Subatomic Physics

388 Introduction to Gauge Theories
ϕ1 = e
�
A·dx, ϕ2 =
�c s1
e
�
A·dx. (12.16)
�c s2
The interference pattern observed on the screen is determined by the phase difference
of the two waves. If |s1| = |s2|, so that P is located at equal distances from
the two slits, the phase difference δϕ is
δϕ = ϕ1 − ϕ2 = e
�� � �
A·dx − A·dx
�c s1
s2
= e
�
A·dx =
�c
e
Φ. (12.17)
�c
Thus, even though there is no magnetic field along the paths of the electrons, they
show interference effects that depend on, and vary with the vector potential A,
which therefore acquires a physical reality that was absent in classical mechanics.
The effect occurs because the local phase at two space–time points is connected
by the potential. The importance of potentials in quantum theory was stressed by
Aharonov and Bohm (2) and the phase difference dependence on the vector potential
A has been observed. (3)
It was shown by Berry (4) that the Aharonov–Bohm effect is a special case of a
geometric phase present for any system transported adiabatically (slowly) around
a closed circuit. The phase can be made visible by beating the system that is made
to go around the circuit with the same system made to go straight to the detector;
another way is to examine the superposition of stationary spin states of a system
of particles, such as neutrons, before and after they have completed a closed path,
as in a helical magnetic field. Berry makes a classical analogy to a body moving
around a closed path on a curved surface. Thus, if a matchstick is taken around a
closed path on a plane, without rotating it, it points in the same direction at the
end as at the start. If, however, it is taken around a path on a sphere, such as from
the North pole of the Earth to the equator, then taken to a different longitude and
returned to the North pole, it ends up pointing along a different longitude at the
end than at the start. Like the quantum mechanical effect, the change in direction
only depends on geometrical factors.
12.3 Gauge Invariance for Non-Abelian Fields
The electromagnetic field is a simple example of a gauge field. If we are to include
the weak interactions, then there are two problems that need to be solved. The first
one is that both neutral and charged vector bosons are required. The second one
is that the weak bosons W + and Z 0 are massive, whereas we showed that gauge
invariance requires massless fields. We tackle the first problem in this section.
3 R. G. Chambers, Phys. Rev. Lett. 5, 3 (1960).
4 V. M. Berry, Proc. R. Soc. London A392, 45 (1984); Sci. Amer. 259, 46 (December 1988).