Quantum Theory: The General Formalism

Franklin E. Schroeck

We present an alternative to the usual formalism for quantum field theory by generalizing the Fock-Cook formalism. We illustrate the method by applying it to a generalization of a familiar model of quantum field theory.

Harald Atmanspacher A

Two key concepts of quantum theory, complementarity and entanglement, are considered with respect to their significance in and beyond physics. An axiomatically formalized, weak version of quantum theory, more general than the ordinary quantum theory of physical systems, is described. Its mathematical structure generalizes the algebraic approach to ordinary quantum theory. The crucial formal feature leading to complementarity and entanglement is the non‐commutativity of observables.The ordinary Hilbert space quantum mechanics can be recovered by stepwise adding the necessary features. This provides a hierarchy of formal frameworks of decreasing generality and increasing specificity. Two concrete applications, more specific than weak quantum theory and more general than ordinary quantum theory, are discussed: (i) complementarity and entanglement in classical dynamical systems, and (ii) complementarity and entanglement in the bistable perception of ambiguous stimuli.

Matthew Leifer, Robert W. Spekkens

Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Classical probability theory is independent of causal structure, whereas the conventional quantum formalism requires causal structure to be fixed in advance. In this paper, we develop the formalism of quantum conditional states, which unifies the description of experiments involving two systems at a single time with the description of those involving a single system at two times. The analogies between quantum theory and classical probability theory are expressed succinctly within the formalism and it unifies the mathematical description of distinct concepts, such as ensemble preparation procedures, measurements, and quantum dynamics. We introduce a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning. Conditioning a quantum state on a classical variable is the correct rule for updating quantum states in light of classical data, regardless of the causal relationship between the classical variable and the quantum degrees of freedom, but it does not include the projection postulate as a special case. We show that previous arguments that projection is the quantum generalization of conditioning are based on misleading analogies. Since our formalism is causally neutral, conditioning provides a unification of the predictive and retrodictive formalisms for prepare-and-measure experiments and leads to an elegant derivation of the set of states that a system can be steered to by making measurements on a remote system.

Andrei Yu. Khrennikov