Stability Implies Chance

Stability Peter Free Implies Chance P. KIRSCHENMANN University Amsterdam De Boelelaan 1105 The Netherlands Abstract. Among the philosophical issues raised by quantum theory, the alleged indeterminism of quantum-physical phenomena, as opposed to the determinism of classical physics, has stood at the center of discussion. There are various other much-debated problems which are directly or indirectly related to this issue or, more generally, the probabilistic character of quantum theory. Less attention has been paid to the questions of the stability of atoms and the specificity of their properties, which quantum theory was primarily designed to resolve. I argue that the indeterminism of quantum theory becomes less intriguing and can be better comprehended when seen in the light of these questions or briefly, that "stability implies indeterminism, or chance." Chance, in this respect, appears as an objective and legitimate mode of succession of otherwise stable and discrete well-determined states of certain physical systems. Philosophicaldiscussion of quantum theory has tended to focus on a number of specific issues, many of which have to do with alleged or genuine differences between classical and quantum physics (see, e.g., Bunge and Kalnay, 1975). The single most intriguing feature of quantum theory has undoubtedly been the alleged indeterminism of quantum-physical phenomena, as contrasted with the determin ism of classicalphysics,and various other features and assertions of the theory which are very directly related to it. One can list here the assertions implied by the so-calleduncertainty relations, which are very general theorems of the part of the theory that can appropriately be called "quantum statics". They already sufficeto highlight the essentially or even irreducibly probabilisticcharacter of the entire theory. This peculiar feature is also apparent in the probabilistic (more precisely, stochastic) assertions of quantum dynamics, the part of the theory which accounts for processes, evolution of states and state transitions, in quantum-physical systems. Familiar examples of such processesare radioactive decay and electron scattering. As is well-known, it was in view of the latter processthat Born established the probabilisticinterpreta tion of the function representing quantum-physical states (see Jammer, 1966, Sect. 6.1). Another much-discussedfeature here is the fact that the probabilities in question are non-classical:there is interferenceamong the probability amplitudes occurring in the state functions. Quite apart from the abundance of particular probabilistic statements in quantum theory, the indeterminism can already be detected in the assertion that quantum-physical systems undergo "guantum -73- 28 P. P. KIRSCHENMANN jumps", inasmuch contrast In to as the quantum they classical theory, quantization, points mechanics by the to play in with the "duality role an say, issues indeterminism and particle" on the notion regarding individuality or greater would one surveys If has a mainly problem, and, of by be these close specificity of the state basic in idea of equations is of underscored principles on which have come with properties been classical which of it, of the given one major to its about the distinction from possibly divergent issue their the concerns peculiar the statistics. question debate problem has significance was that quantum-mechanical discussed and controversial question and philosophical that This quantum-physical of a quantum logic. the notice was less the of about entities connected as of system A degree the questions time. theory. such problem a physical has lesser involve further from or dependency related of and can greater theory, quantum-physical one connection a mentioned raises issues quantum to the comparison, focused, of the distinct moreover, in of space identity been development and continuous. the difference symmetry presumed issues which attention, which the This quantum the and all counterpart, assumptions logic are through between and are of observer, Practically quantum-theoretical Far difference mechanics. states, processes is introduced considerations much-debated alleged of wave classical discrete physical discontinuity essential , its assuming all 5 theory. other measurement. of that classical algebraic quantum phenomena capable of up and, particular Various only discreteness which quantum are presupposition Vol. of the the various in question of the of specificity forms of quantum largely early the of neglected, stages stability their matter physics been of of properties to which the atoms or atoms the could combine. It may attention, one of thing, primary great properties To be of these is sure, and it of and even may overly But have stability is other though seemed and many fact has which "indeterministic theory that, specificity, that explained and after physics it could designed had no -74- than to successfully longer be For their have itself and such been also as philosophical more by "quantum familiar them. answered of notice. account apparently for one specific particular should by account there satisfactory predicted, of hand, surprising. and been beginnings other without account -rather primarily has including a novel the philosophical the the less philosophers of events" was since goes with due specificity On objects easily us elements it nature Ch. ‡U). physical understandable phenomena its and circumstance provided theoretical the of this received stability 1968, render familar uncommon not nature's philosophers of theory unfamiliar jumps" can has of Peterson, permanence quantum by of e.g., which issue question concerns features. intrigued facts, an this the (see, reasons the that because philosophy enough new surprising only the Western are be if Further, the question concern. the No. 2 Stability Implies Chance 29 Finally, there is a general tendency to regard change as being in