CHAPTER 4
Instantons in Abelian Systems
In the previous chapter we saw that the existence of a mass gap is the
most important property of gauge and spin systems. In this chapter we
shall discuss a specific mechanism for gap generation, which is
especially important in Abelian systems and will also play some role in
nonabelian ones.
4.1 Instantons in Quantum Mechanics and the Ising Model
Let us describe the symmetry properties in the quantum mechanical
system described by the action (for imaginary time):
2
a /■)
(4.1)
with A
The only interest of this model for us is that it provides the
simplest demonstration of a phenomenon present in more complicated
systems. The point we intend to examine is that this model in any finite
order of perturbation theory has apparently broken symmetry, whereas
in reality the symmetry is restored. To see this we expand:
(p= ± ^ - ^ X
(4.2)
Expanding the action near, say, the left-hand minimum we have:
dtj^ f +
(4.3)
We see that we have (for small A) almost harmonic oscillations near the
bottom of the left-hand well. The left-right symmetry cp-^ —cp is
broken and this will remain so in any finite order in A. At the same time
we know from quantum mechanics that the ground state of this theory
is described by an even (/^-function and therefore is nondegenerate, with
49
DOI: 10.1201/9780203755082-4
Instantons in Abelian Systems
In the previous chapter we saw that the existence of a mass gap is the
most important property of gauge and spin systems. In this chapter we
shall discuss a specific mechanism for gap generation, which is
especially important in Abelian systems and will also play some role in
nonabelian ones.
4.1 Instantons in Quantum Mechanics and the Ising Model
Let us describe the symmetry properties in the quantum mechanical
system described by the action (for imaginary time):
2
a /■)
(4.1)
with A
The only interest of this model for us is that it provides the
simplest demonstration of a phenomenon present in more complicated
systems. The point we intend to examine is that this model in any finite
order of perturbation theory has apparently broken symmetry, whereas
in reality the symmetry is restored. To see this we expand:
(p= ± ^ - ^ X
(4.2)
Expanding the action near, say, the left-hand minimum we have:
dtj^ f +
(4.3)
We see that we have (for small A) almost harmonic oscillations near the
bottom of the left-hand well. The left-right symmetry cp-^ —cp is
broken and this will remain so in any finite order in A. At the same time
we know from quantum mechanics that the ground state of this theory
is described by an even (/^-function and therefore is nondegenerate, with
49
DOI: 10.1201/9780203755082-4