Three Attempts at Two Axioms for Quantum Mechanics

Daniel Rohrlich

The axioms of nonrelativistic quantum mechanics lack clear physical meaning. In particular, they say nothing about nonlocality. Yet quantum mechanics is not only nonlocal, it is twice nonlocal: there are nonlocal quantum correlations, and there is the Aharonov-Bohm effect, which implies that an electric or magnetic field h e r e may act on an electron t h e r e. Can we invert the logical hierarchy? That is, can we adopt nonlocality as an axiom for quantum mechanics and derive quantum mechanics from this axiom and an additional axiom of causality? Three versions of these two axioms lead to three different theories, characterized by "maximal nonlocal correlations", "jamming" and "modular energy". Where is quantum mechanics in these theories?

K. Heyde, John L. Wood

Nicolaas P. Landsman

Adam M. Brandenburger, Pierfrancesco La Mura

The Heisenberg uncertainty principle is one of the most famous features of quantum mechanics. However, the non-determinism implied by the Heisenberg uncertainty principle --- together with other prominent aspects of quantum mechanics such as superposition, entanglement, and nonlocality --- poses deep puzzles about the underlying physical reality, even while these same features are at the heart of exciting developments such as quantum cryptography, algorithms, and computing. These puzzles might be resolved if the mathematical structure of quantum mechanics were built up from physically interpretable axioms, but it is not. We propose three physically-based axioms which together characterize the simplest quantum system, namely the qubit. Our starting point is the class of all no-signaling theories. Each such theory can be regarded as a family of empirical models, and we proceed to associate entropies, i.e., measures of information, with these models. To do this, we move to phase space and impose the condition that entropies are real-valued. This requirement, which we call the Information Reality Principle, arises because in order to represent all no-signaling theories (including quantum mechanics itself) in phase space, it is necessary to allow negative probabilities (Wigner [1932]). Our second and third principles take two important features of quantum mechanics and turn them into deliberately chosen physical axioms. One axiom is an Uncertainty Principle, stated in terms of entropy. The other axiom is an Unbiasedness Principle, which requires that whenever there is complete certainty about the outcome of a measurement in one of three mutually orthogonal directions, there must be maximal uncertainty about the outcomes in each of the two other directions.

Changyu Huang, Yong-Chang Huang, Jia-Min Song

This chapter demonstrates the deduction of quantum physics and, in particular, presents its new applications to eigenvalue problems. Specifically, it derives three of the five axiom presumptions of quantum mechanics (QM), for example, deducing the Schrödinger equation in two general ways. It transforms these three axiom presumptions into three theorems of QM, not only resolving the difficulty in understanding them but also yielding new theories and discoveries. For instance, this paper addresses the puzzle of the origin of wave-particle duality, derives operators, eigenvalues, and eigenstates, deduces commutation relations for coordinate and momentum as well as for time and energy, and reveals that QM is essentially a generalized (mechanics) theory of the complex square root of the (real density function of) classical statistical mechanics. The realization that QM is a generalized theory of the complex square root of classical statistical mechanics represents both a groundbreaking discovery and a revolutionary advancement in physics. This insight profoundly influences philosophical perspectives on the development of modern physics, resolves numerous puzzles in QM and quantum information, and provides QM with a scientifically solid foundation. This foundation eliminates the need for basic axiom presumptions and dispels the so-called “strange incomprehensible properties” of QM, as both classical statistical mechanics and its complex square root that have been scientifically validated. Additionally, this chapter uncovers the reason for not taking the time derivative of spatial coordinates in the Schrödinger equation. Consequently, it offers a solution to the puzzle of the origin of first quantization and primarily deduces quantum physics without the strange and incomprehensible properties currently associated with it.

Paul Busch, PEKKA J. LAHTI, Juha-Pekka Pellonpää +1 more