Notice that by dividing through equation (4-17) with to assuming the t,-s are in
decreasing size order we see that the j dependence of G is entirely of the form
G(j) A B + Cne a^"-N B* + ( Czj) (4-18)
where AT is the energy gap between the ground state and the n-th excited state and
A, B and the C,'s are constants depending on the overlap of the states |0), Q|) with each
other and the eigenstates of T. By judiciously choosing states |Q), Q|) and going near
the continuum limit (equivalently choosing large enough N and j) we see that the lowest
energy gap of the theory can be read off as the exponential coefficient in the j dependence
of G(j).
A very important fact is that the full 2M state quantum system that describes the
lattice at each time has a large redundancy in terms of the field theory we are simulating.
Let us for example consider the simple case of a propagator in the two languages. On the
periodic lattice this propagator can be represented by a solid line at any of the M different
spatial points. But in the field theory there is no such labelling of which of the M different
propagators to chose from. The propagator state is really the linear combination of all
the states on the space-slice with one upturned spin somewhere. In the more complicated
cases where there are a number of up spins, again the field theory does not distinguish
between where the spin combination lies but only between the different combinations
of down spins in a row and the order of these. In other words the field theory state
is the cyclicly symmetric linear combination of the space-slice states. Notice that the
transition operator T preserves the cyclicly symmetric sector, so if it were possible to
project out the non-cyclicly symmetric contributions by a choice of 1|) and y|) we would
obtain a non-polluted transition amplitude. The problem is that on the lattice only M
of the 2M states at each time-slice are available, namely the "pure" states with each spin
either up or down. Out of these there are only two purely cyclicly symmetric states,
the all-spins-down and the all-spins-up, and the all-spins-down is forbidden as described