120
5. Quantum Field Theory I: The Free Scalar Field
This equation shows that, when written in terms of the normal coordinates,
the total energy contains no couplings terms of the form Q 1 Q 2 ; indeed, the
energy has the remarkable form of a simple sum of two independent uncoupled
oscillators, one with characteristic frequency ω 1 , the other with frequency ω 2 .
The energy (5.23) has exactly the form appropriate to a system of two noninteracting ‘things’, each executing simple harmonic motion: the ‘things’ are
actually the two modes. Modes do not interact, whereas the original atoms do!
Of course, this decoupling in the expression for the total energy is reflected in
the decoupling of the equations of motion for the Q variables:
∂V (Q 1 , Q 2 )
¨
mQ r = −
r = 1, 2.
(5.24)
∂Q r
It is most important to realize that the modes are non-interacting by virtue
of the fact that we ignored higher than quadratic terms in V (q 1 , q 2 ). Although
the simple change of variables (q 1 , q 2 ) → (Q 1 , Q 2 ) of (5.12) does remove the
q 1 q 2 coupling, this would not be the case if, say, cubic terms in V were to
be considered. Such higher order ‘anharmonic’ corrections would produce
couplings between the modes – indeed, this will be the basis of the quantum
field theory description of particle interactions (see the following chapter)!
The system under discussion had just two degrees of freedom. We began
by describing it in terms of the obvious degree of freedom, the physical displacements of the two atoms q 1 and q 2 . But we have learned that it is very
illuminating to describe it in terms of the normal coordinate combinations
Q 1 and Q 2 . The normal coordinates are really the relevant degrees of freedom. Of course, for just two particles, the choice between the q r ’s and the
Q r ’s may seem rather academic; but the important point – and the reason
for going through these simple manipulations in detail – is that the basic idea
of the normal mode, and of normal coordinates, generalizes immediately to
the much less trivial N -atom problem (and also to the field problem). For N
atoms there are (for one-dimensional displacements) N degrees of freedom,
and if we take them to be the actual atomic displacements, the total energy
will be
N
∑
E =
1 mq˙
2 + V (q 1 , . . . , q r )
(5.25)
2
r
r=1
which includes all the couplings between atoms. We assume, as before, that
the q r ’s are small enough so that only quadratic terms need to be kept in V (a
constant is as usual irrelevant, and the linear terms vanish if the q r ’s are the
displacements from equilibrium). In this case, the equations of motion will be
linear. By a linear transformation of the form (generalizing (5.12))
N
∑
Q r =
a rs q s
(5.26)
s=1
5. Quantum Field Theory I: The Free Scalar Field
This equation shows that, when written in terms of the normal coordinates,
the total energy contains no couplings terms of the form Q 1 Q 2 ; indeed, the
energy has the remarkable form of a simple sum of two independent uncoupled
oscillators, one with characteristic frequency ω 1 , the other with frequency ω 2 .
The energy (5.23) has exactly the form appropriate to a system of two noninteracting ‘things’, each executing simple harmonic motion: the ‘things’ are
actually the two modes. Modes do not interact, whereas the original atoms do!
Of course, this decoupling in the expression for the total energy is reflected in
the decoupling of the equations of motion for the Q variables:
∂V (Q 1 , Q 2 )
¨
mQ r = −
r = 1, 2.
(5.24)
∂Q r
It is most important to realize that the modes are non-interacting by virtue
of the fact that we ignored higher than quadratic terms in V (q 1 , q 2 ). Although
the simple change of variables (q 1 , q 2 ) → (Q 1 , Q 2 ) of (5.12) does remove the
q 1 q 2 coupling, this would not be the case if, say, cubic terms in V were to
be considered. Such higher order ‘anharmonic’ corrections would produce
couplings between the modes – indeed, this will be the basis of the quantum
field theory description of particle interactions (see the following chapter)!
The system under discussion had just two degrees of freedom. We began
by describing it in terms of the obvious degree of freedom, the physical displacements of the two atoms q 1 and q 2 . But we have learned that it is very
illuminating to describe it in terms of the normal coordinate combinations
Q 1 and Q 2 . The normal coordinates are really the relevant degrees of freedom. Of course, for just two particles, the choice between the q r ’s and the
Q r ’s may seem rather academic; but the important point – and the reason
for going through these simple manipulations in detail – is that the basic idea
of the normal mode, and of normal coordinates, generalizes immediately to
the much less trivial N -atom problem (and also to the field problem). For N
atoms there are (for one-dimensional displacements) N degrees of freedom,
and if we take them to be the actual atomic displacements, the total energy
will be
N
∑
E =
1 mq˙
2 + V (q 1 , . . . , q r )
(5.25)
2
r
r=1
which includes all the couplings between atoms. We assume, as before, that
the q r ’s are small enough so that only quadratic terms need to be kept in V (a
constant is as usual irrelevant, and the linear terms vanish if the q r ’s are the
displacements from equilibrium). In this case, the equations of motion will be
linear. By a linear transformation of the form (generalizing (5.12))
N
∑
Q r =
a rs q s
(5.26)
s=1