On Wednesday, October 16, 2019 at 5:50:32 PM UTC-5, Philip Thrift wrote:
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> On Wednesday, October 16, 2019 at 5:33:53 PM UTC-5, Lawrence Crowell wrote:
>>
>> 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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> Though the above may be correct mathematically, *this is a misleading and
> incomplete presentation of the path integral*.
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> Nowhere above are probabilities referred to!
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> On the other hand, see
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> http://muchomas.lassp.cornell.edu/8.04/Lecs/lec_FeynmanDiagrams/node3.html
> http://www.johnboccio.com/research/quantum/notes/Feynman-Prob.pdf
>
> etc.
>
> @philipthrift
>
By restricting the domain of integration the partition function or integral
will compute a probability for paths in that restricted domain.
Feynman diagrams can be computed entirely from Greene's functions without
reference to path integration.
LC
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