Foundation of Physics, Vol. 11, Nos. 9/10, 1981
Stochastic Foundation for Microphysics. A Critical Analysis J. C. Aron ~ Received April 17, 1980 The stochastic scheme proposed in a previous paper as subjacent to quantum mechanics is analyzed in the light of the diJficulties and criticisms encountered by similar attempts. It is shown that the limitation of the domain where the theory is valid gives a reply to the eriticisms, but restricts its practical usefulness to the description o f basic features. A stochastic approach of the hadron mass spectrum, allowing the scheme to emerge in the domain of experimental verification (to be worked out in a later paper) is outlined. The model is found not to be in disagreement with Bell's argument opposed to hidden variables; a same origin is suggested for the difficulties encountered in both domains. The views proposed are compared with the Copenhagen interpretation: common points and divergences are analyzed.
1. I N T R O D U C T I O N 1°1.
The ambition of a deeper conceptual grounding for quantum mechanics has encouraged various attempts to revise its foundations. Formal analogies between the stochastic and quantum laws, as well as their common probabilistic character, have led many authors, in the last 20 years, to hope that stochastic processes could yield the wished new basis. It was natural to start from the conservative quantum fluid defined by p and v (where p is the density of probability and pv the velocity density of mean value) and to associate with it a stochastic fluid submitted to classical or semiclassical laws. It has been proved possible to derive from such considerations the Schr6dinger equation. Institut Henri Poincar6, 11 rue P, et M. Curie, Paris, France. 699 0015-9018/81/1000-0699503,00/0 © I981 Plenum Publishing Corporation
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However, deep differences between the two domains have led in this way to difficulties not yet overcome. In particular the quantum fluid does not obey the same law of superposition as a classical one: in this latter the densities add linearly, in the former the linear law applies to the amplitude, i.e., to the square root of the density. Actually none of the stochastic approaches has so far succeeded in making quite clear how the component waves add in an interference process. This point and others (questions on the trajectory of the particle, and on the nature of the stochastic medium supposed to be coupled with it) have led an increasing number of physicists to prefer another approach: stochastic electrodynamics, where the particles are assumed to interact with a zero point electromagnetic field. One may think that there is no incompatibility between the two ways of investigation, admitting that the basic equations of quantum mechanics and those of quantum electrodynamics need both a deeper foundation. Here we limit our study to the former domain, and examine how it is possible to face the mentioned difficulties. In an earlier paper (1) we had developed a stochastic interpretation of quantum mechanics, and derived from the laws of diffusion the nonrelativistic and relativistic quantum equations. In subsequent papers, (2'3) we extended this early work: on the one hand to an analysis of the foundations of relativity, (2) on the other hand to a model of the Schr6dinger zitterbewegung and the plane monochromatic wave, derived from the hydrodynamical representation of the Dirac equation/3) A third paper (4) will deal with the hadron spectroscopy, in connection with the stochastic model (as explained below in Section 6). The present paper is critical, and can be considered both as an introduction and as a justification of the papers mentioned: its aim is to see in what directions the extension of the stochastic approach seems possible. This study will lead to severe limitations, as regards the description of the experimental situations accounted for by the quantum formalism. Thus, the question arises: is our analysis only a selfcriticism, or does it claim to discourage every ambition of building on a stochastic basis something broader than the quantum theory? Imposing restrictions to the future research is always imprudent. What we can do is to compare diffiulties encountered in different domains, to see if they appear to have a common origin, which would suggest that they should have a general significance. 1°2°
This has been done in the present paper, whose plan is the following one. In Section 2 we recall works of different authors using stochastic
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processes to derive the quantum laws, and the main features of Ref. 1. In Section 3 we discuss the interpretation of Ref. 1 and study the objections encountered by similar attempts; we show that our scheme is valid if the local variations are small at the scale of the basic length and time constants. In Section 4 we compare this limitation with another difficulty: Bell's proof that hidden variables are incompatible with quantum mechanics; we propose a common interpretation. Next, after some epistological remarks (Section 5) on the "metaphysical" character of problems which cannot be directly tested by experiment, we give (Section 6) an outline of features derived from the scheme and subject to experimental verifications. Sections 7 and 8 present an overall view; we show that the outlook is more encouraging than it may appear at first sight on account of the limitations imposed.
2. T H E S T O C H A S T I C A P P R O A C H .
OBJECTIONS ENCOUNTERED
We consider a Wiener process dX = u dt + kdw
(1)
involving a drift velocity u and a diffusion constant D = lk = h/2m o
(2)
Introducing the density of probability Po, we obtain the Fokker-Planck equation: C3po/Bt +
div(P0U) = ½k A p o
(3)
If we define a new velocity v = u - lk Vpo/p o =
u -4- u I
(4)
Eq. (3) can be rewritten as an equation of continuity for the fluid with density P0 and velocity v: ~po/?t +
div(p0v ) = 0
(5)
It is tempting to identify (or at least to associate) this fluid with the quantum conservative fluid defined in terms of the wave function ~, = a exp(i~o/h) by its density a 2 and its velocity - V ¢ / r n o. The dependence of v on P0 in Eq. (4) already gives rise to difficulties if one considers a probabilistic fluid. We defer this point and pass to the derivation of the Schr6dinger equation.
