Quantum Physics
p In quantum mechanics probability as a measure of possibility is numbered among the main laws, i.e. possibility is integrally linked with actuality, while probability cannot fail to be objective, and reflected in the corresponding concepts of quantum mechanics. It is possible in quantum mechanics, of course, to indicate the eigenvalues of the operators of quantities from the wave function if this wave function is the eigenfunction of the operators, and to 180 calcuiate the mean values of operators or the probabilities of their eigenvalues, knowing the wave function, if it is not their eigenfunction. This also means that probability should not be approached in quantum mechanics as something subjective and isolated from reality. Let us consider this last point in greater detail.
p Imagine a flux consisting of a large number of electrons hitting a screen with two slits from the left; the electrons passing through the slits then form a diffraction pattern on the right on a photographic plate. The essence of this wellknown imaginary experiment is this: the components of the flux passing through the slits interfere with each other. If one assumes that the electrons are classical particles, the diffraction pattern would mean that electrons sometimes annihilate one another and at other times arise from nothing, but that is excluded by the laws of conservation. It remains to assume that the electron interferes with itself,^^47^^ in other words that the electron possesses wave properties simultaneously with particle ones. But then the probabilities of an electron hitting such-and-such places of the photographic plate cannot be interpreted in the sense that they are the result of ignorance of certain details of its motion: from the point of view of classical theory it is absurd that the probability of an electron reaching a certain spot on the photographic plate through two apertures is zero, while for one aperture (it is irrelevant which one) the probability of an electron hitting the same place has a definite nonzero value.
p The dual corpuscular-wave nature of micro-particles determines that probability in quantum mechanics is by no means the result of mere application of the propositions of the theory of probability to the motion of particles of classical type. Quantum mechanics resembles statistical mechanics in that both theories accept probability in the initial state. Quantum mechanics also resembles the general theory of stochastic processes since both theories accept probability when there is a transition from one state to another. As regards its theoretical foundation, however, quantum mechanics is by no means identical to the statistical theories of classical physics, and the wave function used in it to calculate the probabilities of the eigenvalues of the corresponding operators differs radically from the probability distribution function in classical statistical theory.
181p What is the nature of quantum probabilities? Or what are the grounds for probability in quantum mechanics?
p From what we have said above, the answer suggests itself. The grounds for the probabilities of the values of quantum quantities are in the unity and identity of the opposing particle and wave properties of micro-objects. These probabilities are not therefore something alien to the laws of atomic phenomena, arising as a result of ignorance of certain circumstances, but are a necessary element of these laws. From that angle the wave function pertains to a single microobject; at the same time it characterises those properties that a micro-object possesses in the given conditions and those properties that it will possess in other conditions which exclude the given ones. This fully accords with the concept of quantum probability, since the probability of one behaviour of a micro-object, or another, is not brought in from outside but is internally connected with the properties the micro-object possesses at a given moment of its existence.
p In studying laws of atomic motion that involve probabilities, quantum mechanics, of course, solves the problem of the transition of possibility into reality and of the realisation of probability laws. This question cannot be sidestepped in quantum mechanics, if only because experimental determination of the numerical values of probability becomes necessary, without which quantum mechanics would not be a physical theory.
p As concerns the philosophical aspect of the matter, possibility is not cognised directly but in a mediated way, through the cognition of reality. The mediated way of cognising possibility corresponds to the possible’s being the inner of actuality, i.e. exists owing to actuality, its other, and not in itself. From the standpoint of physics this means that, for the passing over from possibility to reality, from the probability of a process or of the value of a physical quantity to the realised, either unlimited repetition of the conditions in which realisation occurs or an unlimited number of phenomena representing the possibilities being realised in’the given conditions are necessary; in short, it is necessary to introduce the concept of statistical ensemble into quantum theory. This is justified by the fact that quantum theory deals with physical phenomena that arise not as 182 a result of the action of separate individuals but of vast aggregates of them (spectra, a-radioactivity).
p Dialectical ideas of the unity of actuality and possibility, and also of transformation of possibility into actuality, have thus found broad application in quantum mechanics, which cannot be understood correctly without them.
