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

[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.

 

[malawi_glenn]

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

 

[zenith8]

In Quantum mechanics ALL paths contributes, and it is meaningless to ask which path it took.

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.

 

[malawi_glenn]

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:

FORMULAS

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.

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?

 

[Count Iblis]

When will you stop with these personal assaults?

Compute the amplitude:

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

using the path integral formalism.

 

[Demystifier]

an electron, for example, takes every possible path in the universe

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.

 

[Demystifier]

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.

 

[Tac-Tics]

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.

 

[Demystifier]

In Quantum mechanics ALL paths contributes,

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.)

and it is meaningless to ask which path it took.

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.

 

[haushofer]

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.

 

[malawi_glenn]

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.

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?

 

[Demystifier]

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.

 

[Hans de Vries]

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.

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

 

[Demystifier]

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.

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

 

[malawi_glenn]

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.

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?

 

[Demystifier]

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?

I'm sure we can.

 

[malawi_glenn]

I'm sure we can.

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.

 

[Demystifier]

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.

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.

 

[Demystifier]

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.

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.

 

[malawi_glenn]

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.

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

 

[friend]

Yes, I've also been told that every possible path in the Path Integral includes paths that are faster than light, and paths backwards in time. Is this true? Thanks.

 

[malawi_glenn]

Yes, I've also been told that every possible path in the Path Integral includes paths that are faster than light, and paths backwards in time. Is this true? Thanks.

https://www.physicsforums.com/showthread.php?t=321019

 

[jambaugh]

...then you can compute the propagator using only that single 'quantum' track rather than Feynman's infinite number of trajectories.
...
It's all a question of knowing the correct path/trajectory. Not a lot of people know this..

But of course if you physically block some of the other paths you do not get the same dynamics. OK so you (re)interpret this with pilot waves. But then you can't really phenomenologically separate the electron from this pilot wave. So how can you then say "the electron took one path" when in fact the electron system (particle plus pilot wave) does not?

What's more there is no unique path for situations such as a symmetric double-slit trajectory. You rather get equal contributions form two paths. Does the electron split in half?

Other than satisfying some emotional need to recast quantum physics in terms of a classical object world picture what does tacking on pilot waves add to the physics?

Remember CI doesn't so much assert the absence of e.g. pilot waves or even multiple worlds. It rather insists on agnosticism about these theological speculations. It is the same as SR's agnosticism about the unobservable luminiferous aether, or science's general agnosticism about God & friends. Assertions about the nature of reality beyond the observable is a departure from the domain of science.

Finally your reasoning about the "proper path" is not much different from say my arguing that since the area under a smooth curve can be calculated by picking a specific point on that curve and calculating the rectangular area under a constant function with that value means that "we could say that the function really is a constant defined at this point". (Note my analogy addresses the form of reasoning not the relationship between the actual systems.)

Note as well that the Feynman many paths is understood to be a method of calculation and not an ontological assertion. You may hear it said that "the particle really takes all possible paths" but this is usually either a naive misreading of Feynman or more likely they're being loose with the language in the same sense as someone speaking about the "value of a function at infinity" when it is understood that they mean the limit of its value as one approaches infinity.

 

[malawi_glenn]

Maybe we should start a new thread where we can throw in this "interpretation" "formulation" issue again? I am definitely open for such a "battle" again, but not at the cost of this thread, I think it will be come to blurry overwhelming for the OP :-)

 

[zenith8]

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.

Hi Demystifier,

I think you're missing the point actually, for several reasons (notation from my earlier post #3):

(1) In the Feynman path integral method, computing the propagator by summing over all possible paths is only half of it. The Feynman propagator K^F is a many-to-many mapping i.e. all points are linked by all possible paths. So the full \Psi({\bf x}_1,t_1) is found from Huygen's principle by summing the contributions coming from all possible start points - you multiply the amplitude at {\bf x}_0,t_0 by the transition amplitude K^F for 'hopping' to {\bf x}_1,t_1. Then you sum (integrate) over all {\bf x}_0:

\Psi({\bf x}_1,t_1)=\int K^F({\bf x}_1,t_1; {\bf x}_0,t_0) \Psi({\bf x}_0,t_0)\;\mathrm{d} x_0

In the pilot-wave method you achieve the same end as in the path integral method - the computation of \Psi given the initial value - in a quite different and conceptually simpler manner with two spacetime points connected by at most a single path. The two steps in Feynman's approach (propagator then Huygens) are thus condensed into one (propagator). \Psi is generated from its initial form by a single-valued continuum of trajectories.