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2.1. Previous Works This derivation has been obtained simply by Kershaw (5) for a stationary process. For the general case several authors (6-8) introduce a quantum potential. In addition to the abovementioned difficulty, one can object its artificial character. Other authors °) take up for the kinetic energy ½rnou2+ l m 0 u~, which is hard to accept, the usual rule for addition of velocities being linear and not quadratic. The same remark applies to similar alterations in the form of the kinetic energy. (~°) Nelson's approach (1~) is based on the acceleration vector y, obtained by combining second derivatives of the forward and backward velocities; next he admits that under the external force F one has F = too,/. This process is formally elegant; one can object with Kracklauer ~12) that the physical significance of the probabilistic regression towards the past is not clear. Nelson's work has been developed by several authors: Guerra and Ruggiero (13) have pointed out its connection with the Euclidean formulation of quantum field theory; others (Yasue, (~4) Roy (~5~) have studied its relativistic generalization, and also applied it to the problem of field quantization.(14)
2.2. The Scheme Proposed The description of the stochastic scheme and the derivation of Schr6dinger's equation stand in Ref. 1. Here we give in the Appendix a revised presentation where the distinction of the two levels is emphasized. At the lower level there is a stochastic fluid (Fo), nonconservative because its velocity u is an average velocity where diffusion effects are ignored. At the higher level we find the conservative fluid (F), to be identified with the quantum fluid. The basic fluid is (F0), to which corresponds the physical potential U. As to (F), made up with corrected quantities, it is a fictitious fluid to which corresponds a fictitious potential U'; the classical laws [equation of conservation (5), and dynamical relations (A.6) and (A.8) in Appendix], concern (F) and not (F0).
3. DISCUSSION 3.1. Study of Objections
3.1.1. Gilson, (~6~ using the Feynman integral, has proved that the Fokker-Planck equation and the Schr6dinger equation are only consistent if
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the diffusion coefficient is identically zero. This proof is in contradiction with our deduction, where we show the consistence of these equations, and the conflict does not concern the physical aspect of the scheme, but the sheer mathematical process. There are two reasons for this discrepancy: first the use of the Feynman integral applies to small values of t, whereas our approach assumes 00 infinitesimal, i.e., t ~> 00. Secondly, the density p = gt*qj is different from the density P0 considered for the basic fluid (F0) (see Eq. (39) in Ref. 1]. 3.1.2.
Among the objections against the stochastic approach, one finds sometimes this one: it does not clarify what was obscure in quantum mechanics, and cannot explain the source of the randomness. (17) On the first point we comment in Sections 7 and 8. We discriminate two domains, one for the stochastic formalism, the other for the quantum formalism. This does not abolish the obscurity in the foundations of quantum mechanics, but gives suggestions for its origin: they join partially the interpretation of the Copenhagen school, and, nevertheless, are different enough to yield new incidences on the Einstein-Podolsky-Rosen (EPR) paradox. As for the source of the randomness, see Section 8. 3.1.3.
Another difficulty is the nature of the fluid (Fo); is it physical or formal? A first objection against it the physical interpretation is its indeterminacy; we have discussed it in Ref. 1 (p. 181). A second objection (~8) is grounded on the dependence of the velocity (v or u) on the preparation, namely on the initial value of P0 at each point; this is unusual in a classical stochastic process. Actually we show in the next subsection that the scheme is valid in a very narrow domain; we conclude in Section 8 that the problem of the physical nature of the fluid is practically not that much relevant.
3.2. The Limitation of the Scheme
3.2.1.
