QuantumPhysics.dvi
wang
(Wang)
#1
a famous MIT/SLAC experiment. Theoretically, asymptotic freedom was found to be a
generic property of non-Abelian gauge theories (with not too manyfermions), in a famous
calculation by Gross, Wilczek and Politzer (1973). Thus, perturbation theory can be used
also for the strong interactions, provided the energies involved are large.
10.1.2 Convergence of the expansion for finite-dimensional systems
One would hope, ideally, that the perturbative expansions of energy and state in (10.3)
form a convergent Taylor series. Is this really the case? When the corresponding quantum
problem has a finite-dimensional Hilbert space, andH 0 ,H 1 may be represented byN×N
matrices (and|En〉by an N-dimensional column matrix), the perturbative expansion of
(10.3) will in fact be convergent, with some finite radius of convergence. This is because the
energy eigenvalues are solutions to the characteristic equation,
det (En−H 0 −λH 1 ) = 0 n= 1,···,N (10.5)
and the eigenvalues depend analytically on the parameterλ. On the other hand, for quantum
systems with infinite-dimensional Hilbert spaces, the situation is more complicated, and the
λ-dependence may not be analytic.
10.1.3 The asymptotic nature of the expansion for infinite dimensional systems
It is instructive to consider how perturbation theory is carried outin the path integral in order
to shed light on the convergence issues. For a simple one-dimension quantum mechanical
system given by the following Hamiltonian or, equivalently, Lagrangian
H =
p^2
2 m
+
1
2
mω^2 q^2 +λV(q)
L =
1
2
mq ̇^2 −
1
2
mω^2 q^2 −λV(q) (10.6)
withV(q) given, for example, byq^4 , the path integral for the partition function assumes the
form,
∫
Dqexp
{
1
̄h
∫β ̄h
0
dtL(q,q ̇)
}
(10.7)
This path integral is complicated, but for the sake of understanding the perturbative expan-
sion, we shall truncate it to the contributions of just thet-independent functionsq(t), i.e.
constantq. This gives an ordinary integral, which is of the general form,
I(ω,λ) =
∫+∞
−∞
dqexp{−ω^2 q^2 −λq^4 } (10.8)