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 asserted by P. S. Laplace in a celebrated passage (Théorie analytique des probabilités, 3rd ed., Paris 1820, p. VII); it affirms that when the coordinates (position) and the momentum (velocity) of every point of a system are given at a given instant, the state of the system at that instant is completely determined; any further condition is either incompatible with the preceding ones or already contained within them. Consequently, given the state of a system at a certain instant, the values of the positions, velocities and their functions relative to the system at that instant are fixed, as is the result of any measurement of these quantities performed at that instant. If, moreover, the forces acting on the points are known, given the state of the system at a certain instant, its state at every other past or future instant is still precisely determined, so that the result of any measurement at any instant is predictable with arbitrarily great precision (v. DETERMINISM).
In 1927, W. Heisenberg, in a famous note, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik (in Zeitschrift für Physik, 43 [1927], pp. 172 sq.), 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
\[\Delta x \Delta p_x \geq \frac{h}{4\pi}, \Delta y \Delta p_y \geq \frac{h}{4\pi}, \Delta z \Delta p_z \geq \frac{h}{4\pi}; \text{ where } \Delta x \Delta y \Delta z \geq \frac{h^2}{4\pi^2}\]
\(\Delta x \Delta y \Delta z\) are the indeterminacies of the coordinates, \(\Delta p_x \Delta p_y \Delta p_z\) those of the respective momenta, and \(h\) is Planck’s constant.
The principle of indeterminacy is a consequence of the commutation relations in matrix calculus and was demonstrated by E. Schrödinger also through the principles of wave mechanics; physically it depends on the unavoidable perturbation that observation produces in the phenomenon. It is therefore not due to experimental errors, but expresses a physical law, as indicated by Planck’s constant.
Since the state of a system is defined by the set of conditions such that every other is either contained within them or incompatible with them, the principle of indeterminacy alters the classical concept of the state of a system, inasmuch as it does not allow 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 as long as it remains isolated, and so long as this condition persists, the laws of quantum mechanics allow its exact development at every future instant to be foreseen. If, however, we make measurements on it, its isolation generally ceases and its state changes: hence, once the state of a system is determined in quantum mechanics, it is generally not possible to calculate the result of a further observation; it is only possible to calculate the possible results of each observation with their respective probabilities. Generally these possible results constitute a continuous or discrete infinity. This is precisely physical indeterminism. These concepts apply to elementary phenomena, whereas for macroscopic phenomena, where an enormous number of particles are involved, their measurements are averages and determinism then holds.
Many interpret the principle of indeterminacy as the negation of an objective physical determinism, i.e., 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, trans. by G. Crescenzi, Turin 1950, p. 205 sq.); but this entails serious philosophical difficulties. For, whereas the indeterminate is potency and the determined is act, “agere... est per se proprium actus, inquantum est actus” (Summa Theol., I, q. 115, a. 1) and “propria forma uniuscuiusque, faciens ipsum esse in actu, est principium operationis ipsius” (ibid., I-II, q. 179, a. 1, ad 1). Therefore every reality acts insofar as it is determined by a form. This form can be: either a psychic representation, which, if of a spiritual order in an intelligent subject, can be the principle of a free action, i.e., self-determined and indeterministic; or a real form. In non-knowing beings, e.g., in the atomic world, action proceeds from a real, determined, necessary form; it will therefore be a 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, even though we cannot verify it because of the imprecision of our observations, an imprecision given by the principle of indeterminacy.