need of explana tion-and deeper philosophical comprehension, especially in the case of stochastic changes-whereas the stable and fixed features of nature can easily be taken for granted. Several writers have with some emphasis pointed out this neglect (see, e.g., Hawkins, 1964, pp. 172f; Weisskopf, 1972, pp. 41ff.), a neglect which has been unfortunate. If a theory is designed to solve particular problems one should not, in interpreting it, lose sight of its explanatory objectives. If the major success of quantum theory lies in its power of explaining important aspects of nature's stability and specificity, then references to these features, it would seem, should be relevant to any interpretation of this theory. I am of course far from suggestingthat a reorientation toward the questions of stability and specificitywill solve all interpretational problems of quantum physics. What I wish to claim is that the so-called indeterminism of quantum-physical phenomena can lose much of its perplexing character if it is seen in the context of these questions. We lose our perplexity about peculiar aspects of a situation once we realize that, given certain other, less intriguing particularities of the situation, the origin ally perplexing aspects cannot be any other than they actually are. Let us apply this general idea to the situation in quantum theory and its peculiar aspects which I have indicated. In attempts to comprehend this theory, then, it would be helpful to see that inasmuch as it gives an account of nature's familiar stability and specificityit must include elements of indeterminism. This would mean, as regards quantum-physical systems themselves, that if they are to exhibit the particular stability and specificityof their properties, as they do, then the evolu tion of their states or the state transitions cannot be but stochastic. There can, of course, be little hope that one could show that, briefly speaking, stability and specificity necessarily imply indeterminism or stochasticity. Quite generally, it seemspreposterous to attempt to prove the necessity of anything in nature. What one can hope for, however, is that it be possibleto find good reasons in support of the claim that this stability implies indeterminism,or chance. In the sequel I shall outline several major arguments to that effect and discusssome possible objections. The most convincingargument for the claim that "stability implies chance," one should think, is furnished by quantum theory itself. After all, it is the theory which successfullyaccounts for both the remarkably stable and the perplex ing probabilistic features of quantum-physical systems. In the matrix formulation of that theory, the fundamental equations are the commutation relations holding between dynamical variables, especially the non-commutation relations between canonically conjugate variables. On the basis of the latter, quantum theory is capable of accounting for the discrete energy states of atoms, hence their stability as well as their specificproperties, as these are displayed in the discretenessof their -75- 30 P. P. KIRSCHENMANN Vol. 5 spectra. For atoms interacting with radiation fields, the non-diagonal elements of the energy matrix, once they are interpreted as transition probabilities between these discrete states, account for the intensities and polarizations of the spectral lines. Quite obviously,then, the account of stability entails indeterminism in the transitions between the so-called stationary states of atoms. Inasmuch as the latter, characterized by a discrete set of energy values, are the only states the atom is capable of assuming, a continuous transition between them is precluded. The kind of indeterminism expressed by the so-called uncertainty relations, which is a probabilistic feature of quantum statics, is an even more direct result of the quantum-theoretical account of the stability of atoms. The commutation relations which are needed to explain this particular stability are also essential to the derivation of the uncertainty relations. One can thus say that both the stability of atoms and this kind of indeterminism flowfrom the same commonsource (cf., e.g., d'Abro, 1951, pp. 959f.). If quantum mechanics is taken to have the final say in the matters at issue, the relationships pointed out between its account of atomic stability and its indeterministic features are inevitable. But we know that various objections have been raised against the theory as a whole. It was mainly because the theory introduces probabilities in its account of processesor in its account of the relation between various properties of one or more quantum-physical systems that it has been regarded as provisional or incomplete. What has been at the root of most of these objections is the idea that a complete account of quantum-physical pheno mena must in essential respects be similar to the deterministic descriptions of physical processes known from classical physics. The question is whether there is good reason to believe that such an approach will be successful. The determinism of classical physics is based on the idea of continuity: physical processes are described as continuous functions of time. Now, it should be obvious enough that the particular stability of atoms is incompatible with continuous transitions from one of its stationary states to another. Atomic stability implies discreteness of these states and discontinuity in state transitions, which therefore cannot be governed by laws complying with the ideal of classical determinism, differential equations. Once the discrete stationary states of atoms are fixed, the imposition of such laws would mean an over-determination, which in general would involve a contradiction (cf. Holing, 1971, p. 138). The laws governing the state transitions can at best be probabilistic. One may retort that classical physics also accounts for various phenomena of stability, which seems to show that