(2) You don't need to know the wave function over all space to compute the propagator. You just need the second derivative of the wave function - or more accurately the second derivative of its amplitude (for the quantum potential) and of its phase (for the \nabla\cdot{\bf v} in the Jacobian J) at the points along the track. In a practical numerical method, these can be calculated by sending a particle down the trajectory {\bf v} = \nabla S (i.e. following the streamlines of the quantum probability current) and then evaluating the required derivatives numerically using finite differencing or whatever. There is a whole community of physical chemists (believe it or not) who do precisely this to solve chemistry problems.

(3) The point of my original post was anyway not to claim the superiority of the de Broglie-Bohm method over Feynman's method. I was just trying to make a conceptual point i.e. isn't it interesting that you can achieve exactly the same end result (calculating the wave function at some point in the future) by either Feynman's summing over the integrals along the infinite number of all possible trajectories in the entire universe to calculate the propagator then summing over all possible propagators so calculated for the infinite number of all possible different starting points or by er.. de Broglie-Bohm's integrating along one single track. And the latter track must be the actual path of the particle (for the quantum probability current to make sense). From this point of view it doesn't matter whose method is more numerically tractable. It's just highly interesting.

In actual fact Feynman's paths are mathematical tools for computing the evolution of \Psi, while (if particles actually exist) one among the de Broglie-Bohm paths is the actual motion of the particle as deduced from the equations of QM, which exists in addition to \Psi. Keep in mind that path integrals are not exclusive to QM; one can write any linear field equation (e.g. Maxwell) in terms of path integrals.

Note also that in one philosophical interpretation of quantum mechanics, the 'sum over histories' interpretation, the path integral is taken to be fundamental and reality is viewed as a single indistinguishable 'class' of paths which all share the same events. Given the above, you have to say Hmm to that..

 

[zenith8]

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.

Amen to that.. A point so obviously true (and one which I have made repeatedly in previous posts) that I quote it in full above just so people can read it again.

 

[zenith8]

Compute the amplitude:

<zenith8 stops assaults(t)|zenith8 is assaulting(t=0)>

using the path integral formalism.

I'm afraid that integral will give you zero, since in the post #3 referred to I clearly wasn't assaulting him, I was disagreeing with him. Unfortunately, I guess it's hard to tell the difference.

However, as he told me in a previous thread:

I am a science advisor, recognized for my many and high quality repsonses here by the moderators, and you are..?

and

also I was second in the Poll of 2008 years Physics Guru.

so I guess we should just stop disagreeing. We obviously don't stand a chance..

PS: Does anyone know who was first in the Poll? Perhaps we should try to get him on our side..

 

[malawi_glenn]

Amen to that.. A point so obviously true (and one which I have made repeatedly in previous posts) that I quote it in full above just so people can read it again.

and again you do no bother to include my reply or the fact that me and demystifer than came to an agreement.

OP asked about the FEYNMAN path integral, not the BOHM. Why should I answer something else than what the OP asked for?

And of course you didn't bother about replying to jambaugh's answer which is probably the best summary of how "unphysical" the Bohm approach is.

 

[zenith8]

James Baugh,

You make some interesting points, but your comments about Bohm's interpretation don't make much sense. And while I agree with malawi_glenn that this stuff doesn't belong in this thread, let me briefly take issue for the record (1 post only).

What's more there is no unique path for situations such as a symmetric double-slit trajectory. You rather get equal contributions form two paths. Does the electron split in half?

No, it doesn't split in half. There is an electron, and there is a wave. The electron goes through one slit, following a unique spacetime trajectory. The wave goes through both and sets up an interference pattern in itself - just like any wave would. The particle trajectories (which are influenced by the form of the wave) end up being clumped into bunches by the time they reach the screen. When the electron hits the screen, you get a little green dot.

