7
Quantum Field Theory III: Complex Scalar
Fields, Dirac and Maxwell Fields;
Introduction of Electromagnetic Interactions
In the previous two chapters we have introduced the formalism of relativistic
quantum field theory for the case of free real scalar fields obeying the Klein–
Gordon (KG) equation of section 3.1, extended it to describe interactions
between such quantum fields and shown how the Feynman rules for a simple
Yukawa-like theory are derived. It is now time to return to the unfortunately
rather more complicated real world of quarks and leptons interacting via gauge
fields – in particular electromagnetism. For this, several generalizations of the
formalism of chapter 5 are necessary.
First, a glance back at chapter 2 will remind the reader that the electromagnetic interaction has everything to do with the phase of wavefunctions,
and hence presumably of their quantum field generalizations: fields which are
real must be electromagnetically neutral. Indeed, as noted very briefly in
section 5.3, the quanta of a real scalar field are their own antiparticles; for
a given mass, there is only one type of particle being created or destroyed.
However, physical particles and antiparticles have identical masses (e.g. e
− and
e
+ ), and it is actually a deep result of quantum field theory that this is so (see
section 4.2.5, and the end of section 7.1). In this case for a given mass m, there
will have to be two distinct field degrees of freedom, one of which corresponds
somehow to the ‘particle’, the other to the ‘antiparticle’. This suggests that we
will need a complex field if we want to distinguish particle from antiparticle,
¯
even in the absence of electromagnetism (for example, the (K
0 , K
0 ) pair). Such
a distinction will have to be made in terms of some conserved quantum number
(or numbers), having opposite values for ‘particle’ and ‘antiparticle’. This
conserved quantum number must be associated with some symmetry. Now,
referring again to chapter 2, we recall that electromagnetism is associated with
invariance under local U(1) phase transformations. Even in the absence of
electromagnetism, however, a theory with complex fields can exhibit a global
U(1) phase invariance. As we shall show in section 7.1, such a symmetry
indeed leads to the existence of a conserved quantum number, in terms of
which we can distinguish the particle and antiparticle parts of a complex
scalar field.
In section 7.2 we generalize the complex scalar field to the complex spinor
(Dirac) field, suitable for charged spin1 particles. Again we find an analogous
2
183
Quantum Field Theory III: Complex Scalar
Fields, Dirac and Maxwell Fields;
Introduction of Electromagnetic Interactions
In the previous two chapters we have introduced the formalism of relativistic
quantum field theory for the case of free real scalar fields obeying the Klein–
Gordon (KG) equation of section 3.1, extended it to describe interactions
between such quantum fields and shown how the Feynman rules for a simple
Yukawa-like theory are derived. It is now time to return to the unfortunately
rather more complicated real world of quarks and leptons interacting via gauge
fields – in particular electromagnetism. For this, several generalizations of the
formalism of chapter 5 are necessary.
First, a glance back at chapter 2 will remind the reader that the electromagnetic interaction has everything to do with the phase of wavefunctions,
and hence presumably of their quantum field generalizations: fields which are
real must be electromagnetically neutral. Indeed, as noted very briefly in
section 5.3, the quanta of a real scalar field are their own antiparticles; for
a given mass, there is only one type of particle being created or destroyed.
However, physical particles and antiparticles have identical masses (e.g. e
− and
e
+ ), and it is actually a deep result of quantum field theory that this is so (see
section 4.2.5, and the end of section 7.1). In this case for a given mass m, there
will have to be two distinct field degrees of freedom, one of which corresponds
somehow to the ‘particle’, the other to the ‘antiparticle’. This suggests that we
will need a complex field if we want to distinguish particle from antiparticle,
¯
even in the absence of electromagnetism (for example, the (K
0 , K
0 ) pair). Such
a distinction will have to be made in terms of some conserved quantum number
(or numbers), having opposite values for ‘particle’ and ‘antiparticle’. This
conserved quantum number must be associated with some symmetry. Now,
referring again to chapter 2, we recall that electromagnetism is associated with
invariance under local U(1) phase transformations. Even in the absence of
electromagnetism, however, a theory with complex fields can exhibit a global
U(1) phase invariance. As we shall show in section 7.1, such a symmetry
indeed leads to the existence of a conserved quantum number, in terms of
which we can distinguish the particle and antiparticle parts of a complex
scalar field.
In section 7.2 we generalize the complex scalar field to the complex spinor
(Dirac) field, suitable for charged spin1 particles. Again we find an analogous
2
183