stability is compatible with classical deter minism. However, these classical phenomena of stability are not of the same kind as those found in quantum physics. To take an example, the stable equilibrium of a mechanicalparticle under certain kinds of forces is still such that an ever so slight -76- No. 2 Stability Implies Chance 31 additional force on the particle will make it abandon its equilibrium state. An atom, on the other hand, will not necessarily pass from one of its states to the next when some additional energy is offeredto it, as the findings about the photo electric effect show. This plainly means, of course, that state transitions are discontinuous when they occur. Theoretically, the particular nature of atomic stability is evident from the fact that it cannot be represented as an equilibrium under forces, but has to be accounted for by special quantum conditionswhich are founded on the aforementioned commutation relations. Thus, stable configura tions of this kind cannot be and indeed have not been explained within the theoretical framework of classical physics. In classical mechanics, for instance, the existence of stable particles, permanent material objects, or continua with fixed properties was simply taken for granted. (Since no physical theory can explain everything, quantum theory must also take certain elements for granted: the quantum-mechanical account of the atom does not explain the nature and properties of its constituents, stable nucleons and electrons.) That classical mechanics in combination with the theory of electro-magnetism is inadequate to the task of explaining the particular stability of atoms and the existence of discrete optical spectra is well-knownfrom the early theoretical models of the atomic structure. An atom constructed according to Rutherford's model would not be stable. Bohr, who realized this problem in its full significance,was led to the conclusionthat it was necessary to introduce non-classicalideas in order to solve it. One of the assumptions Bohr put forward was just this, that an atom can only assume stationary states corresponding to discrete values of its energy, and that any change of its energy can only occur discontinuouslyby a complete transition between such states. It is impossible, of course, to rule out entirely that the transitions between stationary states could not in some sense be continuous, or that there could not be some underlying continuity. Yet it seems clear that, should these transitions turn out to be continuous processes after all, they can hardly be of the classical type. The particular nature of atomic stability and the discreteness of the energy emitted or absorbed by atoms which results in their specificline spectra constitute sufficient evidence against such a possibility. However, one may grant that atomic stability and specificityimply discrete states and discontinuous state transitions and still hold at the same time that these transitions occur in some deterministic way. It is perfectly possible that discrete states follow upon one another in a strictly deterministic fashion. To be sure, this kind of determinism would be essentially different from the continuous temporal determinism of classical physics. We know, however, that quantum - physical transitions, when they occur in larger numbers, either exhibit statistical regularities or are responsible for particular features such as the brightness of -77- 32 P. P. KIRSCHENMANN Vol. 5 individual spectra lines; and it is in terms of transition probabilities that quantum theory accounts for these regularities and such phenomena. Any attempted deterministic account of the transitions, it would seem, would in turn have to involve an explanation of these probabilities of transitions. Yet, there are strong arguments for saying that a purely deterministic account would not succeed. One can think here of von Neumann's argument, which he thought showed that theories employing "hidden variables" and meant to account for the facts of quantum physics were impossible (von Neumann, 1955, pp. 295-328). Although his argument does not prove quite as much, it is relevant to the present discussion. Von Neumann showed that it follows from quantum mechanics that even in the case of "homogeneous" ensembles of quantum-physical systems not all physical quantities are "dispersion free". Since the notion of dispersion is a probabilistic concept, this means that the probabilistic element cannot be eliminated from the theory. Von Neumann went on to argue that the laws of quantum mechanics could never be re-derived from a "hidden variable" theory. This claim is certainly correct if such a theory is assumed to be purely deterministic (i.e., one from which the concept of probability is absent). The last point can be made, I think, in a quite general way. A probabilistic or stochastic theory can never be reconstructed on the basis of a purely deterministic theory, since no probabilistic statement is derivable from deterministic statements. One would first have to introduce the concept of probability and connect it with the theory before the latter can supply an account of probabilistic statements. One can, of course, introduce equivalent notions instead, e.g. that of random fluctuations, as it has been done in certain versions of "hidden variable" theories. Quite generally, therefore, probabilistic statements cannot be eliminated by way of replacing them with deterministic statements. And to the extent that the stochastic nature of quantum theory is well-established, the indeterminism that appears in it will be a feature of any theory of atomic phenomena . I have emphasized the fact that quantum theory successfully accounts for the stability of atomic systems and the specificity of their properties , e.g., their discrete line spectra. The notion of chance transitions between their states is often looked upon as nothing but an indeterministic element of the theory , which