All of the process developed in Ref. 1 is based on the infinitesimal character of 00. The Fokker-Planck equation is written to order zero in 00, its relativistic extension [Ref. 1, Eq. (31)] to order one. The infinitesimal character of the length L 0 associated with 00 results, in the nonrelativistic case, from L 0 = k/c (with k finite), in the relativistic case from L 0 = cOo
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whence L 0 is of the same order as 00. Thus, 00 and L 0 are infinitesimal at the scale of the times and lengths considered; this means that the system should not have sensible variations on L 0 and 0o, or else the scheme breaks down. We have noted this when studying the Dirac fluid (Ref. 1, p. 179). Now all practical problems involve local (in space and time) violations of the constraint of smooth variation: at the point where the system is in contact with a macroscopic object (measuring device, mirror, Young slits, etc...), or at the instant when a particle is created or annihilated. We must already consider our scheme as an idealization. This point will be developed below and commented on in Section 7. 3.2.2. We look at the problem from another angle. Given an initial state qJ~n= ~'in (x, 0), we derive q/(x, t) from Schr6dinger's equation. With ~/in we associate a fluid (Fin) whose density and velocity are Pin = q/*nv/~n and v i n = - l / m o Vrpi.. With ~(xt) we associate similarly a fluid (F) whose density and velocity are p = ~t*~, and v = -1/m o • V~0. On what conditions can we claim that (Fin) and (F) are derived from physical fluids (Fo,in) and (F0), respectively, (Fo) being obtained from the evolution of (F0,in) with time? Do the definitions of qJ~n and gt yield physical quantities for the densities and velocities? We examine two simple examples. First we choose the initial situation as resulting from an approximate measurement with a Gaussian uncertainty; we have: ~'in = exp(--x2/a2) exp [-imo/h, (v. x)] The calculation, developed by Darwin, ~19) shows that the wave packet spreads, its dimensions growing like: al=(a2+ht/moa) v2. Should we attempt to describe stochastically this evolution? Such an effort would certainly be meaningless, since the initial conditions refer to the result of a measurement and not to intrinsic features. The reduction of a wave packet by an observation gives rise to a similar remark. As a second example we consider the action of a totally reflecting mirror, on which light falls normally. The fluid (Fin) is stationary; the elements of (Fo,in) have velocities with periodic space variations. If there is a physical fluid, it seems probable that its elements have average velocities approaching +e or --e, according to if they have or have not undergone the reflection. Thus, the stationary (or quasi-stationary) fluid (F) [or (F0) ] should describe an average situation; (F) and (Fo) are not real fluids. Can we imagine a situation where (F0) would be physical? This would be an ideal situation where the conditions of smooth variation would be
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satisfied. As soon as we introduce a macroscopic body, these conditions hold no longer in its vicinity. The fluids are formal, as explained above; what appears mathematically, in the case of the mirror, as an addition of waves is physically the macroscopic action that breaks the model (see further comments in Sections 7 and 8). 3.2,3. A large number of criticisms raised against the stochastic interpretation give rise to the same reply: they refer to situations which stand beyond the approximation of the scheme. For instance Claverie and Diner (2°) have remarked that the stochastic model based on diffusion accounts poorly for the behavior of a harmonic oscillator in an electromagnetic field, which is better described by stochastic electrodynamics. Indeed this problem involves quantum transitions, and the exchange of photons with the field, which exceed the ideal conditions defined above. Stochastic electrodynamics, which considers stochastic processes in phase space, works at a higher level than ours, where no phase space representation is possible because the elements of our basic fluid have no definite velocity.
4. H I D D E N V A R I A B L E S A N D D E T E R M I N I S M 4.1. Hidden Variables
The ambition of a pictorial scheme subjacent to the quantum formalism is, in its principle, akin to the ambition of a subjacent determinism defined by hidden variables. The limitations encountered in the stochastic description of quantum laws by stochastic processes should be compared with the obstacles opposing the development of hidden variables theories. In our scheme, if the corpuscle can be identified with a definite stochastic element, its position, its mean velocity, and the orientation of its spin vector s are hidden variables. This holds in the ideal conditions where our scheme is valid, and where, at least from the conceptual point of view, the quantum formalism cannot be qualified complete. Von Neumann, (24) and then Jauch and Piron (22) and others, have on the contrary attempted to prove its complete character. We do not think it necessary to discuss here their proofs, recognized as unsatisfactory by Bell, ~23) followed by Belinfante. ~24) We have shown elsewhere (28) that: 1. yon Neumann's reasoning is unconvincing, for he attempts to prove the nonexistence of dispersionless subensembles by applying to them a formalism which we only know by experiment to be valid for the real quantum systems;
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the argumentation of Jauch and Piron being divided as follows: a.
if a system has hidden variables, any two propositions are
compatible b.
experiment shows that there are incompatible propositions
C.
consequently, hidden variables cannot exist,
(a) and (b) are both true, with different definitions of the propositions and of their compatibility, so that (c) is not derivable from (a) and (b). Both proofs are now superseded by that given by Bell, (26) which appears much more solid, and which we will have to examine in relation to our scheme. Bell (followed by Wigner (27~) considers a pair (say A and B) of spin - 5 t particles formed in the singlet spin state and moving freely in opposite directions. Assuming a set of hidden variables called 2, he writes the expectation value of the product of two different spin components, concerning A and B, respectively, which should be equal to the quantum expectation value. Calling a and b the unit vectors along the two directions, one should have:
fd2p(2)A(a,,~)B(b, 2)= ((o,.
a ) . (o z • b)) = - { a . b)
(6)
where A(a, 2) and B(b, 2) are the two spin components. Bell's reasoning proves that Eq. (6) leads to a contradiction. To comment on this result, we remark that such a case of correlated spin values is obtained by a transformation of the type:
C~A+B where C is a spinless particle. Should this transformation be a smooth process (in the acceptation defined in Section 3.2) one would expect (supposing the ideal scheme valid at the start) that the hidden parameters involved in the description of C would yield parameters for the description of A and B. The reasoning of Bell and Wigner proves that the transformation is not a smooth process, similarly as it has been shown for the other practical situations (see the examples in Section 3.2). Such a feature is in full agreement with the observations previously presented in this article. This does not imply nonlocality, nonseparability, instantaneous transmission to a distant point of the measurements of an apparatus; there can be a deterministic correlation between A and B, but it cannot be expressed by a relation of the type (6). What is broken is not the conceptual determinism, but its transcription into practical calculations; determinism is not usable.
