Physical interpretation of Feynman path integral

[Feynman]

But integration over a sheme or path (i think) is a deduction from homotopy theory and algebic topology?

 

[selfAdjoint]

But integration over a sheme or path (i think) is a deduction from homotopy theory and algebic topology?

? Integration derives from measure theory. There is a link to homotopy, but I don't think homotopy is prior.

 

[Mike2]

As I recall, \phi (x,t) is the wave function of 1st quantization. And I believe \Psi (x,t) is called the "field" of second quantization. Are there any other names for \Psi (x,t), for example, amplitude of something, field of something? Thanks.

The integration over D \hat \phi is over all of "functional space", over all possible changes in the function \phi (x,t) that gets you from the starting \phi_i (x,t) to the ending \phi_f (x,t). It would appear that there is no geometry involved with this space, right? I mean, there would have to be a metric associated with this functional space in order to have geometry, right?

Just a moment... There is an "inner product" defined on these spaces, aren't there? This inner product determines amplitudes, right? So would this inner product define a metric for this space? What is this space called, Hilbert space, Fock space, ...? Thanks

 

[Feynman]

But Mr selfAdjoint I have an article which talk about the importance of the homotopy to construct the Feynman path integral

 

[Feynman]

But Mr Mike2 How we can define this measure?
And i think that the differential and algebric geometry has an important role on construction of these measure

 

[selfAdjoint]

But Mr selfAdjoint I have an article which talk about the importance of the homotopy to construct the Feynman path integral

Citation please? Link if possible?

 

[Feynman]

ok Mr selfAdjoint But this article is on french!

 

[dextercioby]

That's irrelevant.Please make a link to the article...

 

[Feynman]

ok dextercioby , but i don't know how i send this article.

 

[selfAdjoint]

ok dextercioby , but i don't know how i send this article.

If the article is online, go to it, copy the URL (in the address box at the top of your screen), then come here and paste the url into the reply window. PF doesn't even need all the link apparatus, it automatically puts url tags before and after every url it recognizes.

 

[Feynman]

This is the article

 

[Feynman]

so?please help me

 

[Feynman]

Who can i browse this article ?

 

[selfAdjoint]

Who can i browse this article ?

What article? Did you have a link?

 

[Feynman]

Finnally this is the adrres of the article:
lpt1.u-strasbg.fr/kenoufi/MEMOIRES/magistere.pdf

 

[Feynman]

so no answer?

 

[Hans de Vries]

Finnally this is the adrres of the article:
http://lpt1.u-strasbg.fr/kenoufi/MEMOIRES/magistere.pdf

The paper discusses the path-integral in so-called "multiple connected"
spaces, as opposed to "single connected" spaces. The homotopy aspect
here is not important to understand the principle of the path integral.
Elementary introduction to grasp the ideas:

"QED, the strange theory of light and matter from Feynman"

Feynman deriving Schrödingers equation from the Path-integral:

"Space-time approach to non-relativistic Quantum Mechanics"
(Rev.Mod Phys. 20, 367-387 (1948)

Which you can find reprinted here in "Feynman's Thesis"


This brand new book which was published just last month is edited by
Laurie M. Brown who wrote a breakthrough paper together with Feynman
in 1952 (!) "Radiative Corrections to Compton Scattering"
The important follow-up papers from Feynman are:

"The theory of positrons (1949)" , start of the relativistic path integral.

Followed by:

"Space-time approach to Quantum Electro Dynamics, (1949)"
All the above papers can be found in the collection:

"Selected Papers of Richard Feynman"

(Again with Laurie M. Brown as editor)Regards, Hans

 

[straycat]

Hi
What is the physical interpretation of Feynman path integral?
Thanks

Here's a little one-page summary of the FPI that I wrote after studying the Feynman and Hibbs text for a while. Go to the files section of my yahoo group and download the file: FPI.pdf.

http://groups.yahoo.com/group/QM_from_GR/files/



David

 

[Feynman]

For Hans de Vries , so wath is the physical difference between the multiple connected and single connected and where the path integral entered in this case?

 

[Hans de Vries]

For Hans de Vries , so wath is the physical difference between the multiple connected and single connected and where the path integral entered in this case?

