151
6.1. Interactions in quantum field theory: qualitative introduction
What about the quantum mechanics of two coupled nonlinear oscillators?
In the same way, the general state is assumed to be a superposition
∑
|r ¯> =
c r,n1n2 |n 1 >|n 2 >
(6.11)
n1 ,n2
of states of arbitrary numbers of quanta of the unperturbed oscillator Hamiltonians H ˆ 0(1) and H ˆ 0(2) . States of the unperturbed system contain definite
numbers n 1 and n 2 , say, of the ‘1’ and ‘2’ quanta. Perturbation calculations of
the interacting system will involve matrix elements connecting such |n 1 >|n 2 >
′
′
states to states |n 1 >|n 2 > with different numbers of these quanta.
All this can be summarized by the remark that the typical feature of
quantized interacting modes is that we need to consider processes in which
the numbers of the different mode quanta are not constants of the motion.
This is, of course, exactly what happens when we have collisions between
high-energy particles. When far apart the particles, definite in number, are
indeed free and are just the mode quanta of some quantized fields. But, when
they interact, we must expect to see changes in the numbers of quanta, and
can envisage processes in which the number of quanta which emerge finally
as free particles is different from the number that originally collided. From
the quantum mechanical examples we have discussed, we expect that these
interactions will be produced by terms like φ ˆ3 or φ ˆ4 , since the free – ‘harmonic’
2
– case has φ ˆ2 , analogous to ˆ
q in the quantum mechanics example. Such
terms arise in the solid state phonon application precisely from anharmonic
corrections involving the atomic displacements. These terms lead to nontrivial phonon–phonon scattering, the treatment of which forms the basis of
the quantum theory of thermal resistivity of insulators. In the quantum field
theory case, when we have generalized the formalism to fermions and photons,
the nonlinear interaction terms will produce e
+ e
− scattering, q¯ q annihilation
and so on. As in the quantum mechanical case, the basic calculational method
will be perturbation theory.
As remarked earlier, the trouble with all these ‘real-life’ cases is that they
involve significant complications due to spin; the corresponding fields then
have several components, with attendant complexity in the solutions of the
associated free-particle wave equations (Maxwell, Dirac). So in this chapter
we shall seek to explain the essence of the perturbative approach to quantum
field dynamics – which we take to be essentially the Feynman graph version
of Yukawa’s exchange mechanism – in the context of simple models involving
only scalar fields; Maxwell (vector) and Dirac (spinor) fields will be introduced
in the following chapter. The route we follow to the ‘Feynman rules’ is the one
first given (with remarkable clarity) by Dyson (1949a), which rapidly became
the standard formulation.
Before proceeding it may be worth emphasizing that in introducing a ‘nonharmonic’ term such as φ ˆ3 and thus departing from linearity in that sense,
we are in no way affecting the basic linearity of state vector superposition in
quantum mechanics (cf (6.11)), which continues to hold.
6.1. Interactions in quantum field theory: qualitative introduction
What about the quantum mechanics of two coupled nonlinear oscillators?
In the same way, the general state is assumed to be a superposition
∑
|r ¯> =
c r,n1n2 |n 1 >|n 2 >
(6.11)
n1 ,n2
of states of arbitrary numbers of quanta of the unperturbed oscillator Hamiltonians H ˆ 0(1) and H ˆ 0(2) . States of the unperturbed system contain definite
numbers n 1 and n 2 , say, of the ‘1’ and ‘2’ quanta. Perturbation calculations of
the interacting system will involve matrix elements connecting such |n 1 >|n 2 >
′
′
states to states |n 1 >|n 2 > with different numbers of these quanta.
All this can be summarized by the remark that the typical feature of
quantized interacting modes is that we need to consider processes in which
the numbers of the different mode quanta are not constants of the motion.
This is, of course, exactly what happens when we have collisions between
high-energy particles. When far apart the particles, definite in number, are
indeed free and are just the mode quanta of some quantized fields. But, when
they interact, we must expect to see changes in the numbers of quanta, and
can envisage processes in which the number of quanta which emerge finally
as free particles is different from the number that originally collided. From
the quantum mechanical examples we have discussed, we expect that these
interactions will be produced by terms like φ ˆ3 or φ ˆ4 , since the free – ‘harmonic’
2
– case has φ ˆ2 , analogous to ˆ
q in the quantum mechanics example. Such
terms arise in the solid state phonon application precisely from anharmonic
corrections involving the atomic displacements. These terms lead to nontrivial phonon–phonon scattering, the treatment of which forms the basis of
the quantum theory of thermal resistivity of insulators. In the quantum field
theory case, when we have generalized the formalism to fermions and photons,
the nonlinear interaction terms will produce e
+ e
− scattering, q¯ q annihilation
and so on. As in the quantum mechanical case, the basic calculational method
will be perturbation theory.
As remarked earlier, the trouble with all these ‘real-life’ cases is that they
involve significant complications due to spin; the corresponding fields then
have several components, with attendant complexity in the solutions of the
associated free-particle wave equations (Maxwell, Dirac). So in this chapter
we shall seek to explain the essence of the perturbative approach to quantum
field dynamics – which we take to be essentially the Feynman graph version
of Yukawa’s exchange mechanism – in the context of simple models involving
only scalar fields; Maxwell (vector) and Dirac (spinor) fields will be introduced
in the following chapter. The route we follow to the ‘Feynman rules’ is the one
first given (with remarkable clarity) by Dyson (1949a), which rapidly became
the standard formulation.
Before proceeding it may be worth emphasizing that in introducing a ‘nonharmonic’ term such as φ ˆ3 and thus departing from linearity in that sense,
we are in no way affecting the basic linearity of state vector superposition in
quantum mechanics (cf (6.11)), which continues to hold.