40
GAUGE FIELDS AND STRINGS
The last model with global symmetry is the principal chiral field. Its
Lagrangian is:
^ = Z T
' 9 y ) + ~ l (Tr(^;'
+ C.C.)
(3.28)
y ^
^ y&
We see that it describes a set of symmetric tops (in the case geSU{2)).
The Hamiltonian has the form:
y à
y ^d y + ô) + C.C.}
(3.29)
Here ly is an operator of left rotations which, in the case of the SU{2)
group, can be expressed in terms of derivatives in Euler angles (see any
book on quantum mechanics). Eigenvalues of are again /(/ + 1) but
the degeneracy is (21 -h 1)^ due to the fact that the symmetry group of
(3.29) is SU(2)® SU(2) (body and frame rotations in quantum mechanics of the top). The quantum numbers of elementary excitations are
such that they transform by fundamental representations of both
groups. In the case of SU(2) it is a vectorlike excitation of SO(4)
SU(2)(S) SU(2). Again, this conclusion will be confirmed by exact
results.
3.3 Gauge Symmetries
We have seen that in the strong coupling region all systems with global
symmetries look roughly the same. In all cases we had massive pointlike excitations which propagate through the lattice. In the NonAbelian cases with Q ) = 2 this picture remained valid even for small
coupling.
In the case of a gauge system the strong coupling region is again
rather insensitive to the type of symmetry. However, gauge invariance
introduces qualitatively new features to this region which will be
discussed now.
Let us look first at the small jS expansion for the partition function in
the Z 2 case. Since each term in the energy (1.46) is associated with a
plaquette, the result of the expansion can be presented as a collection of
plaquettes, such that at each link an even number of plaquettes meet.
Therefore we have something like closed surfaces instead of closed
paths for a nongauge system. The contribution of a given surface to the
partition function is given by (tanh jS)^ where A is the number of
plaquettes or, in other words, the area of the surface. For more
GAUGE FIELDS AND STRINGS
The last model with global symmetry is the principal chiral field. Its
Lagrangian is:
^ = Z T
' 9 y ) + ~ l (Tr(^;'
+ C.C.)
(3.28)
y ^
^ y&
We see that it describes a set of symmetric tops (in the case geSU{2)).
The Hamiltonian has the form:
y à
y ^d y + ô) + C.C.}
(3.29)
Here ly is an operator of left rotations which, in the case of the SU{2)
group, can be expressed in terms of derivatives in Euler angles (see any
book on quantum mechanics). Eigenvalues of are again /(/ + 1) but
the degeneracy is (21 -h 1)^ due to the fact that the symmetry group of
(3.29) is SU(2)® SU(2) (body and frame rotations in quantum mechanics of the top). The quantum numbers of elementary excitations are
such that they transform by fundamental representations of both
groups. In the case of SU(2) it is a vectorlike excitation of SO(4)
SU(2)(S) SU(2). Again, this conclusion will be confirmed by exact
results.
3.3 Gauge Symmetries
We have seen that in the strong coupling region all systems with global
symmetries look roughly the same. In all cases we had massive pointlike excitations which propagate through the lattice. In the NonAbelian cases with Q ) = 2 this picture remained valid even for small
coupling.
In the case of a gauge system the strong coupling region is again
rather insensitive to the type of symmetry. However, gauge invariance
introduces qualitatively new features to this region which will be
discussed now.
Let us look first at the small jS expansion for the partition function in
the Z 2 case. Since each term in the energy (1.46) is associated with a
plaquette, the result of the expansion can be presented as a collection of
plaquettes, such that at each link an even number of plaquettes meet.
Therefore we have something like closed surfaces instead of closed
paths for a nongauge system. The contribution of a given surface to the
partition function is given by (tanh jS)^ where A is the number of
plaquettes or, in other words, the area of the surface. For more