No contradiction: the integral is over all paths, that satisfy the boundary conditions, but that do *not* solve the classical equations of motion. For such paths the constraints that are relevant for physical states, precisely, do not apply. So it's wrong to expect that trajectories, that are not solutions of the equations of motion, respect any such constraints. Nevertheless, these paths do contribute to the transition amplitude and if one tries to only include the classical solution-the uniquely defined path, that solves the equations of motion, subject to the boundary conditions, one gets wrong results; whereas, properly taking them into account, as the path integral prescription states,  can be checked and has been checked experimentally any number of times, in all possible situations, whether it's non-relativistic quantum mechanics or quantum field theory.  Once more, history is fascinating, but science does move beyond what was written at any one time. Cf. http://www.amazon.com/Integrals-Quantum-Mechanics-Oxford-Graduate/dp/0198566751 for instance. 

Dear Sofia, can it be so K(rA, rC)=0 for all rC > rA+c (tC - tA)? Then no contradiction is there. Please acknowledge this. Bye!

To Remi and Dmitri,

First of all thanks for your comments. Both your proposals are interesting.

I read a good part of the book of Feynman and Hibbs (F&H) - by the way, an excellently clear book - and strangely enough, there was no thought on such a problem as I ask about.

Let me say an additional worry of mine. In the coefficient of the kernel K in many cases (e.g. free particle, particle in harmonic potential or in constant field) appears the rest-mass of the particle. Though, at high velocities, the rest-mass should be replaced by a relativistic mass. Though, neither in the initial steps when F&H constructed the idea of kernel, was there any concern for such a problem.

Then, could it be that actually, F&H don't mean ∞ to be as far as to consider relativistic velocities? I.e. could it be that their theory is just strictly non-relativistic?

With thanks again. 

The formula (1) seems to refer to non-relativistic quantum mechanics. A path integral approach to relativistic quantum mechanics is not that straightforward: Since particles can be created and annihilated, paths may split and rejoin, and even form loops in space-time. Hence it is more common  to start from functional integral formulations.

Although each term of the resulting Feynman diagram expansion may be given some kind of (multi-)path integral interpretation, involving virtual particles (which do not satisfy the classical energy-momentum relation for a particle of definite mass), there would be one type of integral for each topologically different diagram. And that may still not provide the complete result, since there may be non-perturbative contributions as well. 

There is a subset of cases, corresponding to linear quantum field theories, where a single path integral expression (for the propagator) is enough. It differs in several respects from the non-relativistic expression. But the main point of Sofias question remain: The integral will involve paths which sometimes exceed the speed of light, and which may even move backward in time (reinterpreted as an antiparticle motion forward in time).

As Remi points out, in a stationary phase approximation only paths corresponding to classical motion (of a single particle or antiparticle) should normally survive. But there are exceptions for external field configurations leading to 'Klein paradox'-type situations, where spontaneous creation of particle-antiparticle pairs becomes possible. A nice discussion can be found in A. Hansen and F. Ravndal, Phys. Scr. 23, 1030 (1981). 

Nevertheless, causality is maintained in relativistic quantum dynamics!

folk.uio.no/finnr/talks/Klein-Paradox.pdf

Beautiful notes by Ravndal! The historical part is most interesting.

``The path integral theory admits faster than light velocities''. Indeed it does, but there is nothing strange about that: the ``path integral theory'' is strictly equivalent to the non-relativistic Schroedinger equation. And the latter, of course, allows for faster than light propagation, although with very low probability: if you start out with a wave function strictly located at the origin (or having a small compact support around the origin, if you wish to avoid delta functions). Then at arbitrarily short time the wave function is different from zero everywhere. So if you have an arbitrarily distant detector, say further than cT, it has a nonzero probability of detecting the particle at time T. Of course, this is a point in which the Schroedinger equation, both in its traditional formulation and in the path integral formalism, are wrong. Fixing this starts out with the Dirac equation, the path integral formulation of which does not contain faster than light paths.

Leyvraz> Fixing this starts out with the Dirac equation, the path integral formulation of which does not contain faster than light paths.

I would like to see a demonstration of that.

@Kare: I was thinking of the Feynman checkerboard model. I did not remember that it was only for one dimension. Sorry for the carelessness.

I was only vaguely aware of this model; it is not clear to me if (or how) it can reproduce standard quantized Dirac field theory in 1+1 dimensions.

The most useful (and used) propagator in QFT is the Feynman propagator, which do not vanish outside the light-cone. Hence, any kind of path integral representation for it must include faster-than-light paths. There are other propagators, which vanish outside the light cone, but I am not sure how useful they are for QFT applications (or if they have any computable path integral representations).

No path is an event. 

