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The impossibility of experimental testing of the hidden variables theory
Volume 137, number 4,5
PHYSICS LETTERS A
15 May 1989
THE IMPOSSIBILITY OF EXPERIMENTAL TESTING OF THE H I D D E N VARIABLES THEORY Vladan P A N K O V I C
Laboratoryfor TheoreticalPhysics,Instituteof Physics,Facultyof Sciences, 21000NoviSad, Yugoslavia Received 23 February 1989; accepted for publication 15 March 1989 Communicated by J.P. Vigier
The experimental testing of the theories of hidden variables which consider the quantum mechanical formalism as consistent, but just incomplete, has been proved impossible.
Let there be given a homogeneous q u a n t u m ensemble in the pure q u a n t u m state I~u). We measure some observable A which possesses a discrete and nondegenerate spectrum with eigenvectors the quantum states I~u,), n = 1, 2 . . . . . This measurement dehomogenizes a given ensemble into q u a n t u m subensembles in the pure q u a n t u m states I~',) with corresponding statistic weights pn = [ ( ~u, I ~u) 12, for n = 1, 2, .... Let us assume that HV is a theory o f hidden variables which considers the q u a n t u m mechanical formalism consistent at the q u a n t u m level of precision, but just incomplete ~1, intending, therefore, to complete it to a more complete theory by the introduction of the hidden variables 2. At the subquantum level of precision, these variables dehomogenize the quantum ensemble in the q u a n t u m state I ~u), in the sense that for any two q u a n t u m objects, 1 and 2, of this ensemble, the first can be described by the completed state (Iq/), 2~) and the second by the completed state (I~u), 22), so that one can distinguish 1 from 2 for 21~22. Besides, HV should be a deterministic theory, whereas the quantum mechanics, Q, is a statistical theory. Now, let us suppose that q u a n t u m objects of the homogeneous ensemble in the pure q u a n t u m state I~u) are exposed to the mentioned measuring pro-
cess one at a time, whereupon a nonhomogeneous ensemble of N measured quantum objects is formed with Nn denoting the number of q u a n t u m objects in the I ~n) post-measuring state, where n = 1, 2, .... Thereby, the following is satisfied N= ~Nn,
w h e r e N ~ > ~ 0 a n d n = l , 2 .....
(1)
n
IfN-+oo, i.e. if N>> 1 we speak of the quantum mechanical limit and the quantum ensemble o f N measured q u a n t u m objects, whereupon
Nn=pnN,
where n = 1, 2 .....
(2)
is fulfilled with a very good approximation. Let us suppose, however, that N is a large number yet considerably smaller than required by the quantum mechanical limit, so that there exists at least some n for which (2) has not been satisfied to the required degree of approximative accuracy. In this case, the impossibility of a more precise determination of the functional dependence between Nn and pnN by means of the quantum mechanical formalism may, from the aspect of HV, be regarded as incompleteness o f Q. It is then possible to presume the existence of the following kind of a more precise functional dependence in HV:
Nn=pnN ( p(A, I q/~), 2) d 2 ,
for N~ <<.p.N,
An
~ Such is, for example, Gudder's theory of contextual hidden variables. See ref. [ I ]. 158
(3) 0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 137, number 4,5
PHYSICS LETTERS A
and
N./N=p.FA.,
N. f p(A, Iv/.), 2) d2=p.N, forN.>~p.N,
(4)
An
15 May 1989
forN. <~p.N,
and
N./N=p.Fx. l , forN.>~p.NandFA.:/:O.
for n = 1, 2 ..... where A. is the space of hidden variables which, for the given N., correspond to the q u a n t u m objects in the q u a n t u m state ]~u.), d2 is the differential measure o f the space o f hidden variables under the condition that 2 are continuous variables, p(A, [~.), 2) is the density o f the distribution function of the hidden variables which can be made explicitly dependent on A (i.e. on the process of measuring), [~u.) and 2, as, for instance, in Gudder's theory of contextual hidden variables ~2, or, as, for instance, in the standard Einstein [ 2 ] or Bell [ 3 ] approach to the hidden variables, depending on 2 only, so that its dependence on A and [gt.) is only fictitious. The integral fA. is the 2 hidden variables average within some space A., which in the case that 2 are discrete variables, can be replaced by an adequate summation without any influence on the generality o f further conclusions. It is, also, necessary to state that A. increases if N. increases, whereupon A ° stands for the space of hidden variables in the case when N. and N are very large numbers, i.e. in the case of q u a n t u m mechanical limit. From the standpoint of mathematical statistics, the real non-negative number, the factor of precision FA.= jp(A,[q/.),2)d2>~0,
for n = l, 2, ... ,
(8)
(9)
Let us suppose now that we also perform next a new measurement o f the described type (a>~0), so that the n u m b e r N increases to N + a and the number N. to N.+a., where a. stands for the number of newly measured quantum objects which are in the [ ~. ) post-measuring state, where n = l, 2 ..... so that, on the grounds of (6), results FA. ~
(N.+a.)/(N+a)>~N./N,
forN.<~p.N,
(10)
forN.>~p.N,
(11)
and
(N~+a.)/(N+a)<~NJN,
for n-- 1, 2, .... Furthermore, since any natural number x, y, z, t for which x/y>_.z/t is satisfied, (x+z)/ (y+ t) >iz/t is satisfied too, and vice versa, then from (10) and ( 11 ), for n = 1, 2 ..... we obtain
aJa>~(N.+a.)/(N+a),
forN.<.p.N,
(12)
forN.>~p.N.
