An Algebraic Way of Simultaneously Analyzing Both Einstein, Podolsky and Rosen Paper and Bohr’s Reply to It

An Algebraic Way of Simultaneously Analyzing Both Einstein, Podolsky and Rosen Paper and Bohr’s Reply to It

N. Gurappa

Velammal Engineering College, Chennai-600066, Tamil Nadu, India.

Received (Day Month Year) Revised (Day Month Year)

Abstract – In their celebrated paper titled Can quantum mechanical description of physical reality be considered complete?, Einstein, Podolsky and Rosen (EPR) showed for the first time the existence of Spooky action-at-a-distance. Though the result of their paper is un- questionable, but the conclusion of the same became sensational because of its challenge to quantum mechanical formalism whether its complete or not in describing the physical reality of Nature. Bohrs physical and philosophical reply to that conclusion justified the completeness of quantum mechanics. Here, a simple algebraic way is presented for the results of these two classic papers in such a way that the actual reason behind why quan- tum world necessarily exhibits the action-at-a-distance and how Bohr defended against the incompleteness of the quantum formalism will become clear.This approach naturally reveals what physical assumption of EPR went wrong while considering the entangled quantum system and also provides the missing mathematical argument in Bohrs reply.

Keywords: EPR Paradox; Quantum Entanglement; Spooky action-at-distance. PACS Nos.: 03.65.Ta, 03.65.Ud

1. INTRODUCTION

   Quantum entanglement is a natural consequence of the quantum formalism whose existence in Nature is experimentally confirmed 1,2,3,4,5,6,7,8, thanks to Bells in- equalities 9 . Nature is indeed quantum mechanically spooky. Any measurement on a remote quantum particle has an instantaneous influence on its entangled partner which can, in principle, be separated to a distance as much as the known dimen- sions of the Universe. This present paper is an algebraic effort to harmoniously mix and analyze the essence of both Einstein, Podolsky and Rosen (EPR) paper 10,11 and Bohrs reply 12,13,11 to it and hence to pin-down the necessity for the Nature to maintain such a quantum entanglement or spooky action-at-a-distance as one of Her fundamental requirements.

   Consider the crucial extracts from page:140 of EPR paper 10 , We see therefore that, as a consequence of two different measurements performed upon the first sys-

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   tem, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in

   consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems. Thus, it is possible to assign two different wave functions to the same reality (the second system after the interaction with the first) and from Bohrs short note 12 , It is true that in the measurements under consideration any direct mechanical interaction of the system and the measuring agencies is excluded, but a closer examination reveals that the procedure of measurements has an essential influence on the conditions on which the very definition of the physical quantities in question rests. Aim of the present paper is to distill out the physically relevant essence of these two extracts algebraically.

   In quantum mechanics, Heisenbergs uncertainty relation states that the posi- tion and momentum of a quantum particle can not be precisely and simultaneously measured which is a consequence of the canonical commutation relation [x, p] = i h/(2 ); where, x is the position operator, p is the momentum operator. In fact, this commutation relation is the very basis for the entire quantum formalism. The main purpose of EPR argument was to show that in the case of two entangled particles, it becomes possible to si- multaneously measure both the position and momentum of both the particles which invalidates the correctness of uncertainty relation and hence the corresponding basic commutation relation itself. Hence, they concluded by questioning the completeness of quantum mechanics in describing the reality of Nature. The present analysis ex- poses what classical physics assumption of EPR will not hold to be true in the case of quantum mechanics as pointed out by Bohr. Also, the missing mathematical argument for Bohrs reply is provided. A simple proof is given to show that the conserved quantities are responsible for the existence of the spooky action in the quantum world.

2. CONSERVED QUANTITIES AND THE SPOOKY ACTION-AT-A-DISTANCE

   Consider the EPR case of two entangled quantum particles in one-dimension. They had interacted initially at some position, x0 , for a very brief time (like the case of elastic collision) and became entangled. When they are separated to a large distance, then it is

   assumed that there are no more physically known interactions acting between them.

   Now, make the position measurement of particle-1 accurately. Hence, the out- come of its momentum measurement becomes uncertain. On the particle-2, perform the momentum measurement and hence its position gains a huge uncertainty. This is in perfect agreement with the position-momentum commutation relations as given below:

   [x , p ] = [x , p ] = i h/(2 )

   ] = i~ and [x2 , p2 ] = h/(2 ). This violation can not be acceptable as mentioned earlier. To circumvent this situation, it becomes necessary to accept that x1 measurement necessarily influences the distant out come, p2 , even though the particles are no more interacting. Einstein called this kind of influence as spooky because such an action-at-a-distance is not at all visible at a mere glance at the quantum formalism.

   The above equations (1), (2) and (3) were considered by Bohr in the footnote of his reply to the EPR paper 13 . But actually he missed out the following observation

   1 1 2 2

   (1)

   needed to explain the EPR correlations:

   In the case of entangled particles, x1 and p2 and

   An algebraic way of simultaneously analyzing both Einstein, Podolsky and Rosen paper… 3

   where, x1 and x2 are position operators and p1 and p2 are momentum operators of particle-1 and particle-2, respectively.

