15
1.3. Particle interactions in the Standard Model
The final excited field state is defined by the presence of one quantum (photon)
of the appropriate energy.
We obviously cannot stop here (‘Electrons behave just like light’). All the
particles of the SM must be described as excitation quanta of the corresponding quantum fields. But of course Feynman was somewhat overstating the
case. The quanta of the electromagnetic field are bosons, and there is no limit
on the number of them that can occupy a single quantum state. By contrast,
the quanta of the electron field, for example, must be fermions, obeying the
exclusion principle. In chapter 7 we shall see what modifications to the quantization procedure this requires. We must also introduce interactions between
the excitation quanta, or equivalently between the quantum fields. This we
do in chapter 6 for bosonic fields, and in chapter 7 for the Dirac and Maxwell
fields thereby arriving at QED, our first quantum gauge field theory of the
SM.
One reason the Lagrangian formulation of classical field (or particle) physics
is so powerful is that symmetries can be efficiently incorporated, and their connection with conservation laws easily exhibited. The same is even more true
in qft. For example, only in qft can the symmetry corresponding to electric
charge conservation be simply understood. Indeed, all the quantum gauge
field theories of the SM are deeply related to symmetries, as will become clear
in the subsequent development.
In some cases, however, the symmetry – though manifest in the Lagrangian
– is not visible in the usual empirical ways (conservation laws, particle multiplets, and so on). Instead, it is ‘spontaneously (or dynamically) broken’. This
phenomenon plays a crucial role in both QCD and the GSW theory. An aid to
understanding it physically is provided by the analogy between the vacuum
state of an interacting qft and the ground state of an interacting quantum
many-body system – an insight due to Nambu (1960). We give an extended
discussion of spontaneously broken symmetry in Part VII of volume 2. We
shall see how the neutral bosonic (Bogoliubov) superfluid, and the charged
fermionic (BCS) superconductor, offer instructive working models of dynamical symmetry breaking, relevant to chiral symmetry breaking in QCD, and to
the generation of gauge boson masses in the GSW theory.
The road ahead is a long one, and we begin our journey at a more descriptive and pictorial level, making essential use of Yukawa’s remarkable insight
into the quantum nature of force. In due course, in chapter 6, we shall begin to see how qft supplies the precise mathematical formulae associated with
such pictures.
1.3.2 The Yukawa theory of force as virtual quantum
exchange
Yukawa’s revolutionary paper (Yukawa 1935) proposed a theory of the strong
interaction between a proton and a neutron, and also considered its possible
extension to neutron β-decay. He built his theory by analogy with electromag
1.3. Particle interactions in the Standard Model
The final excited field state is defined by the presence of one quantum (photon)
of the appropriate energy.
We obviously cannot stop here (‘Electrons behave just like light’). All the
particles of the SM must be described as excitation quanta of the corresponding quantum fields. But of course Feynman was somewhat overstating the
case. The quanta of the electromagnetic field are bosons, and there is no limit
on the number of them that can occupy a single quantum state. By contrast,
the quanta of the electron field, for example, must be fermions, obeying the
exclusion principle. In chapter 7 we shall see what modifications to the quantization procedure this requires. We must also introduce interactions between
the excitation quanta, or equivalently between the quantum fields. This we
do in chapter 6 for bosonic fields, and in chapter 7 for the Dirac and Maxwell
fields thereby arriving at QED, our first quantum gauge field theory of the
SM.
One reason the Lagrangian formulation of classical field (or particle) physics
is so powerful is that symmetries can be efficiently incorporated, and their connection with conservation laws easily exhibited. The same is even more true
in qft. For example, only in qft can the symmetry corresponding to electric
charge conservation be simply understood. Indeed, all the quantum gauge
field theories of the SM are deeply related to symmetries, as will become clear
in the subsequent development.
In some cases, however, the symmetry – though manifest in the Lagrangian
– is not visible in the usual empirical ways (conservation laws, particle multiplets, and so on). Instead, it is ‘spontaneously (or dynamically) broken’. This
phenomenon plays a crucial role in both QCD and the GSW theory. An aid to
understanding it physically is provided by the analogy between the vacuum
state of an interacting qft and the ground state of an interacting quantum
many-body system – an insight due to Nambu (1960). We give an extended
discussion of spontaneously broken symmetry in Part VII of volume 2. We
shall see how the neutral bosonic (Bogoliubov) superfluid, and the charged
fermionic (BCS) superconductor, offer instructive working models of dynamical symmetry breaking, relevant to chiral symmetry breaking in QCD, and to
the generation of gauge boson masses in the GSW theory.
The road ahead is a long one, and we begin our journey at a more descriptive and pictorial level, making essential use of Yukawa’s remarkable insight
into the quantum nature of force. In due course, in chapter 6, we shall begin to see how qft supplies the precise mathematical formulae associated with
such pictures.
1.3.2 The Yukawa theory of force as virtual quantum
exchange
Yukawa’s revolutionary paper (Yukawa 1935) proposed a theory of the strong
interaction between a proton and a neutron, and also considered its possible
extension to neutron β-decay. He built his theory by analogy with electromag