Quantum Mechanics for Mathematicians

formalism. It also appears to solve our problem with operator ordering ambi-
guities, since the effect of products ofPandQoperators at various times can
be computed by computing path integrals with variouspandqfactors in the
integrand. These integrand factors commute, giving just one way of producing
products at equal times of any number ofPandQoperators.
Unfortunately, we know from the Groenewold-van Hove theorem that this
is too good to be true. This expression cannot give a unitary representation of
the full group of canonical transformations, at least not one that is irreducible
and restricts to what we want on transformations generated by linear functions
qandp. Another way to see the problem is that a simple argument shows
that by canonical transformations any Hamiltonian can be transformed into a
free particle Hamiltonian, so all quantum systems would just be free particles
in some choice of variables. For the details of these arguments and a careful
examination of what goes wrong, see chapter 31 of [77]. One aspect of the
problem is that for successive values ofjthe coordinatesqtj orptj have no
reason to be close together. This is an integral over “paths” that do not acquire
any expected continuity property asN→∞, so the answer one gets can depend
on the details of the discretization chosen, reintroducing the operator-ordering
ambiguity problem.
One can intuitively see that there is something disturbing about such paths,
since one is alternately at each time interval switching back and forth between
aq-space representation whereqhas a fixed value and nothing is known about
p, and apspace representation wherephas a fixed value but nothing is known
aboutq. The “paths” of the limit are objects with little relation to continuous
paths in phase space, so while one may be able to define the limit of equation
35.2, it will not necessarily have any of the properties one expects of an integral
over continuous paths.
When the Hamiltonianhis quadratic in the momentump, theptjintegrals
will be Gaussian integrals that can be performed exactly. Equivalently, the
kinetic energy partKof the Hamiltonian operator will have a kernel in position
space that can be computed exactly (see equation 12.9). As a result, theptj
integrals can be eliminated, along with the problematic use of alternatingq-
space andp-space representations. The remaining integrals over theqtj are
then interpreted as a path integral over paths not in phase space, but in position
space. One finds, ifK=P
2
2 m

〈qT|e−

~iHT

|q 0 〉=

lim

N→∞

(

i 2 π~T

Nm

)

N 2

√

m

i 2 π~T

∏N

j=1

∫∞

−∞

dqtje

~i∑Nj=1

(m(q

tj−qtj− 1 )^2

2 T/N −V(qtj)TN

)

In the limitN→∞the phase of the exponential becomes

S(γ) =

∫T

0

dt(

1

2

m( ̇q^2 )−V(q(t)))