PHYSICAL INDETERMINISM. – Physical indeterminism depends on the principle of indeterminacy, the foundation of atomic mechanics, due to W. Heisenberg.
In classical mechanics determinism holds, already summarized by P. S. Laplace in a passage that has remained famous (Théorie analytique des probabilités, 3e ed., Paris 1820, p. VII); it states that when the coordinates (position) and momentum (velocity) of every point of a system at a given instant are assigned, the state of the system at that instant is completely determined; every further condition is either incompatible with the preceding ones or already contained in them. Consequently, given the state of a system at a certain instant, the value of the positions, velocities, and their functions relating to the system at that instant is fixed, as is also the result of every measurement of these quantities carried out at that instant. If, moreover, the forces acting on the points are known, then, given the state of the system at a certain instant, its state at every other past or future instant is likewise specified; hence the result of any measurement at any instant can be predicted with arbitrarily great precision (v. DETERMINISMO).
In 1927, W. Heisenberg, in a famous paper, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik (in Zeitschrift für Physik, 43 [1927], pp. 172-83g.), established the principle of indeterminacy: it is conceptually impossible to determine the position and momentum of a particle at a given instant with a precision greater than that indicated by the relations
Δₓ Δₚ ≥ frac{h}{4π}, Δ_y Δₚ ≥ frac{h}{4π}, Δ_z Δₚ ≥ frac{h}{4π}; where Δₓ Δ_y Δ_z
are the indeterminacies of the coordinates, Δₚ, Δₚ, Δₚ those of the respective momenta, and h is the constant of M. Planck.
The principle of indeterminacy is a consequence of the commutation relations in matrix calculus and was also demonstrated by E. Schrödinger through the principles of wave mechanics; physically, it depends on the unavoidable disturbance that observation produces in the phenomenon. It is therefore not due to experimental errors, but expresses a physical law, as indicated by the constant of M. Planck.
Since the state of a system is defined by the set of conditions such that every other condition is either contained in them or incompatible with them, the principle of indeterminacy alters the classical-mechanical concept of the state of a system, insofar as it does not permit the exact measurement of the positions and, simultaneously, of the momenta of the points. In quantum mechanics, the state of a system is defined by a complex of all those measurements that are mutually compatible, according to the principle of indeterminacy. The system maintains its state so long as it remains isolated and, while this condition persists, the laws of quantum mechanics make it possible to predict exactly its development at every future instant. If, however, we carry out measurements in it, its isolation generally ends and its state changes: therefore, once the state of a system has been determined in quantum mechanics, it is generally impossible to calculate the result of a further observation; it is possible only to calculate the possible results of each observation together with their respective probabilities. In general, these possible results constitute a continuous or discrete infinity. This is precisely physical indeterminism. These concepts apply to elementary phenomena, whereas in macroscopic phenomena, in which particles intervene in very large numbers, their measurements are an average, and determinism then holds.
Many interpret the principle of indeterminacy as the denial of an objective physical determinism, that is, of a univocal, predetermined causality, in elementary particles, where pure chance would reign, albeit with tendencies toward the various possibilities (cf. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Berlin 1932; A. Eddington, Sur le problème du déterminisme, Paris 1934, p. 11; E. Persico, Analisi del determinismo fisico, in Fondamenti logici della scienza, Turin 1947; L. De Broglie, Fisica e microfisica, translated by G. Crescenzi, Turin 1950, p. 205-89g.); but this entails serious philosophical difficulties. Indeed, while the indeterminate is potency and the determinate is act, «agere... est per se proprium actus, inquantum est actus» (Sum. Theol., 1a, q. 115, a. 1) and «propria forma uniuscuiusque, faciens ipsum esse in actu, est principium operationis ipsius» (ibid., 2a-2ae, q. 179, a. 1, ad 1). Thus every reality acts insofar as it is determined by a form. This may be either a psychic representation which, if spiritual in nature in an intelligent subject, may be the principle of a free action, that is, self-determined and indeterministic; or a real form. In a non-knowing being, for example, in the atomic world, action proceeds from a real, determined, necessary form; it will therefore be necessary, deterministic action. This excludes objective indeterminism in the atomic world.
The principle of indeterminacy indicates only the impossibility of measuring a phenomenon without disturbing its process. The atomic world is indeed deterministic, although we cannot ascertain this because of the imprecision of our observations, an imprecision given by the principle of indeterminacy.