You can find the meaning of "simply connected" and "multiply connected"
in any textbook on complex analysis. Simply connected means that you
can deform each possible path between two points into any other.
Otherwise they are multiply connected

However, This is hardly relevant for studying the path-integral in a four
dimensional world. Maybe you should ask the author here:

http://dcwww.camp.dtu.dk/~kenoufi

Since the paper is his first years master's thesis.

(When he talks about multiply connected paths, he means paths in 3D space)

-

Regards, Hans

...

Grumpy?

 

[Feynman]

So Hans de Vrie, i understood from you that this is a topology argument,
Ok i agrre this idea, but the problem is still here : What is the physical and not the mathematical signification of path integral

 

[Feynman]

And where is the signification between the topological argument and the physical interpretation?

 

[Feynman]

So no ideas please?

 

[Feynman]

mr Hans de Vries, i have looked the memory of Dr Knoufi on path integrals,
But in this memory , it is a mathematical study for path integrals,
I still search a physical interpretation

 

[Hans de Vries]

mr Hans de Vries, i have looked the memory of Dr Knoufi on path integrals,
But in this memory , it is a mathematical study for path integrals,
I still search a physical interpretation

It's a bit difficult to respond to the somewhat vague questions without
knowing your background.The simplest idea behind the path-integral is to do the same in general
what Huygens principle did for the propagation of light. This states that
light goes every but most paths cancel each other out. Leaving the
path in the direction of the wave front. This is the idea which Feynman
brings forward in his public lectures on QED.

The most important physical idea behind the path-integral is the principle
of least action. The classic Lagrangian becomes the measure of the
phase changes over the trajectory of the particle in Quantum Mechanics.

A important offshoot comes from the way how the functional integrals
are solved in the path-integral formalism. They are not solved directly
but expanded into series of which the individual terms can be solved.

What Feynman did was to associate to each such term an interaction
with virtual particles. Each term is associated with a so-called Feynman
diagram which represents the interaction.Regards, Hans

 

[Feynman]

Ok very good Hans de Vries , i undrstood you.
But i will posed a new question : so why we are obliged to create a new mathematical measure for the path integrale (physically) please

 

[Feynman]

In other term, Why we don t use Lebesgue measure ?
What is the physical utility of feynmann measure?

 

[starfield]

robphy, the lectures at http://www.vega.org.uk/series/lectures/feynman/ had some problem opening...! it was saying 'Failed to open stream. No Such File or Directory"

 

[robphy]

I don't know what is wrong... try entering by
http://www.vega.org.uk/video/series/5 or
http://www.vega.org.uk/video/subseries/8
...or else send them an email.

 

[Feynman]

It is interresant , thank you robphy.\
Do you think it is really an open problem>?

 

[Feynman]

So, My question has no exact solution .
I replay my question : First if you should speak mathematiquelly Why we are obliged to create a new measure (wich it Feynman measure), Secondelly What this measure and this Feynman integrals represent physically (Energy , momentum ,...?)?
thanks

 

[Feynman]

So no idea?
help me please

 

[samalkhaiat]

So no idea?
help me please

I am impressed, you created this thread 18 months ago!
You could have read three or four books on Path Intrgral in less than a year! You could have become an expert on the subject in 18 months!


sam

 

[Feynman]

samalkhaiat It is an open problem!

 

[Perturbation]

So, My question has no exact solution .
I replay my question : First if you should speak mathematiquelly Why we are obliged to create a new measure (wich it Feynman measure), Secondelly What this measure and this Feynman integrals represent physically (Energy , momentum ,...?)?
thanks

We start with an amplitude and identify it as a sum of paths weighted by a pure phase term

\langle x_1|e^{-iHT/\hbar}|x_2\rangle =\sum_{\mbox{all paths}}e^{i\cdot phase}

Since there are an infinity of paths between the end points we can convert the summation into a functional integral

\sum_{\mbox{all paths}} \longrightarrow \int {\cal D}x

To choose the phase term in the exponential we can identify the classical path x_{cl} by the method of stationary phase

\left.\frac{\delta(phase)}{\delta x}\right|_{x_{cl}}=0

But the classical path is one that satisfies the principle of least action

\left.\frac{\delta S}{\delta x}\right|_{x_{cl}}=0

Where S is the action. Thus it seems sensible to use the action (over h-bar) as the phase term, for which one can gain some confidence in by using this approach in the double slit experiment. Thus our path integral becomes

\langle x_1|e^{-iHT/\hbar}|x_2\rangle =\int {\cal D}xe^{i\int d^4x{\cal L}/\hbar}

That the equality holds involves more justification that I'm not going to give.

