38
GAUGE FIELDS AND STRINGS
Hamiltonian version. In order to obtain it we introduce a large
anisotropy in the “time” direction:
- S = /ioZcos((Pj,,,-(? , ,+ ,)
y
+ PlY. COS(cp,,,, - (py + s j
(3.19)
(Here j belongs to the ^ — 1-dimensional lattice). In the limit of large
j?o we can replace (3.19) by:
- S = c o n s t -
- i ? i Z cos( Comparing (3.20) with (1.10) we see that it is just the action for
imaginary time of the system of coupled rotators sitting at the lattice
sites and described by the angles {(Py). In the Minkowskian time their
lagrangian is:
^ = Z ^2 f d ? )' ^
'
Passing to the Hamiltonian by the standard procedure we obtain:
W = ,77 Z
Z cos( y
y, 6
L
1 d
i ^(Py
(3.22)
(Notice the analogy with (3.4)). Now the strong coupling limit
^
corresponds to neglect of the potential energy in (3.22):
■ exp( i z » , V y
^ Z
(3.23)
Here {riy} is an arbitrary set of integers; the ground state corresponds to
all fiy = 0. The first excited state is obtained by taking some one of the n,
riy^= + i, and has a mass gap. The reason for this is simple—in our
approximation all the rotators are decoupled. The elementary excitation we have described is just the excitation of a single rotator placed at
the point J q- As before taking into account the potential energy causes
hopping of the excitation from j to j -h 6:
(3.24)