# A silly question I'm sure about Feynman's many paths

1. Jul 13, 2009

### jaketodd

A silly question I'm sure about Feynman's "many paths"

I've been reading about Feynman's many-paths idea. And I've read that according to his idea, an electron, for example, takes every possible path in the universe and they cancel out (the arrows pointing different directions) to a definite path. But if the electron took every possible path, then wouldn't they all cancel and the electron would go nowhere since the arrows point in every direction and would cancel completely to no path? Don't be too hard on me.

2. Jul 13, 2009

### malawi_glenn

Re: A silly question I'm sure about Feynman's "many paths"

This was a very oversimplification view on Feynman path integrals, I assume this was not mentioned in a physics textbook?

The thing you mention is the classical regime of the path integrals, that only ONE path contribute. In Quantum mechanics ALL paths contributes, and it is meaningless to ask which path it took.

Each path is weighted by exp(iS/hbar), where S is the classical action. You are thinking of the paths as vectors, which is not true. In the classical regime, the action S is much larger than hbar - the typical QM scale - and hence the path integral will receive a fast oscillating phase making that paths will interfere and cancel out and only the classical path (where S is extremum) will contribute.

http://en.wikipedia.org/wiki/Path_integral_formulation

http://www.quantumfieldtheory.info/Path_Integrals_in_Quantum_Theories.htm [Broken]

Last edited by a moderator: May 4, 2017
3. Jul 13, 2009

### zenith8

Re: A silly question I'm sure about Feynman's "many paths"

Here we go again with your 'meaningless' stuff.

The main point about Feynman's theory is to calculate the propagator (essentially, the thing that enables you to calculate the wave function at some space point and time in the future, given the wave function at some space point and time now).

It's very interesting to note that if you subscribe to the view that electrons have trajectories (i.e. the de Broglie-Bohm interpretation) and you use the obvious trajectory implied by the quantum formalism, then you can compute the propagator using only that single 'quantum' track rather than Feynman's infinite number of trajectories. Perhaps the OP won't be able to follow the meaning of the equations, but he can certainly appreciate the similarity between the following formulae for the propagators:

BOHM

$$K^Q({\bf x}_1,t_1;{\bf x}_0,t_0) = \frac{1}{J(t)^ {\frac{1}{2}} } \exp\left[{\frac{i}{\hbar}}}\int_{t_0}^{t_1}L(t)\;dt\right]$$

FEYNMAN

$$K^F({\bf x}_1,t_1;{\bf x}_0,t_0) = N \sum_{all paths} \exp \left[\frac{i}{\hbar}\int_{t_0}^{t_1}L_{cl}(t)\;dt \right]$$

In the Feynman case the propagator linking two spacetime points is calculated by linearly superposing amplitudes $$e^{iS/\hbar}$$ obtained by integrating the classical Lagrangian $$L_{cl}(t)={\frac{1}{2}}mv^2-V$$ associated with the infinite number of all possible paths connecting the points.

In the de Broglie-Bohm pilot-wave approach, you achieve the same effect by integrating the quantum Lagrangian $$L(t)={\frac{1}{2}}mv^2-(V+Q)$$ along precisely one path (the one the electron actually follows). Here Q is the potential associated with the quantum force (the particle being pushed by the wave function).

It's all a question of knowing the correct path/trajectory. Not a lot of people know this..

Note finally that knowing this elevates the de Broglie-Bohm theory from being an 'interpretation' to a mathematical reformulation of quantum mechanics equivalent in status to Feynman's.

4. Jul 13, 2009

### malawi_glenn

Re: A silly question I'm sure about Feynman's "many paths"

I will, as I have stated many times, go for the default, Copenhagen interpretation/formulation is default and implicit when one says QM.

He specifically asked also for Feynman approach, not Bohm..

And it is not "my" meaningless stuff, it is the meaningless stuff inherited in the Copenhagen formulation of QM used by the majority of physics community. When will you stop with these personal assaults?

