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

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Feynman's many-paths concept suggests that an electron takes every possible path, with each path contributing to a probability amplitude that can cancel out others. This cancellation leads to a definitive probability distribution for the electron's position, rather than implying that the electron travels nowhere. The discussion highlights that while all paths contribute in quantum mechanics, the classical path becomes significant due to the action's extremum, which dominates the contributions. The conversation also touches on the de Broglie-Bohm interpretation, which posits that a single trajectory can compute the same propagator as Feynman's approach, emphasizing the mathematical equivalence of different interpretations. Ultimately, the path integral formulation serves to illustrate the wave-like nature of quantum mechanics rather than pinpointing actual particle paths.
  • #31


zenith8 said:
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.
 
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  • #32


malawi_glenn said:
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?
 
  • #33


zenith8 said:
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 :-P

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 :-)
 
  • #34


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

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!
 
  • #35


zenith8 said:
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
 
  • #36


zenith8 said:
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
 
  • #37


Hans de Vries said:
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:
jambaugh said:
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).
 
  • #38


zenith8 said:
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:

Code: 
---------------- <-- 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
 
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  • #39


Hans de Vries said:
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:

Code: 
---------------- <-- 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.. :smile:

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.
 
  • #40


zenith8 said:
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
 
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  • #41


Hans de Vries said:
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.
 
  • #42


zenith8 said:
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
 
  • #43


Hans de Vries said:
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.
 
  • #44


zenith8 said:
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
 
  • #45


Hans de Vries said:
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.
 
  • #46


zenith8 said:
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
 
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  • #47


Hans de Vries said:
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?
 
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  • #48


Hans de Vries said:
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? :wink:
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.
 
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  • #49


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? :-)
 
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  • #50
jambaugh said:
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
 
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  • #51


jambaugh said:
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.
Let me ask you a personal question:
Do you believe that the universe existed even before humans (or animals) have started to observe it? Or are you completely agnostic about that?
 
  • #52
zenith8 said:
(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.
You mean like this?
http://prola.aps.org/abstract/PRL/v82/i26/p5190_1
 
  • #53
Demystifier said:
You mean like this?
http://prola.aps.org/abstract/PRL/v82/i26/p5190_1

Exactly like that, yes. I think Wyatt wrote a book on it as well.

PS: Thanks for filling in for me while I was asleep..
 
  • #54


zenith8 said:
PS: Thanks for filling in for me while I was asleep..
It was my pleasure. :smile:

By the way, what do you think about #49?
 
  • #55


Demystifier said:
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? :-)

You ask, what do I think of #49 (quoted above)?

I don't know, I think Wald's quote is a bit glib.

The fact of the matter is that we all know that QM provides statistical data on the results of experiments - one hardly needs the 'Copenhagen interpretation' to tell you that.

So basically there is a group of people who are interested in why QM does this, and there is another group who don't care (the instrumentalist people who use QM to build things - fair enough), and there is a third very vocal group who not only don't care but actively try to suppress any attempt to find out. Why they do this is beyond me. It makes rational discussion about - for example - the meaning of paths in the various kinds of path integral (the subject of this thread as defined by the OP) almost impossible without having these people throw their toys out of their pram.

In the end, I prefer some quotes about the Copenhagen interpretation from the Cambridge lecture course I referred to earlier:

"A philosophical extravaganza dictated by despair." (Schroedinger)

"It is now well-known that Copenhagen cannot be reconstructed as a coherent philosophical framework - it is a collection of local, often contradictory, arguments embedded in changing theoretical and sociopolitical circumstances.. ..riddled with vaccillations, about-faces and inconsistencies." (historian Mara Beller)

"One would expect proponents of Copenhagen were in possession of some very strong arguments, if not for inevitability, at least for high plausibility. Yet a critical reading reveals that all the far-reaching, or one might say far-fetched, claims of inevitability are built on shaky circular arguments, on intuitively appealing but incorrect statements, on metaphorical allusions to quantum 'inseparability' and 'indivisibility' that have nothing to do with quantum entanglement and nonlocality."
(historian Mara Beller)

"[Copenhagen QM is] an idea for making it easier to evade the implications of quantum theory for the nature of reality" [David Deutsch, albeit for the wrong reasons]

By the way, have you noticed we seem to be the last two left standing. Don't tell me we've won an argument for once?
 
  • #56


zenith8 said:
By the way, have you noticed we seem to be the last two left standing. Don't tell me we've won an argument for once?
It's too early to say. Let us give them 24 hours more. :smile:
 
  • #57


Pardon my absence I had a long reply but had to be somewhere yesterday. As you (zenith8) point out this is not the thread to continue the long debate on interpretations. I intend to start a new thread to continue it.

BTW Dr. Beller is a better historian than student of QM. One expects that as CI or any interpretation is developed there will be contradictions over time and difficulty in expressing revolutionary concepts. You see the same in the unfolding of the aetherless relativistic theory of electromagnetism. Note Poincare derived the E=mc^2 formula, and other researchers had various other versions E=2/3mc^2 etc.

Again the comparison between SR and CIQM has important parallels. SR rejects the aether not because it is disproved but because it is irrelevant. This is a positivist position exactly in keeping with the CI of QM which rejects the objective state of reality for the same reason. You can find legion "kooks" out there claiming to "disprove Einstein" with their pet reinvention of the aether. They make the same predictions as SR and can't understand why they are not raised up on the shoulders of the physics community for their brilliant insight. I see Bohm's interpretation and Everett's as important for the development of CI as they point out that (up to a point) ontological interpretations are possible. I see it as the same as the emergence within mathematics of geometry as an abstract topic instead of a natural science. The demonstrations of models of Geometry sans the parallel postulates show that Euclidean geometry is not inevitable but rather one of a larger class. These models did not supplant the Euclidean case as natural science their appearance rather show that natural science is not the proper context to study geometry. One is in the context of logic rather than science and one is thus free to choose axioms freely (within the constraint of consistency).

Likewise the various divergent ontological interpretations of QM point out the inappropriateness of viewing QM in the context of ontology. It is a description of phenomena not of reality. This is unarguable and that is the essence of CI.

Oops... I intended to make a quick comment but got worked up. I'll save further comments for another thread.
 
  • #58


jambaugh said:
These models did not supplant the Euclidean case as natural science their appearance rather show that natural science is not the proper context to study geometry.
But we have a very physical theory - general relativity - that tells us that geometry may be an important part of a natural science. In particular, it is possible to decide by an experiment whether the universe is curved or not.

By the way, you haven't answered my questions in #51.
 
  • #59


jambaugh said:
Pardon my absence I had a long reply but had to be somewhere yesterday. As you (zenith8) point out this is not the thread to continue the long debate on interpretations. I intend to start a new thread to continue it.

Go on, we're waiting. Or did they already move it to Philosophy?

Zenith
 

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