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
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/everything-list/1bca82ef-eeb5-4a2c-9f08-c4c61139ef70%40googlegroups.com.
