Basic path integral

[Lawrence Crowell]

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

[Philip Thrift]

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.johnboccio.com/research/quantum/notes/Feynman-Prob.pdf

etc.

@philipthrift

 

[Philip Thrift]

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

 

[Lawrence Crowell]

On Wednesday, October 16, 2019 at 5:50:32 PM UTC-5, Philip Thrift wrote:

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 

[Philip Thrift]

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

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  associated with every history leading to the event.
3. The quantum amplitude associated with a given history  is the product of the amplitudes  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  , 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  so that then  , where the  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  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  from the Davisson-Germer and G.P. Thomson experiments into the formulation, for instance, would be to ascribe a quantum amplitude of  for the propagation of a particle with momentum  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