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Revisiting the First Postulate of Quantum Mechanics: Invariance and Physically Reality

Christian DE RONDE, César Massri

Published Dec 8, 2014• 2 citations
OpenAlex

Abstract

In this paper we derive a theorem which proves that the physical interpretation implied by the first postulate of quantum mechanics (QM) is inconsistent with the orthodox formalism. In order to expose this inconsistency we will analyze how the concept of ‘physical system’ is built within classical theories through the notion of invariance and explain in what sense a vector in Hilbert space is not capable of fulfilling these same mathematical conditions. Through an analysis of the mathematical formalism we derive a No Dynamical Invariance (NDI) theorem which proves that, contrary to what is claimed in the first postulate, a vector in Hilbert space cannot be interpreted coherently as the state of a physical (quantum) system. We conclude the paper by analyzing the consequences of the NDI theorem with respect to several ongoing debates in QM.

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In this paper we derive a theorem which proves that the physical interpretation implied by the first postulate of quantum mechanics (QM) is inconsistent with the orthodox formalism. In order to expose this inconsistency we will analyze how the concept of ‘physical system’ is built within classical theories through the notion of invariance and explain in what sense a vector in Hilbert space is not capable of fulfilling these same mathematical conditions. Through an analysis of the mathematical formalism we derive a No Dynamical Invariance (NDI) theorem which proves that, contrary to what is claimed in the first postulate, a vector in Hilbert space cannot be interpreted coherently as the state of a physical (quantum) system. We conclude the paper by analyzing the consequences of the NDI theorem with respect to several ongoing debates in QM.
Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to observations or reality in classical physics. Consequently, interpretation of quantum mechanics and its formalism should involve proper physical mathematics, physical logic and classical physics, which is not the case from the Copenhagen interpretation. Here, we shall revisit all the postulates of quantum mechanics with proper physics and physical logic and reconstruct them to establish the less-complex interpretation of quantum mechanics with additional new postulates from classical physics. A new quantum mechanical postulate (Postulate 9) is also introduced to understand the meaning of the forbidden gap between discrete energy levels and/or electron orbitals in atoms that is valid for molecules and condensed matter based on quantum mechanics that is technically well-defined and experimentally observable. Here, we tackle all the problems arise from the Copenhagen interpretation systematically by revising and/or extending them with proper classical and quantum physics without violating established experiments and without proposing ideas that violate physical reality.
In this work we analyze the deep link between the 20th Century positivist re-foundation of physics and the famous measurement problem of quantum mechanics. We attempt to show why this is not an "obvious" nor "self evident" problem for the theory of quanta, but rather a direct consequence of the empirical-positivist understanding of physical theories when applied to the orthodox quantum formalism. In contraposition, we discuss a representational realist account of both physical 'theories' and 'measurement' which goes back to the works of Einstein, Heisenberg and Pauli. After presenting a critical analysis of Bohr's definitions of 'measurement' we continue to discuss the way in which several contemporary approaches to QM --such as decoherence, modal interpretations and QBism-- remain committed to Bohr's general methodology. Finally, in order to expose the many inconsistencies present within the (empirical-positivist) presuppositions responsible for creating the quantum measurement problem, we show how through these same set of presuppositions it is easy to derive a completely analogous paradox for the case of classical mechanics.
In this paper we analyze the definition of quantum superpositions within orthodox Quantum Mechanics (QM) and their relation to physical reality. We will begin by discussing how the metaphysical presuppositions imposed by Bohr on the interpretation of QM have become not only interpretational dogmas which constrain the limits of the present Orthodox Line of Research (OLR), but also how these desiderata implicitly preclude the possibility of developing a physical representation of quantum superpositions. We will then continue analyzing how most interpretations of QM argue against the existence of superpositions. Firstly, we will focus on those interpretations which attempt to recover a classical representation about ''what there is'', and secondly, we will concentrate on the arguments provided by Dieks and Griffiths who, staying close to the orthodox formalism, also attempt to ''get rid of the ghost of Schrodinger's cat''. Contrary to the OLR, we will argue -based on our definition of Meaningful Physical Statements (MPS)- that from a representational realist perspective which stays close to the orthodox Hilbert space formalism, quantum superpositions are not only the key to the most important -present and future- technological and experimental developments in quantum information processing but also, they must be considered as the kernel of any interpretation of QM that attempts to provide a physical representation of reality. We will also argue that the price to pay for such representational realist development must be the abandonment of the (dogmatic) idea that 'Actuality = Reality'.
In this paper we analyze the definition of quantum superpositions within orthodox Quantum Mechanics (QM) and their relation to physical reality. We will begin by discussing how the metaphysical presuppositions imposed by Bohr on the interpretation of QM have become not only interpretational dogmas which constrain the limits of the present Orthodox Line of Research (OLR), but also how these desiderata implicitly preclude the possibility of developing a physical representation of quantum superpositions. We will then continue analyzing how most interpretations of QM argue against the existence of superpositions. Firstly, we will focus on those interpretations which attempt to recover a classical representation about "what there is", and secondly, we will concentrate on the arguments provided by Dieks and Griffiths who, staying close to the orthodox formalism, also attempt to "get rid of the ghost of Schrodinger's cat". Contrary to the OLR, we will argue -based on our definition of Meaningful Physical Statements (MPS)- that from a representational realist perspective which stays close to the orthodox Hilbert space formalism, quantum superpositions are not only the key to the most important -present and future- technological and experimental developments in quantum information processing but also, they must be considered as the kernel of any interpretation of QM that attempts to provide a physical representation of reality. We will also argue that the price to pay for such representational realist development must be the abandonment of the (dogmatic) idea that 'Actuality = Reality'.
We discuss the hypothesis that the debate about the interpretation of the orthodox formalism of quantum mechanics (QM) might have been misguided right from the start by a biased metaphysical interpretation of the formalism and its inner mathematical relations. In particular, we focus on the orthodox interpretation of the congruence relation, '=', which relates equivalent classes of different mathematical representations of a vector in Hilbert space, in terms of metaphysical identity. We will argue that this seemingly "common sense" interpretation, at the semantic level, has severe difficulties when considering the syntactic level of the theory.
The principle of superposition of states requires that the states of a dynamical system form a linear vector space. This hypothesis of linearity has usually been regarded as a fundamental postulate in quantum mechanics, of a kind that cannot be explained by classical concepts. Indeed, Dirac [2, p. 14] comments that "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation."