125
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
vibrations for this electron field? We do not answer this question just as we did
not for the photon. We postulate a relativistic quantum field for the electron
which obeys some suitable wave equation – in this case, for non-interacting
electrons, the Dirac equation. The field is expanded as a sum of Fourier
components, as with the electromagnetic field. Each component behaves as
an independent oscillator degree of freedom (and there are, of course, an
infinite number of them); the quanta of these oscillators are electrons.
Actually this, though correctly expressing the basic idea, omits one crucial
factor, which makes it almost fraudulently oversimplified. There is of course
one very big difference between photons and electrons. The former are bosons
and the latter are fermions; photons have spin angular momentum of one
(in unit of ħ), electrons of one-half. It is very difficult, if not downright
impossible, to construct any mechanical model at all which has fermionic
excitations. Phonons have spin-1, in fact, corresponding to the three states of
polarization of the corresponding vibrational waves. But ‘phonons’ carrying
spin1 are hard to come by. No matter, you may say, Maxwell has weaned
2
us away from jelly, so we shall be grown up and boldly postulate the electron
field as a basic thing.
Certainly this is what we do. But we also know that fermionic particles,
like electrons, have to obey an exclusion principle: no two identical fermions
can have the same quantum numbers. In chapter 7, we shall learn how the
idea sketched here must be modified for fields whose quanta are fermions.
5.2 The quantum field: (ii) Lagrange–Hamilton
formulation
5.2.1 The action principle: Lagrangian particle mechanics
We must now make the foregoing qualitative picture more mathematically
precise. It is clear that we would like a formalism capable of treating, within
a single overall framework, the mechanics of both fields and particles, in both
classical and quantum aspects. Remarkably enough, such a framework does
exist (and was developed long before quantum field theory): Hamilton’s principle of least action, with the action defined in terms of a Lagrangian. We
strongly recommend the reader with no prior acquaintance with this profound approach to physical laws read chapter 19 of volume 2 of Feynman’s
Lectures on Physics (Feynman 1964).
The least action approach differs radically from the more familiar one
which can conveniently be called ‘Newtonian’. Consider the simplest case,
that of classical particle mechanics. In the Newtonian approach, equations
of motion are postulated which involve forces as the essential physical input;
from these, the trajectories of the particle can be calculated. In the least
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
vibrations for this electron field? We do not answer this question just as we did
not for the photon. We postulate a relativistic quantum field for the electron
which obeys some suitable wave equation – in this case, for non-interacting
electrons, the Dirac equation. The field is expanded as a sum of Fourier
components, as with the electromagnetic field. Each component behaves as
an independent oscillator degree of freedom (and there are, of course, an
infinite number of them); the quanta of these oscillators are electrons.
Actually this, though correctly expressing the basic idea, omits one crucial
factor, which makes it almost fraudulently oversimplified. There is of course
one very big difference between photons and electrons. The former are bosons
and the latter are fermions; photons have spin angular momentum of one
(in unit of ħ), electrons of one-half. It is very difficult, if not downright
impossible, to construct any mechanical model at all which has fermionic
excitations. Phonons have spin-1, in fact, corresponding to the three states of
polarization of the corresponding vibrational waves. But ‘phonons’ carrying
spin1 are hard to come by. No matter, you may say, Maxwell has weaned
2
us away from jelly, so we shall be grown up and boldly postulate the electron
field as a basic thing.
Certainly this is what we do. But we also know that fermionic particles,
like electrons, have to obey an exclusion principle: no two identical fermions
can have the same quantum numbers. In chapter 7, we shall learn how the
idea sketched here must be modified for fields whose quanta are fermions.
5.2 The quantum field: (ii) Lagrange–Hamilton
formulation
5.2.1 The action principle: Lagrangian particle mechanics
We must now make the foregoing qualitative picture more mathematically
precise. It is clear that we would like a formalism capable of treating, within
a single overall framework, the mechanics of both fields and particles, in both
classical and quantum aspects. Remarkably enough, such a framework does
exist (and was developed long before quantum field theory): Hamilton’s principle of least action, with the action defined in terms of a Lagrangian. We
strongly recommend the reader with no prior acquaintance with this profound approach to physical laws read chapter 19 of volume 2 of Feynman’s
Lectures on Physics (Feynman 1964).
The least action approach differs radically from the more familiar one
which can conveniently be called ‘Newtonian’. Consider the simplest case,
that of classical particle mechanics. In the Newtonian approach, equations
of motion are postulated which involve forces as the essential physical input;
from these, the trajectories of the particle can be calculated. In the least