(13)
and
a./a<.(N.+a.)/(N+a),
If A~ is the space o f the hidden variables ofa. newly measured quantum objects, for n = l, 2 .... , then from (12) and ( 13 ), if p. ~ 0, we obtain
An
(5) represents the distribution function of the hidden variables which is nondecreasing in the sense that, f o r n = l , 2 .... ,
(A. = A; ) ~ (FA. ~
(6)
(7)
Expression (7) is, also, the normalizing condition for the density o f the distribution function of the hidden variables. On the grounds of (3), (4) and (5) follows, for n = 1, 2, ..., ~2 See ref. [ 1].
(14)
and
F;.~ <~Fy..t, forN.>_.p.N,
(15)
both of which yield the following expression,
FA'~>~FA".
and FA o - 1 .
FA~>~FA,~, forN. <~p~N,
(16)
On the other hand, for n = 1, 2, ..., A~ c A~ follows from the condition a,~N~+a,. Hence, on the grounds o f (6), we obtain
FA.~FA., for n = 1, 2 ..... a n d p . ~ 0 .
(17)
Expressions (16) and (17) are only then satisfied simultaneously if FA~ = FA~, for n = 1, 2, ..., andp~ # 0 ,
(18)
159
Volume 137, number 4,5
PHYSICS LETTERS A
which represents the condition of theoretical consistency of HV. This condition is met only ifFA, in fact does not depend on Nn and N which is only possible if N + a, i.e. a, is large enough for the q u a n t u m mechanical limit, but this contradicts the initial assumption. This contradiction does not oppose the existence of HV itself, but the possibility of describing, by means of HV, those ensembles of q u a n t u m objects that are less numerous than the q u a n t u m ensembles. This remark also refers to Ballentine's statistical interpretation of quantum mechanics [4 ]. On the other hand, all measurement results in q u a n t u m ensembles are, with the required degree of approximative accuracy, in accordance with the predictions of Q on the grounds o f the assumption about the consistency of Q. Consequently, HV cannot be experimentally verified in this case in the sense of its distinguishing from Q ~3 Finally, it is possible that expressions ( 1 0 ) - ( 18 ) are incorrect because HV, itself, is incomplete in the sense that for a n u m b e r a, when a and N are not large enough for the q u a n t u m mechanical limit, there is no precise functional dependency between an and pna, or between Nn and pnN, or between Nn+an and pn(N+a). Then, we may assume the existence of a new and better founded theory of hidden variables, ~3This idea has been presented, without a firm mathematical proof by Peres and Zurek [5].
160
15 May 1989
H V ' , which should make HV complete similarly as Q is made complete by HV. Nevertheless, following the described completion algorithm, it is possible to prove in an analogous way that H V ' is consistent only for sufficiently numerous q u a n t u m ensembles on which, however, it is not experimentally verifiable in the sense of its distinguishing from Q and HV, and it is, therefore, possible to presume the existence o f a new, even more complete theory o f hidden variables, HV", which should make H V ' complete, and so ad infinitum ~4. The author is deeply grateful to Professors I. Ivanovir, F. Herbut, D. Kapor, M. Durdevir, G. Dull6 and O. Mihailovir, for critical remarks, illuminating discussion and permanent support. ~4 One explicit way of building up this infinite succession of theories of hidden variables is given in ref. [ 6 ].
References [ 1] S.P. Gudder, J. Math. Phys. 11 (1970) 431. [ 2 ] A. Einstein, B. Podolskyand N. Rosen, Phys. Rev. 47 ( 1935) 777. [ 3 ] J.S. Bell, Physics 1 (1965) 195. [4] L.E. Ballentine, Rev. Mod. Phys. 42 (1970) 358. [ 5 ] A. Peres and W.H. Zurek, Am. J. Phys. 50 (1982) 807. [6] V. Pankovi~, Phys. Lett. A 133 (1988) 267.
