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

[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

 

[Demystifier]

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?

 

[Demystifier]

(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

 

[zenith8]

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..

 

[Demystifier]

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

It was my pleasure.

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

 

[zenith8]

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?

 

[Demystifier]

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.

 

[jambaugh]

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.

 

[Demystifier]

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.

 

[zenith8]

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

 

[jambaugh]

Sorry it took so long. (Busy busy busy!)https://www.physicsforums.com/showthread.php?p=2274454#post2274454"

I got to run.

 

[maverick_starstrider]

Lol. I love how this post has gone on for 4 pages and the OP has not replied since the first post and probably have no idea what was being said since the second post.

 

[jaketodd]

Ok, I have not read all four pages of posts so this may have already been addressed. From what I did read, the paths the particle takes (Feynman) are probability waves (probability given by amplitude) and the amplitudes interfere, some cancel out. I remember the words of one post: "a probability with a direction." And I remember someone else saying that's a good interpretation.

My question now is: These waves with amplitude travel in different directions, so they are not superimposed on each other, so how can they cancel out or interfere at all? So I would say that the electron does not take all these many paths. Instead I would favor the idea that it is a singular wave of probabilities that propagates through space and manifests as a particle according to a formula that takes into account how many thing(s) it runs into (the more, the more likely for collapsing to a particle and sooner), how far away from the source of the electron wave those thing(s) are (the closer, the more likely the manifestation of a particle there), and some degree of randomness.

 

[jambaugh]

Ok, I have not read all four pages of posts so this may have already been addressed. From what I did read, the paths the particle takes (Feynman) are probability waves (probability given by amplitude) and the amplitudes interfere, some cancel out. I remember the words of one post: "a probability with a direction." And I remember someone else saying that's a good interpretation.

My question now is: These waves with amplitude travel in different directions, so they are not superimposed on each other, so how can they cancel out or interfere at all? So I would say that the electron does not take all these many paths. Instead I would favor the idea that it is a singular wave of probabilities that propagates through space and manifests as a particle according to a formula that takes into account how many thing(s) it runs into (the more, the more likely for collapsing to a particle and sooner), how far away from the source of the electron wave those thing(s) are (the closer, the more likely the manifestation of a particle there), and some degree of randomness.

Much of how you are characterizing this is interpretation dependent. What QM says operationally is that the wave functions add linearly and thence interference patterns in the wave function of e.g. an electron can be setup e.g. via double slit experiment. If you then in that experiment measure the position of an electron prior to and posterior to "passage through the double slits" and in addition configure the experiment so as not to allow measurement of "which slit the electron passed through" then you will get a probabilistic prediction for the final position measurement expressed using the interfering wave function.

When you do multiple experiments you can confirm the probabilistic prediction via the distribution pattern of the many electron position measurements. You thus see an interference pattern in the distribution of electron position measurements.
This is all the theory predicts. We argue about what the theory implies vis-à-vis interpretation debates which also delve into the semantics of what we mean by "the electron".

Is the electron some type of field modeled by our wave-function said field behaving non-locally when it collapses during a position measurement? Is the electron a point particle guided by a Bohm pilot wave represented by the wave-function? Is the electron a point particle passing through different slits in different sub-universes? Is "an electron" shorthand for a systematic class of phenomena wherein a certain mass and charge leaves one device and enters another and should we view questions such as "which slit the electron passed through" as ill posed given that in this instance no measurements are being made which would distinguish cases? Different interpretations answer this in different ways.

Pick your favorite.