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Journal of Mathematical Physics

Is quantum mechanics exact?

Anton Kapustin

Published Jun 1, 2013• 17 citations
Publisher

Abstract

We formulate physically motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to quantum mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory, respectively. We show that even if the second group of axioms is dropped, there are no deformations of quantum mechanics which preserve the kinematic axioms. Thus, any theory going beyond quantum mechanics must represent a radical departure from the usual a priori assumptions about the laws of nature.

DOI
10.1063/1.4811217

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We formulate physically motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to quantum mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory, respectively. We show that even if the second group of axioms is dropped, there are no deformations of quantum mechanics which preserve the kinematic axioms. Thus, any theory going beyond quantum mechanics must represent a radical departure from the usual a priori assumptions about the laws of nature.
We formulate physically-motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to Quantum Mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory respectively. We show that even if the second group of axioms is dropped, there are no deformations of Quantum Mechanics which preserve the kinematic axioms. Thus any theory going beyond Quantum Mechanics must represent a radical departure from the usual a priori assumptions about the laws of Nature.
The paper proposes a new approach in the foundations of Quantum Mechanics.It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models.Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...).It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations.Moreover this approach sheds a new ligth on the issues of determism, and interacting systems.In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
The list of basic axioms of quantum mechanics as it was formulated by von Neumann includes only the mathematical formalism of the Hilbert space and its statistical interpretation. We point out that such an approach is too general to be considered as the foundation of quantum mechanics. In particular in this approach any finite-dimensional Hilbert space describes a quantum system. Though such a treatment might be a convenient approximation it can not be considered as a fundamental description of a quantum system and moreover it leads to some paradoxes like Bell's theorem. I present a list from seven basic postulates of axiomatic quantum mechanics. In particular the list includes the axiom describing spatial properties of quantum system. These axioms do not admit a nontrivial realization in the finite-dimensional Hilbert space. One suggests that the axiomatic quantum mechanics is consistent with local realism.
The paper proposes a new approach in the foundations of Quantum Mechanics.It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models.Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...).It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations.Moreover this approach sheds a new ligth on the issues of determism, and interacting systems.In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
Quantum mechanics is a cornerstone of modern physics, yet its fundamental origin and mathematical foundation remain elusive. In this paper, we derive the complete structure of quantum mechanics from a purely classical field theory. By gauging the double cover of the Lorentz symmetry group of space–time, the resulting equations of motion naturally yield all postulates of quantum mechanics, including the emergence of the Planck constant. This approach provides a coherent interpretation and demonstrates that quantum mechanics can be viewed as a mathematically consistent classical field theory.