130
5. Quantum Field Theory I: The Free Scalar Field
Now, although this looks like the familiar classical equation of motion
for the position of the oscillator – and recovering it from the Lagrangian
formalism is encouraging – we must be very careful to appreciate that this is
an equation stating how an operator evolves with time. Where the quantum
particle will actually be found is an entirely different matter. By sandwiching
(5.65) between wavefunctions, we can at once see that the average position of
the particle will follow the classical trajectory (remember that wavefunctions
are independent of time in the Heisenberg formulation). But fluctuations
about this trajectory will certainly occur: a quantum particle does not follow
a ray-like classical trajectory. Come to think of it, neither does a photon!
In the original formulations of quantum theory, such fluctuations were generally taken to imply that the very notion of a ‘path’ was no longer a useful
one. However, just as the differential equations satisfied by operators in the
Heisenberg picture are quantum generalizations of Newtonian mechanics, so
there is an analogous quantum generalization of the ‘path-contribution to the
action’ approach to classical mechanics. The idea was first hinted at by Dirac
(1933, 1981, section 32), but it was Feynman who worked it out completely.
The book by Feynman and Hibbs (1965) presents a characteristically fascinating discussion – here we only wish to indicate the central idea. We ask:
how does a particle get from the point q(t 1 ) at time t 1 to the point q(t 2 ) at
t 2 ? Referring back to figure 5.4, in the classical case we imagined (infinitely)
many possible paths q i (t), of which, however, only one was the actual path
followed, namely the one we called q c (t) which minimized the action integral
(5.38) as a functional of q(t). In the quantum case, however, we previously
noted that a particle will no longer follow any definite path, because of quantum fluctuations. But rather than, as a consequence, throwing away the whole
idea of a path, Feynman’s insight was to appreciate that the ‘opposite’ viewpoint is also possible: since unique paths are forbidden in quantum theory, we
should in principle include all possible paths! In other words, we take all the
trajectories on figure 5.4 as physically possible (together with all the other
infinitely many ways of accomplishing the trip).
However, surely not all paths are equally likely: after all, we must presumably recover the classical trajectory as ħ → 0, in some sense. Thus we must
find an appropriate weighting for the paths. Feynman’s recipe is beautifully
simple: weight each path by the factor
iS/ħ
e
(5.66)
where S is the action for that particular path. At first sight this is a rather
strange proposal, since all paths – even the classical one – are weighted by a
quantity which is of unit modulus. But of course contributions of the form
(5.66) from all the paths have to be added coherently – just as we superposed
the amplitudes in the ‘two-slit’ discussion in section 2.5. What distinguishes
the classical path q c (t) is that it makes S stationary under small changes of
path: thus in its vicinity paths have a strong tendency to add up constructively, while far from it the phase factors will tend to produce cancellations.
5. Quantum Field Theory I: The Free Scalar Field
Now, although this looks like the familiar classical equation of motion
for the position of the oscillator – and recovering it from the Lagrangian
formalism is encouraging – we must be very careful to appreciate that this is
an equation stating how an operator evolves with time. Where the quantum
particle will actually be found is an entirely different matter. By sandwiching
(5.65) between wavefunctions, we can at once see that the average position of
the particle will follow the classical trajectory (remember that wavefunctions
are independent of time in the Heisenberg formulation). But fluctuations
about this trajectory will certainly occur: a quantum particle does not follow
a ray-like classical trajectory. Come to think of it, neither does a photon!
In the original formulations of quantum theory, such fluctuations were generally taken to imply that the very notion of a ‘path’ was no longer a useful
one. However, just as the differential equations satisfied by operators in the
Heisenberg picture are quantum generalizations of Newtonian mechanics, so
there is an analogous quantum generalization of the ‘path-contribution to the
action’ approach to classical mechanics. The idea was first hinted at by Dirac
(1933, 1981, section 32), but it was Feynman who worked it out completely.
The book by Feynman and Hibbs (1965) presents a characteristically fascinating discussion – here we only wish to indicate the central idea. We ask:
how does a particle get from the point q(t 1 ) at time t 1 to the point q(t 2 ) at
t 2 ? Referring back to figure 5.4, in the classical case we imagined (infinitely)
many possible paths q i (t), of which, however, only one was the actual path
followed, namely the one we called q c (t) which minimized the action integral
(5.38) as a functional of q(t). In the quantum case, however, we previously
noted that a particle will no longer follow any definite path, because of quantum fluctuations. But rather than, as a consequence, throwing away the whole
idea of a path, Feynman’s insight was to appreciate that the ‘opposite’ viewpoint is also possible: since unique paths are forbidden in quantum theory, we
should in principle include all possible paths! In other words, we take all the
trajectories on figure 5.4 as physically possible (together with all the other
infinitely many ways of accomplishing the trip).
However, surely not all paths are equally likely: after all, we must presumably recover the classical trajectory as ħ → 0, in some sense. Thus we must
find an appropriate weighting for the paths. Feynman’s recipe is beautifully
simple: weight each path by the factor
iS/ħ
e
(5.66)
where S is the action for that particular path. At first sight this is a rather
strange proposal, since all paths – even the classical one – are weighted by a
quantity which is of unit modulus. But of course contributions of the form
(5.66) from all the paths have to be added coherently – just as we superposed
the amplitudes in the ‘two-slit’ discussion in section 2.5. What distinguishes
the classical path q c (t) is that it makes S stationary under small changes of
path: thus in its vicinity paths have a strong tendency to add up constructively, while far from it the phase factors will tend to produce cancellations.