35.3 Quantization and path integrals
After use of the Legendre transform to pass to a Hamiltonian system, one then
faces the question of how to construct a corresponding quantum theory. The
method of “canonical quantization” is the one we have studied, taking the posi-
tion coordinatesqjto operatorsQjand momentum coordinatespjto operators
Pj, withQjandPjsatisfying the Heisenberg commutation relations. By the
Stone von-Neumann theorem, up to unitary equivalence there is only one way
to do this and realize these operators on a state spaceH. Recall though that the
Groenewold-van Hove no-go theorem says that there is an inherent operator-
ordering ambiguity for operators of higher order than quadratic, thus for such
operators providing many different possible quantizations of the same classical
system (different though only by terms proportional to~). In cases where the
Legendre transform is not an isomorphism, a new set of problems appear when
one tries to pass to a quantum system since the standard method of canonical
quantization will no longer apply, and new methods are needed.
There is however a very different approach to relating classical and quantum
theories, which completely bypasses the Hamiltonian formalism, just using the
Lagrangian. This is the path integral formalism, which is based upon a method
for calculating matrix elements of the time evolution operator
〈qT|e−
~iHT
|q 0 〉in the position eigenstate basis in terms of an integral over the space of paths
that go from positionq 0 to positionqTin timeT(we will here only treat the
d= 1 case). Here|q 0 〉is an eigenstate ofQwith eigenvalueq 0 (a delta-function
atq 0 in the position space representation), and|qT〉hasQeigenvalueqT. This
matrix element has a physical interpretation as the amplitude for a particle
starting atq 0 att= 0 to have positionqTat timeT, with its norm-squared
giving the probability density for observing the particle at positionqT. It is
also the kernel function that allows one to determine the wavefunctionψ(q,T)
at any timet=Tin terms of its initial value att= 0, by calculating
ψ(qT,T) =∫∞
−∞〈qT|e−
~iHT
|q 0 〉ψ(q 0 ,0)dq 0Note that for the free particle case this is the propagator that we studied in
section 12.5.
To try and derive a path-integral expression for this, one breaks up the
interval [0,T] intoNequal-sized sub-intervals and calculates
〈qT|(e−
Ni~HT
)N|q 0 〉If the Hamiltonian is a sumH=K+V, the Trotter product formula shows
that
〈qT|e−
~iHT
|q 0 〉= lim
N→∞
〈qT|(e−
Ni~KT
e−
Ni~V T
)N|q 0 〉 (35.2)