Real quantum cybernetics

Volume 121, number 6

PHYSICS LETFERS A

4 May 1987

REAL QUANTUM CYBERNETICS Gerhard GROSSING Atominstitut der Osterreichischen Universitäten, Schuttelstrasse 115, A-1020 Vienna, Austria Received 25 November 1986; revised manuscript received 19 February 1987; accepted for publication 2 March 1987

It is shown on the basis of quantum cybernetics that one can obtain the usual predictions of quantum theory without ever referring to complex numbered “quantum mechanical amplitudes”. Instead, a very simple formula for transition and certain

conditional probabilities is developed that involves real numbers only, thus relating intuitively understandable and in principle directly observable physical quantities.

1. Introduction

2. Quantum cybernetics and organizational coherence

It is possible that all the basic principles of quanturn theory are easy to understand. As will be shown here, the question of when to take a “coherent” (“incoherent”) sum of amplitudes (probabilities) to calculate transition probabilities is not a necessary one, since there exists a simpler formula that provides the usual quantum mechanical results without posing such a question. The proposed theory, named “quantum cybernetics” provides both the usual predictions of quantum theory and a description of the measurement process [1]. The main purpose of this paper is to show the possibility of doing quantum physics without ever talking about complex amplitudes. Instead, we will show that one can easily arrive at the usual probabilistic predictions of quantum theory by using real numbers only. Moreover, these real numbers always represent quantities which are intuitively understandable and in principle directly obervable. Thus, we shall review in section 2 the basic results of ref. [I] and introduce the notion of organizational coherence. In section 3 we shall formulate and prove a theorem stating that realnumbers representing the basic physical quantities of quantum cybernetics are sufficient for a complete description of quantum systems. To give a concrete example, section 4 discusses “EPR-interferometry” in the framework of real quantum cybernetics.

In quantum cybernetics, a “particle” characterized by its typical oscillatory frequency cv is considered to be a discrete excited state ofthe vacuum which is in phase with the “wave”-like oscillation of the surrounding vacuum. Since the latter oscillation may change, for example due to a change in the setting of an apparatus, the particle oscillation itself will react accordingly and change its frequency as well. Thus, one can consider a quantum system as a feedback system with a given reference signal that compensates disturbances relative to the reference point, i.e. the particle’s frequency cv. The waves, then, act as mediators between particle and the surrounding objects, thus constituting a feedbackloop through the particle’s environment. Denoting with a(x, t) the (real numbered) amplitude of a quantum system and with P= a2 its probability distribution, one gets with the Boltzmann principle S~ k in P~

S/h = 5e/1’~in a2,

(1)

where S and are action constants and entropy, h and k are Planck’s and Se Boltzmann’s [1,2]. Inserting (1) into the continuity equation for the conserved probability current JP

a~j~ 8~(Fô~s)0, =

=

(2)

provides 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

259

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2=0

DP= Ea

PHYSICS LETFERS A

(3)

and EJS/h=—(ô~,S/h) (&US/h).

(4)

Note that eq. (2) in combination with the Hamilton—Jacobi—Bohm equation ~2 =E~+ c2( VS)2 + c2 /12 Da/a (the last term on the r.h.s. representing an additional contribution to the rest mass, often called “quantum potential”) is an alternative way of expressing the Klein—Gordon and Dirac equations (see ref. [3]). Our new relations (3) and (4) can now be interpreted in the framework as outlined above. Eq. (3) shows that P= a2 obeys a d’Alembert equation, and it has simple solutions of the form a2 = C0S2 0,

(6)

where 0 can be chosen as 0= S/h = k~x’~. Assuming that P can also be expressed as p= aa’, a’ ~ a, one verifies that a simple solution of Gaa’ = 0 is a product ofan incoming and an outgoing wave aa’ =cos 01

