# Feynman's Sum Over Paths? (n00b question)

Gecko
what exactly happens in this theory? is it a list of possible paths that the particle COULD take, or is it the paths the particle DID take? also, wouldnt there be an infinite amount of paths? how do these cancel out?

## Answers and Replies

Staff Emeritus
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Feynmann's perspective is that the particle does take all the paths. But they are not "observable" so you can't tell, and argumants may arise. Yes there are infinitely many paths in the general case, although in particular experimental situations they reduce the number of possible paths to just a few. For how they can all cancel out, I highly recomment you read Feynmann's little book QED.

Feynman
But the problem is bigger .
the problem is how we can define this integral?

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Feynman said:
But the problem is bigger .
the problem is how we can define this integral?

That's a tricky mathematical issue.Physics books dealing with path integral do not explore the depth of this notion.I didn't either,though i was taught QFT by a guy who graduted math.He simply said:"it's not an ordinary integral".
That's physics,basically it uses mathematical results without questioning the way those results have been gotten to.

Feynman
But actually, the scientists try to define the path integral with algebric topology and homotopy

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For the sake of physical theory (meaning QFT:SM,its supersymmetrical generalizations and supergravity),it's not WHY does this path intergral quantizatin works (which has to do with its mathematical grounds and their connections to physics),it matters that IT WORKS,and so far,gives results corresponding to measured values.I dont know wether (super)string quantization via path integral poses problems with what lies behind this fundamental concept (mathematics,I mean).If it does,then we,as theorists,should enlarge our mathematical knowledge with algebraic topology and homotopy just to KNOW what lies behind path integrals.

It's the for the reason of discovering path integrals,that Stephen Hawking thinks of Richard P.Feynman as the theorist of all time.

String theory is actually better defined mathematically in a sense than regular field theory, since it usually deals with 2d conformal field theory. Anyone who has ever worked on the subject knows that treating 4 dimensional lorentzian manifolds + quantum mechanics has unusually hard difficulties that aren't present in lower dimensions (and sometimes higher dimensions).

As a simple example, 2d field theories are nearly always resumable, so we have nice analytic results that make for interesting roundtable talks.

masudr
Feynman and path integrals

Feynman DID NOT invent path integrals. Path integrals are a way to sum a function which values every point in n-space when taking a particular path. That's what they are. Feynman used a different kind of integral, and the terminology's confused - the two types of path integrals don't mean the same thing.

The "reason" Feynman's approach works is 'cos he tried to generalise the classical mechanics path approach (Lagrangian) to quantum theory. And it worked. More or less.

And finally, to the n00b:

1. We first decide we wanna know the probability that a particle goes from one point to another.

2. For each of the possible paths it could take through spacetime, we assign a little arrow (a complex number on the complex plane). And when we say all the paths are added, we put the little arrows tip-to-toe. The reason an infinity of paths can cancel is that if one arrow points to the left and the other points to the right, they add up to make nothing. That's how infinity paths cancel.

3. We find the length of the arrow of whatever's left over, and then we square it and then we get the probability of the particle going from that one point to the other.

We have not calculated the probability of a particle taking a particular path. We have calculated it going from one point to the other. If we want to know the probability of it going from A to C to B, we need to do those parts separately, then multiply and so on.

Do read QED, it's brilliant. Forget what Stephen Hawking thinks (well not everything he thinks) I think Feynman's one of THE greatest theorists for his path integral formulation of quantum mechanics.

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maudr said:
Feynman DID NOT invent path integrals. Path integrals are a way to sum a function which values every point in n-space when taking a particular path. That's what they are. Feynman used a different kind of integral, and the terminology's confused - the two types of path integrals don't mean the same thing.

You're thinking of line integrals or contour integrals. Integrating a value along a path. You're right that they're not the same thing as what Feynmann used, but the term "path integral" does not mean line integral, it means Feynman's integrals over paths with a measure that's like "d(path)". You're not integrating along the paths but across different paths.

Technically, though, you're right. Feynmann didn't invent them, Dirac did. Feynmanns insight was the big introduction of path integrals into pjysics.

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masudr said:
Do read QED, it's brilliant. Forget what Stephen Hawking thinks (well not everything he thinks).I think Feynman's one of THE greatest theorists for his path integral formulation of quantum mechanics.

Thanks for quoting me. Technically, though, you're right. Feynmann didn't invent them, Dirac did. Feynman's insight was the big introduction of path integrals into pjysics.

And i got Pierre Ramond's book to show that Feynman didn't take those integrals out of the blue. caribou
I was watching Feynman's second lecture last night in which he talks all about the little arrows masudr mentions, here in the context of reflection, refraction, etc.:

http://www.vega.org.uk/series/lectures/feynman/index.php [Broken]

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