What is a path integral? It is an accounting of all amplitudes or paths
from the state |i> to |f> at t = 0 and t = T respectively. The amplitude
for this is <f|i>, and to compute the paths we consider the completeness
sums 1 = ∫|q(t)><q(t)| over configuration space and a product of them as
<f|i> = <f| ∫|q(δt)><q(δt)| ∫|q(2δt)><q(2δt)| … ∫|q(t + δt)><q(t + δt)| …
∫|q(T - δt)><q(T - δt)|i>
which we write in the more compact formatting for the initial and final
states configuration variables, or eigenstates of the position operator
<f|i> = <f| Π_{n=1}^N ∫|q(nδt)><q(nδt)| |i> = Π_{n=1}^N∫<q(nδt)|q((n+1)δt)>
Now in the terms <q(nδt)|q((n+1)δt)> I insert the momentum completeness sum
*1* = ∫|p(nδt)><p(nδt)| so that
<q(nδt)|q((n+1)δt)> = <q(nδt)|*1*|q((n+1)δt)> =
<q(nδt)|p(nδt)><p(nδt)|q((n+1)δt)>.
Fourier theory tells us <q(nδt)|p(nδt)> = (1/2π)e^{ip(dq/dt)δt}and that
<q(nδt)|p(nδt)><p(nδt)|q((n+1)δt)> = (1/2π)e^{ip(dq/dt) - iHδt}
where the Hδt is from the time translation of q. This is then
<f|i> = ∫d[δq]Π_{n=1}^N (1/2π)e^{ip(dq/dt) – Hδt} = (1/2π)∫D[q]e^{∫(ipdq -
iHdt)
or Z = (1/2π)∫D[q]e^{iS}, where the upper case D just represents an
integration from a product of integrations. The action comes from the
Lagrangian L = pdq/dt - H and S = ∫Ldt.
That's all folks! There is nothing mysterious about path integrals! There
is nothing that makes them contrary to any quantum interpretation or that
makes them render a proof of one. The ideology of Dowker and others amount
to an auxiliary axiom or physical postulate that is in addition to be basic
idea of a path integral. There is nothing I did above that is not straight
up plain vanilla quantum mechanics. Things get a little more funny with
QFT, but there is nothing outside of QFT in the nature of a path integral.
LC
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