Other than satisfying some emotional need to recast quantum physics in terms of a classical object world picture what does tacking on pilot waves add to the physics?

Accusations that we are overly emotional are just boring.

The objects in Bohm's theory are very far indeed from being classical objects - they are as 'quantum' as anything else in QM. The don't behave classically, that's the point.

The pilot-wave is not 'tacked on' to anything. It's just the wave function. (One should perhaps distinguish between the wave field - the thing that actually exists - and the wave function - the mathematical representation of the former).

The whole theory is just perfectly ordinary quantum mechanics. It follows from taking seriously the word 'particle' - which is used all the time in QM with a sort of 'not really' attached to it. Simply say that the particle exists all the time, rather than jumping into existence when you measure where it is, and the whole thing follows. There is no extra maths, or extra concepts, other than assigning objective reality to both the objects that one normally discusses as part of QM.

Remember CI doesn't so much assert the absence of e.g. pilot waves or even multiple worlds. It rather insists on agnosticism about these theological speculations. It is the same as SR's agnosticism about the unobservable luminiferous aether, or science's general agnosticism about God & friends. Assertions about the nature of reality beyond the observable is a departure from the domain of science.

Now, in the 21st century, you can detect the physical existence of the wave by matter wave optics. You can 'detect' the physical existence of particles because you can take photographs of them, manipulate them, trap them, isolate them for as long as you want, with any number of different kinds of experiment and get the same results.

In the 1920s, when you could do none of the above, it made sense to restrict the implications of QM only to the results of macroscopic observations (in terms of the CI), because the above technology did not exist.

Stating that all questions about what happens in between observations are 'meaningless' should be restricted to instrumental practical applications of the theory, and not be used to answer questions about 'meaning' (which is what Demystifier and I are complaining about).

Finally your reasoning about the "proper path" is not much different from say my arguing that since the area under a smooth curve can be calculated by picking a specific point on that curve and calculating the rectangular area under a constant function with that value means that "we could say that the function really is a constant defined at this point". (Note my analogy addresses the form of reasoning not the relationship between the actual systems.)

I agree, to some extent, and your point is?

Note as well that the Feynman many paths is understood to be a method of calculation and not an ontological assertion. You may hear it said that "the particle really takes all possible paths" but this is usually either a naive misreading of Feynman or more likely they're being loose with the language in the same sense as someone speaking about the "value of a function at infinity" when it is understood that they mean the limit of its value as one approaches infinity.

I agree with this. So when malawi_glenn says:

In Quantum mechanics ALL paths contributes, and it is meaningless to ask which path it took.

in reply to a questions about 'meaning' without making clear that it is just a mathematical tool and that the paths are not meant to be actual physical paths, you agree that he is confusing the OP?

 

[malawi_glenn]

I'm afraid that integral will give you zero, since in the post #3 referred to I clearly wasn't assaulting him, I was disagreeing with him. Unfortunately, I guess it's hard to tell the difference.

However, as he told me in a previous thread:

and

so I guess we should just stop disagreeing. We obviously don't stand a chance..

PS: Does anyone know who was first in the Poll? Perhaps we should try to get him on our side..

No you hunt me like I was some kind of decease, am I the only one stating that the question is meaningless? Then why are you explicitly saying that I am the one coming with these meaningless stuff? Any introductory QM book will tell you that such question are meaningless (in the CI, which is default).

So? I write MANY good posts, does that mean that ALL are good and world class? What has this to do with the topic? It is SOO off topic, why are you doing this again and again and again?

Why are you so obsessed with spreading the Bohm gospel even though the OP didn't asked for it? He clearly asked about FEYNMAN...

 

[malawi_glenn]

I agree with this. So when malawi_glenn says:

in reply to a questions about 'meaning' without making clear that it is just a mathematical tool and that the paths are not meant to be actual physical paths, you agree that he is confusing the OP?

Why just don't wait til the OP replies and see what is going to happen? Are you mind readers and know that he will become confused?

