From formula (4.78)-(4.80) we obtain
68
GAUGE FIELDS AND STRINGS
1
/c„/c„ k„k
-^(H^(k)H,(-k)} = d^,
k^
k}
+ k}
_ c
+ k^
(4.81)
This formula implies that there are no massless particles in the theory
and instead we have a massive scalar particle with a small mass M.
Analogously we find the result
The qualitative explanation of the above result is the following. In our
system there is a finite density of pseudoparticles with long range
interaction and their random fields spoil the correlation. From the
strong coupling expansion we know that the correlation length is
nonzero also for > 1. It is reasonable to assume that there is no phase
transition in this system and that the confinement regime continues to
weak coupling. As we shall show in the next chapter, this is indeed true.
Here we shall discuss another implication of the result.
First of all let us notice that since we showed that the 0(2) system for
^ = 3 is disordered (has a mass gap) the same must be true for
nonabelian systems. Indeed, let us take the case of SU{2) and constrain
the
so that they lie in 0(2) c: SU{2). One should expect that this
constraint increases the order in the system and if the constrained
system is disordered the unconstrained one must be even more disordered and have an even larger gap. There is no doubt that this
statement is correct but a rigorous proof has yet to be given.
As we turn to ^ = 4 the picture described above changes. Threedimensional cubes embedded into four-dimensional space have four
different orientations (analogously to the three different orientations for
squares embedded in 3-space). Hence we expect that the number q in
(4.60) for ^ = 4 will have a directional index
To check this,
let us write down the decomposition analogous to (4.61) for the
numbers N*,«/? Ihe case ^ = 4:
(4.83)
Here we have used the notation for the lattice derivatives, e.g.:
d^rrio =
« - m,_
(4.84)
68
GAUGE FIELDS AND STRINGS
1
/c„/c„ k„k
-^(H^(k)H,(-k)} = d^,
k^
k}
+ k}
_ c
+ k^
(4.81)
This formula implies that there are no massless particles in the theory
and instead we have a massive scalar particle with a small mass M.
Analogously we find the result
The qualitative explanation of the above result is the following. In our
system there is a finite density of pseudoparticles with long range
interaction and their random fields spoil the correlation. From the
strong coupling expansion we know that the correlation length is
nonzero also for > 1. It is reasonable to assume that there is no phase
transition in this system and that the confinement regime continues to
weak coupling. As we shall show in the next chapter, this is indeed true.
Here we shall discuss another implication of the result.
First of all let us notice that since we showed that the 0(2) system for
^ = 3 is disordered (has a mass gap) the same must be true for
nonabelian systems. Indeed, let us take the case of SU{2) and constrain
the
so that they lie in 0(2) c: SU{2). One should expect that this
constraint increases the order in the system and if the constrained
system is disordered the unconstrained one must be even more disordered and have an even larger gap. There is no doubt that this
statement is correct but a rigorous proof has yet to be given.
As we turn to ^ = 4 the picture described above changes. Threedimensional cubes embedded into four-dimensional space have four
different orientations (analogously to the three different orientations for
squares embedded in 3-space). Hence we expect that the number q in
(4.60) for ^ = 4 will have a directional index
To check this,
let us write down the decomposition analogous to (4.61) for the
numbers N*,«/? Ihe case ^ = 4:
(4.83)
Here we have used the notation for the lattice derivatives, e.g.:
d^rrio =
« - m,_
(4.84)