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arXiv:1303.3935 · Journal of Physics Conference Series

Quantum mechanics from invariance principles

Florin Moldoveanu

Published Jul 3, 2015• 1 citation
arXivHTMLPDF

Abstract

Quantum mechanics is an extremely successful theory of nature and yet it lacks an intuitive axiomatization. In contrast, the special theory of relativity is well understood and is rooted into natural or experimentally justified postulates. Here we introduce an axiomatization approach to quantum mechanics which is very similar to special theory of relativity derivation. The core idea is that a composed system obeys the same laws of nature as its components. This leads to a Jordan-Lie algebraic formulation of quantum mechanics. The starting assumptions are minimal: the laws of nature are invariant under time evolution, the laws of nature are invariant under tensor composition, the laws of nature are relational, together with the ability to define a physical state (positivity). Quantum mechanics is singled out by a fifth experimentally justified postulate: nature violates Bell inequalities.

DOI
10.1088/1742-6596/626/1/012067

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Quantum mechanics is an extremely successful theory of nature and yet it lacks an intuitive axiomatization. In contrast, the special theory of relativity is well understood and is rooted into natural or experimentally justified postulates. Here we introduce an axiomatization approach to quantum mechanics which is very similar to special theory of relativity derivation. The core idea is that a composed system obeys the same laws of nature as its components. This leads to a Jordan-Lie algebraic formulation of quantum mechanics. The starting assumptions are minimal: the laws of nature are invariant under time evolution, the laws of nature are invariant under tensor composition, the laws of nature are relational, together with the ability to define a physical state (positivity). Quantum mechanics is singled out by a fifth experimentally justified postulate: nature violates Bell inequalities.
Quantum theory (QT) is usually formulated in terms of abstract mathematical postulates involving Hilbert spaces, state vectors and unitary operators. In this paper, we show that the full formalism of QT can instead be derived from five simple physical requirements, based on elementary assumptions regarding preparations, transformations and measurements. This is very similar to the usual formulation of special relativity, where two simple physical requirements—the principles of relativity and light speed invariance—are used to derive the mathematical structure of Minkowski space–time. Our derivation provides insights into the physical origin of the structure of quantum state spaces (including a group-theoretic explanation of the Bloch ball and its three dimensionality) and suggests several natural possibilities to construct consistent modifications of QT.
We use the relativity principle to complete axiomatic reconstructions of quantum mechanics (QM) via information-theoretic principles that are based on Darrigol's "discreteness" requirement or its equivalent, e.g., Brukner & Zeilinger's Information Invariance & Continuity or Khrennikov's quantum action invariance principle. In this approach to the quantum reconstruction program (QRP), the Hilbert space kinematics of QM is derived most fundamentally from the experimentally motivated postulate of "discreteness," rendering QM a principle theory as defined by Einstein. Special relativity is also a principle theory, since its Lorentz transformation kinematics is derived from the experimentally motivated light postulate. While special relativity has a compelling fundamental principle (relativity principle) to account for its experimentally motivated light postulate, QRP has not produced a compelling fundamental principle to account for its experimentally motivated "discreteness" requirement. We complete QRP by showing how the "discreteness" requirement is necessitated by the relativity principle and Planck's radiation law, just like the light postulate is necessitated by the relativity principle and Maxwell's equations. Accordingly, the fundamental explanans for time dilation, length contraction, quantum superposition, and entanglement is the relativity principle. Phenomenologically speaking, the relativity principle is "the equality of all perspectives." Thus, quantum entanglement isn't evidence for the violation of intersubjective agreement or the need to abandon factive or objective models of reality, as in relational interpretations of QM. It is evidence that reality is not a fragmented collage of different subjective experiences from different perspectives, it's a comprehensive and coherent integration of those different subjective experiences per the equality of all perspectives.
Quantum theory (QT) is usually formulated in terms of abstract mathematical postulates involving Hilbert spaces, state vectors and unitary operators. In this paper, we show that the full formalism of QT can instead be derived from five simple physical requirements, based on elementary assumptions regarding preparations, transformations and measurements. This is very similar to the usual formulation of special relativity, where two simple physical requirements-the principles of relativity and light speed invariance-are used to derive the mathematical structure of Minkowski space-time. Our derivation provides insights into the physical origin of the structure of quantum state spaces (including a group-theoretic explanation of the Bloch ball and its three dimensionality) and suggests several natural possibilities to construct consistent modifications of QT.
We present a group-theoretic reconstruction of quantum mechanics in which the standard formalism emerges from the representation theory of symmetry groups rather than being introduced axiomatically. Starting from physically motivated symmetry principles, we show how the conventional postulates of quantum mechanics including the Hilbert space structure of states, observables as generators of symmetry transformations, the Born probability rule, unitary time evolution, the composition of multipartite systems, and the measurement framework can be recovered within a unified representation-theoretic framework. We then examine the measurement problem by interpreting wave-function collapse as the scattering of quantum states, thereby providing an account of the apparent loss of information. Finally, we investigate the origin of Planck's constant, proposing that it arises as the conserved charge of the Lorentz symmetry group. In this formulation, ℏ\hbarℏ is an invariant that relates Lie algebra generators to observable quantities and fixes the scale of quantum action. This framework offers a conceptually unified perspective on the foundations of quantum mechanics, connecting its postulates, measurement theory, and the emergence of the quantum of action to a common symmetry-based origin while identifying conditions under which the conventional quantum formalism is recovered.