All paths contribute MATHEMATICALLY one should perhaps ADD to that, adding things and clarify is much better than claiming "OH NOW YOU ARE CONFUSING THE OP!" As I told you in another thread (or PM) we are doing this together, that I also wrote in this thread to demystifier. The OP has most probably NEVER HEARD that there is more ways to formulate and interpret QM. So why not just ADDING that - mention the BOHM formulation etc and that physicists mainly talk about QM with CI implicitly given as default. Surely the conversation will both be much more professional, mature and perhaps most important of all - the OP will receive answers which primarly are there for his sake - now this thread looks again like a battle between CI's and Bohm-ones where we either are assualting each other or arguing which interpretation is the most superior, thus the discussion has turned the focus to US - the guys who already know this crap =/

My approach to answering is to give a just one way to see it and make sure to make room for other guys to answer, or just give a good link to something on WWW. But I seldom write an entire essay and post it here - I find the discussion interesting - like a seminar. That is at least how I view the forums - it's more of a seminar room than a lecture hall.

 

[zenith8]

Why are you so obsessed with spreading the Bohm gospel even though the OP didn't asked for it? He clearly asked about FEYNMAN...

Now, now. Take a stress pill, malawi...

I am using Bohm to make a point about Feynman. The difference between the Bohm path integral and the Feynman path integral makes an obvious and important point about the latter, which is relevant to the OP's question (read my posts carefully).

You react like you've been shot when anyone tries to use an interpretation of QM other than Copenhagen to answer a question. Remember that now Bohr is actually dead the meaning of QM is an open question. We ought to be able to use all the tools available to us in understanding the quantum world.

We are just (gently) trying to make the point to you that using the one interpretation that denies 'meaning' on principle is perhaps the wrong one to use when people ask what QM means. I mean, maybe I'm slightly mad, but is that really controversial?

 

[malawi_glenn]

Now, now. Take a stress pill, malawi...

I am using Bohm to make a point about Feynman. The difference between the Bohm path integral and the Feynman path integral make an obvious and important point about the latter, which is relevant to the OP's question (read my posts carefully).

You react like you've been shot when anyone tries to use an interpretation of QM other than Copenhagen to answer a question. Remember that now Bohr is actually dead the meaning of QM is an open question. We ought to be able to use all the tools available to us in understanding the quantum world.

We are just (gently) trying to make the point to you that using the one interpretation that denies 'meaning' on principle is perhaps the wrong one to use when people asks what QM means. I mean, maybe I'm slightly mad, but is that really controversial?

No I just came home from the gym, LOADS of adrenaline - I could kill someone now

My point is that since the OP asked about FEYNMAN path integral, one should either WAIT til he replies or just say "in addition to what malawi_glenn told you, we have also BOHM bla bla bla.." instead of using personal attacks or whatever. You are also doing many off topic things, like invoking what I have written in other threads etc.

It would also be nice to state a thing like "what malawi_glenn means by that 'all paths contribute in QM' is mathematical contrubition' " or similar, just not "NO YOU ARE CONFUSING THE OP".

I remember when I was fresh in QM and had those questions, I was not confused with the CI, what confused me when I asked something was that sometimes this Bohm-thing went up totally un-announced = very annoying (for me).

From what I could read from the OP's post was that he asked how Feynman path integral WORKED, how the paths cancel and so on. And since (I say this again again) the OP asked for FEYNMAN path integral, I simply think that one should stick with that - I gave him the Feynman answer. If someone wants to ADD that there are more ways to formulate it and so on, please feel free - but do not think I am the confuser (the OP should be the one who judge) I think adding more and more ways and interpretations this and reformulations that is confusing (from my own experience ofcourse - you Zentih told me that your students are left consfused after your QM classes..LOL :-)

 

[zenith8]

No I just came home from the gym, LOADS of adrenaline - I could kill someone now

It's only a point of view, Malawi - calm down. I thought your issue was that other people were being aggressive?

I remember when I was fresh in QM and had those questions, I was not confused with the CI.

"Whoever is not confused by QM has not understood it." [Bohr]

Didn't you send me that last week?

from my own experience of course - you Zenith told me that your students are left confused after your QM classes..LOL [spelling corrected]

Never has Bohr seemed so apposite!

