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
>
It might help to learn some basic path integral theory as well:
Path Integrals for Stochastic Processes: An Introduction
https://books.google.com/books/about/Path_Integrals_for_Stochastic_Processes.html?id=vKStkQEACAAJ
This book provides an introductory albeit solid presentation of path
integration techniques as applied to the field of stochastic processes. The
subject began with the work of Wiener during the 1920's, corresponding to a
sum over random trajectories, anticipating by two decades Feynman's famous
work on the path integral representation of quantum mechanics. However, the
true trigger for the application of these techniques within nonequilibrium
statistical mechanics and stochastic processes was the work of Onsager and
Machlup in the early 1950's. The last quarter of the 20th century has
witnessed a growing interest in this technique and its application in
several branches of research, even outside physics (for instance, in
economy).
The aim of this book is to offer a brief but complete presentation of the
path integral approach to stochastic processes. It could be used as an
advanced textbook for graduate students and even ambitious undergraduates
in physics. It describes how to apply these techniques for both Markov and
non-Markov process. The path expansion (or semiclassical approximation) is
discussed and adapted to the stochastic context. Also, some examples of
nonlinear transformations and some applications are discussed, as well as
examples of rather unusual applications. An extensive bibliography is
included. The book is detailed enough to capture the interest of the
curious reader, and complete enough to provide a solid background to
explore the research literature and start exploiting the learned material
in real situations.
@philipthrift
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