The functional integral formalism can be made use of in calculating correlation functions and the Feynman rules for field theories, where we replace the measure Dx by D(field) and the integration now runs over the field configurations.

 

[Jimmy Snyder]

What is the physical interpretation of Feynman path integral?

In the beginning of the book "QFT in a Nutshell", by A. Zee, there is a description of the physical interpretation. I will give a poor outline of the discussion, but it is no substitute for reading it in Zee's words.

In the double slit experiment, the intensity of the light at a given point on the target screen is a sum of two products. The first product is the intensity of the light that went through one of the slits times the probability that the photon went through that slit. The second product is a similar one using the other slit.

If there were three slits, the the intensity at a given point would be a sum of three products and if you increase the number of slits, you increase the number of products to be summed.

If there is a second screen with slits in it placed between the first screen and the target, then you can calculate the intensity of the light reaching each slit in the second screen and use that to calculate the intensity of the light reaching the target. You can add more screens, and the calculations are of the same type.

Add more and more screens, each one with more and more holes. Add so many holes to each screen, that the screens aren't really there at all. That is, the intensity of light at each point on the target screen is the path integral from the source to that point.

 

[Feynman]

jimmysnyder : thanks for your information,
But your are talking without the relativistic effect to the light on the paths.
So if we take the relativistic effect on the light, how can the geometric form of the paths become?

 

[ninki]

So Hans de Vrie, i understood from you that this is a topology argument,
Ok i agrre this idea, but the problem is still here : What is the physical and not the mathematical signification of path integral

Feynman, I am jumping in here, but perhaps this will help.
When an electron is directed at the reverse side of your CRT it is intuitive that the electron takes but a single path, which is a straight line, to the spot targeted by the electronics manipulating the electromagetic elements that aim the electron.
Now simply draw other non-straight lines above and below the single straight line. These squiggly lines are those that the electron may take in arriving at the targeted spot on the screen of the CRT. However, these other lines all conveniently, "cancel" and voila, the electron goes where it was intended by those engineers that designed the CRT circuitry.
As far as I can tell Feynman, the one that is presently deceased, never intended that the electron in the CRT case actually take those squiggly paths, he had something else in mind.
Consider a stern-gerlach transition experiment (R. Feynman, "Lectures on Physics" Vol. III Chapter V.). A spin-1 particle can take one of three possible paths through the inhomogeneous magnetic field of the segment

1. ,
2. "up",
3. "horizontal" or
4. "down".

Now the particle, the atom, can only take one of the three possible trajectories, or paths. However, the magnetic field induces elements of the spin state that are not locally expressed, to expand out, and actually take each of the other two trajectories that the particle does not take (but would have taken if polarized as such). This is not as confusing as it may seem.
Consider the spin-1 particle as being able to generate each of the thee possible states very rapidly. When the particle, the atom, reaches the magnetic field the spin state that is currently generated becomes the polarized, or the observed state. The unexpressed spin states remain nonlocal (meaning that '0' does not mean 'off', like a light switch turns the room light off. 0 means nonlocal, or unobserved, and one cannot assign without more, any physical significance, meaning any observable significance, to the nonlocal state). While the spin generator is generating the spin states, there are always two states that are nonlocal and that simply wait their turn until they are generated. Being generated here means that the nonlocal state is made obervable.
Now, when one sums over the possible paths that the "spin states" may take, and if we assign a probability that the particle will be in one of these states, the probability function will always sum as,
1/3(up) + 1/3(horizontal) + 1/3(down) = 1,
but the particle state, the observed state will be in only one of these states at any time while in thje magnetic field. Before polarization the spin state is a rapidly changing function. The spin state time line history looks like the following (assume 100 = up, 010 = horizontal and 001 = down):
100 010 001 100 010 001 100 010 001 100 010 001 ...
and so on, where the '1' means the current observed state, the '0' the nonlocal (or nonobserved) state. After polarization the partcle spin state time line history looks like the following (assume the 100 state is the polarized state):
100 100 100 100 100 100 100 100 ...
and so on. Summing over the paths, the 1's and 0's, is merely summing the probabilities that the particle must be in one of the allowed number of possible states, which the sum will always equal one.
Ninki

 

[Feynman]

So did the relativistic effect change the form and the physical sens of feynman path integral?