 

[malawi_glenn]

It's only a point of view, Malawi - calm down. I thought your issue was that other people were being aggressive?


"Whoever is not confused by QM has not understood it." [Bohr]

Didn't you send me that last week?


Never has Bohr seemed so apposite!

It is not that one posts about Bohm that freaks me out, it is that you think that I am the only physicist alive who takes CI as default and constantly post totally irrelevant things like what I have written elsewhere which has nothing to do with the OP question.

No it was Feynman who said it I think, Bohr said "if someone says he has understood quantum mechanics, he has not understood quantum mechanics".

I checked, Feynman said "Anyone who is not SHOCKED by QM has not understood it" :-) I kinda like mixed those two quotes into one LOL

 

[Hans de Vries]

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]


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.

How do you want to account for multi path interference with a single track?


Regards, Hans

 

[zenith8]

How do you want to account for multi path interference with a single track?Regards, Hans

Hi Hans,

See my earlier reponse to the following question by jambaugh:

What's more there is no unique path for situations such as a symmetric double-slit trajectory. You rather get equal contributions form two paths. Does the electron split in half?

No, it doesn't split in half. There is an electron, and there is a wave. The electron goes through one slit, following a unique spacetime trajectory. The wave goes through both and sets up an interference pattern in itself - just like any wave would. The particle trajectories (which are influenced by the form of the wave) end up being clumped into bunches by the time they reach the screen. When the electron hits the screen, you get a little green dot.

Repeat the operation a million times with electron trajectories starting in different positions, and the pattern of little green dots looks like the interference pattern of the guiding wave (since the particles are distributed like the square of the wave function).

 

[Hans de Vries]

The particle trajectories (which are influenced by the form of the wave) end up being clumped into bunches by the time they reach the screen. When the electron hits the screen, you get a little green dot.

Some interference patterns would seem to allow such clumped bunches but
in other cases this seems impossible. How would you for instance explain
a "last pico second destructive interference" just before the particle hits
the screen.

For instance, A detector screen with two pairs of parallel plates:

---------------- <-- detector screen
      /  \
     / /\ \
    / /  \ \
___/ /____\ \__

Two 2D wave functions (each one between a pair of plates) come together at
the detector plate. The electron can be detected everywhere on the line where
the 2D wave functions come together. (This line is orthogonal to your screen)

The particle in the Bohmian interpretation travels all the way to the screen
between one pair of plates without any influence from the wave function
between the other pair.

Then, in the last picosecond the two wavefunctions interfere and determine
where the particle might impact...

There's no way for the electron to be influenced by the other wave function
in such a way that it's gradually directed to the right area in advance. Regards, Hans

 

[zenith8]

Some interference patterns would seem to allow such clumped bunches but
in other cases this seems impossible. How would you for instance explain
a "last pico second destructive interference" just before the particle hits
the screen.

For instance, A detector screen with two pairs of parallel plates:

---------------- <-- detector screen
      /  \
     / /\ \
    / /  \ \
___/ /____\ \__

Two 2D wave functions (each one between a pair of plates) come together at
the detector plate. The electron can be detected everywhere on the line where
the 2D wave functions come together. (This line is orthogonal to your screen)

The particle in the Bohmian interpretation travels all the way to the screen
between one pair of plates without any influence from the wave function
between the other pair.

Then, in the last picosecond the two wavefunctions interfere and determine
where the particle might impact...

There's no way for the electron to be influenced by the other wave function
in such a way that it's gradually directed to the right area in advance.


Regards, Hans

Well, a picosecond is a long time in the quantum world..

I'm not sure I understand your point though. While traveling down channel 1 the particle will be influenced only by wave function 1.

If wave function 2 in the other channel never overlaps with wave function 1, then it is not an interference experiment and the particle will go all the way to the screen influenced only by wave function 1.

If wave function 2 does overlap with wave function 1 in the region in front of the screen, even for a short time, then their superposition will have a different shape to either of the components, and the trajectory of the particle will be affected. Now clearly it will be less affected than in the standard double-slit experiment configuration, where there are no 'channels', and so?