No contradiction: the integral is over all paths, that satisfy the boundary conditions, but that do *not* solve the classical equations of motion. For such paths the constraints that are relevant for physical states, precisely, do not apply. So it's wrong to expect that trajectories, that are not solutions of the equations of motion, respect any such constraints. Nevertheless, these paths do contribute to the transition amplitude and if one tries to only include the classical solution-the uniquely defined path, that solves the equations of motion, subject to the boundary conditions, one gets wrong results; whereas, properly taking them into account, as the path integral prescription states,  can be checked and has been checked experimentally any number of times, in all possible situations, whether it's non-relativistic quantum mechanics or quantum field theory.  Once more, history is fascinating, but science does move beyond what was written at any one time. Cf. http://www.amazon.com/Integrals-Quantum-Mechanics-Oxford-Graduate/dp/0198566751 for instance. 

Dear friends,

In formulating their theory of path integrals, F&H speak of trajectories. But, in QM, the concept of continuous trajectory for a particle does not exist at all. Even before examining the velocities, as in my question, don't we have an inconsistency from the very beginning?

Next, Kåre opened a very interesting issue. He said:

"Since particles can be created and annihilated, paths may split and rejoin, and even form loops in space-time."

Going with his idea, let's admit that to the path integral can contribute two trajectories that are identical between tA and tC, however, between tC and tB they differ. The meaning is that going with their common part, there exists some time, let's name it tD, at which the derivative dr/dt from the left (t < tD) is unique, while from the right (t > tD) takes two different values. In the classical mechanics such a thing is not allowed. But in QM, where there is no concept of continuous trajectory with unique dr/dt at each time, such a possibility exists. However, if a trajectory splits, why does that imply that an additional particle is created? Even in the 1st quantization the trajectories are allowed to have a non-unique value for dr/dt (uncertainty principle).

So, why should we have in the above case a creation of particle? And what happens with the energy and linear momentum conservation? Do they still hold?

In the path integral formalism position and momentum are classical variables. Quantum fluctuations are described by the interference effects produced by the *classical* paths that *don't* satisfy the classical equations of motion. While Feynman and Hibbs is wonderful for the insights, the calculations are assumed to be done, first and many times.Then one sees that the transition amplitudes do, indeed, come out as they should. 

Energy and momentum conservation hold, because the transition amplitude is invariant under the change of variables that the translation in time and space imply. 

All these issues are described in detail in Zinn-Justin's book and in any lecture notes, e.g. http://web.mit.edu/dvp/www/Work/8.06/dvp-8.06-paper.pdf

The path integral for a relativistic particle is, of course, also, known, e.g. http://www.phys.columbia.edu/~kabat/strings/Spring08/handout2.pdf

(Relativistic invariance implies that a finite number of particles can't interact, a result that was pointed out by Leutwyler, http://link.springer.com/article/10.1007%2FBF02749856#page-1 )

Stam,

what you say is amazing. It seems contradictory with QM. In the path integral formalism the trajectories are classical? If so it is, then its proof of the formalism is wrong!!!!

Let me tell you that I also had the impression of classical trajectories, as you say, but I thought that maybe I misunderstand something.

Then, what happens here, is the formalism wrongly proved? Please make light in the issue!

My kind regards!

The formalism is correct and correctly proved. It's not contradictory with quantum mechanics, it's another formalism for describing the interference effects of quantum mechanics and calculating the transition amplitudes. It does identify the quantum effects in a certain way and it is possible to prove that this is consistent. There are *many* ways of calculating things and the result matters, not the method itself. 

Stam,

While you were sending your comment, I was posting my additional comment. So, you say

"In the path integral formalism position and momentum are classical variables."

But, that's illegal ! It's against the uncertainty principle ! What's going here?

About quantum fluctuations, I didn't arrive at that issue, I am still stuck with the problem no. 1, i.e. how can one speak, in QM, of trajectories. F&H didn't construct their formalism for classical QM, where trajectories are a correct concept, but exactly for QM where trajectories don't exist.

So, let's try to find an answer to this problem, and then we could go on.

Third, does it seem to you that one should read a book for getting an answer to a qualitative question like mine? I read the F&H book. If people should read one more book, and one more book, for finding an answer, what for is the RG? Books are for details, for mathematical elaborations, not for major conceptual questions.

Kind regards.

Stam,

your words

"The formalism is correct and correctly proved. It's not contradictory with quantum mechanics, it's another formalism"

are just as telling me, "don't worry, it's O.K." But you know that this is not enough. As to "the result matters, not the method itself", I have difficulty in believing that you would accept the conclusion of a wrong proof.

However, I think that I found the answer why Feynman can use trajectories: because they are NOT classical. So, I have to disagree with your saying

"In the path integral formalism position and momentum are classical variables."

Please see: in classical mechanics, in which indeed particles move on continuous trajectories, the position r(t) has to be continuous and admit a 1st derivative, dr/dt, finite and unique at any point on the trajectory. To the contrary, in Feynman's formalism, as Kåre pointed out, a trajectory can split, or trajectories can join. In fact, a trajectory can split into an infinite number of paths, because, as formula (1) shows, rC sweeps all the space . The meaning of this infinite splitting is that given a fixed point rA where the particle may be at a given time tA, at a later time tC the particle can be anywhere in the space. Thus, the velocity is undefined - uncertainty principle.

So, at least this doubt is removed. And thank you for leading to this issue, it was worthy to elucidate.

Kind regards

If analyzed in detail, the path integral constructs a continuous up and down jumping between configuration space and Fourier space. In Fourier space the displacement generator works in a multiplicative fashion. If this is applied in configuration space to a coherent location swarm, which implements a stochastic hopping path, then this results in a series of multiplication terms that represent the sequence of the hops. The swarm is considered to be described by a continuous location density distribution, which is supposed to have a Fourier transform. As a result the swarm owns a displacement generator and at first approximation it can be considered to move as one unit. The swarm is regenerated in a cyclic fashion and it is assumed that during that cycle the displacement generator can be considered to be approximately constant.

We place the happening in the realm of a Hilbert space. Now the n-th hop is represented by three factors:

A. The inner product between the eigenvector that corresponds to the start location of the hop and the eigenvector that represents the displacement generator p. This represents the jump into Fourier space.

B. The factor that represents the displacement. This factor is e to the power of the the inner product of the displacement generator p and the displacement vector that represents the hop.

C. The inner product between the eigenvector that represents the displacement generator p and the eigenvector that corresponds to the end location of the hop. This represents the jump back to configuration space.

In the representation of the subsequent hop the inner product that corresponds to the end point of the previous hop and the inner product that corresponds to the start of the current hop compensate each other and become a unity factor. Further the factors of the contributions in Fourier space can be combined. This means that the exponent becomes a summation.

When this is done for all subsequent hops then the summation represents the displacement of the center of the full swarm.

This interpretation does not imply all possible paths, but only the hopping path that is represented by the coherent location swarm. The location density distribution that describes the swarm corresponds to the squared modulus of the wave function of the object for which the swarm represents the spatial map of its hectic hopping behavior,

If this interpretation is correct, then the suggestion that the path integral integrates over all possible paths is wrong. It only integrates over the hopping path that is generated by a mechanism that controls the recurrent regeneration of the location swarm that characterizes the corresponding elementary object. The mechanism obviously uses a stochastic process that implements its activity.

An example would be a Poisson process, which is coupled to a binomial process, which is implemented by a Gaussian spatial spread function.

http://www.e-physics.eu/FoundationOfAMathematicalModelOfPhysicalReality.pdf

Sofia> So, why should we have in the above case a creation of particle? And what happens with the energy and linear momentum conservation? Do they still hold?

Maybe one could turn the question around: It is an experimentally observed fact of nature that particles can be created and annihilated. F.i., a neutron, classically moving along a single path, will at some random point decay into a proton plus an electron plus an  antineutrino, a configuration which must be described by three paths. Apparently we don't have a proper description for that in Classical Mechanics (but such a description can probably be constructed by extending Classical Mechanics). However, such a description comes naturally in QFT, by giving a path integral interpretation of Feynman diagrams. And, in the momentum space representation of such diagrams, energy and momentum is conserved. At the cost of having particles which do not obey the energy-momentum relation of classical particles: p_0^2 - p_x^2 - p_y^2 -p_z^2 will  mostly differ from (mc)^2.

As for the nature of the paths I think it is a bit tricky. In the conventional configuration space path integrals of Feynman, there are no damping factors which exclude very wildly behaved paths. But, by going to imaginary time one arrive at the mathematically well behaved path integrals of Wiener and Kac, where only continuos (nowhere differentiable) paths contribute (have non-zero measure). Maybe the real-time version can be made mathematically respectable by introducing an i\epsilon factor, with finite \epsilon, and taking the limit \epsilon-> 0^+ later. But I am not aware of any proofs along such lines.

Phase space path integrals are even more tricky, since they will not have support on continuous paths --- hence they should probably not even be called paths. But I once heard a talk by John R. Klauder, where he indicated how one may arrive at a respectable formulation by considering a particle in a strong magnetic field, eventually going to the limit of an infinity strong field (which can turn two configuration space dimensions into two phase-space dimensions -- with quantum non-commuting coordinates).

In non-relativistic quantum mechanics there's no particle creation or annihilation, so it's not useful to bring such processes in the discussion. And while terms like ``non-differentiable'' may sound imposing and erudite, it's useful to focus on the subject at hand:

The statement of the Feynman path integral is very simple: One considers all possible paths between an initial position and a final position. For each path it's possible to compute the action. This is a number associated to each path and is its phase, in Planck constant units. One sums all these contributions and obtains the transition amplitude. 

There are three, related, subtleties: are all paths, of fixed endpoints, differentiable? How should one take them into account? And how should the sum be carried out? 

The answers to all three, in fact, are provided by the realization that this is a quantum field theory in one dimensional spacetime. Therefore the usual Feynman rules apply, for perturbative calculations and the usual lattice constructions and Monte Carlo methods apply for non-perturbative calculations. 

``While you were sending your comment, I was posting my additional comment. So, you say

"In the path integral formalism position and momentum are classical variables."

But, that's illegal ! It's against the uncertainty principle ! What's going here?''

Actually, there are several different kinds of path integrals. Stam seems to be referring to path integrals in phase space. These are indeed somewhat ill-behaved, as a consequence of the uncertainty principle.

Feynman & Hibbs refer to a path integral in configuration space. This means they consider all paths going from x0, t0 to x1, t1. All paths means all continuous paths. In general, these are nondifferentiable. Indeed, the typical path looks like the limit of a curve that at every time increment dt moves randomly up or down by a change in position dx, in the limit where  dt->0 and (dx)^2/dt remains constant. So one can picture nonrelativistic QM in terms of paths, but the particles following these paths do not have well defined velocities. Further, what actually happens depends on the combined effect of what happens to the particle as it goes through all paths, in much the same way as in the two-slit experiment.

The equivalence between path integrals and ordinary nonnrelativistic QM is straightforward. So the path integral shows us, how one might view nonrelativistic QM in terms of paths in classical configuration space. The paths themselves, however, are highly nonclassical, for starters because they generally have no velocity at any point.

And again, there is  no consistency issue with faster than light propagation: that happens both in nonrelativistic QM in its ordinary formulation, and in the equivalent paths integral formalism. How path integrals in QFT avoid this problem is something I do not understand.

Dear Kåre,

I want to concentrate on a statement of yours (in some message) which is especially interesting. You said,

"each path integrated over is a reasonably decent one (continuous, although nowhere differentiable)."

How do we know that it is continuous? If we would examine that path with "some sort of a magnifying glass" we would discover that between two neighbor points P and P', there are intermediate points which may be even far from the segment PP'. So, how can we connect the points P and P' by some line and get a continuous trajectory? Even worse, a super-magnifying glass may show that between neighbor points revealed by the magnifying one, there are other points wildly positioned.

Let me tell you the motivation behind my question. I have a great pleasure and admiration for the so-called Hardy's paradox. A main conclusion of it, is that particles cannot follow continuous trajectories. I stress, continuous, which is a weaker property than possessing a derivative.

With thanks again,

Sofia

Sofia> How do we know that it is continuous?

It has been proven, I think first by Wiener, that only such paths have non-zero measure in the Wiener process. And the proof extends to the Kac integral, which is the imaginary time version of the non-relativistic Feynman path integral. I cannot recall ever having studied the proof in depth, but I think it is mostly an exercise in gaussian integrals. The situation you envisage gets vanishing probability when the points P and P' get closer and closer -- or rather when the times t and t' attached to these points gets closer and closer.

I am not sufficiently familiar with Hardy's paradox, or its Tel Aviv resolution, to say anything sensible about it. But keep in mind that the properties I mentioned refer to the imaginary time path integrals of Wiener and Kac, not the real time path integrals of Feynman.

Kåre,

It's not Tel-Aviv here, it's Jerusalem, but besides that it's also München, Zurich, Boston, Trieste, and others. There is a long list of famous researchers who deal with the foundations of QM, and draw from this so-called paradox the same conclusion, namely, that to admit continuous trajectories for particles is problematic. To put it in short, if at a certain time t we detect a particle at one position r, nothing guarantees us that immediately before or after t, one would have detected the particle in the immediate neighborhood of r. By the way, this is what makes Bohm's mechanics problematic.

Now, I see what you say by Wiener proof, I am not sure whether it answers my question s.t. if you can give a reference, I'd be glad to see that proof.

With thanks in advance,

Sofia

Sofia> It's not Tel-Aviv...

I mixed up some history (demonstrating my incompetence on the matter), and stand corrected. 

I was thinking of the resolution by Toronto quantum physicists Jeff Lundeen and Ephraim Sternberg, using weak measurements ideas of Tel Aviv physicist Yakir Aharonov.

http://phys.org/news/2009-01-physicists-paradox-quantum-theory.html

Dear Kåre,

I know Hardy's paradox so well that in the middle of the night if I am awaken up, I can say its demonstration. It was love at first sight. And I am not the only one, I told you what said Henri Stapp. Moreover, Stapp, Shimony and Unruh had a long debate (over a couple of years) on this paradox. Asher Peres invited them to include me too, but the truth is that the paradox was NEVER solved, and the debate faded away without any conclusion. Karen Berndl, a student of Detlef Dürr (who is one of the champions of the Bohmian mechanics), gave an explanation that doesn't explain much - that Hardy's thought experiment is contextual - if you'd be interested in details, I'd give you them gladly. 

I am aware of the Lundeen & Steinberg experiment (with something called "weak measurements"). They did all sort of measurements but to solve the paradox it's IMPOSSIBLE, unless one accepts one of two ideas, each one hard to accept:

A) There exists, for the propagation of the wave-function, a PREFERRED FRAME of coordinates. I.e. the wave-function is obeyed by a quantum system, only in a certain frame, and not in the other frames.

B) the concepts of particles and continuous trajectories for them, are wrong in QM.

The first idea quarrels with the special relativity (SR), because the SR says that there exists no preferred frame. By the way, Nicolas Gisin did a couple of experiments in looking for NON-preferred frames, i.e. for a violation of the wave-function, but he found nothing.

The second idea clashes especially with the Bohmian interpretation and mechanics. But, in fact, it leads to more puzzles, quite hard to summarize in a post on RG.

So, to conclude, Hardy's paradox is the most difficult conceptual problem of QM, it reveals how problematic is von Neumann's postulate of collapse.

Well, I apologize for the long comment, and kind regards from me,

Sofia

This discussion is becoming very illuminating; I hope it goes on for a while. 

It is possible to accept that an elementary particle is always somewhere, thus with a well defined position and at the next instance it is somewhere else, while any future location is yet unknown. It means that the (point-like) particle is hopping around such that its subsequent locations form a stochastic hopping path and at the same time the landing locations form a coherent location swarm. If this swarm can be characterized by a continuous location density distribution and is recurrently regenerated, then the swarm not only characterizes the particle, It can also go together with a near to continuous motion of the particle. What is needed is the spatial and temporal averaging of the swarm and defining the effective location of the particle as the average center of the swarm. The location density distribution corresponds to the squared modulus of the wave function of the particle. In this way the swarm corresponds to a position and a momentum of the owner of the wave function. If the particle is detected, then that will occur at the last location in the hopping path. After that the swarm and the corresponding wave function collapse. If the location density distribution owns a Fourier transform, then the swarm also corresponds to a wave package. It also means that at first approximation the swarm moves as one unit. Usually, on movement, wave packages tend to disperse, but since the wave package is only attached to the swarm by an artificial mathematical trick, in practice the wave package does not disperse, but together with the swarm it is recurrently renewed. Still the described system can produce interference patterns as if the particle also represented a wave package. In fact such interference pattern would be nothing else than a potential detection pattern. Even the most sophisticated observation device will not observe any waves.

Something must generate the stochastic hopping path and the corresponding swarm. That is performed by a controlling mechanism, which uses a stochastic process in order to generate the locations. The mechanism must ensure the spatial and dynamic coherence of the swarm. No current physical theory describes such mechanisms, because physical theories do not look into what can exist underneath the wave function. Such analysis will sin against the scientific method. It is in no way verifiable by experiments.

Hi all,

I also got into the same trouble with PI in my first acquaintance with it. The problem raised by Sofia needs to be addressed.

By the way how many of the respondents here  thought about this issue before coming across this thread? Everyone?

The notion of paths in space are as non-local as the non-relativistic wave function and hence violate relativity.

Regards.

Path integrals were used to solve the Bohr magneton problem using various Riemann zeta functions.

In the end it is all constructed to make QM appear super accurate. But it all begins to look a little like the epicycles used to solve the planetary paths while maintaining Earth at the center of the Universe.

In truth it throws up even more paradoxes.

There are many contradictions to Special Relativity and to Feynman's Path Integral formulation throughout Physics, not just between the two theories.  The bigger problem in Physics is that the Theories and Experiments are so vast that no one person knows how all of it works, and I would say, no group does either.   Just look at the idea of Tensor from Einstein's "Meaning of Relativity",  obviously this idea is very vague, as many things that aren't Tensors,  just projections, have an impact on an observers Universe.  I would go so far as to say the need to have Observers, Events and Tensors, is just a patch over how the Physics of Relativity, and for that matter Electromagnetic theory actually works.   I would think it is well worth the time, to have available a massive review of the older theories, as there is a problem in the details of these theories, given how ideas have evolved since then.

Richard,

I am a retired physicist. Since 2009 I am running a personal research project that has as it target to investigate the current physical theories and compare them with a mathematical model, which is founded on the vision of John von Neumann on the applicability of Hilbert spaces in the description of quantum physical systems. This vision was put in the 1936 paper of Garret Birkhoff and John von Neumann on quantum logic and its relation to Hilbert spaces. You can find the results of my research on my personal e-print library and on my website. "Foundation of a Mathematical Model of Physical Reality" gives an overview of the latest results. Other papers are older or represent excepts from the overview paper. My vision on the path integral is given in chapter 29 of the current version of this document.

The papers on my website appear as .pdf files and are at nearly the same url available as .docx files. Just change pdf into docx. None of the papers on my website are copyright documents. Thus you may use the contents of these articles. 

http://vixra.org/author/j_a_j_van_leunen

http://www.e-physics.eu

@Richard,

Special relativity works very well (except some time slippage)

Agreed have re-examined the tensor it breaks down at the event horizon (see links 1 & 2)

Have re-examined QM it breaks down also (see link 3)

Article The formulation of Dynamic Newtonian advanced gravity, DNAg

Book Everything is Physics Book 1. Understanding physics at the f...

It is always good to realize that QP and GRT must concern the same particles and the same fields. All fields obey the same differential calculus but encounter different start and boundary conditions. These conditions are set by the particles and some larger objects, such as black holes.

Most theories ignore the influence of ordering and if they do, then they talk about symmetries without explaining the origin of the symmetry. All theories ignore the mechanisms that control dynamics by preventing that the whole system turns into dynamical chaos. 

 No contadiction whatsoever. Feynman's path integral stems from the work of frenchman (Bachelier) done under supervision of H.Poincare at the beginning of 20th century.Then, the idea was picked up by N.Wiener and, subsequently, by Mark Kac who, incidentally worked at Cornell U.at the time when Feynman was there an assistant professor(thanks to H.Bethe with whom he worked at Los Alamos). So, if you are in doubt, please go to the original sourses 

What theory realistically, survives more than 100 years, few - not GTR nor QM. Even Newton and Darwin needs a bit of tweeking nowadays to make it work

what if either amplitude, K(rA, rC) or K(rC, rB), are zero or are vanishing/interfering etc.? so no 'jumping between rA to rC etc. happens in a finite time with velocities greater than the speed of light...  

Very interesting question Asher,

and personally, I like it. Let me try to answer, but as you'll see below, we don't know much.

If the particle jumps at a faster-than-light velocity from rA to rC, then, a moving frame can be found, in which one and the same particle can be found simultaneously in rA and rC. That would contradict the energy conservation. One particle cannot be two particles.

However, if we place a detector at rA, from the point of view of a frame in which the particle visits rC AFTER visiting rA, we disturb the free behavior of the particle. The uncertainty principle says that if the position is determined with precision, the velocity becomes totally undetermined. I suppose that "undetermined" means, still, below the light velocity. Though, the main point is that the indetermination in velocity happens because we disturbed the free behavior of the particle, fixing it to one point, rA.

In short, we cannot know how would have travelled our particle if it were left to travel freely.

Best regards,

Sofia

of course we can not know what path it took -- that's the heart and soul flesh and blood... of QM!

Well, Sofia.I am working now on the problem of demonstration that [t,H]=ih in QM. I've got this result rigorously already. It is surely implying the uncertainty principle between E and t. This result exist in literature on QM but the explanation is very different from what I am having.I am leaving to the researchers of the Research Gate the opportunity to figure out what is the physics behind [t.H]=ih.

@Arkady

Already figured

Because E = hn = hf =ih.dphi/dt,

where n is the number of quanta contained in a quantum system per unit time, and f is the frequency.

QED

Article The formulation of harmonic quintessence and a fundamental e...

This is what is known publicly. However, there is much more to this which is forthcoming...

Dear Sofia D. Wechsler,

In fact you are right, there is a contradiction. The contradiction is resulted from adopting the relativity the objectivity and then continuity same as in classical physics. Objectivity and continuity resulted in relativity by the reciprocity principle. But when computing the proper time, in this case we consider the Lorentz  length contraction in one side for one observer and we ignoring the length contraction in the other side, for the other observer. In this case we violate the reciprocity principle in Lorentz transformation when computing the proper time. But although of violating the reciprocity principle of the Lorentz transformation when computing the proper time, we propose objectivity and continuity same as in classical implicitly, and that is wrong! In order to consider objectivity and continuity same as in classical, we must consider the length contraction in both sides same as in the reciprocity principle in Lorentz transformation, but that will lead to remain the paradoxes in SRT, and thus physics appears to move according to the paradoxes. 

So, in order to solve the problem, we must consider space is invariant and quitting the reciprocity principle, and thus disappearing all the paradoxes in SRT. And in this case objectivity and continuity are not exist in nature. And the new transformation which expressing about the invariance of space expressing about the Copenhagen school view point of the phenomenon. According to this transformation light have no aberration and only fields and the constancy of light be suitable with Galilean transformations. This is exactly what I reached in my transformation in my paper  http://vixra.org/abs/1509.0059

x=R^2(x'-vt'), t=R^2(t'-vx'/c^2), y=Ry' and z=Rz'

Here R is the Lorentz factor.

http://vixra.org/abs/1509.0059

In fact motion in macro and micro world must be defined always according to 4-D and because of that the motion always  as observed globally  must be governed by the wave-particle duality and the uncertainty principle in micro  and macro. The problem here when you make a measurement at a certain point in space for the moving object, in this case the wave-function which describes the motion is collapse, and in this case it is translating from 4-D to 2-D, and in this case the uncertainty principle plays the rule.

That interprets the double slit experiment. As in my paper, Figs 1 & 2, every motion in constant speed v for micro and macro, are governed by two pictures are separated in space and time because of the effect of time dilation, but these two pictures are entangled with each others by the energy moment four vectors. According to these two pictures the motion are governed by the wave-particle duality. For me on the ground, I see one picture now according to my time on the ground, for example when the plane arrives Paris now for me. But there is another picture is hidden relative to me, which is related to where is the plane now for the observer on the moving plane according to his time. Here t' not equal to t (my time), and thus also x' must not equal to x. This interprets for you hidden variable theory.

Dear Sofia D. Wechsler,

You said:

If the particle jumps at a faster-than-light velocity from rA to rC, then, a moving frame can be found, in which one and the same particle can be found simultaneously in rA and rC. That would contradict the energy conservation. One particle cannot be two particles.

That impossible to be happened locally, but it happens according to observation. Existing an object in two states at the same time for an observer outside the system was verified experimentally by a team of scientists that has succeeded in putting an object large enough to be visible to the naked eye into a mixed quantum state of moving and not moving.  

Scientists supersize quantum mechanics, Nature,

doi:10.1038/news.2010.130.

That explained in my paper http://vixra.org/abs/1509.0059 in figs 3&4, and that explaining quantum entanglement and tunnelling. In this case there is no any violation of energy conservation, because light speed is not exceeded locally. According to Lorentz transformation there is no any interpretation for that. But according to my transformation Quantum mechanics be shown to be an effect of a relativistic system.

Sofia,

Sorry, but you are wrong considering the limits of the integral as infinite if you only consider the propagator between two points rA and rB. The path integral is an integral functional, i.e., a number obtained by integrating between

the end points of a certain path which are kept fixed. Thus it is necessary to fix to points, then you need to make a choice to take +infinity, -infinity or rA or rB.

On the other hand this definition is purely non relativistic because the time is always assumed a parameter universal independent of the system of coordinates, but obviously it is possible to do a relavistic definition of the path integrals and it is done in most of the books devoted to QFT.

No, Daniel,

I am not wrong. First of all it's Feynman's formula, not mine, and second, the intermediate point rC may be outside the line connected the two points and even far away from them.

Though, indeed, since all the treatment is non-relativistic, it seems that Feynman considered the light velocity as practically infinite, and, as implied by that, a Galilean space-time.

Best regards,

Sofia

Sofia,

In Feynman's formula, the extremals of the integral are the fixed points of the infinite curves which act as boundaries,in your case rA and rB (no infinities), that if you want can contain the rC point in one of the infinite curves that pass for the infinite. No problem, but such interpretation is no natural and contributes with a negligible value to the whole amplitude of probability for the particle in motion.

Although it seems Galilean relativistic formalism obviously, due to the propagators that you use that do not depend of time and they correspond to Green's functions of Schrodinger equation (diffusion differential equation). The discussion that you do has induced me to think that you find deeper problems that I couldn't see. The relativistic formalism of the path integrals is well known in representations as the one of Schwinger with a 4-dimensional Euclidean space-time (time pure imaginary quantity).

Integration differs from summation by the facts that integration involves the definition of the derivative of the manifold that is being integrated. In multiple dimensions, the integration is affected by the choice of the involved parameter space. In multiple dimensions, multiple choices for the parameter spaces are possible and often multiple parameter spaces play a role in the same formula. Typical examples are the divergence theorem and the Stokes theorem.

The path integral can better be interpreted as a summation of the effects of a sequence of products. Due to the small steps the products of the corresponding terms can be converted into summations of the contributions. That is why the summation can be approached by an integral.   The factors in the product represent hops and these hops are represented by a jump from configuration space to momentum space, a factor representing the displacement that corresponds to the hop and a jump back from momentum space to configuration space.

This procedure can handle a stochastic hopping path that is represented by a coherent hop landing location swarm. At first approximation the swarm moves as one unit and can be viewed as the representation of the investigated object.

In the sequence of the factors, the jumps back to configuration space of the previous factor and the jump into the momentum space of the next factor compensate each other, such that only the products of the hop displacement factors remain. This product can be converted in a sum and results in the representation of the displacement of the complete swarm as a single unit.. 

The coherent location swarm can be characterized by a continuous location distribution and this location distribution can be considered as the squared modulus of the wave function of the investigated object.

http://www.e-physics.eu/TheGeneralizedStokesTheorem.pdf

Newton and Einstein were two legendary alchemists.

They studied all their life what they did not understand: Gravitation!

When they were young they stolen from others theories, but after people started to watch them and they were not able to steal anything, they did not discover anything.

General relativity, LQG, String theory, Quantum gravity theories are wrong theories because are limited to the speed of light and do not explain Gravitation.

“I am the first who Understood and Explained Gravitation with high speed gravitons v = 1.001762 × 10^17 m/s, with Negative Impulse, Negative Mass and Negative Energy”                                                       Adrian Ferent

https://www.researchgate.net/publication/287331589_Ferent_Gravitation_theory

Adrian,

take it easy.

"Newton and Einstein were two legendary alchemists.  . . .  When they were young they stolen from others theories, but after people started to watch them and they were not able to steal anything, they did not discover anything."

Is anybody aware of any lawsuit against Newton and Einstein for having stolen ideas from other researchers?

Your theory may be quite a discovery, but there is symmetry in the world. Exactly as you say that Newton and Einstein discovered nothing, some people may say that you discovered nothing. I mean, some people may disagree with what you say.

Therefore, some moderation would be a good idea.

Both Newton and Einstein left behind not only their science but many quotations. Should these people be plagiators in science, they still would be famous for their quotations which definitely put them way above others.

I have many quotations too, on my theories before Conclusions!