p Let us now summarise Heisenberg’s point of view on possibility and actuality in quantum mechanics by means of the following scheme:
p Physical Knowledge
p of Atomic Processes
p Symbolic representation of state by Description of state by means means of the wave function
p of classical concepts
p A wave equation of dynamic type
p Statistical laws
p Knowledge of ‘actuality’ is incomplete knowledge
p The wave function governing pos- A description by means of sibility symbolically character- classical concepts inevitably ises the state of an atomic particle contains a ‘subjective’ element, completely
p i.e. a statement about the ob-
p server (instrument) The characterisation of atomic processes by means of the wave function is a complete, objective one, i. e. contains no references to the observer (instrument).
p The transition from possibility to reality (actuality), or from the mathematics of the atom to its physics, is made by introducing classical concepts or references to the observer (instrument) into quantum theory.
p Thus, according to Heisenberg, the possibility of an event is something lying between the idea of an event and the actual event. The statistical element, however, is introduced simply because atomic processes cannot be described by any means other than classical concepts, which are only applicable to atomic processes with limited accuracy. In other words, Heisenberg believed that quantum theory dealt not with nature as such but with nature subjected to the effect of human methods of investigation; that is why statistics comes into quantum theory.
p For a critical review of these views of Heisenberg’s we refer the reader to the preceding sections; here we should discuss other topics of our theme.
p In analysing determinism in quantum physics the thesis of the infinity of nature, the inexhaustibility of matter and 183 of any of its particles, and the infinity of matter in depth and breadth is very important. This infinity is composed of an infinite number of finite objects at various stages ( elementary particles, atomic particles, macroscopic bodies, cosmic formations—to sketch the division roughly), the transition from one stage to another being transitions of quantity into quality and vice versa.
p Taking this thesis as our guide we can say that knowledge of an object (or process) is knowledge of it as an element of a certain whole (e.g. the atom is an element of the molecule) and at the same time knowledge of it as a certain whole (the atom consists of a nucleus and electrons). One-sided development of the first aspect leads to a tendency to explain all nature’s phenomena from knowledge of elementary phenomena (the atomistic approach). One-sided development of the second aspect leads to a tendency to explain elementary phenomena from knowledge of the whole (the integral approach).
p In classical physics the atomistic approach came to the fore and found its extreme expression in the mechanistic picture of the world. In relativistic physics an integral approach began to arise though it did not receive final expression. Both approaches are interwoven in atomic and quantum physics, the link between the atomistic and the integral aspects in investigating nature becoming more and more organic and inseparable with the development of quantum physics. Thus, in modern conceptions, the atom not only cannot be reduced to the sum of nucleons and electrons but these structural units of it themselves differ in many of their properties from electrons, protons and neutrons in the free state. The modern theory of elementary particles provides even more striking examples of unity of the atomistic and integral approaches (see Chapter VII).
p In the transition from the whole to its elements and vice versa, when the appropriate round of phenomena is studied, it becomes crucially important whether the whole is an aggregate of a large number of objects to which the statistical method can be applied. If the situation is such, statistics and its related theory of probability are necessarily brought into the theory of the sphere of phenomena being studied, with all the conclusions and generalisations following from this. The transition from macroscopic phenomena to molecular or atomic ones, for example, and to elementary 184 processes, and the reverse transition from elementary phenomena to macroscopic ones are impossible without statistics and the theory of probability. Philosophical problems of law and causality, necessity and chance, actuality and possibility therefore inevitably arise in both macroscopic and microscopic physics (which we discussed above).
p Let us note, in connection with the question raised about the atomistic and integral aspects of the approach to cognising physical phenomena, that both of them affect understanding of statistical regularities. Let us also recall that the atomistic approach leads in its extreme expression to mechanistic Laplacian determinism and the denial of objective random events. The truth, however, as follows from the laws of dialectics, lies at the juncture of the two approaches, and consists in the fact that dynamic and statistical regularities are actually inseparable, that macroscopic and microscopic phenomena are linked by transitions and are in fact united. Our whole book is essentially concerned with this point; here we shall consider its related topics from the standpoint of this section.
p It would be trivial to state that it is impossible to extract information about the behaviour of an individual molecule or a single atom from phenomenological (classical) thermodynamics. On the other hand, knowledge of the positions and momenta of all the molecules of a gas would not by any means lead to or be a substitute for knowledge of the gas’s temperature. The truth was found (and its discovery marked a great advance in physics) when the thermodynamic internal energy of a system, which depends on temperature and other macroscopic parameters, was identified with the mean statistical value of the kinetic energy of microscopic particles, which depends on their velocities and mutual position. This identification of the dynamic and the statistical, which combined the properties and microstructure of macroscopic bodies into something united, made it possible to validate thermodynamics in a profound way, to overcome the drawbacks of classical thermodynamics (including such a philosophical flaw as the formal possibility of assuming ’thermal death’), and to explain and discover new facts (e.g. the fluctuation of thermodynamic quantities).
p Within the limits of classical physics, however, the principles relating to probability concepts are, of course, only an addition to the basic laws, and the probabilities of 185 various possible events do not figure in the content of these laws. It cannot be otherwise from the angle of classical mechanics since, in it, a particle that moves with constant velocity in space and time remains everywhere and always identical to itself and alters its velocity only when acted on by other particles (when the number of such actions is large enough and they are mutually independent, we have movement similar to Brownian motion). The theory of relativity did not add anything new, compared to classical physics, on the nature of statistical laws and of probability in physics, and only quantum theory, as we know, introduced radical changes into this problem.
p This can be explained as follows. Quantum theory posed the task of revising the concept of a particle identical to itself—the most important concept of pre-quantum physics. In quantum mechanics a moving particle is no longer considered identical to itself since it has not merely particle properties but dual particle and wave ones. The relativity of the concept of a particle identical to itself becomes more definite in relativistic quantum mechanics according to which matter and field do not exist separately, and the law of conservation of the number and kind of particles ceases to operate. The concept of the inseparability of particles and fields accordingly becomes filled with new meaning, which is expressed by the wave function, which becomes an operator in quantum field theory. In quantum field theory, of course, the probability interpretation of the wave function is not only not removed but undergoes further development (which we will not discuss).
p As quantum field theory develops, the boundaries of the applicability of the concept of a self-identical particle are being made more precise, the relativity of the concept is being brought out more completely and more deeply, and the idea of the reciprocal transformability of particles is coming to the fore and subordinating the idea of the selfidentity of a particle to itself.
p The statement that the idea of transformability does not exist in classical and relativistic theory is not, of course, true; to disprove such statements, it is sufficient to give an example from the law of the conservation and transformation of energy. It was assumed in pre-quantum theories, however, that the matter studied by physics ultimately consisted of constant elements of one kind or another (the 186 material points and empty space of classical mechanics; the continuous field of Maxwell’s theory and the theory of relativity). Quantum physics eliminated the idea of the ultimate invariable bricks of the Universe; it regards elementary particles as mutually transformable, and the law of reciprocal transformation of elementary particles forms the foundation of the theoretical building of modern physics.
p The theory of elementary particles has not yet been built, and the job of building it is at the centre of attention of present-day physics. There are still only the sketches of a theory; we shall recall here Heisenberg’s programme for a unified theory of matter, but shall not go into it. Let us just note that it would have to remove the difficulties of quantum relativistic physics (the divergent expressions that figure in it in place of finite values observed in experiment for a number of quantities), unify microscopic particles and fields, theoretically deduce the properties of elementary particles known from the experiment and the types of transformation of elementary particles. Let us, in conclusion, consider certain aspects of the problem of causality and determinism in connection with the theory of the so-called scattering matrix formulated by Heisenberg, in particular, to eliminate divergencies in the theory of particles and fields.
