# 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?

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Feynman said:
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

Mike2 said:
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 wich 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

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

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#### Feynman

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

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Feynman said:
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

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Feynman said:
What article? Did you have a link?

#### Feynman

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

#### Hans de Vries

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Feynman said:
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"

http://www.amazon.fr/exec/obidos/ASIN/0691024170/qid=1133806210/sr=1-2/ref=sr_1_8_2/403-4743641-8596453

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"
http://www.amazon.fr/exec/obidos/ASIN/9812563806/qid=1133806672/sr=1-1/ref=sr_1_0_1/403-4743641-8596453

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"
http://www.amazon.fr/exec/obidos/ASIN/9810241313/qid=1133807324/sr=1-2/ref=sr_1_0_2/403-4743641-8596453
(Again with Laurie M. Brown as editor)

Regards, Hans

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#### straycat

Feynman said:
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?:grumpy:

#### Hans de Vries

Gold Member
Feynman said:
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?:grumpy:

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 [Broken]

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

Feynman said:
:grumpy:.....
Grumpy?

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#### 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

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

Gold Member
Feynman said:
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

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

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