5. Jul 13, 2009

### Count Iblis

Re: A silly question I'm sure about Feynman's "many paths"

Compute the amplitude:

<zenith8 stops assults(t)|zenith8 is assulting(t=0)>

using the path integral formalism.

6. Jul 13, 2009

### Demystifier

Re: A silly question I'm sure about Feynman's "many paths"

This is certainly not true. Electron either takes exactly one path (as in the Bohmian interpretation) or does not take any path at all (as in the Copenhagen interpretation). What is true in both interpretations is that the wave function can be calculated such that certain quantity is mathematically calculated over all paths and that results obtained from different paths are summed up. However, this mathematical method for calculating the wave function (or more precisely, the propagator) has nothing to do with actual particle paths. In fact, this mathematical method can be used for solving any first-order linear differential equation, which, in general, may have nothing to do with quantum mechanics and particles.

Last edited: Jul 13, 2009
7. Jul 13, 2009

### Demystifier

Re: A silly question I'm sure about Feynman's "many paths"

Zenith8, even though, as you probably know, I am also a supporter of the Bohmian interpretation, I must criticize the assertion that in the Bohmian approach one can calculate the propagator with one path only. Namely, to do that one must first know the quantum potential. But to calculate the quantum potential, one must first know the wave function. But to know the wave function, one must first calculate the wave function by a more conventional method, e.g., by the path-integral method that involves a sum over ALL paths with classical action. In this sense, the Bohmian approach with one path only does not simplify the calculation of the propagator.

8. Jul 13, 2009

### Tac-Tics

Re: A silly question I'm sure about Feynman's "many paths"

Forum drama aside....

OP: When Feynman says "cancel out" he doesn't mean the paths themselves. He is talking about the amplitudes.

For each possible path an electron could take, there is an amplitude assigned to it.

An amplitude is a quantum probability, given as a complex number. Instead of saying something has a 50% chance of happening, you say it has an amplitude of $$\frac{1}{\sqrt{2}} e^{i \frac{\pi}{4}}$$. We use complex numbers because they are convenient, but intuitively, you should think of it as a "probability with a direction". When you square the length of an amplitude (its modulous or "absolute value"), you get the classical probability for it.

The key difference between a probability and an amplitude is how they add up. When you add classical probabilities, your chances always increase. If I have a 1% chance to win the lottery (and those are fantastic odds, btw!), I always increase my chances by buying more tickets. In QM, though, because amplitudes have a direction associated with them, adding them together doesn't always increase their length. In other words, when you play the quantum lotto, buying more tickets might increase you chances, but it could also lower them as well.

So back to physics. An electron moves. Each possible path has an amplitude. When you sum together the amplitudes for each path, you find that most of them end up canceling out. The paths that don't dictate the probability distribution of finding the particle in a given place. The details are encoded into the formulas given by the above posters, but I thought I'd put the layman's explanation out there, since it sounds like you aren't quite ready for the ugly details yet.

9. Jul 13, 2009

### Demystifier

Re: A silly question I'm sure about Feynman's "many paths"

Such a claim is at best misleading. First, such a claim is meaningfull only within one (among many) formulations of QM. Second, many phenomena in classical physics (such as classical Brownian motion) can also be calculated with the path-integral method, so would you say that in classical physics all paths contribute as well? (Of course you wouldn't.)

Generally, it is not meaningless to ask this question, unless you assume that you know what is the correct interpretation of QM. But nobody knows that yet (even though many prefer some interpretations over the others), so this question is still meaningless and legitimate. The correct answer may be - neither (we do not know yet), but even if we knew that it was the right answer, the question would still be meaningfull.

10. Jul 13, 2009

### haushofer

Re: A silly question I'm sure about Feynman's "many paths"

I also had a question about this here a time ago; maybe it's best to think about these paths in configuration space and think in terms of waves, not in terms of pinpointed particles.

