QuantumPhysics.dvi

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)