Do you have a particular experimental result in mind which has some bearing on this, or is it just a thought experiment? How would you expect the Bohmian result to differ from that of standard QM (which I'm not sure it can, since the particles are only following the streamlines of the quantum probability current)?.

I apologize if I'm being slow - I'm just not getting the point.

 

[Hans de Vries]

If wave function 2 does overlap with wave function 1 in the region in front of the screen, even for a short time, then their superposition will have a different shape to either of the components, and the trajectory of the particle will be affected.

It's a thought experiment,

The particle will travel towards some point at the line at the
detector plate where the two 2D wave functions interfere.

Then, at the last moment, it finds itself in a valley of the
interference pattern with a very small chance of detection.

The particle should then take a 90 degrees sharp turn, either
upwards or downwards, and then land at a position of the
line where there is a high probability of detection.

This can be very far away, say half the length of the vertical
line on the detector screen. This might add 50% or so to the
total path length, but the particle would need to bridge this
gap almost instantaneously.Regards, Hans

 

[zenith8]

Then, at the last moment, it finds itself in a valley of the
interference pattern with a very small chance of detection.

Hi Hans,

The particle should then take a 90 degrees sharp turn, either
upwards or downwards, and then land at a position of the
line where there is a high probability of detection.
Regards, Hans

An important point might be that - given the mathematical form of the trajectory equation - the particles cannot travel through the nodal surfaces in the wave function (i.e. hypersurfaces on which the wave field is zero). If a trajectory is heading straight for one, then it will indeed make a sharp turn in the opposite direction on close approach to the node. I mean, I can show you plots if you like..

I'm still not sure what your point is..? Is the Bohmian result supposed to differ from the orthodox QM one? Are the trajectories not believable?

Why are you bringing time into this (with your talk of 'picoseconds' and 'bridging gaps almost instantaneously'). Are you making some inferences about how long the process takes ('time of flight'! Not allowed in orthodox QM.)? The Bohmian particle can in fact speed up and slow down - v=\nabla S / m where S is the phase of the wave function for the current particle position(s) - if that helps.

 

[Hans de Vries]

Why are you bringing time into this (with your talk of 'picoseconds' and 'bridging gaps almost instantaneously'). Are you making some inferences about how long the process takes ('time of flight'! Not allowed in orthodox QM.)? The Bohmian particle can in fact speed up and slow down - v=\nabla S / m where S is the phase of the wave function for the current particle position(s) - if that helps.

Well, the wavelength represents a momentum (and thus velocity) so, with a
certain given uncertainty, one could determine the time of impact.

If the particle, at the moment that it reaches the detector, is very far away
from the region with a high probability then it's difficult to see how it gets
there in the time allowed by the uncertainty. As I said, the gap it has to
bridge might add 50% or so to the total path length.


Regards, Hans

 

[zenith8]

Well, the wavelength represents a momentum (and thus velocity) so, with a
certain given uncertainty, one could determine the time of impact.

If the particle, at the moment that it reaches the detector, is very far away
from the region with a high probability then it's difficult to see how it gets
there in the time allowed by the uncertainty. As I said, the gap it has to
bridge might add 50% or so to the total path length.

Regards, Hans

Hi Hans,

Ah, now I understand your problem! I don't think you can use uncertainty arguments in pilot-wave theory in that way.

In a momentum eigenstate - for example - one would normally say that there is a definite momentum but the position is completely unknown (or even 'does not exist'). Unfortunately in pilot-wave theory the particle has a definite position and velocity at all times (given the position x the momentum is just given as \nabla S at x). The uncertainty principle thus doesn't have the implication normally ascribed to it. In particular it has no bearing whatsoever on the actual properties of a single particle in a single experiment, but only on the statistical scatter of results in an ensemble of similar experiments.

So in pilot-wave theory the actual momentum is only uncertain because the initial position is. I could go into more detail, but this is not the place for that. Does this shed any light on your problem?

And surely doubling the path length just means that it has to double its velocity to get there in the same time, which is well within its capabilities. Trust me, there can be no conflict with orthodox QM here, by definition.