C05 02

4 May 1987

recognises is aa four-gradient ô’~S, this will bethat thewhenever source ofthere waves with wave fronts 5= const. This holds for particles with momenta k~= ä’25th as well as for any other four-gradient ô~S appearing in the dynamics of quantum systems. In particular, it holds for both Klein—Gordon and Dirac particles, since it is derived from eq. (2). Having outlined the conceptual framework of quantum cybernetics one can now use eq. (1) to formulate a “general action principle”. For organizationa! coherence, the minima! action principle 65=0 will be the determinant of particle motion. In genera!, however, it is possible that 6S=2h6a/a. (8) This means that 6S= 65vL dt= 0 is only valid as long as there is no new source of wavesa (or, respectively, of gradients &~S) along the integration area elf. Obviously, eq. (8) describes the quantum mechanical measurement process where a change in the experimental situation implies the existence of a new source. Now note that whenever there exists a nonvanishing source term, the relative variation 6S for the particle (with P~,= exp (S/h)) and for the wave (with a~=cos(S/h))are identical:

=~[cos(Ø 1 +02)+cos(Ø1 —02)],

(7)

such that (6) can also be interpreted as a standing wave where incoming and outgoing waves oscillate completely in phase with each other. We describe the situation of such a harmonic oscillation (as in (6)) by the term “complete organizational coherence”, while in (7) the organizational coherence is only partial. If the phase difference 01—02 is an odd integer multiple of ‘it, then we speak of “organizational separation” since the waves will have no possibility to interact. Note also that for appropriate redefinition of 0 as 0’ = ~(0i+ 02) one can transform (7) into 2 (7’) aa’ = co~ where ~

0~=0’—~0’, 02=0+~0. Eq. (4) is readily seen to represent Huygens’ principle. Since the waves a can also be described by their fronts of constant phase S=/10 = const, eq. (4) describes the propagation of these wave fronts. One 260

6SIh=öP~/P0=6a~,./a~.

(9)

We thus see how information about the particle’s environment produces a change in the overall organization as represented by the quantity S. Moreover, a change in the particle’s oscillation (like in an melastic collision) will have the same effect for the wave associated with it. This, again, corresponds to the mutual constitution of the wave- and particle-frequencies in our cybernetic model of quantum systems. Therefore, it will generally not be sufficient to talk about a wave with its associated particle entering a specific situation to be described. Rather, for a cornplete description of a quantum process one has to start with all the elements taking part in that process (such as particles and specific settings of apparati), study the organizational coherences (i.e. which parts, if any, oscillate in phase with each other), and describe particle motion as resultant of the phase relations determined by the experimental situation. In the “ideal” case ofcomplete organizational coherence, standing waves will emerge representing the

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harmonic oscillation of particle and apparatus (source plus detector). We now want to find out which energy Evacuum we can associate with our postulated basic frequency of the whole quantum system. Of course, the energy of the particleat rest is given as usual by E= hw0. However, what causes the harmonic oscillation of particle and wave is an overall excillation of the vacuum determined by the amplitude a=~.fP=exp(S/2h)=exp(Evacuumt/h).

(10)

It follows from (10) that Evacuum = hw0/2

(11)

4 May 1987

shift contributes to the overall oscillation, it will have to be included in the sum of all phase contributions. Note, however, that we are talking here about “averaged” phase contributions from path-sets 1 or 2, as opposed to counting the phase contributions of all possible individual paths in each set as it is done in the path integral formalism of Feynman and Hibbs [4]. It is therefore necessary to work out the ring integral !#dS=

!J

h

h

dS+

dS h

(13)

2

to obtain the correct probabilities. (Note that the

is the zero-point energyof the vacuum associated with a particle of frequency w0. Thus, it is the undulatory nature of the zero-point energy which is responsible for the wave equations (3) and (4). Since the minimally resolvable energy then is L~Emin= *w0/2 and the minimally resolvable time is &min = i/COo, we get the uncertainty relation

elements dS in the integrals on the r.h.s. have opposite signs.) Thus, we can formulate the following (first version of the) Theorem on organizational coherence. The total probability P~0~ for the transition of a particleat point 1 in spacetime to a point 2 in spacetime is exclu-

L~E ~t ~ /2.

sively determined coherence of the elements by the constituting degree of organizational the quantum

(12)

system (particle plus apparatus) and is given by 3. Calculation of quantum mechanical probabilities without complex amplitudes In the following discussion, we shall mostly consider the possibility of particles travelling from a point 1 to a point 2 in spacetime along two possible sets of paths, each governed by 8S= 0 from point 1 to point 2,tobut one can number easily extend this scheme by induction any larger of possible sets. Further, we shall for simplicity restrict ourselves to two spatial dimensions (discussing interference effects, EPR problems, and the like). To obtain transition probabilities in quantum cybernetics, one may use eq. (1), i.e. P=exp(S/h). However, in order to calculate the proper function S, one has to find out the degree of organizational coherence determining the effective relationsbetween the relevant elements ofthe processunder study. The most important requirement is to take into account all possible paths when calculating the organizational coherence, because one could have, for examplc, a phase shift b.S/h on one path which is missing on the other. Clearly, if the path involving this phase

P,0~(1 2) —~

=N

exp

(_ ~

~),

(14)

h

where the minus sign is chosen for convenience and N is a normalization factor. We now calculate the action function S by Integrating eq (4)~ 2co2

=

—

h~ ~~

dX~dx

=

—

1 jj ff h

~

—-—

dx,. dx

(15) Introducing the “proper time” interval s of the partide as S2 = C2 T2,

we get

s

=

—

~= 2ic/cü,

(16)

r r (th)2

jJ

~2

= ~

(17)

To calculate the ring integral (13) we now have to take into account the opposite signs of the elements dS. Thus, 261

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4 May 1987

?r4~ £S=o/

\ss~o 5A

)

\~“~

const

Fig. 1. Schematic representation of the transition of a particle from point 1 to point 2 in spacetime via two possible sets ofpaths, each set governed by a minimal action principle. One sees that the total phase gradient at point2 is given by ô’~Ø2= ÔMØA + &“ØB.

fdS _~—~-

Th~

Fig. 2. Geometric relation between phase gradients and enclosed angles providing eq. (22).

point 1, we get from (19) the probability for a parti=

_jd~ns+ 1dlns=ln~-’-.

(18)

s~

SI

P

2(Ø 10, (1—+2)=s~/s~=k 8p 02/&M01

paths as

S2

Inserting (18) into (14) provides a second version ofthe “theorem on organizational coherence”:

= ~~02

cletravellingfrompoint 1 topoi~t2viatwopossible

a~

1)[k

+2k(0A)k(0B)cosO],

2(Ø 2)/k

=

2(OA)+k2(0B)

P(l—~2)=k2(0

(21)

1)

1,

(19)

where 0, denotes the phase at point i, i= 1 or 2. It may seem that so far we have not gained much, since in (19) the path dependence has apparently dropped out. Moreover, (19) only tells us that upon conservation of four-momentum k~(Ø)= oM0 the probability of the particle to arrive at some point 2 characterized by k~’(02), where I k~(02)12 =

where bS/h. Here we have used the fact theÔ~0B the angle a0 = between two phase gradients ô’0A and

is

L~.S

f

dS

~

(22)

—

—

I k~(Ø

as can be seen from fig. 2. Note that only in the simplest cases the orientation ofthe phase gradients corresponding to the momenta kM(O) = &u0 =

1)12, is one. However, to illustrate the usefulness of the foregoing procedure, let us consider the following examplc for the transition of a particle from point 1 to point 2 via two possible sets of paths, each set governed by a minimal action principle 6S=0 (see fig.

~9~(h—‘ .N dS) will coincide with the orientation of the macroscopically observed momenta kg’. Generally, the latter will be completely determined by the minimal action principle with points 1 and 2 as integration limits, i.e. using eq. (17),

1).

6 JdS=0_*6 j’(ds)2~6 J(c2dt2_dx2)=0.

On the statistical average, assuming the waves on the two paths to have equal amplitudes, the vectors ô’~Ø1clearly are composed in such a way that ~0, =~0A +&20B, i= 1 or 2. (20) Ifwe are interested in the final distributions at point 2 only, and assuming we have plane waves at starting 262

2

I

2

I

2

1

Clearly, (21) describes the double slit interference pattern forpoints 2 on ascreen afterparticles inphase with plane waves have left point 1 and passed a double slit: putting I k(OA) I = I k(OE) I = Ik(Ø1) I one can rewrite (21) as

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P(l—*2)

=

PHYSICS LETTERS A

~(l +cos 0).