11. Jul 13, 2009

### malawi_glenn

Re: A silly question I'm sure about Feynman's "many paths"

Given the status of the OP, and that I always choose Copenhagen formulation, all paths will contribute weighted by the exponent of i times the action divided by hbar. Saying so does not rule out that Path integral formalism can be used for classical phenomenon, such as Brownian motion. I only tried to give an answer from the pragmatic point of view which is often the first view one starts with.

Same with the "meaningless", I always answer in Copenhagen, that is default and is what is taught in first classes in QM, which I hope and persume that the OP will do one one day.

I wonder what has made the OP really confused? This whole discussion of the different interpretations and formulations of QM perhaps?

12. Jul 13, 2009

### Demystifier

Re: A silly question I'm sure about Feynman's "many paths"

Malawi_glenn, it is fine to be pragmatic and to assume the Copenhagen approach when one asks a practical question.
However, jaketodd clearly does not ask a practical question, but a conceptual one. A conceptual question requires a different type of answer. If you are not interested in conceptual questions (because you find them irrelevant or whatever) then you should leave answering such questions to others. Otherwise, you confuse non-expert readers who cannot easily distinguish between practical and conceptual questions and answers.

13. Jul 13, 2009

### Hans de Vries

Re: A silly question I'm sure about Feynman's "many paths"

Indeed.

Technically it's not about "the paths which the particle takes" but about how
the wavefunction propagates. All the paths arise because each point of the
wavefunction acts as a new source for propagation.

In this regard the path integral does not depend on the interpretation of QM.

Regards, Hans

Last edited: Jul 13, 2009
14. Jul 13, 2009

### Demystifier

Re: A silly question I'm sure about Feynman's "many paths"

I agree. The path integrals tell us about the wave aspects of quantum phenomena and do not much help to understand their particle aspects.

15. Jul 13, 2009

### malawi_glenn

Re: A silly question I'm sure about Feynman's "many paths"

OP clearly asked about FEYNMANS "many-path idea", and I gave him the answer which can be found by reading about Feynmans own explanations of his "idea". Feynmans approach was "pragmatically conceptual" one could say. If someone ask about Feynmans way, why should I give him/her Bohms way?

Now to do this fair, one COULD mention that there are other ways to see this than Feynmans way of explaning this, like you guys did - introducing Bohmian way - but can we do this without personal assaults?

16. Jul 13, 2009

### Demystifier

Re: A silly question I'm sure about Feynman's "many paths"

I'm sure we can.

17. Jul 13, 2009

### malawi_glenn

Re: A silly question I'm sure about Feynman's "many paths"

So instead of calling my answer "confusing" one can say that the answer is not complete, since there are a lot more ways both to interpret QM and formulate it. And that the answer is accordance to the mainstream, and there included Feynmans original, interpretation and formulation.

But what we all agree on is that one should not think the paths as paths in space, like vectors, but rather something like "probability with direction" as Tac-Tics formulated it, which actually was the main question of the OP, how this "path-assignment" works in detail and conceptually.

18. Jul 13, 2009

### Demystifier

Re: A silly question I'm sure about Feynman's "many paths"

In fact, I think Feynman originally believed that, in some weird sense, particles really DO take all these paths at once. But later he gave up of such a weird interpretation and accepted the path-integral method, as most physicists today do, merely as a calculation tool.

19. Jul 13, 2009

### Demystifier

Re: A silly question I'm sure about Feynman's "many paths"

Well, it is certainly not my intention to insult you. Still, the claim that paths should be understood as "probabilities with direction" is still confusing to me. It's nothing personal against you, but such an explanation is confusing to me.

20. Jul 13, 2009

### malawi_glenn

Re: A silly question I'm sure about Feynman's "many paths"

Well by assault I was mainly considering Zenith, who thinks that I am the only person in the world that uses Copenhagen as default.

Yeah, but that is Toc-Tics explanation, I found it good - but is perhatps that I am so used with this probabilistic chat around QM. What "is happening" (on the mathematical level) is that there is destructive interference..

Maybe one should wait til/if the OP returns, QM tends to become very mathematically