p Heisenberg suggested a new formalism—a scattering matrix for the subatomic spatial and temporal domain with its extremely short distances and durations—instead of the Schrodinger equation which determines the values of the wave function at an arbitrary moment of time from its values at the initial moment of time (the so-called Hamiltonian formalism, which assumes the continuum of space and time). The scattering matrix is an operator that transforms the wave function of particles before scattering into their wave function after scattering. If the scattering takes place in the subatomic domain, the state corresponding to the first wave function and that corresponding to the second wave function should be separated by a time interval greater than the temporal subatomic domain. Then the equation "t+oo = 5i|5_oo (where ty is the wave function and 5 is the scattering matrix) expresses the connection between the values of the wave function at a moment of time remote in the past t——oo and its values in the remote future t =+°o-
p From this point of view the transfer of the interaction in such subatomic space and time domain occurs at a velocity 187 greater than that of light and the time sequence of events, it would seem, and the connection of cause and effect are violated (as was assumed by Heisenberg).
p There is no need, however, to draw odious conclusions against materialism from such an interpretation of processes occurring in the subatomic domain. Several authors, and Heisenberg himself (in his programme for a unified theory of matter) assume the existence of a minimum length (of the order of the radius of a light atomic nucleus 10 ^^13^^ centimetre), a universal constant that is part of the basic laws of nature. This assumption may mean that a geometry founded on the principle of continuity alone (mathematical continuum) is insufficient for describing the spatial properties of matter at subatomic level (the same must also be said about the temporal properties of subatomic processes, which, it must be assumed, cannot be described just by the continuity of time). As a result, we arrive at the following conclusion: discontinuity, being intrinsically connected with continuity, must come into the space and time concepts that relate to the subatomic domain, and the space and time concepts of physics should be revised accordingly.
p But in that case the concept of velocity for subatomic domain would also need to be altered. It will have to reflect the discontinuity of space and time, and will therefore receive a new content compared to the concept of velocity employed for the macroscopic and atomic domains. Then the concept ’a velocity greater than that of light’ may have a sense that is by no means relativistic. What we have said does not cancel the proposition of the theory of relativity that the value of the velocity of light is a limiting value; it only expresses the idea that the future theory will give a deeper substantiation of this fundamental proposition of the theory of relativity by determining the boundaries of its applicability.
p The theory of the scattering matrix, incidentally, should not necessarily pose the problem of causality on the plane discussed above. A scattering matrix can be constructed in such a way that the construction assumes a condition of microscopic causality by which the physical action, i.e. signal, cannot be propagated at a velocity greater than that of light in ultrasmall spatial and temporal domains (just as in the domain of large space-time scales). This construction could be used to deduce the so-called dispersion relations, 188 study of which provides important information on the nature of reciprocally interacting elementary particles.
p Since dispersion relations contain directly observable quantities, they make it possible to examine a large deal of experimental material. Checking their validity at high energies enables us to determine the limits of applicability of the notion about point interaction and consequently to confirm or disprove the validity of the condition of causality in the special form we are concerned with here. Such verification of the validity of dispersion relations is understandably of enormous significance for both physics and philosophy.
p We would like to add that the most general formulation of the condition of causality employed in quantum field theory is that given by Bogolyubov and Shirkov: ’Any event occurring in a system can influence the system’s evolution only in the future and cannot influence its behaviour in the past. ’^^48^^ This formulation leaves a possibility for generalising the Lorentz transformations, not to mention that it excludes the assumption of velocities higher than that of light.
p In connection with what has been said above, the concept of a particle identical to itself should also undergo radical change in as much as its transformability is beginning to play a main role. From this angle a particle before scattering is not identical with a particle after scattering. A theory that has to reflect the processes allowing not only for the continuous nature of space, time and motion but also for their discontinuous nature, the appearance and annihilation of elementary particles, the transformation of matter into field and vice versa undoubtedly cannot make do with the concepts and principles of classical and modern physical theories; it must develop new concepts and principles, preserving the results of all the preceding work of physics as a limiting case.
In quantum field theory and the physics of elementary particles, in which the idea of the reciprocal transformability of elementary particles engenders (or rather should engender) their whole theoretical content, new physical concepts and statements are thus needed, and not rejection of the objective reality of the physical world, and objective necessity and causality in nature. The so-called virtual processes, which witness in fact not to an imaginary violation of, say, the 189 law of the conservation of energy but to the need for a further development of the basic physical concepts and principles about the deeper levels of infinite matter, point in their own way to the same thing.
Notes