   Now, according to the EPR assumption, the two particles were treated as in- dependent systems like in the case of classical physics because at the time of mea- surement they were no longer interacting with each other by any known physical interactions. It implies the following commutation relations:

   [x1 , p2 ] = [x2 , p1 ] = 0 (2)

   It can be easily seen that the relative position operator, x0 = x1 x2 , cor-

   responding to the distance of separation between the two particles and the total momentum operator, Pcm

   = p1 + p2 , corresponding to the center-of-mass motion of the combined two particle system commute with each other,

   [x0 , Pcm ] = [x1 x2 , p1 + p2 ] = [x1 , p1 ] [x2 , p2 ]

   = 0 (3)

   and therefore, they can be measured accurately and simultaneously.

   So, from the accurate knowledge of x1 , p2 , x0 and Pcm , one can know x2 and p1 as accurately as desired, which is a violation of the uncertainty principle. Even in this case, the validity of uncertainty relation requires that somehow the mea- surement of x1 should not allow for the accurate measurement of p2 . It means that the measurement of x1 must have an influence on the other particles outcome. But we have [x1 , p2 ] = 0 implying that its possible to simultaneously and accurately measure both x1 and p2 since heir corresponding operators commute with each other. Therefore, x1 (x2 ) and p2 (p1 ) can be measured accurately which, in turn, is the violation of the original commutation relations [x1 , p1

   x2 and p1 should not com- mute with each other respectively, in order to maintain the original commutation relations, [x1 , p1 ] = [x2 , p2 ] = h/(2 ) and also the law of total momentum conservation. Since, the center-of-mass and relative degrees of freedom are always independent of each other, one has

   [x0 , Pcm ] = 0

   = [x1 x2 , p1 + p2 ]

   = [x1 , p1 ] [x2 , p2 ] + [x1 , p2 ] [x2 , p1 ]

   = [x1 , p2 ] [x2 , p1 ] (4)

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   Therefore, the commutators [x1 , p2 ] and [x2 , p1 ] need not be equal to zero indepen- dently, but they should always be equal to each other i.e.,[x1 , p2 ] = [x2 , p1 ]. Now, its easy to see that the EPR assumption about the well-separated entangled par- ticles as two independent and mutually non-influencing systems will not hold to be true within the quantum formalism and that was precisely the whole point Bohr explained with suitable physical examples.

   Without loss of generality, by taking the total conserved momentum Pcm to be

   equal to zero,

   p1 + p2 = 0 (5)

   and finding the commutator with x1 and x2 respectively, one has

   [x1, p1 + p2] = [x1, p1] + [x1, p2] = 0 (6)

   and

   which yield, [x2, p1 + p2] = [x2, p1] + [x2, p2] =

   (7)

   [p1, x2] = [p2, x1] = h/(2 )

   (8)

   In Eq. (8), if [p1 , x2 ] = [p2 , x1 ] = 0 as anticipated by EPR and accepted by Bohr in his reply, then it implies from Eq. (6) and Eq. ( 7) that either [x1 , p1 ] = [x2 , p2 ]

   = 0 or the law of conservation of total momentum is not valid in the quantum world. But, both the conclusions are physically unacceptable. Therefore, in the case of entangled particles, x1 will not commute with p2 and as well x2 with p1 , like the case of unentangled free particles in such a way that the actual commutation relations, [x1, p1] = [x2, p2] = h/(2 ) and the law of conservation of momentum were unaffected in the quantum mechanical description of the Nature, leaving no room to claim that quantum mechanics is incomplete. So, this simple explanation is sufficient to show that quantum mechanics is spooky. The entire Bohrs reply to EPR paper revolves around the explanation of the commutation relations [p1, x2] = [p2, x1] = h/(2 ) which he did not explicitly consider, but only with some suitable experimental situations.

3. SUMMARY

I algebraically studied the crucial results of both EPR paper and Bohrs reply to it. The EPR idea of treating two distant particles, which had interacted initially

.

and no more interacting by any known physical

mechanism, as two independent systems like in the case of classical physics can not hold to be true in the quantum world as pointed out by Bohr. This aspect was elegantly shown with the use of simple commutation relations and also, the missing mathematical argument for Bohrs reply is provided. In this approach, it becomes extremely transparent that

even the quantum world, whose physical phenomena are based on the canonical quantum commutation relations, is bound to obey the conservation laws which naturally give

An algebraic way of simultaneously analyzing both Einstein, Podolsky and Rosen paper… 5 raise to the existence of Einsteins spooky action-at-a-distance among the entangled particles.

To visualize physically How entangled particles are able to communicate even though they are well- separated? (or) How actually the spooky-action-at-a- distance is becoming possible?, will necessarily depend on the physical reality of the Schrodinger wave function. In a recent paper, I gave a new non- dualistic inter- pretation for the quantum formalism where the Schrodinger wave function is in- terpreted as an instantaneous resonant spatial mode 14 . This new interpretation unambiguously resolves the well- known paradoxes and puzzles in quantum mechan- ics at a single quantum level and provides a natural mechanism for the Einsteins spooky action-at-a- distance. Nature seems to be perfectly stubborn not to violate the conservation laws. Certainly, this instantaneous spooky action is not carried out by some physical carriers which require the exchange of energy and momentum between entangled particles because, we know that energy-momentum transport cant be superluminal and is bound to the Cosmic speed limit in accordance with the special theory of relativity. The main purpose for the existence of the spooky action in the quantum world seems to maintain strictly the conservation laws even in the absence of exchange interactions.

ACKNOWLEDGMENTS

I would like to thank S H S Hussainsha for useful discussions.

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