124
5. Quantum Field Theory I: The Free Scalar Field
written as a sum of individual mode energies. We note that the Fourier
amplitude A r acts as a normal coordinate. Comparing (5.36) with (5.27), we
see that the string behaves exactly like a system of independent uncoupled
oscillators, the only difference being that now there are an infinite number
of them, corresponding to the infinite number of degrees of freedom in the
continuous field φ(x, t). The normal coordinates A r (t) are, for many purposes,
a much more relevant set of degrees of freedom than the original displacements
φ(x, t).
The final step is to apply quantum mechanics to this classical field system. Once again, the total energy is equivalent to that of a sum of (infinitely
many) mode oscillators, each of which has to be quantized. The total energy
eigenvalue has the form (5.29), except that now the sum extends to infinity:
∞
∑
1
E =
(n r + )ħω r .
(5.37)
2
r=1
The excited states of the quantized field φ ˆ (x, t) are characterized by saying
how many phonons of each frequency are present; the ground state has no
phonons at all. We remark that as e → ∞, the mode sum in (5.36) or (5.37)
will be replaced by an integral over a continuous frequency variable.
We have now completed, in outline, the programme introduced earlier,
ending up with the quantization of a ‘mechanical’ system. All of the foregoing, it must be clearly emphasized, is absolutely basic to modern solid state
physics. The essential idea – quantizing independent modes – can be applied to an enormous variety of ‘oscillations’. In all cases the crucial concept
is the elementary excitation – the mode quantum. Thus we have plasmons
(quanta of plasma oscillations), magnons (magnetic oscillations), . . . , as well
as phonons (vibrational oscillations). All this is securely anchored in the
physics of many-body systems.
Now we come to the use of these ideas as an analogy, to help us understand
the (presumably non-mechanical) quantum fields with which we shall actually
be concerned in this book – for example the electromagnetic field. Consider a
region of space containing electromagnetic fields. These fields obey (a threedimensional version of) the wave equation (5.30), with c now standing for
the speed of light. By imposing suitable boundary conditions, the total electromagnetic energy in any region of space can be written as a sum of mode
energies. Each mode has the form of an oscillator, whose amplitude is (see
(5.31)) the Fourier component of the wave, for a given wavelength. These
oscillators are all quantized. Their quanta are called photons. Thus, a photon
is an elementary quantum of excitation of the electromagnetic field.
So far the only kind of ‘particle’ we have in our relativistic quantum field
theoretic world is the photon. What about the electron, say? Well, recalling
Feynman again, ‘There is one lucky break, however – electrons behave just
like light’. In other words, we shall also regard an electron as an elementary
quantum of excitation of an ‘electron field’. What is ‘waving’ to supply the
5. Quantum Field Theory I: The Free Scalar Field
written as a sum of individual mode energies. We note that the Fourier
amplitude A r acts as a normal coordinate. Comparing (5.36) with (5.27), we
see that the string behaves exactly like a system of independent uncoupled
oscillators, the only difference being that now there are an infinite number
of them, corresponding to the infinite number of degrees of freedom in the
continuous field φ(x, t). The normal coordinates A r (t) are, for many purposes,
a much more relevant set of degrees of freedom than the original displacements
φ(x, t).
The final step is to apply quantum mechanics to this classical field system. Once again, the total energy is equivalent to that of a sum of (infinitely
many) mode oscillators, each of which has to be quantized. The total energy
eigenvalue has the form (5.29), except that now the sum extends to infinity:
∞
∑
1
E =
(n r + )ħω r .
(5.37)
2
r=1
The excited states of the quantized field φ ˆ (x, t) are characterized by saying
how many phonons of each frequency are present; the ground state has no
phonons at all. We remark that as e → ∞, the mode sum in (5.36) or (5.37)
will be replaced by an integral over a continuous frequency variable.
We have now completed, in outline, the programme introduced earlier,
ending up with the quantization of a ‘mechanical’ system. All of the foregoing, it must be clearly emphasized, is absolutely basic to modern solid state
physics. The essential idea – quantizing independent modes – can be applied to an enormous variety of ‘oscillations’. In all cases the crucial concept
is the elementary excitation – the mode quantum. Thus we have plasmons
(quanta of plasma oscillations), magnons (magnetic oscillations), . . . , as well
as phonons (vibrational oscillations). All this is securely anchored in the
physics of many-body systems.
Now we come to the use of these ideas as an analogy, to help us understand
the (presumably non-mechanical) quantum fields with which we shall actually
be concerned in this book – for example the electromagnetic field. Consider a
region of space containing electromagnetic fields. These fields obey (a threedimensional version of) the wave equation (5.30), with c now standing for
the speed of light. By imposing suitable boundary conditions, the total electromagnetic energy in any region of space can be written as a sum of mode
energies. Each mode has the form of an oscillator, whose amplitude is (see
(5.31)) the Fourier component of the wave, for a given wavelength. These
oscillators are all quantized. Their quanta are called photons. Thus, a photon
is an elementary quantum of excitation of the electromagnetic field.
So far the only kind of ‘particle’ we have in our relativistic quantum field
theoretic world is the photon. What about the electron, say? Well, recalling
Feynman again, ‘There is one lucky break, however – electrons behave just
like light’. In other words, we shall also regard an electron as an elementary
quantum of excitation of an ‘electron field’. What is ‘waving’ to supply the