165
6.3. Applications to the ‘ABC’ theory
numbers (see (6.46)). We can rewrite (6.78) in more suggestive form by noting
that
ˆ
ˆ
ˆ
A ˆ
[ˆ a, B] = <0|[ˆ a, B]|0> = <0|a ˆB|0> = <0| ˆ B|0>.
(6.79)
Thus the vev of a product of four operators is just the sum of the products
of all the possible pairwise ‘contractions’ (the name given to the vev of the
product of two fields):
<0|A ˆ B ˆ C ˆ D ˆ |0> = <0|A ˆ B ˆ |0><0|C ˆ D ˆ |0> + <0|A ˆ C ˆ |0><0|B ˆ D ˆ |0> + <0|A ˆ D ˆ |0><0|B ˆ C ˆ |0>.
(6.80)
This result generalizes to the vev of the product of any number of operators;
there is also a similar result for the vev of time-ordered products of operators,
which is known as Wick’s theorem (Wick 1950), and is indispensable for a
general discussion of quantum field perturbation theory.
Consider then the application of (6.80), as generalized to ten operators,
to the vev in (6.74). The only kind of non-vanishing contractions are of the
†
form <0|a ˆ i a ˆ |0>. Thus the contractions of A-, B- and C-type operators can be
i
considered separately. As far as the C-operators are concerned, then, we can
immediately conclude that the only surviving contraction is
<0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>.
(6.81)
This quantity is, in fact, of fundamental importance: it is called the Feynman
propagator (in coordinate space) for the spin-0 C-particle. We shall derive
the mathematical formula for it in due course, but for the moment let us
understand its physical significance. Each of the φ ˆ C ’s in (6.81) can create
or destroy C-quanta, but for the vev to be non-zero anything created in the
‘initial’ state must be destroyed in the ‘final’ one. Which of the times t 1 and
t 2 is initial or final is determined by the T -ordering symbol: for t 1 > t 2 , a Cquantum is created at x 2 and destroyed at x 1 , while for t 1 < t 2 a C-quantum
is created at x 1 and destroyed at x 2 . Thus the amplitude (6.81) may be
represented pictorially as in figure 6.2, where time increases to the right, and
the vertical axis is a one-dimensional version of three-dimensional space. It
seems reasonable, indeed, to call this object the ‘propagator’, since it clearly
has to do with a quantum propagating between two space–time points.
We might now worry that this explicit time-ordering seems to introduce a
Lorentz non-invariant element into the calculation, ultimately threatening the
Lorentz invariance of the S ˆ -operator (6.42). The reason that this is in fact not
the case exposes an important property of quantum field theory. If the two
points x 1 and x 2 are separated by a time-like interval (i.e. (x 1 − x 2 )
2 > 0),
then the time-ordering is Lorentz invariant; this is because no proper Lorentz
transformation can alter the time-ordering of time-like separated events (here,
the events are the creation/annihilation of particles/antiparticles at x 1 and
x 2 ). By ‘proper’ is meant a transformation that does not reverse the sense of
time; the behaviour of the theory under time-reversal is a different question
altogether, discussed earlier in section 4.2.4. The fact that time-ordering is
6.3. Applications to the ‘ABC’ theory
numbers (see (6.46)). We can rewrite (6.78) in more suggestive form by noting
that
ˆ
ˆ
ˆ
A ˆ
[ˆ a, B] = <0|[ˆ a, B]|0> = <0|a ˆB|0> = <0| ˆ B|0>.
(6.79)
Thus the vev of a product of four operators is just the sum of the products
of all the possible pairwise ‘contractions’ (the name given to the vev of the
product of two fields):
<0|A ˆ B ˆ C ˆ D ˆ |0> = <0|A ˆ B ˆ |0><0|C ˆ D ˆ |0> + <0|A ˆ C ˆ |0><0|B ˆ D ˆ |0> + <0|A ˆ D ˆ |0><0|B ˆ C ˆ |0>.
(6.80)
This result generalizes to the vev of the product of any number of operators;
there is also a similar result for the vev of time-ordered products of operators,
which is known as Wick’s theorem (Wick 1950), and is indispensable for a
general discussion of quantum field perturbation theory.
Consider then the application of (6.80), as generalized to ten operators,
to the vev in (6.74). The only kind of non-vanishing contractions are of the
†
form <0|a ˆ i a ˆ |0>. Thus the contractions of A-, B- and C-type operators can be
i
considered separately. As far as the C-operators are concerned, then, we can
immediately conclude that the only surviving contraction is
<0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>.
(6.81)
This quantity is, in fact, of fundamental importance: it is called the Feynman
propagator (in coordinate space) for the spin-0 C-particle. We shall derive
the mathematical formula for it in due course, but for the moment let us
understand its physical significance. Each of the φ ˆ C ’s in (6.81) can create
or destroy C-quanta, but for the vev to be non-zero anything created in the
‘initial’ state must be destroyed in the ‘final’ one. Which of the times t 1 and
t 2 is initial or final is determined by the T -ordering symbol: for t 1 > t 2 , a Cquantum is created at x 2 and destroyed at x 1 , while for t 1 < t 2 a C-quantum
is created at x 1 and destroyed at x 2 . Thus the amplitude (6.81) may be
represented pictorially as in figure 6.2, where time increases to the right, and
the vertical axis is a one-dimensional version of three-dimensional space. It
seems reasonable, indeed, to call this object the ‘propagator’, since it clearly
has to do with a quantum propagating between two space–time points.
We might now worry that this explicit time-ordering seems to introduce a
Lorentz non-invariant element into the calculation, ultimately threatening the
Lorentz invariance of the S ˆ -operator (6.42). The reason that this is in fact not
the case exposes an important property of quantum field theory. If the two
points x 1 and x 2 are separated by a time-like interval (i.e. (x 1 − x 2 )
2 > 0),
then the time-ordering is Lorentz invariant; this is because no proper Lorentz
transformation can alter the time-ordering of time-like separated events (here,
the events are the creation/annihilation of particles/antiparticles at x 1 and
x 2 ). By ‘proper’ is meant a transformation that does not reverse the sense of
time; the behaviour of the theory under time-reversal is a different question
altogether, discussed earlier in section 4.2.4. The fact that time-ordering is