortunately, despite this
‘gauge’ property, no consistent quantum field theory of general relativity is
known.
In order to proceed further, we must now discuss how such (gauge) ideas
are incorporated into quantum mechanics.