(21’)

Having so far used a “particle representation” of probability, eq. (1), we shall now derive our result (21) in an even simpler manner using a “wave representation”. Remembering that wave and particle are supposed to oscillate in phase with each other when in complete organizational coherence such that for the standing waves’ phases i~Ø=0,one obtains P= cos2i~Ø= 1. Recalling now that ~ 0, x, 2x,...

4 May 1987

The solutions of (26) can, of course, be written in terms of complex numbers cv such that

cv (x,k) x exp(iØ), where 0 is an arbitrary phase factor, while the general solution will be

J

(27) (x) = d4kF(x, k)~(x,k). Thus, we can compare the two existing notations and we find for I I=I I the identity

P, 01 =

describes partial organizational coherence and organizational separation for ~Ø=x /2, we obtain the Theorem organizational coherencefor two aalternative paths.onRedefining the scale of 0 in such way that the phase difference i~o = 01—02 (phase at point 1 minus phase at point 2) is distributed equally on both paths, we obtain for equal (real) amplitudes on both sides

k

I ~(x, k) 12

8~z0i= #‘ 02 ~02 Notet9”that the “~0i unit vector ~

= 1. = 1Y’Ø/ I

(28) already representing a statistical average can equivalently be represented by a vector in the complex plane defined by the phase 0= k~x’~ such that we obtain the equivalence

I

—

—

(23)

cv(x)=Neiø4~,w(x)=Nn11, (29) where N is a normalization factor such that (28) is

(For the case ofpath-dependent phase factors, as, e.g., in the Aharonov—Bohm effect, one has to use the more general form of (23) involving the ring integral.) Naturally, for a double slit experiment with point 1 representing the source and point 2 a location on a screen behind the double slit, we again obtain with the identity (22):

fulfilled. Therefore, we obtain for two different possible paths A and B and equal real amplitudes W ~ +Wn) ~(e~+e’~)

P(l—~2)=cos2(~L~0)~cos2(~ ~d5)

(30) such

that

with

k(OA)=k(0B)= ~k(Ø 1) and

P(l—’2)=~(l+cosz~).

(24)

So, we see that one can calculate the double slit interference pattern without using complex amplitudes. How can this be understood in connection with the latter? To show the transition to the orthodox quantum theory, let us ignore organizational coherence, and rewrite eq. (3) as follows:

na/a =

—

8’1a8,~a/a2 =

—

&0a~0

~0=0A—0B:

2 = I ~ e’~(l+e~0) 2 =~(l+cos AØ),

IWI

~

(3la)

=

(&OA+ô~0B)(ô,~0A+öP0B)

—

(ô~Ø8~Ø1)

= ~(1+

cos i~Ø). (31b)

Thus, up to a factor N, we associate with each quan(25)

cv a unit four-vector ñ’~,or, respectively, a complex number exp ( iØ) as

It follows that for a single wave a (i.e. not for the standing wave representing organizational coherence) we obtain the Klein—Gordon equation

the equivalence of using either the real or the cornplex formalism. Now we want to give a more general proof of our

(0+ m ~c2//1 2) a = 0.

theorem coherence. Weequivalence can restrict ourselvesontoorganizational showing the validity of the

=

—

k/z( 0)k~,(0).

tum mechanical wave function

(26)

givenbyeq. (28). The results (31a) and (31b) show

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(31) for single solutions cv of the Klein—Gordon equation, since the integration (27) takes place in real phase space, once cv is substituted by a real function. Now, the most general solution of (26) for two alternative paths can be written as a linear combination oftwo solutions (the proofforn alternatives follows simply by induction):

cv(x, k) =N’[f(x, k)

ei~1

0~,

(32)

+g(x, k) e~2] e’ where 03 is an arbitrary phase factor we shall ignore in the following since it drops out when calculating P= Icv12. Putting F(x,k)=g(x,k)If(x,k), cv basically becomes

cv(x, k) =N[ 1 +F(x, k) e

4 May 1987

P(l—~2)=a12a21=alaaap...ap2a2p...aflaaaI. (36) We have seen that the superposition principle fol-

lows from the theorem on organizational coherence. We know, however, that the coherent sum of state cv is only applicable for bosons while for fermions one has to calculate differences. Although the theory presented here does not explain quantum mechanical spm, one can easily show the amplitude subtraction law for fermions.

