ATTEMPT AT A SYNTHESIS
281
particle excitation in the two-dimensional Ising model. That does not
mean of course that the 3d Ising string can be considered as a collection
of noninteracting 2d Ising particles. The difference is that in the process
of development the string does not break (as a consequence of gauge
invariance) and hence when some link moves it creates extra portions of
string, necessary for continuity. But this continuity is the only source of
interaction. Such a system will be called a free string.
Let us now summarize our conclusions. We have proved that the 2d
Ising model is equivalent to the problem of free particle propagation.
This particle carries internal index a, or, more geometrically, an arrow
which will be finally identified with spin. In the process of propagation of the particle the spin rotates due to spin-orbit interaction, but the
total angular momentum is conserved. The classical vector
on the
lattice does not correspond to any definite value of the spin but,
according to (10.66) and (10.67) only the spin 1/2 part of the wave
function has long range correlations in the critical region. It is not
surprising therefore that as a result we obtained the Dirac equation.
Turning now to the 3d case we discovered that the Ising model is
described by the propagation of a closed string with internal degrees of
freedom distributed on the links. These degrees of freedom are precisely
the same as in the previous case, so we can say that we have a spin
density distributed along the string. Our major result was the conclusion that the string moves piecewise as a 2d-Ising particle, and there is
only implicit interaction following from continuity.
The most difficult problem now is to find a continuum limit for the
equation (10.78). In the 2d case such a problem was solved trivially, by
solving first the lattice equation (10.60) and taking the limit jS ->
Unfortunately the equation (10.78) on the lattice is completely
hopeless. The best thing we can do is to guess on physical grounds what
kind of system it describes in the critical region. We shall do this by
looking at things the other way around. Namely we discuss a continuum string model, which behave piecewise as a free Dirac particle, and
has a very good chance to describe the critical region of equation
(10.78).
Let us consider the NSR-string, the wave functional of which is
annihilated by the supercurrent and energy momentum tensor. The
supercurrent condition can be presented as:
± ^ ;)i0 > = 0
If we replace
by (l/i)(<5/5x^) and introduce:
281
particle excitation in the two-dimensional Ising model. That does not
mean of course that the 3d Ising string can be considered as a collection
of noninteracting 2d Ising particles. The difference is that in the process
of development the string does not break (as a consequence of gauge
invariance) and hence when some link moves it creates extra portions of
string, necessary for continuity. But this continuity is the only source of
interaction. Such a system will be called a free string.
Let us now summarize our conclusions. We have proved that the 2d
Ising model is equivalent to the problem of free particle propagation.
This particle carries internal index a, or, more geometrically, an arrow
which will be finally identified with spin. In the process of propagation of the particle the spin rotates due to spin-orbit interaction, but the
total angular momentum is conserved. The classical vector
on the
lattice does not correspond to any definite value of the spin but,
according to (10.66) and (10.67) only the spin 1/2 part of the wave
function has long range correlations in the critical region. It is not
surprising therefore that as a result we obtained the Dirac equation.
Turning now to the 3d case we discovered that the Ising model is
described by the propagation of a closed string with internal degrees of
freedom distributed on the links. These degrees of freedom are precisely
the same as in the previous case, so we can say that we have a spin
density distributed along the string. Our major result was the conclusion that the string moves piecewise as a 2d-Ising particle, and there is
only implicit interaction following from continuity.
The most difficult problem now is to find a continuum limit for the
equation (10.78). In the 2d case such a problem was solved trivially, by
solving first the lattice equation (10.60) and taking the limit jS ->
Unfortunately the equation (10.78) on the lattice is completely
hopeless. The best thing we can do is to guess on physical grounds what
kind of system it describes in the critical region. We shall do this by
looking at things the other way around. Namely we discuss a continuum string model, which behave piecewise as a free Dirac particle, and
has a very good chance to describe the critical region of equation
(10.78).
Let us consider the NSR-string, the wave functional of which is
annihilated by the supercurrent and energy momentum tensor. The
supercurrent condition can be presented as:
± ^ ;)i0 > = 0
If we replace
by (l/i)(<5/5x^) and introduce: