_{1}

It is in general accepted that the concept of continuous trajectories for particles is at odds with the relativistic quantum mechanics. Namely, when examining the evolution of entangled quantum objects according to frames of coordinates in relative move-ment, one gets contradictory trajectories. Such a situation is typically derived from the famous “Hardy’s paradox”. However, it is argued here that if the rationale ignores the principle of quantum contextuality, as happens typically when using Hardy’s thought-experiment, the conclusion—rejection of the assumption of trajectories—is questionable. The issue is exemplified by an additional example: the 101 property of spin 1 bosons implies conflicting trajectories when the singlet state of two such bosons is examined according to frames in relative movement. It is concluded here that in the absence of a rationale which doesn’t violate the quantum contextuality, there are no sufficient arguments for refuting the possibility of a substructure of the quantum mechanics, consisting in particles following continuous trajectories.

The formalism of the quantum mechanics (QM) proved itself able to explain many phenomena in the microscopic world. Though, when the question is asked whether the QM is an ultimate theory, with no substructure, it is difficult to give a definite answer. The measurement process was not explained in entirety, until now, with the QM formalism. For the final step of the measurement, i.e. with a macroscopic apparatus, J. von Neumann introduced the postulate of wave-function collapse. Next, the nonlocal correlations appearing in the phenomenon of entanglement seem to imply that results of present measurements depend on data of future experiments.

Different attempts were done for eliminating the collapse postulate. A widely known attempt is due to L. de Broglie [

However, the Bohmian mechanics was proved unfit for explaining the evolution of identical quantum particles. As proved by P. Ghose [

Though, a particular formula in some formalism may be replaced, eventually, within an improved formalism. Therefore, the present article poses a more general problem: leaving aside any particular formula, is the very idea of particles following continuous trajectories, wrong?

By continuous trajectories it’s understood here that a particle doesn’t jump from a region in space, to another region, space-separated from the former by a gap in which the wave-function is null.

Since 1992 when L. Hardy published his famous “paradox” [

The present article brings an additional example of contradiction between trajectories. The analysis is done on the singlet states of spin 1 bosons, using the 101 property of these bosons. According to different frames of coordinates the wave-function of the system evolves differently, s.t. one infers a different pair of trajectories for the entangled particles. However, at an attentive examination, one can see that what changes from frame to frame is the quantum context: while one of the bosons―let’s name it

The principle of quantum contextuality states the following:

Given an operator

For the proof of this principle see for instance [

Berndl and Goldstein [

The present text analyses the consequences of the contextuality on the supposed substructure. The path followed by

Given that the reasoning with moving frames predicts contradictory trajectories, the question arises what exactly is wrong here: the concept of trajectories, or the use of moving frames in combination with ambiguous contexts?

If a concept is wrong, it should lead to contradictions also when judging within one single frame. So, for ruling out the concept of trajectories for particles, a proof which reasons in one single frame of coordinates is needed. Whether such a proof can be done, it’s for the moment an open question.^{3} Anyway, as long as no such proof is available, a substructure of the QM, with particles following continuous trajectories, is not excluded.

The next sections are organized as follows: Section 2 presents the main physical properties to be used in the text. Section 3 examines the evolution of a singlet of bosons according to the QM, in a rest-frame―the frame in which the setups are at rest. Section 4 does the analysis in the light of the hypothesis of trajectories, first according to the rest-frame, then, according to a frame in movement with respect to the rest-frame. A contradiction is shown to appear, and is explained as arising from making predictions for one and the same result in the presence of differing contexts. Section 5 contains discussions and stresses questions that remain open. Section 6 contains conclusions. The Appendixes detail part of the calculi done in the text.

We are going to work, for simplicity, with a system of units in which

The properties of spin 1 bosons relevant here were presented in [

Since the spin 1 bosons have the total squared spin equal to 2, given any three directions in space mutually orthogonal two by two―let’s call them X, Y, Z―the squared spin-projections

Two spin 1 bosons can form the entangled state

where Q is an arbitrary direction in space, and

In this text, each product of states is ordered with the state of boson

Four properties will be extensively used in the next sections:

(a) The state (1) is invariant at the change of the direction Q. That can be easily proved by substituting in (1) the equations (A2) with arbitrary angles θ and φ―see Appendix A, or the proof in [

(b) The expression of the state (1) reveals the nonlocal property of this state, that if one particle is tested on some direction whatsoever, the other particle behaves as if it got on that direction, the same squared spin projection.

(c) If two directions are mutually perpendicular,

(d) For

It has to be noticed that the opposite implication is not true.

About the property (b), attention should be paid to the fact that this property acts as long as the entanglement is preserved. If the entanglement is broken, i.e. if it is truncated to a single product of independent states of the two bosons, the nonlocal correlation between particles is destroyed.

Two spin 1 bosons,

Note 1: with the today techniques, the SG merger doesn’t restore the wave-packets that entered the splitter, with high-fidelity. Also, these apparatuses are long, which is inconvenient too. High-precision, much smaller apparatuses are desirable. However, what is examined here is a thought-experiment, technical difficulties being left aside hoping for future improvements.

By rest-frame is understood here the frame of coordinates according to which both Alice’s and Bob’s setups, are at rest.

Note 2: in the present proof we work with the squared spin-projections. However, an SG doesn’t split a wave-packet according to the squared spin projections, but only according to the simple spin projections. We overcome the problem by placing a detector only on the central beam exiting a splitter, and/or on the beam exiting a merger. If the central wave-packet exiting a splitter

The experiment is supposed to be performed in a dynamical way: a detector is placed on a beam after the respective wave-packet exited the SG apparatus.

as one can check by setting in (1)

In continuation we describe the main steps of the evolution of the wave-function according to the rest-frame, the frame of coordinates by which all the setups are at rest.

When the boson

One can see that if the observables

Next, when the boson

Finally the boson

On this wave-function are done measurements. One can see that three combina- tions of outcomes are possible:

In this section we assume a substructure of the QM, consisting in particles traveling along definite trajectories. For distinguishing between these hypothetical particles and the bosons

Let’s remind that we speak here of a dynamical experiment, as said in the beginning of the Section 3. Therefore, the correlation between responses of the two bosons, if measurements were done, should have existed between the paths picked by the hypothetical particles at the exit of the splitters. Indeed, the assumption of continuous trajectories implies that the hypothetical particles don’t jump from one path to another.

In the former section was shown that if the observables _{0}. It’s these trials on which we focus or attention in this section.

The wave-function (5) shows that if _{0}, _{+}, or d_{?}. The state (6) is tested experimentally and the trajectories obtained above determine the outcome

Let’s consider a frame of coordinates in movement with respect to the rest-frame. Passing from one frame to another one, moving with some velocity V with respect to the former, entails changes in the electromagnetic field and in the spin direction [

The velocity V is also supposed to be tuned so as the boson

Since we consider the cases in which _{0}-implication marked in _{0} the outcome is

The outcome

obtained by retaining in (8) the terms with

It is opposite to the predictions (7). Obviously, the reason behind this contradiction is the changed context: the outcome

It is not the purpose of this article to advocate for a substructure of the QM, but to claim that this substructure cannot be refuted in the absence of a proof that won’t ignore the quantum contextuality―desirably, a proof within one and single frame of coordinates.

Since the substructure discussed here is at odds with reasoning by moving frames, because of the ambiguity in quantum context entailed by the latter, that means the substructure requires a preferred frame of coordinates. The theory of relativity holds that there is no preferred frame, and the relativistic QM deals by definition with the transformation of wave-functions from one frame to another. If particles follow trajectories, these trajectories should transform covariantly from one frame to another. But the previous sections showed a clash between trajectories found by different frames, not a covariant transformation. This is an argument against the above substructure.

However, entanglements have properties outside the relativistic theory of QM: instantaneous mutual influence at a distance between entangled particles is not a prediction of this theory. It isn’t produced by any type of field that transforms according to the Lorentz transformations, but by the superposition of multiparticle amplitudes which occurs in multiparticle interferometry, a phenomenon alien to the physics of the macroscopic objects.

Besides, if one denies the existence of a preferred frame, one comes to a harder problem: by the time-axis of the rest-frame, the boson

However, by the time-axis of the moving frame, Section 4.2, it’s the boson

The problem posed in this text was whether it is possible to assume a substructure of the QM, consisting in particles following defined, continuous, trajectories. It was argued that as long as there is no proof against trajectories within a single frame of coordinates―more exactly, a proof that does not rely on ambiguous quantum context― there is no solid ground to refute this substructure.

The analysis discussed the general possibility of trajectories for particles, in the sense that it did not restrict itself to any particular formalism for trajectories, as for example Bohm’s formalism. This generality is important because, while Bohm’s formalism was proved incompatible with the experiment, a different formalism may eventually be proved compatible. The only restriction admitted here was continuity of the trajectories, in the sense that a particle doesn’t jump over regions where the wave-function is null.

Wechsler, S. (2017) Which Proof We Have against Continuous Trajectories for Particles? Journal of Modern Physics, 8, 68-81. http://dx.doi.org/10.4236/jmp.2017.81006

The general expression of the squared spin-projection operator for an arbitrary direction Q is

Here,

For the calculi that follow, one can take in

From this, one can deduce further the following angles and sine values:

, (A6)

, (A7)

, (A8)

In

In

For obtaining the eigenfunctions of

The direction C makes with the axis Z an angle

Thus,

The direction D makes an angle

The direction

The following scalar products are needed in the text and can be calculated from the above sets of vectors:

Identities:

The change in the electromagnetic field when passing from the lab frame to a frame moving with a velocity V is given by the Lorentz transformation,

where

No electric field acts on the bosons in the lab,

The spin of a particle also changes when passing from one frame to another. Let’s denote as S_{0} the spin of the particle according the frame in which the particle is at rest.^{4} In a frame moving with a velocity U with respect to the rest-frame of the particle, the spin is [

The bosons in this text are considered non-relativistic particles, s.t. their absolute velocities

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