PHYSICS LETTERS A
15 May 1989
THE IMPOSSIBILITY OF EXPERIMENTAL TESTING OF THE H I D D E N VARIABLES THEORY Vladan P A N K O V I C
Laboratoryfor TheoreticalPhysics,Instituteof Physics,Facultyof Sciences, 21000NoviSad, Yugoslavia Received 23 February 1989; accepted for publication 15 March 1989 Communicated by J.P. Vigier
The experimental testing of the theories of hidden variables which consider the quantum mechanical formalism as consistent, but just incomplete, has been proved impossible.
Let there be given a homogeneous q u a n t u m ensemble in the pure q u a n t u m state I~u). We measure some observable A which possesses a discrete and nondegenerate spectrum with eigenvectors the quantum states I~u,), n = 1, 2 . . . . . This measurement dehomogenizes a given ensemble into q u a n t u m subensembles in the pure q u a n t u m states I~',) with corresponding statistic weights pn = [ ( ~u, I ~u) 12, for n = 1, 2, .... Let us assume that HV is a theory o f hidden variables which considers the q u a n t u m mechanical formalism consistent at the q u a n t u m level of precision, but just incomplete ~1, intending, therefore, to complete it to a more complete theory by the introduction of the hidden variables 2. At the subquantum level of precision, these variables dehomogenize the quantum ensemble in the q u a n t u m state I ~u), in the sense that for any two q u a n t u m objects, 1 and 2, of this ensemble, the first can be described by the completed state (Iq/), 2~) and the second by the completed state (I~u), 22), so that one can distinguish 1 from 2 for 21~22. Besides, HV should be a deterministic theory, whereas the quantum mechanics, Q, is a statistical theory. Now, let us suppose that q u a n t u m objects of the homogeneous ensemble in the pure q u a n t u m state I~u) are exposed to the mentioned measuring pro-
cess one at a time, whereupon a nonhomogeneous ensemble of N measured quantum objects is formed with Nn denoting the number of q u a n t u m objects in the I ~n) post-measuring state, where n = 1, 2, .... Thereby, the following is satisfied N= ~Nn,
w h e r e N ~ > ~ 0 a n d n = l , 2 .....
(1)
n
IfN-+oo, i.e. if N>> 1 we speak of the quantum mechanical limit and the quantum ensemble o f N measured q u a n t u m objects, whereupon
Nn=pnN,
where n = 1, 2 .....
(2)
is fulfilled with a very good approximation. Let us suppose, however, that N is a large number yet considerably smaller than required by the quantum mechanical limit, so that there exists at least some n for which (2) has not been satisfied to the required degree of approximative accuracy. In this case, the impossibility of a more precise determination of the functional dependence between Nn and pnN by means of the quantum mechanical formalism may, from the aspect of HV, be regarded as incompleteness o f Q. It is then possible to presume the existence of the following kind of a more precise functional dependence in HV:
Nn=pnN ( p(A, I q/~), 2) d 2 ,
for N~ <<.p.N,
An
~ Such is, for example, Gudder's theory of contextual hidden variables. See ref. [ I ]. 158
(3) 0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 137, number 4,5
PHYSICS LETTERS A
and
N./N=p.FA.,
N. f p(A, Iv/.), 2) d2=p.N, forN.>~p.N,
(4)
An
15 May 1989
forN. <~p.N,
and
N./N=p.Fx. l , forN.>~p.NandFA.:/:O.
for n = 1, 2 ..... where A. is the space of hidden variables which, for the given N., correspond to the q u a n t u m objects in the q u a n t u m state ]~u.), d2 is the differential measure o f the space o f hidden variables under the condition that 2 are continuous variables, p(A, [~.), 2) is the density o f the distribution function of the hidden variables which can be made explicitly dependent on A (i.e. on the process of measuring), [~u.) and 2, as, for instance, in Gudder's theory of contextual hidden variables ~2, or, as, for instance, in the standard Einstein [ 2 ] or Bell [ 3 ] approach to the hidden variables, depending on 2 only, so that its dependence on A and [gt.) is only fictitious. The integral fA. is the 2 hidden variables average within some space A., which in the case that 2 are discrete variables, can be replaced by an adequate summation without any influence on the generality o f further conclusions. It is, also, necessary to state that A. increases if N. increases, whereupon A ° stands for the space of hidden variables in the case when N. and N are very large numbers, i.e. in the case of q u a n t u m mechanical limit. From the standpoint of mathematical statistics, the real non-negative number, the factor of precision FA.= jp(A,[q/.),2)d2>~0,
for n = l, 2, ... ,
(8)
(9)
Let us suppose now that we also perform next a new measurement o f the described type (a>~0), so that the n u m b e r N increases to N + a and the number N. to N.+a., where a. stands for the number of newly measured quantum objects which are in the [ ~. ) post-measuring state, where n = l, 2 ..... so that, on the grounds of (6), results FA. ~
(N.+a.)/(N+a)>~N./N,
forN.<~p.N,
(10)
forN.>~p.N,
(11)
and
(N~+a.)/(N+a)<~NJN,
for n-- 1, 2, .... Furthermore, since any natural number x, y, z, t for which x/y>_.z/t is satisfied, (x+z)/ (y+ t) >iz/t is satisfied too, and vice versa, then from (10) and ( 11 ), for n = 1, 2 ..... we obtain
aJa>~(N.+a.)/(N+a),
forN.<.p.N,
(12)
forN.>~p.N.
