On Wednesday, October 16, 2019 at 7:22:45 PM UTC-5, Lawrence Crowell wrote:
>
> On Wednesday, October 16, 2019 at 5:50:32 PM UTC-5, Philip Thrift wrote:
>>
>>
>>
>> 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
>>>
>>
>>
>>
>>
>>
>>
>> Though the above may be correct mathematically, *this is a misleading
>> and incomplete presentation of the path integral*.
>>
>> Nowhere above are probabilities referred to!
>>
>> On the other hand, see
>>
>> http://muchomas.lassp.cornell.edu/8.04/Lecs/lec_FeynmanDiagrams/node3.html
>> <http://www.google.com/url?q=http%3A%2F%2Fmuchomas.lassp.cornell.edu%2F8.04%2FLecs%2Flec_FeynmanDiagrams%2Fnode3.html&sa=D&sntz=1&usg=AFQjCNGw3YF43oMwvripjLktdfwGwBZ7TQ>
>> 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
>
Probabilities are part of sum over histories, or path integrals. Green's
functions are used in probability theory and stochastic processes but have
nothing to do with *the probabilistic nature of path integrals.*
http://muchomas.lassp.cornell.edu/8.04/Lecs/lec_FeynmanDiagrams/node3.html
<http://www.google.com/url?q=http%3A%2F%2Fmuchomas.lassp.cornell.edu%2F8.04%2FLecs%2Flec_FeynmanDiagrams%2Fnode3.html&sa=D&sntz=1&usg=AFQjCNGw3YF43oMwvripjLktdfwGwBZ7TQ>
The Sum Over Histories
The Feynman formulation of Quantum Mechanics builds three central ideas
from the de Broglie hypothesis into the computation of quantum amplitudes:
the probabilistic aspect of nature, superposition, and the classical limit.
This is done by making the following three three postulates:
1. *Events in nature are probabilistic with predictable probabilities P.*
2. *The probability P for an event to occur is given by the square of
the complex magnitude of a quantum amplitude for the event, Q. The quantum
amplitude Q associated with an event is the sum of the amplitudes [image:
tex2html_wrap_inline1605] associated with every history leading to the
event.*
3. *The quantum amplitude associated with a given history [image:
tex2html_wrap_inline1605] is the product of the amplitudes [image:
tex2html_wrap_inline1609] associated with each fundamental process in the
history.*
Postulate (1) states the fundamental probabilistic nature of our world, and
opens the way for computing these probabilities.
Postulate (2) specifies how probabilities are to be computed. This item
builds the concept of superposition, and thus the possibility of quantum
interference, directly into the formulation. Specifying that the
probability for an event is given as the magnitude-squared of a sum made
from complex numbers, allows for negative, positive and intermediate
interference effects. This part of the formulation thus builds the
description of experiments such as the two-slit experiment directly into
the formulation. A *history* is a *sequence* of fundamental processes
leading to the the event in question. We now have an explicit formulation
for calculating the probabilities for events in terms of the [image:
tex2html_wrap_inline1605] , quantum amplitudes for individual histories,
which the third postulate will now specify.
Postulate (3) specifies the quantum amplitude associated with individual
histories in terms of *fundamental processes*. A *fundamental process* is
any process which cannot be interrupted by another fundamental process. The
*fundamental
processes* are thus indivisible ``atomic units'' of history. With this
constraint of the choice of *fundamental processes*, individual histories
may always be divided unambiguously into ordered sequences of fundamental
events, which is key to making a consistent prescription for computing the
amplitudes of individual histories from *fundamental processes*. The fact
that the definition of fundamental processes is not very specific is
actually one of the strongest aspects of the Feynman approach. As we will
see, we may sometimes discover that we may lump fundamental processes
together into larger units which make up new fundamental processes. This
procedure is know as renormalization and is one the the great central ideas
in managing the infinities in quantum field theory.
The third postulate builds in the classical limit by allowing recovery of
the classical physics notion that the probability of an *independent* sequence
of events is the product of the probabilities for each event in the
sequence. If we know the sequence of fundamental processes leading to an
event, the only contributing history is that sequence of processes. In such
a case, we have [image: tex2html_wrap_inline1613] so that then [image:
tex2html_wrap_inline1615] , where the [image: tex2html_wrap_inline1617] are
just the probabilities for the individual processes in the sequence, and we
recover the usual classical probabilistic result.
What remains unspecified by these postulates is the specification of a
valid set of fundamental processes and corresponding quantum amplitudes [image:
tex2html_wrap_inline1609] for the phenomena we wish to describe. For this
information, we must rely upon experimental observations. It is at this
point that experimental information is input into the Feynman formulation
much like how we inputted experimental information into our formulation
when we produced the forms for our operators and the Schrödinger Equations.
A great appeal to the Feynman sum over histories approach is that often we
are able to intuit the nature and amplitudes of the fundamental events. A
natural way to build the de Broglie hypothesis [image:
tex2html_wrap_inline1621] from the Davisson-Germer and G.P. Thomson
experiments into the formulation, for instance, would be to ascribe a
quantum amplitude of [image: tex2html_wrap_inline1623] for the propagation
of a particle with momentum [image: tex2html_wrap_inline1625] across a
distance *a*.
Another common way to infer the fundamental events and associated
amplitudes is to determine the amplitudes for fundamental processes from
the requirement that the Feynman formulation always give the same results
as an already established approach, such as Schrödinger formulation. This
latter procedure is referred as *construction of Feynman rules*, and is
also how we determine that the Feynman approach is indeed equivalent to the
other formulations of quantum mechanics. We shall follow this procedure in
the next section.
@philipthrift
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