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4.2. Additional Comment
We come back to our stochastic scheme. Taking for granted the existence of two levels, the higher (experimental) and the subjacent (ideal), asking with Moore~17~: "Can stochastic physics be a complete theory of nature?", and with Graber e t a [ . (18) "Is quantum mechanics equivalent to a classical process?", we must, in agreement with these authors, answer both questions negatively; the higher level is ruled by the quantization formalism, not by the stochastic laws. (This, at least, applies to our work; for further comment see Section 7.) Can we yet hope to establish a link between the two levels? Should we expect the formal analogy between the two processes to melt into a unique theory, as de Broglie has aimed at by his idea of a double solution? ~28) This would be the actual completion of quantum mechanics, ambitioned by hidden variable theorists: the new formalism being broader than the quantum one, would fit new experimental features. This ambition seems strongly limited by Bell's argument: presently, in all practical calculations where the complexity of the experimental situations is involved (we develop this idea below), the quantum formalism remains our only solid basis. Should we consequently discard as "metaphysical" (or "unphysical", as Onofri ~29) says) an effort of clarifying the conceptual domain? 2 Should we (we quote Belinfante t3°)) "retreat to positivism and refuse to consider questions meaningful to which the answers cannot be verified directly by observations?". We suggest in Section 6 an approach derived from the stochastic scheme and leading to experimental verifications. Nevertheless, positivism seems to us to entail such dangerous restrictions that we want to first deal with for a moment the epistemological problem.
5. THE EPISTEMOLOGICAL PROBLEM 5.1. Positivism
Positivism is not new, but before the last 50 years it was mainly supported by those who philosophized on science, not by those who made it: for these latter ones it has always been a powerful stimulus to believe that science was more than a collection of successful recipes allowing to predict 20nofri's argument is that an infinity of transition probabilities density of probability: p(xt) -- l" P(xt/x°
to) p(x° to) dx°
P(xt/xoO ) leads to the same
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experiment, and that striving toward a better understanding of the physical world was, in itself, relevant and fruitful. What could have happened if positivism had been prevailing in 1905? Lorentz has established formulas which account for the strange properties of light. His explanations are rather awkward and intricate; however, it does not matter much, since the formulas agree with experiment. Therefore, little attention would be paid to an obscure researcher who claimed a better description by further considerations on space and time; all the formalism of special relativity would have been developed without him, and without much attention to his metaphysical speculations. Positivism came among the physicists when Heisenberg was facing the quantum states of the spectrum and the enigma of light emission, and decided one should establish relations between what could be measured (i.e., the states) and ignore what could not (i.e., the transition mechanism). This behavior, quite justified by the situation at this time, unduly turned into a general law. Another contributing factor to the positivistic attitude was the puzzling quantum dualism, leading to this behavior: "let us go ahead, we understand later." The physicists went ahead, and built up the quantum formalism. Its efficacy, its mathematical perfection, gave an answer to many practical problems, but this basic one remained: what was lying beneath was still as obscure as ever. Philosophy came over this, the state of not understanding was no longer considered a provisional one, one stood before a rupture with "an age old way of thinking of all mankind. ''(31) There may be a risk of dogmatism in these statements: one may reject lightly as obsolete what does not go with the stream. There is also a risk of confusion: for example, between the effort of revising the foundations of microphysics and the denial of the practical indeterminism, The ambiguity of the word "complete" applied to quantum mechanics encourages such confusions. Therefore, it is perhaps more sensible to stick to a conceptual determinism, even if it remains "metaphysical." About positivism, what is the present situation? Physicists work on quarks and partons, not waiting till their experimental evidence makes them emerge in the "physical" world. Why should we not follow their example?
