Quantum Mechanics for Mathematicians

IfK(P) can be chosen to depend only on the momentum operatorPandV(Q)
depends only on the operatorQ, then one can insert alternate copies of the
identity operator in the forms
∫∞

−∞

|q〉〈q|dq= 1 ,

∫∞

−∞

|p〉〈p|dp= 1

This gives a product of terms of the form

〈qtj|e−

Ni~K(P)T

|ptj〉〈ptj|e−

Ni~V(Q)T

|qtj− 1 〉

where the indexjgoes from 0 toN,tj=jT/Nand the variablesqtjandptj
will be integrated over.
Such a term can be evaluated as

〈qtj|ptj〉〈ptj|qtj− 1 〉e−

Ni~K(ptj)T

e−

Ni~V(qtj− 1 )T

=

1

√

2 π~

e

~iqtjptj^1

√

2 π~

e−

i~qtj− 1 ptj

e−

Ni~K(ptj)T

e−

Ni~V(qtj− 1 )T

=

1

2 π~

e

~iptj(qtj−qtj− 1 )

e−

Ni~(K(ptj)+V(qtj− 1 ))T

TheNfactors of this kind give an overall factor of ( 2 π^1 ~)Ntimes something
which is a discretized approximation to

e

~i∫ 0 T(pq ̇−h(q(t),p(t)))dt

where the phase in the exponential is just the action. Taking into account the
integrations overqtjandptjone should have something like

〈qT|e−

~iHT

|q 0 〉= lim

N→∞

(

1

2 π~

)N

∏N

j=1

∫∞

−∞

∫∞

−∞

dptjdqtje

~i∫ 0 T(pq ̇−h(q(t),p(t)))dt

although one should not do the first and last integrals overqbut fix the first
value ofqtoq 0 and the last one toqT. One can try and interpret this sort of
integration in the limit as an integral over the space of paths in phase space,
thus a “phase space path integral”.
This is an extremely simple and seductive expression, apparently saying that,
once the actionSis specified, a quantum system is defined just by considering
integrals ∫

Dγ e

~iS[γ]

over pathsγin phase space, whereDγis some sort of measure on this space of
paths. Since the integration just involves factors ofdpdqand the exponential
justpdqandh(p,q), this formalism seems to share the same sort of behavior un-
der the infinite dimensional group of canonical transformations (transformations
of the phase space preserving the Poisson bracket) as the classical Hamiltonian