So far we have(3), only considered symmetrical solutions ofeq. e.g. a= cos 3.i~Ø,for “phase wave symmetry” between paths A and B. Naturally, there do exist antisymmetrical waves fulfilling (3) ofthe form a = sin ~

—~‘

(33)

—02>].

(37)

which provide transition probabilities P(l-+2)=a2=~(l—cosAØ).

Therefore we get I cv(x, k) 12 =N2 [(1+ F(x, k) 2 cos(Ø 1 —02)

2(x, k)], +F which for F(x, k) =1 reduces to

(34)

(38)

We immediately see that eq. (38) is the result for P( 1-+2) when applied to fermions, i.e. to antisymmetrical (notalso thebewave-packets!). Of course, the resultwaves (38) can written in the form P( 1—~2)

Icv(x,kH2=N2(l+ cosi~O). Now let us use the real-numbered notation. We obtain

= Iô~0

immediately

=

1~+F(x, k)n21~] 2 [1 + 2F(x, k) cos i.~Ø+ F2 (x, k) 1~

(35) =N having thus proven the general equivalence (29), and therefore our theorem on organizational coherence as formulated in eqs. (28), (14), and (23). We now see that the theorem on organizational coherence describes both the “superposition principlc” for complex numbered quantum mechanical states to be added “coherently” and the situation where one would have to calculate an “incoherent” sum of probabilities. Moreover, it is obvious from the theorem that one can insert various intermediate points between the points 1 and 2 on each path and calculate the corresponding transition probabilities as a multiplication ofthe form

264

Ik~(0

2[k2(0A)+k2(0

1)I

2[ñ’j’+F(x, k)ñ~][ñ P=N

2(ô~0A—&0B)(ôP0A—8~0B) 1L

—2k(0A)k(Ø~)cosL~Ø],

8)

(39)

leads the to notation the usualofeq. amplitude subtraction for using (21), which via (29) law directly fermions. From (39) we derive the Pauli principle: For fermions (i.e. for antisymmetrical waves) there can be no two equal “states” ô’~OA= ~0B atone and the same point 2 in spacetime, since then P( 2) ~ (&‘~0A 8’~0B)= 0. —

4. Application of real quantum cybernetics: EPRinterferometry

The Einstein—Podolsky—Rosen—Bohm version of testing Bell’s inequalities exploits the two-valuedness of spin measurements. Home and Zeilinger [5] have shown recently that when a source emitting particle pairs in momentum anticorrelated states

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4 May 1987

sition andlor conditional probabilities with the mitial wave function ofa source emitting particle pairs,

R~

lw>=~(lRi> L2>+1L1> 1R2>),

(40)

we shall now derive the corresponding probabilities from the theorem on organizational coherence to show the practicality of our real-numbered formalism. First, note that on the basis of the concept of organizational coherence a transition probability P( 1 2) for a single particle is identical with the conditional detection probability for anticorrelated particle pairs P( 112) since both probabilities depend

I

—~

—-..

~

/

/

1

L2”

/

solely on the actual oscillations at points 1 and 2 in spacetime:

Ra S

Psingje particle(l

—+2)

=Pparticle pairsU

12).

Thus, we can apply eq. (22) providing

/

(

/ —

I

(42)

~

aL

(41)

where the integral connects points 1 and 2 via two possible sets of paths, each obeying 6S=0, and we immediately obtain the conditional probabilities

I I

~ =P(L~IL~)=~[l+cos(x1 —X2)], R

P(L~IR~)=P(R~IL)=P(L~IR) Fig. 3. Schematic sketch of an EPR-experiment Mach—Zehnderinterferometers (from ref. [5]).