(13)
and
a./a<.(N.+a.)/(N+a),
If A~ is the space o f the hidden variables ofa. newly measured quantum objects, for n = l, 2 .... , then from (12) and ( 13 ), if p. ~ 0, we obtain
An
(5) represents the distribution function of the hidden variables which is nondecreasing in the sense that, f o r n = l , 2 .... ,
(A. = A; ) ~ (FA. ~
(6)
(7)
Expression (7) is, also, the normalizing condition for the density o f the distribution function of the hidden variables. On the grounds of (3), (4) and (5) follows, for n = 1, 2, ..., ~2 See ref. [ 1].
(14)
and
F;.~ <~Fy..t, forN.>_.p.N,
(15)
both of which yield the following expression,
FA'~>~FA".
and FA o - 1 .
FA~>~FA,~, forN. <~p~N,
(16)
On the other hand, for n = 1, 2, ..., A~ c A~ follows from the condition a,~N~+a,. Hence, on the grounds o f (6), we obtain
FA.~FA., for n = 1, 2 ..... a n d p . ~ 0 .
(17)
Expressions (16) and (17) are only then satisfied simultaneously if FA~ = FA~, for n = 1, 2, ..., andp~ # 0 ,
(18)
159
Volume 137, number 4,5
PHYSICS LETTERS A
which represents the condition of theoretical consistency of HV. This condition is met only ifFA, in fact does not depend on Nn and N which is only possible if N + a, i.e. a, is large enough for the q u a n t u m mechanical limit, but this contradicts the initial assumption. This contradiction does not oppose the existence of HV itself, but the possibility of describing, by means of HV, those ensembles of q u a n t u m objects that are less numerous than the q u a n t u m ensembles. This remark also refers to Ballentine's statistical interpretation of quantum mechanics [4 ]. On the other hand, all measurement results in q u a n t u m ensembles are, with the required degree of approximative accuracy, in accordance with the predictions of Q on the grounds o f the assumption about the consistency of Q. Consequently, HV cannot be experimentally verified in this case in the sense of its distinguishing from Q ~3 Finally, it is possible that expressions ( 1 0 ) - ( 18 ) are incorrect because HV, itself, is incomplete in the sense that for a n u m b e r a, when a and N are not large enough for the q u a n t u m mechanical limit, there is no precise functional dependency between an and pna, or between Nn and pnN, or between Nn+an and pn(N+a). Then, we may assume the existence of a new and better founded theory of hidden variables, ~3This idea has been presented, without a firm mathematical proof by Peres and Zurek [5].
160
15 May 1989
H V ' , which should make HV complete similarly as Q is made complete by HV. Nevertheless, following the described completion algorithm, it is possible to prove in an analogous way that H V ' is consistent only for sufficiently numerous q u a n t u m ensembles on which, however, it is not experimentally verifiable in the sense of its distinguishing from Q and HV, and it is, therefore, possible to presume the existence o f a new, even more complete theory o f hidden variables, HV", which should make H V ' complete, and so ad infinitum ~4. The author is deeply grateful to Professors I. Ivanovir, F. Herbut, D. Kapor, M. Durdevir, G. Dull6 and O. Mihailovir, for critical remarks, illuminating discussion and permanent support. ~4 One explicit way of building up this infinite succession of theories of hidden variables is given in ref. [ 6 ].
References [ 1] S.P. Gudder, J. Math. Phys. 11 (1970) 431. [ 2 ] A. Einstein, B. Podolskyand N. Rosen, Phys. Rev. 47 ( 1935) 777. [ 3 ] J.S. Bell, Physics 1 (1965) 195. [4] L.E. Ballentine, Rev. Mod. Phys. 42 (1970) 358. [ 5 ] A. Peres and W.H. Zurek, Am. J. Phys. 50 (1982) 807. [6] V. Pankovi~, Phys. Lett. A 133 (1988) 267.