6
Quantum Field Theory II: Interacting Scalar
Fields
6.1 Interactions in quantum field theory: qualitative
introduction
In the previous chapter we considered only free – i.e. non-interacting – quantum fields. The fact that they are non-interacting is evident in a number of
ways. The mode expansions (5.129) and (5.155) are written in terms of the
(free) plane-wave solutions of the associated wave equations. Also the Hamiltonians turned out to be just the sum of individual oscillator Hamiltonians
for each mode frequency, as in (5.132) or (5.159). The energies of the quanta
add up – they are non-interacting quanta. Finally, since the Hamiltonians are
just sums of number operators
n ˆ(k) = ˆ
a
† (k)ˆ a(k)
(6.1)
it is obvious that each such operator commutes with the Hamiltonian and is
therefore a constant of the motion. Thus two waves, each with one excitation
quantum, travelling towards each other will pass smoothly through each other
and emerge unscathed on the other side – they will not interact at all.
How can we get the mode quanta to interact? If we return to our discussion of classical mechanical systems in section 5.1, we see that the crucial
step in arriving at the ‘sum over oscillators’ form for the energy was the assumption that the potential energy was quadratic in the small displacements
q r . We expect that ‘modes will interact’ when we go beyond this harmonic
approximation. The same is true in the continuous (wave or field) case. In the
derivation of the appropriate wave equation you will find that somewhere an
approximation like tan φ ≈ φ or sin φ ≈ φ is made. This linearizes the equation, and solutions to linear equations can be linearly superposed to make new
solutions. If we retain higher powers of φ, such as φ
3 , the resulting nonlinear
equation has solutions that cannot be obtained by superposing two independent solutions. Thus two waves travelling towards each other will not just
pass smoothly through each other: various forms of interaction and distortion
of the original waveforms will occur.
What happens when we quantize such anharmonic systems? To gain some
idea of the new features that emerge, consider just one ‘anharmonic oscillator’
149
Quantum Field Theory II: Interacting Scalar
Fields
6.1 Interactions in quantum field theory: qualitative
introduction
In the previous chapter we considered only free – i.e. non-interacting – quantum fields. The fact that they are non-interacting is evident in a number of
ways. The mode expansions (5.129) and (5.155) are written in terms of the
(free) plane-wave solutions of the associated wave equations. Also the Hamiltonians turned out to be just the sum of individual oscillator Hamiltonians
for each mode frequency, as in (5.132) or (5.159). The energies of the quanta
add up – they are non-interacting quanta. Finally, since the Hamiltonians are
just sums of number operators
n ˆ(k) = ˆ
a
† (k)ˆ a(k)
(6.1)
it is obvious that each such operator commutes with the Hamiltonian and is
therefore a constant of the motion. Thus two waves, each with one excitation
quantum, travelling towards each other will pass smoothly through each other
and emerge unscathed on the other side – they will not interact at all.
How can we get the mode quanta to interact? If we return to our discussion of classical mechanical systems in section 5.1, we see that the crucial
step in arriving at the ‘sum over oscillators’ form for the energy was the assumption that the potential energy was quadratic in the small displacements
q r . We expect that ‘modes will interact’ when we go beyond this harmonic
approximation. The same is true in the continuous (wave or field) case. In the
derivation of the appropriate wave equation you will find that somewhere an
approximation like tan φ ≈ φ or sin φ ≈ φ is made. This linearizes the equation, and solutions to linear equations can be linearly superposed to make new
solutions. If we retain higher powers of φ, such as φ
3 , the resulting nonlinear
equation has solutions that cannot be obtained by superposing two independent solutions. Thus two waves travelling towards each other will not just
pass smoothly through each other: various forms of interaction and distortion
of the original waveforms will occur.
What happens when we quantize such anharmonic systems? To gain some
idea of the new features that emerge, consider just one ‘anharmonic oscillator’
149