5.2. Some More Remarks
A new feature equations are found equations, he knew equations, on the
of modern theories seems to us worth mentioning: the before their interpretation. When Maxwell laid down his what meaning he attributed to each term. The quantum contrary, come from algebrai'cal requirements; they
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introduce magnitudes (Dirac invariants, etc...) leaving open a wide field of investigation, as to their physical meaning and their foundation. What can be the tests for a stochastic foundation of quantum mechanics? The obtainment of the Schr6dinger equation is perhaps not the most convincing: it is always possible to derive it from appropriate assumptions, and it is sometimes not easy to distinguish those which are plausible or artificial. A more important point is the basic unification with relativity, whose formalism has been so far simply inserted in the quantum equations: we have outlined this process in Ref. 1 and developed it in Ref. 2. One understands the link of relativity with spin and nonlocalization. More generally, the new description should explain features which up to now have been introduced formally. We see why the eigenvalues of the velocity in the Dirac theory are ±c: they correspond to the actual velocity of free travel. Similarly, we should not be satisfied to account for the Schr6dinger zitterbewegung by the interference of the positive and negative energy waves: we propose a model in Ref. 3, and link it with a model of the plane monochromatic wave in the proper frame. All this is ideal, i.e., far from the experiment. Another new feature of modern physics is that the simplest experiment involves a set of intricate events (dressed particles, emission and absorption, etc...). There remains, however, a domain which is both experimental and open to simplifying abstraction: the mass spectrum. Gilson (32) has anticipated that here the stochastic approach might reach practical success. Let us try to outline a program in this domain.
6. STOCHASTIC INTERPRETATION OF MASS A N D THE H A D R O N MASS SPECTRUM 6.1. Stochastic Interpretation of Mass
Eq. (2) can be rewritten:
mo= h/k
(7)
In a diffusion process the mean free path l is proportional to the diffusion constant ½k. (For instance the self-diffusion of gases C33~ yields l = ~ k # , where ~ is the free travel velocity). Thus, if we follow a given element, mass is proportional to the frequency f = c/l of its collisions with the other elements of the stochastic medium. (The random walk is assumed here for simplicity to be due to collisions; see Section 8.) A change of frame transforms the mass m 0 into rn = m0(1 --Z)2/C2) -1/2, and maintains it propor-
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tional to the density of the elements, and then again to the collision frequency. We show below that, if we consider the total number of collisions per unit time, it is proportional to m~. What is important to note at the moment is the link between the mass and the collisions. This link appears in the model of the Dirac fluid outlined in Ref. 1 (pp. 176-182): the collisions are due to the counter-velocities of the two antagonistic streams (~) and (~,), which give rise to a finite free path instead of the infinite (or very large) free path that characterizes the random fluid with velocity e (or approaching c). Mass is, thus, created by the coupling, in accordance with the following relations: (i/c) 274~O/Ot + S~, 8f)/~xk + (imoe/h ) ~v= 0
(8)
(l/c) 274 ~qJ/~t - S k, ~ z / & ~ + (imoc/h)(~ = 0
(9)
where 274 and S k are the Pauli matrices, and where mass appears as a coupling constant between the two streams. One also understands that the invariant !-Q21, which by Eq. (77) of Ref. 1 acts as a stochastic average of the corrected mass densities moP'~ and moP's, takes the role of half-mass density in the Dirac equations, as pointed out by de Broglie~34~: this feature also appears in the main term mo121c2 of the energy density (see Ref. 1, p. 189, Eq. B4).
6.2. The Hadron Mass Spectrum Before proposing suggestions on the hadron mass spectrum, we want to study from another angle the stochastic process. Let us consider the total number n of collisions per unit time of all the stochastic elements contained in a unit volume. If N is the number of these elements, a calculation similar to that 3 used in the kinetic theory of gases ~33~ gives n proportional to N 2. Besides, since each collision terminates two free paths, the total number of free paths is 2n, and their total length 2nl (in the proper frame) is also the total distance Nc traveled by the N elements in unit time, whence
N = 2nl/c
(10)
Putting first
n = a N 2 = a .4n212c-2,
whence
n=c2/(4al 2)
(11)
3In the theory of gases one has n = (2)- 1/2]-[ N2a2#, where a is the radius of the eiements and is # their average velocity.
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and then: (see 6.1): l = flk
(12)
n = c2/(4al 2) = m~ C2/(4af12h2) = 7m~
(13)
one has by Eqs. (7) and (10)-(12):
(with y = c2/(4afl2h 2) Thus, one obtains the following distinction: whereas the number of collisions per unit time of a given stochastic element is proportional to the mass m 0, the total number of collisions, per unit time, of all the elements contained in the unit volume is proportional to m 2. Now we remark that the choice between a mass formula in m 0 or in rn~ for the hadrons is still in discussion. Gell-Mann and Okubo ~35) have proposed a m0-formula for the baryons, and Feynman eL al. ~36) and others {37'38) have proposed equations containing mo2. The first option, which amounts to following an individual element, leads in our stochastic approach to a picture centered on the corpuscle; the second to an interpretation focused on the extended field. We arrive at the following conclusion: in the choice between a mass formula in m 0 or m 2, the quantum dualism is involved. Taking up a relation in m 02 suggests a sort of democracy between the elements of the stochastic medium which will appear in Sections 7 and 8 in good agreement with the patton model. There are serious reasons to take up a rn~-formula. First, for sake of unity, it is unsatisfactory to use a m0-formula for the baryons, whereas a m~formula is recognized necessary for the mesons. Next the Regge pole theory yields straight trajectories in the (or,m~) plane (dr is the spin). We show in Ref. 4 that the hadron phenomenology accords very well with the formula: m o2 = m ot2 -- A Y + B T ( T + 1) ( Y is the hypercharge, Tthe isospin) (14)
with simple laws of variation of A and B with the spin J: A =~,+OS,
B=(a+(flJ)/(Zl+
1)
Eq. (14), already proposed by Flato and Sternheimer C~s) for the baryons, is more accurate than the Gell-Mann-Okubo formula for the ½ + lower octet and the 3 + lower decuplet; for these two multiplets, m o being constant within each of them, it fits the general SU3-formula established by Okubo, ~39~to the second order of symmetry breaking effect: m0(or m~) = m~(or m'o2) + B T ( T + 1) + C Y 2 + D Y K + E K 2 where K is T ( T + 1)--¼Y 2 (Here C = D = E = 0 . )
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We describe in Ref. 4 the m~2-spectrum and show that all the m 0i2 can be described as linear combinations, with simple integral coefficients, of three basic quanta. This quantization of m~2 entails that of m o, 2 and should be connected with the granular structure of matter (partons). This point will be developed in Ref. 4.
