56
2. Electromagnetism as a Gauge Theory
via the replacement (2.44)
3 . As before, the method clearly generalizes to the
four-dimensional case.
2.6 Comments on the gauge principle in electromagnetism
Comment (i)
A properly sceptical reader may have detected an important sleight of hand in
the previous discussion. Where exactly did the electromagnetic charge appear
from? The trouble with our argument as so far presented is that we could
have defined fields A and V so that they coupled equally to all particles –
instead we smuggled in a factor q.
Actually we can do a bit better than this. We can use the fact that the
electromagnetic charge is absolutely conserved to claim that there can be no
quantum mechanical interference between states of different charge q. Hence
different phase changes are allowed within each ‘sector’ of definite q:
ψ
′ = exp(iqχ)ψ
(2.61)
let us say. When this becomes a local transformation, χ → χ(x, t), we shall
need to cancel a term q∇χ, which will imply the presence of a ‘−qA’ term,
as required. Note that such an argument is only possible for an absolutely
conserved quantum number q – otherwise we cannot split up the states of
the system into non-communicating sectors specified by different values of q.
Reversing this line of reasoning, a conservation law such as baryon number
conservation, with no related gauge field, would therefore now be suspected
of not being absolutely conserved.
We still have not tied down why q is the electromagnetic charge and not
some other absolutely conserved quantum number. A proper discussion of
the reasons for identifying A
μ with the electromagnetic potential and q with
the particle’s charge will be given in chapter 7 with the help of quantum field
theory.
Comment (ii)
Accepting these identifications, we note that the form of the interaction contains but one parameter, the electromagnetic charge q of the particle in question. It is the same whatever the type of particle with charge q, whether it
be lepton, hadron, nucleus, ion, atom, etc. Precisely this type of ‘universality’ is present in the weak couplings of quarks and leptons, as we shall see in
volume 2. This strongly suggests that some form of gauge principle must be
3 Actually the electromagnetic interaction is uniquely specified by this procedure only
1
for particles of spin-0 or 2 . The spin-1 case will be discussed in volume 2.
2. Electromagnetism as a Gauge Theory
via the replacement (2.44)
3 . As before, the method clearly generalizes to the
four-dimensional case.
2.6 Comments on the gauge principle in electromagnetism
Comment (i)
A properly sceptical reader may have detected an important sleight of hand in
the previous discussion. Where exactly did the electromagnetic charge appear
from? The trouble with our argument as so far presented is that we could
have defined fields A and V so that they coupled equally to all particles –
instead we smuggled in a factor q.
Actually we can do a bit better than this. We can use the fact that the
electromagnetic charge is absolutely conserved to claim that there can be no
quantum mechanical interference between states of different charge q. Hence
different phase changes are allowed within each ‘sector’ of definite q:
ψ
′ = exp(iqχ)ψ
(2.61)
let us say. When this becomes a local transformation, χ → χ(x, t), we shall
need to cancel a term q∇χ, which will imply the presence of a ‘−qA’ term,
as required. Note that such an argument is only possible for an absolutely
conserved quantum number q – otherwise we cannot split up the states of
the system into non-communicating sectors specified by different values of q.
Reversing this line of reasoning, a conservation law such as baryon number
conservation, with no related gauge field, would therefore now be suspected
of not being absolutely conserved.
We still have not tied down why q is the electromagnetic charge and not
some other absolutely conserved quantum number. A proper discussion of
the reasons for identifying A
μ with the electromagnetic potential and q with
the particle’s charge will be given in chapter 7 with the help of quantum field
theory.
Comment (ii)
Accepting these identifications, we note that the form of the interaction contains but one parameter, the electromagnetic charge q of the particle in question. It is the same whatever the type of particle with charge q, whether it
be lepton, hadron, nucleus, ion, atom, etc. Precisely this type of ‘universality’ is present in the weak couplings of quarks and leptons, as we shall see in
volume 2. This strongly suggests that some form of gauge principle must be
3 Actually the electromagnetic interaction is uniquely specified by this procedure only
1
for particles of spin-0 or 2 . The spin-1 case will be discussed in volume 2.