I value your sense of humor. I wonder only if your quotes could be collected into volume like this  http://www.amazon.com/The-Quotable-Einstein-Albert/dp/0691026963   so that others would appreciate them too. If not yet, you have to hurry and to publish yours, provided that there will be a reputable publisher who would do the job...

I see this is a quotation on that link: "I have reached an age when, if someone tells me to wear socks, I don't have to."--Albert Einstein

I like this one:  I am truly a "lone traveler" and have never belonged to my country, my home, my friends, or even my immediate family, with my whole heart. In the face of all this, I have never lost a sense of distance and the need for solitude.

http://press.princeton.edu/chapters/s6908.pdf

Heisenberg just used the matrix calculus, which was discovered and widely used before him. So, the Heisenberg picture in QM is just a quotation?

Sofia, with Heisenberg the situation is more intricate. E.g. read my paper http://adsabs.harvard.edu/abs/2006hep.th....8117K  Heisenberg-representing  the upper class of German aristocracy -was initially a student of Max Born in Gottingen. Born subsequently sent him to N.Bohr (Copenhagen, Denmark) where Heisenberg  got G.Kramers as his mentor. With Kramers they made 2 papers https://en.wikipedia.org/wiki/Kramers%E2%80%93Heisenberg_formula    BEFORE quantum mechanics was officially inagurated. Then Heisenberg took a holiday vacation and came up with the paper in which QM was formulated based on his 2 papers with Kramers. This paper was accepted by Bohr as pivotal paper in formulation of new QM. When Kramers complained to N.Bohr that the ideas were his, Bohr was treating Kramers so bad that he ended up in the mad house. When he finally recovered, to smoth things up, Bohr found for Kramers a full professor position in Leiden which was previously occupied by Ehrenfest who committed suicide.  While Kramers was in hospital, Heisenberg send his paper to Paul Dirac who was a student at Cambridge at this time. Dirac made cosmetic changes in Heisenberg's paper and published his own. Kramers was completely by passed. In his book on QM Dirac  very vaguely  somewhere on pages 120-130 (or so) of his book(read my paper) mentions as a footnote work by Heisenberg and Kramers which, in fact, played central role in designing QM. Heisenberg during rest of his life was promoting Dirac and Dirac ws promoting Heisenberg. Kramers was left (with blessings of N.Bohr) completely outside.

 Adrian (Ferent), I have no idea about the amount of money you had inherited from your parents but I can rest assure  you that you have to prepare yourself to use up all these money since there will be no scientific community (and I do not count those who publish on viXra) who will pay any attention to your works. Research Gate is also full of  all kinds of nuts which, however, are not going to help you out  either. By writing your paper in the style you wrote, you  had issued to yourself a death sentence. This is the way science operates from times of Newton and Leibnitz.  Sorry that I cannot be of more help to you.

Adrian, I do not understand the English of your remark "Yesterday asked me a Journal 150$ to publish my paper and I received a lot of them since September." 

LIGO and the gravitational waves: the Biggest and the most Expensive LIE in science!

“They detected what can not be detected: gravitons having the speed of light and positive mass“ Adrian Ferent

LIGO scientists estimate that the black holes for this event were about 29 and 36 times the mass of the sun, and the event took place 1.3 billion years ago. 

It is like SETI, the Search for Extraterrestrial Intelligence an exploratory science that seeks evidence of life in the universe!

What they discovered:

1. They calculated the postive mass of the graviton, my theory explains that this is wong!

If the graviton has positive mass, it has positive momentum this means the Earth pushes you, and we suppose to fly!

“I am the first who understood and explained the Gravitation with high speed gravitons   v = 1.001762 × 1017 m/s, with Negative Momentum, Negative Mass and Negative Energy” Adrian Ferent

2. Black holes have the escape velocity bigger than the speed of light, this means the gravitons with the speed of light can not escape, this means they will never be able to come to Earth, to LIGO!

“How light can’t escape from inside event horizon of Black holes, in the same way the gravitons with the speed of light c = 2.9979 × 108 m/s can't escape from inside the event horizon. Only high speed gravitons, v = 1.001762 × 1017 m/s, can escape from inside the  event horizon of Black holes and keep the galaxy together”  Adrian Ferent

Einstein Gravitation theory: Gravitation is a distortion of space-time.

Ferent Gravitation theory: Gravitation is a force mediated by gravitons.

Einstein Gravitation theory is wrong and Ferent Gravitation theory is right!

http://www.dailymail.co.uk/sciencetech/article-3442022/Major-breakthrough-hunt-gravitational-waves-announced-today-Discovery-finally-prove-Einstein-s-100-year-old-theory-ripples-space-time.html

https://www.researchgate.net/publication/294087093_Ferent_Gravitation_theory

Arkady> I do not understand the English of your [Adrian Ferents] remark "Yesterday asked me a Journal 150$ to publish my paper and I received a lot of them since September

It most likely mean that there is a lot of predatory open access journals who offer to publish anything you write. If you just pay them enough.

https://scholarlyoa.com