7. OVERALL C O M M E N T We give an overall comment, where we emphasize the link between the points which have been dealt with separately: diffusion process, hidden variables, and epistemology. 7°1°
In Section 3 we have studied the limitation of the scheme proposed in Refs. 1 and 2. Here a first point should be made clear: is this only a selfcriticism? Comparing Sections 3 and 4, we notice similar features in the two basic difficulties of quantum mechanics: on one hand the possibility of a clear conceptual representation, and on the other hand the (EPR) paradox and the possibility of solving it by hidden variables. As to the first problem, we have seen our scheme vanish with the appearance of the macroscopic body destroying the smooth variation of the local magnitudes. Turning now to Bell's proof, we observe that the contradiction between hidden variables and quantum mechanics can receive the same interpretation; whence this suggestion: the two obstacles m a k e one. To discuss it, we ask ourselves how many ways we have around this latter problem: (1) Bell's proof is wrong; (2) the proof is right and we explain the paradox by nonlocality, nonseparability, instantaneous transmission.., etc.; (3) the explanation lies in the general limitation that we have suggested. If one admits (1) or (2), this paper can be restrictively considered as a self-criticism. If, however, one sees no reason to reject Bell's proof, not to find satisfactory the explanations so far proposed in the frame of (2), then (3) becomes plausible, which gives a broader significance to the limitation that we have introduced. 7.2. To go a step further in our analysis, we take the example of the diffraction by two slits (Young experiment): the particle localized behind one of the slits has been influenced by the other. The usual interpretation (sudden point-like condensation of an extended wave) is hard to accept. LandC 4°)
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remarks that the matter distribution around the slits has periodicities in space analogous in some way to those of a crystal, and proposes to ignore the wave and to explain diffraction in a purely mechanical fashion. If we accept his interpretation, we must recognize this (which is by no means a criticism): the explanation is not so simple as the wave formalism. This leads us, coming back to our scheme, to distinguish two domains. In the first one, taking no care of the practical conditions of experiment, and ignoring quantization and addition of waves, we build our stochastic formalism; we obtain, if not its physical interpretation, at least its logical consistency (basic distinction of the nonrelativistic and relativistic levels, derivation of the fundamental equations, and model of the spinned fluid). We note that it ceases to be valid when macroscopic bodies are introduced, i.e., in all practical situations; our scheme is ideal. Now we forget it for a moment and notice that with the macroscopic body (mirror, Young slits, etc...) another feature emerges: if we do not want to take into account, in its complexity, the structure of the macroscopic body, we have another way around the problem, we introduce the wave, i.e., a tool of a powerful simplicity, which is devoid of any physical existence. Next we introduce similarly all the quantum formalism, on which we observe the same character: it is simple, efficacious, and devoid of foundation. What we notice of this rupture when passing from one of these domains to the other, is not specifically linked to our scheme; we perceive it as well when following Land6's approach, or following Bell and standing before the discrepancy between the lhs and the rhs of Eq. (6). If it should then have a general character, where should we have to look for its grounding? 7,3.
We think that we find it in the very evolution of microphysics. The crisis that broke out in the 1930's appeared with the quantum dualism; yet it seems to us that its deeper feature is a change in the situation as regards idealization. Idealization has so far been recognized as necessary for the progress of science: no laws could have been established without a method of simplification where secondary effects were ignored. The laws emerged from this method because the alterations produced by the neglected effects were actually small (the radiation pressure of the sun on a projectile, etc...). Now this happened, and should necessarily happen when going deeper in the microscopic domain: the effects that one was tempted to neglect began to be of the same order as the main process. In particular the fiction of the test body, which allows a measurement without perturbing sensibly the object to be measured, vanishes at microscopic scale.
