82
GAUGE FIELDS AND STRINGS
There exists an interesting analogy between k in the gauge systems and
certain quantities in global systems. Let us consider the response of the
global 0(2) system to the external vector field v^:
Q-W[v,] ^
^(p e - S (d ^< p + v^)
Again
is gauge invariant:
If we neglect vortices, for constant we obtain:
^ Psi^)
djc
(5.40)
(5.41)
(5.42)
iPsiP) being some function).
The constant can be removed from the action by the transformation:
(5.43)
but then it appears in the boundary conditions. Formula (5.42) does not
contradict gauge invariance since due to massless excitations in the
system v^{x) appears in the combination:
W
dx
(5.44)
After the phase transition, when a correlation length appears in the
system, we must have:
Ps(P) = 0 for
(5.45)
The analogy between p^ and k is obvious. What is interesting is that if
our 0(2) system describes "^He then psiP) has an interpretation as the
superfluid density. Therefore, the normal phase of "^He correspond to
the confining phase of the gauge system, and the superfluid phase to the
Coulomb phase. Let us show now that Ps is indeed the superfluid
density. To do this we consider a second-quantized Hamiltonian,
describing a Bose liquid:
^ ^ (jc)'F(jc)w(jc - y ) ^ ^ ty)'FCv) d y j dx (5.46)
where m is a shortrange interatomic interaction. Suppose that the walls,
containing our liquid, move with velocity v. The question is whether the
liquid will move with the same velocity (owing to friction with the