using

(like the positronium annihilation radiation) is placed at the center of a double Mach—Zehnder interferometer as depicted in fig. 3, an EPR—Bell type situation arises. The corresponding two-valuedness is the property that a particle is found in either ofthe two outgoing beams (L or R~,and L~or R~in fig. 3) behind the interferometers. The authors have shown that this situation can be made formally analogous to the spin cases, hence is subject to Bell’s theorem, which now applies to a situation involving linear momenta. This result holds for any situation imparting four independently adjustable phase shifts aJ, a~, PL, PR onto the beams. Instead ofstarting the usual procedure ofquantum theory to calculate tran-

=P(R1 IL~)= ~[1 +cos(x1 —X2 +~t)] = [1 C05 (x X2)], (43) where x’ = aL a~,X2 =/JL —flu, which is the same result as that of a longer calculation in ref. [51. Similarly, calculating the probabilities of registering a particle in a primed detector given that its counterpart is detected in an unprimed detector (see fig. 3), we get with (42) ~

—

—

~ P(R~IR1)=P(R~lL1)=P(L~IR1)=P(L~IL1)=1, (44) which again is the same result as that of ref. [5]. 265

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5. Conclusion It was shown for two spatial dimensions that one can arrive at the usual results of quantum mechanics by considering single bosons as real cosine waves and

single fermions as real sine waves representing the zero-point oscillations of the vacuum, and by calculating the various transition and conditional probabilities in real space, i.e. without referring to “complex quantum mechanical amplitudes”. The real numbers involved in these calculations always represent quantities which are intuitively understandable and in principle directly observable. In fact, their introduction makes the distinction between coherent and incoherent sums of quantum mechanical amplitudes (probabilities), the “only mystery” of quantum mechanics [6], unnecessary. Nevertheless, this distinction is completely explainable, once the

4 May 1987

uum” will not exhibit completely smooth wave behavior, but will essentially be characterized by “noisy” fluctuations. Much work, however, has already been done on the stochasticity underlying the simple quantum mechanical/cybernetical processes ([7] and an introduction to more recent work in, e.g.,

ref. [8]), and a further development of quantum cybernetics will have to refer to it. Acknowledgement This work was supported in part by the Bundesministerium fir Wissenschaft und Forschung under contract number Z 1.19.153/3-26/85. Furthermore, I thank S. Fussy, D. Greenberger, J. Summhammer,

and A. Zeilinger for stimulating discussions.

equivalence between complex and real numbered formalism is established.

The basic conceptual framework of this study Is expressed by the notion of organizational coherence between particle(s) and apparatus. As can be clearly seen from our discussion relating real and complex numbered formalisms, and also from section 4, one can say that the problems that have arisen historically with the introduction ofcomplex amplitudes are due to the fact that thereby one has often decomposed organizationally coherent entities into seemingly separate ones, and consequently “found” the “mysterious” non-locality of quantum theory. Relating the model presented here to other existing ones in a more detailed analysis will be the sub-

ject of a forthcoming paper. Moreover, this paper has not discussed the degree of approximation to which the real waves presented here are valid assumptions. In other words, apart from the ideal cases as discussed here, the “real vac-

266

References [1] G. Grossing, Phys. Lett. A 118 (1986) 381; in: Proc. Int. ~onf on Microphysical reality and quantum formalism (Urbino, Italy, 1985), eds. G. Tarozzi and A. van der Merwe (Reidel, Dordrecht), to be published. [2] L. de Broglie, La thermodynamique de Ia particle isolée (Gauthier-Villars, Paris, 1964). [3] L. de Broglie, Non-linear wave mechanics (Elsevier, Amsterdam, 1960). [41R.P. Feynman andA.R. Hibbs, Quantum mechanics and path (McGraw-Hill, New York, 1965). [5) integrals M.A. Home and A. Zeilinger, in: Proc. mt. Conf. on Micro-

physical reality and quantum formalism (Urbino, Italy, 1985), eds. G. Tarozzi and A. van der Merwe (Reidel, Dordrecht), to be published. [6] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman lectures on physics (Addison-Wesley, Reading, 1965).

[7] E. Nelson, Phys. Rev. 150 (1966) 1079. [8] C. Dewdney, P.R. Holland, A. Kyprianidis, Z. Maric and J.P. Vigier, Phys. Lett. A 113 (1986) 359.