would suggest that it could scarcely have any direct connection to anything definite or stable. It is remarkable, therefore, that these chance transitions are themselves essential to quantum-theoretical accounts of various specific permanent properties of matter (which also were taken for granted and left unexplained in classical physics). Among such properties are electrical conductivity, magnetic constants, properties of semiconductors and transistors. Their specificity is due to definite or stable probabilities of transitions between various states of electrons . On the basis of quantum theory as a whole, many other material properties have also received a -78- No. 2 Stability Implies Chance 33 satisfactory explanation, for instance, densities, viscosities, or heat capacities (see Wigner, 1973, p. 374). We have found good reasons for saying that the stability and specificity of atoms require an explanation in terms of discrete atomic states, and that this explanation is not compatible with the assumption that the transitions between such states be regulated by deterministic laws of the type known from classical physics. Given the latter circumstance alone, the transitions as well as quantum - physical processesin general could still be totally chaotic. The fact that they are not, but rather are governed by stochastic laws involving definite probabilities can be considered an additional determination, over and above the determination which fixes the discrete stationary states. In this sense, the addition of a stochastic determination for quantum-physical processes is the most one could and should expect. Any further element of determination which would more completely define the successivestates of such processeswould, as has been noted, in general involve a contradiction. Thus, when taken together with the structural determina tion of quantum theory which fixes the stationary states, the stochastic determina tion of the successionof states turns out to be the fullest possible determinism one can have for quantum-physical systems (cf. Hawkins, 1964, pp. 172-178). The label "indeterminism" for the latter part of the complex of laws governing atomic systemsmust therefore appear to be entirely inappropriate; it seems to be so readily applicable only once one has forgotten about the part which determines stable configurations. We are thus led to admit chance as a legitimate and lawful mode of the succes sion of states of particular physical systems. The following analogy may help render this claim still more plausible. There is a hardly disquieting notion of objective chance which has been proposed by various philosophers, such as Chrysippus, Mill, or Cournot (cf. Bunge, 1951). They interpreted chance occur rences as accidental crossings of encounters of two or more independent determ inistic ("causal") processes. These deterministic processes are precisely of the kind that is known from classical physics. Since such chance encounters between classicallydeterministic processescan only be due to accidental correlations of their initial conditions, this notion of chance reflects nothing else than the well - known circumstance that in classical physics the initial conditions of processes are totally arbitrary and in no way explained by the theories accounting for the processes in question. Thus, these processes appear as self-sufficiententities, temporally fully determined once the values of the relevant variables are fixed at one time. They can then bear only chance relationships to other equally fully determined processes. In this context, chance figures as a legitimate mode of coexistence of temporally well-defined entities, viz., deterministic processes or histories of classical deterministic systems. -79- 34 P. P. KIRSCHENMANN Vol. 5 To draw the analogy now, we have seen that the stationary states of an atom are likewise, not temporally, but configurationally determined once the value of the energy (and those of possibly relevant other quantum numbers) is given. Thus, the states of an atom are structurally fully determined in themselves. The temporal transitions between states, in analogy to the spatial correlations or encounters of temporally determined processes,must therefore also be "accidental", i.e., chance relationships. This analogy, I think, can help remove some of the mystifyingaspects of stochastic transitions and thus make it easier to acknowledge objective chance as a legitimate mode of succession of physical states. It is the mode of state successionwhich alone seems compatible with the particular stability and specificitydisplayed by quantum-physical systems. Bibliography 1. Bunge, M.: 1951, "What is Chance?', Science and Society 15, 209-231. 2. Bunge, M., and A.J. Kalnay: 1975, "Welches sind die Besonderheiten der Quantenphy sik gegeniiber der klassischen Physik?", in R. Haller, J. Gotschl (eds.), Philosophie u nd Physik, Vieweg, Braunschweig, 1975, pp. 25-38. 3. D'Abro, A.: 1951, The Rise of the New Physics, Vol. 2, Dover, New York. 4. Hawkins, D.: 1964, The Language of Nature, W.H. Freeman, San Francisco. 5. Heisenberg, W.: 1958, Physics and Philosophy, Harper & Row, New York. 6. Hulling, J.: 1971, 'Zur Kategorialanalyse des physikalischen Feldbegriffs', in J. Hulling, Realismus and Relativitat, W. Fink, Munich, 1971, pp. 126-139. First publ. in Philosophia Naturalis 10 (1968), 343-356.7 . Jammer, M.: 1966, The Conceptual Development of Quantum Mechanics, McGraw-Hill, New York. 8. Neumann, J. von: 1955, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton. 9. Peterson, A.: 1968, Quantum Physics and the Philosophical Tradition, MIT Press, Cambridge, Mass. 10. Weisskopf, V.F.: 1972, Physics in the Twentieth Century, MIT Press, Cambridge, Mass. 11. Wigner, E.P.: 1973, 'Epistemological Perspective on Quantum Theory', in C.A. Hooker (ed.), Contemporary Research in the Foundations and Philosophy of Quantum Theory, D. Reidel, Dordrecht, 1973, 369-385. -80-