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Is this not the very language of the Copenhagen school? It is indeed, with an important difference. Instead of saying, as it does: the old concept of intelligible reality has vanished and been replaced by a new representation expressed by the quantum formalism, we say: a sort of complementarity has arisen temporarily between intelligibility and practical efficacy. In the two slits experiment the choice, at least in principle, was still possible. In the process C ~ A + B involved in Bell's proof (Section 4) we can no longer choose because the creation or destruction of a particle is (at the moment) unintelligible, and breaks the description by deterministic parameters in the same way as it breaks our scheme. As another aspect of this complementarity one can mention the second quantization for a system of many particles: a degree higher of quantization corresponds to a degree higher of complexity in the experimental situation. This seems to be still not so far from the orthodox interpretation. Actually it is, for instead of stating the quantum formalism is complete, there is nothing beneath (or something metaphysical in which we are not interested) we say there is something beneath, which it is important to search into, even if at the moment it has no practical effects because striving after a deeper foundation must in the long run have practical consequences.
7.4. Here we find the epistemological problem. We have called the positivism in question. Another aspect is the idea that a new theory brings nothing if it does not predict new experiments. Consequently, some authors of the stochastic (or hidden variable) school imagine experiments where quantum mechanics should be wrong. They may succeed, but even if they do not, their work will not necessarily be a failure: the creation of the quark and parton model had more at the start of the ambition to understand better the hadron structure than to predict new experiments. Science does not not always progress by extension, (i.e., accounting for new facts), it may also progress by deepening and unifying.
7.5. In what direction can we expect this deepening? An approach to the hadron spectrum can conciliate the constraint of ideal simplicity with the possibility of experimental verifications. Moore ~41~ has already proceeded in this way, in his stochastic study of the bag model. We have outlined in Section 6 our contribution to hadron spectroscopy. 14) In this domain we notice the previously mentioned conciliation: the mass of a resonance is a
Stochastic Foundation for Microphysics. A Critical Analysis
715
basic magnitude that emerges in a scattering experiment which stands at the higher level of complexity. The proof that the parton peripheral model involves a random walk opens a new field of research. Calling K o the modified Bessel function: co
K°(r) = ~i c°s(rt)(t2 + 1)- 1/2 dt and denoting by Po the mass (physical or conventional) attributed to the parton, the mean free travel 2/~ of the transverse random walk is (4:) given by s~
= ( x ~)
, 2 )= :(h/.oc , =
with A and B expressed as follows (putting z =
Crpo/h):
.00
A = J
z3Kg(z) dz o . oo
= Jo zK~(z) dz One finds A ~ 0, 3 and B ~ 0, 7. Thus f o can be identified with the mean free path l 0 = h/moe of our stochastic scheme if p0 is about half of the particle mass m o. This calculation is not much relevant because the peripheral model is very crude, and because it is not sure that attributing a mass to the parton has a definite significance; however, it is sufficient to give the hope that a unified scheme is possible. 7.6.
Deepening indeed is also unification. A basic unification of quantum mechanics and relativity has not yet been much considered: the relativistic formalism has been simply inserted into the quantum equations. We recall that Ref. 1 and 2 are an effort of unification at the lower stochastic level. The relativistic features [e.g., space-time symmetry in Eq. (31) of Ref. 1, generalizing the Fokker-Planck equation] do not emerge, as usual, from the Lorentz formulas introduced at the start, but directly from an analysis, at a higher degree of approximation, of the diffusion process.
8. C O N C L U S I O N 8.1.
First, have we definite suggestions on the physical interpretation of our scheme?
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It is tempting to explain the two slits experiment by assuming that, when the corpuscle does through one of the slits, the fluid is in contact with the other. If, however, following Land6, one thinks that one can do without the wave, one can also do without a fluid attached to the wave. Then we turn toward the particle structure. We remark that for the electron the mean free path in the rest frame is three orders of magnitude higher than the radius, which makes it hard to consider the stochastic process as physical. On the contrary, these quantities are of the same order for the hadrons, and the physical interpretation becomes all the more plausible that the parton model involves a random walk. 4 Moreover, this model describes'the partons as interacting individually in a scattering or formation process: this agrees with the suggestion presented in Section 6 (to justify a mass formula in rn02) of a sort of democracy between the stochastic elements constituting the particle. The extended medium formed by the grains of matter will in a later stage have to be identified with the fluid (F0) of our stochastic formalism, in the ideal conditions where (F0) can be defined. At the moment the fluid should be considered as a working tool, which allows us to build the formalism unifying the bases of quantum mechanics and relativity. It is not rare that a formalism survives the physical interpretation which yet had been necessary for its elaboration. As regards the cause of the randomness, the parton model involves a fragmentation of the partons (at least in its early version, see Ref. 42). On the other hand we may assume that the randomness extends to the cloud surrounding the hadron: this suggests to explain it by radiation. We leave the choice open. We had mentioned for simplicity collisions in Section 6, but this picture is not essential for the analysis that we have outlined there: the main point is that mass is assumed to be linked to the number of path changes per unit time. 8.2.
