Probability and Determinism in Quantum Theory

Lev Vaidman A

Philip Goyal, Kevin H. Knuth

Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with probability theory? If so, does it extend or generalize probability theory? In this paper, we answer these questions, and precisely determine the relationship between quantum theory and probability theory, by explicitly deriving both theories from first principles. In both cases, the derivation depends upon identifying and harnessing the appropriate symmetries that are operative in each domain. We prove, for example, that quantum theory is compatible with probability theory by explicitly deriving quantum theory on the assumption that probability theory is generally valid.

Yoshifumi Ito

In this article, we established the fundamental principles of new quantum theory. At first, we mention the notion of quantum probability and its basic properties. Next, we propose the axiom of new quantum theory, using the notion of quantum probability. The characteristic property of the quantum state is that the quantum state is determined by the L2-density of the quantum probability distribution of the position variables and the momentum variables of the quantum system. At last, we derive the Schrödinger equations by the reasonable method based on the axiom of new quantum theory. Here we use the variational principle that the real L2-density determining the quantum state of the quantum system, which is, in practice, realized physically, should be determined among the admissible L2-densities so that the energy expectation value takes its stationary value.

Luigi Accardi

Veeravalli S. Varadarajan

M. D. Srinivas

The problem of defining conditional probabilities and the notion of statistical independence in quantum theory is analyzed. It is shown that (unlike in classical probability theory) the conditional probabilities of a given set of events can be determined only if the sequence of all the experiments performed on the system is also specified. Such a specification is necessary also for the concept of statistical independence to become physically meaningful.