 

[Hans de Vries]

And surely doubling the path length just means that it has to double its velocity to get there in the same time, which is well within its capabilities. Trust me, there can be no conflict with orthodox QM here, by definition.

But it can't double its velocity. Only when the particle reaches the
interference zone near the screen then it knows if it's in a valley or
in a high probability zone of the interference region. It can't go back
in time and adjust its velocity accordingly.


Regards, Hans

 

[zenith8]

But it can't double its velocity. Only when the particle reaches the interference zone near the screen then it knows if it's in a valley or in a high probability zone of the interference region. It can't go back in time and adjust its velocity accordingly.

Sorry Hans darling, you're really not making sense now. If the particle is 'near the screen' then there must be a finite amount of time left before it hits it. In that time the particle can increase its velocity and change direction. Why does it need to 'go back in time' in order to adjust it's velocity? It just gets accelerated by the local wave field.

 

[Hans de Vries]

Sorry Hans darling, you're really not making sense now. If the particle is 'near the screen' then there must be a finite amount of time left before it hits it. In that time the particle can increase its velocity and change direction. Why does it need to 'go back in time' in order to adjust it's velocity? It just gets accelerated by the local wave field.

The required speed may well be higher as the speed of light. That's why I said that it
has to do so virtually instantaneously.

The whole issue is that, in this experiment, there is no guiding wave that leads the
Bohmian particle gradually to a region of high probability. The interference occurs just
before the particle is about to hit the screen. Consequently, If the particle has ended
up at a low probability valley of the screen then is has to make a sharp 90 degrees
turn to move parallel along the screen, at a speed faster then light, to a region of high
probability. All of this seems highly unlikely.

With all sympathy for the Bohmian approach. I thinks that it is (unfortunately) not that simple.Regards, Hans

 

[Demystifier]

The required speed may well be higher as the speed of light. That's why I said that it has to do so virtually instantaneously.

In Bohmian mechanics, the particle velocity can exceed the velocity of light. And no, it does not contradict relativity. It can be shown that effective mass squared may become negative in relativistic Bohmian mechanics, and you probably know that velocities higher than the velocity of light are compatible with relativity in one allows negative squared masses (tachyons).

And yet, it can be shown within Bohmian mechanics that if you MEASURE the velocity of the particle, then you cannot obtain a velocity larger than the velocity of light. Thus, superluminal Bohmian velocities do not contradict experimental data.

Does it help?

 

[Demystifier]

Consequently, If the particle has ended
up at a low probability valley of the screen then is has to make a sharp 90 degrees
turn to move parallel along the screen, at a speed faster then light, to a region of high
probability. All of this seems highly unlikely.

Now, who is thinking classically?
It would be, of course, unlikely in classical mechanics. But Bohmian mechanics IS NOT classical mechanics.

Indeed, it occurs very frequently that the same persons first accuse Bohmians for attempting to restore classical mechanics in quantum phenomena, and then use classical reasoning by themselves to provide an argument against the Bohmian interpretation. It is really difficult to me to understand thinking of such persons.

 

[Demystifier]

By the way, I've seen nice citations on weirdness of QM in this thread, so here is another one:
"If you really believe in quantum mechanics, then you can't take it seriously."
(Robert Wald)

In this spirit, I would say that Copenhagenians are those who really believe in QM, while Bohmians are those who take it seriously. Or is it just the opposite? :-)

 

[Demystifier]

Other than satisfying some emotional need to recast quantum physics in terms of a classical object world picture what does tacking on pilot waves add to the physics?

Pilot waves offer a possible answer to the question: What happens when measurements are NOT performed?
Copenhagen interpretation does not provide ANY answer to this question.

There is one additional use of pilot waves. Even if you are familiar with the formalism of QM that allows you to calculate probabilities of possible measurement outcomes, it is easier to think about that stuff if your formal knowledge is enriched by an intuitive understanding as well. Pilot waves offer a useful intuitive picture of QM, even if this is not what really happens in nature. In fact, I have never met a practical quantum physicist who does not use any intuitive picture of QM.

Besides, there is even a purely practical-calculation use o pilot waves:
http://prola.aps.org/abstract/PRL/v82/i26/p5190_1