Stochastic electrodynamics (SED) has been left out in this paper. Important works have been produced in this domain by Boyer, ~43) Claverie and Diner, t44) Marshall, (45) de la Pena-Auerbach and Cetto, (45) Santos, (47) Surdin ~48) and others. It is certainly a fascinating task to give a new basis to quantum electrodynamics, where the technical efficacy has so far prevailed on the conceptual perfection. Here we stand at the higher level, where the 4 The development of quantum chromodynamics leads to a distinction between the valence elements (quarks) and the sea (gluons). One should examine if the random walk concerns all the grains of matter, or only the gluons. The latter view would suggest an approach to stochastic chromodynamics somewhat similar to stochastic electrodynamics.
Stochastic Foundation for Microphysics. A Critical Analysis
717
simplest experiment involves multiple and complicated actions. Thus, the removal of the infinities by the revision of the foundations might be a fardistant outlook. 8.3.
Coming back to the hadron structure, we recall that, in Section 5 of Ref. 2, quoting the creators of the quark model, we had mentioned their feeling that the laws of mechanics might not apply to the subparticles. Now, in our scheme we find similar features: in a stochastic medium supposed to be made of several elements, mass is defined by Eq. (7) as a stochastic property of the whole medium, and has no meaning for the individual elements (See also subsection 2.2 of the present paper, and Sections 3 and 5 of Ref. 2). All this gives the hope of new relations to emerge between the diffusion process and the hadron substructure. Thus, even if the limitations that we have encountered could not be eliminated, there would remain a broad field of research for the stochastic foundation of microphysics.
APPENDIX. THE DERIVATION OF EQUATION: A NEW PRESENTATION
THE
SCHRODINGER
A.1. Starting from the fluid (F0) with density P0 and mean velocity u, one defines a new fluid (F) whose velocity v is given by Eq. (8) of Ref. 1, its density p being obtained from Eq. (39) of Ref. 1. The fluid (F) will be shown possible to identify with the quantum fluid whose density and velocity are defined in terms of the wave function ~ = a e x p ( i c p / h ) , as p = a2
(A.1)
v = -Vq~/rn o
(A.2)
We denote as follows the quantities involved in (F0) and (F): Position of an element Density Velocity Energy of an element Action function for an element Potential
82511119110/I-5
(V0)
(F)
x P0 u w0 ... (see below) U
x P v w ~w) ... (see below)
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Aron
For the energy wo in (F0) we take up the classical expression (e is the charge): w o = mo c2 +½too u2 + e U (A.3) The first empty compartment in our table is the function in (Fo) to be compared with the action function 2(F ) of the quantum fluid, defined as ~(F) = m0 v2
-
-
W
(A.4)
The stochastic fluid (F0) is not governed by the Lagrange-Hamilton formalism; however, we can conventionally define a function ~(Fo) by the correspondence: po~(rol(X, U) = p~@(F)(X, V) (A.5) The second empty compartment is the potential in (F). We remark that (F) is a fictitious fluid derived, as explained above, from the basic fluid (Fo). Thus, the physical potential U applies to (F0); with the fictitious element of (F) we can associate a fictitious potential U', such that W = mo ¢2 -}- ½mo v2 + eU'
(A.6)
Now we make the following dynamical assumptions: I.
The function ~(Fo)(X, U) has for u = 0 the classical expression: ~(Fo)(X, O) = --rn o c 2 -- e U
II.
(A.7)
The element of (F) moves according to the classical law: m o dv/dt = -cVU'
(A.8)
A.2 Expression of w in the case u = 0. By Eqs. (A.3)--(A.7), and using Eq. (39) of Ref. ! with t--+ t - 500, 1 one obtains Eq. (45) of Ref. 1, whence the expression (47) of the energy density. A.3 Expression of w in the general case (u 4= 0). With the basic quantities at our disposal: k, m o, p, and v, to order zero in 00 we can define two velocities: v and vl = k V p / p . We should expect for w in the general case a quadratic expression which reduces to Eq. (47) of Ref. 1 for v = - ½ v l ; i.e. (to order zero in 00): 1 2 w = m o e2 + 7rnov + eU-
½mok2(ApV2)/p 1/2
+ m0(v + l g l ) . (a7 ~-/~Vl) where a and fl are dimensionless constants.
Stochastic Foundation for Mierophysles, A Crltical Analysis
719
F o r v a n d U c o n s t a n t the average value of w should be moC2+ 1 2 ~moV + e U ; this yields a = f l = 0 , and Eq. (47) of Ref. 1 valid for u : / : 0 . Next, using Eq. (A.8), one derives the Schr6dinger e q u a t i o n as explained in Ref. 1.
ACKNOWLEDGMENTS The a u t h o r is i n d e b t e d to P. Claverie, T. W. M a r s h a l l , a n d L. Pesquale for fruitful discussions. He is also grateful to one of the referees for his attentive reading a n d his constructive criticism.
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