Why and whence the Hilbert space in quantum theory?

Brezhnev, Yu. V.

doi:10.1016/j.geomphys.2023.104779

Cited by 2 publications

(2 citation statements)

References 24 publications

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Section: Introductionmentioning

confidence: 99%

“…One of the last derivations of the Hilbert space formalism is Brezhnev [14], where also other derivations are referred to. [14] is based upon an unlimited validity of the superposition principle. As a contrast, in the present approach the operators associated with theoretical variables are derived first, and a ket vector is only seen as a valid state vector if it is an eigenvector of a physically meaningful operator.…”

Section: Introductionmentioning

confidence: 99%

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An alternative foundation of quantum theory

Helland

2023

Preprint

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A new approach to quantum theory is proposed in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the discrete case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is shown here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. This simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. The essential use of inaccessible variables can be avoided by basing the approach on some simple category theory.The interpretation inferred from the proposed foundation here may be called a general epistemic interpretation of quantum theory. A special case of this interpretation is QBism; it also has a relationship to several other interpretations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An alternative foundation of quantum theory

Helland

2023

Preprint

View full text Add to dashboard Cite

show abstract

An Alternative Foundation of Quantum Theory

Helland

2023

Found Phys

View full text Add to dashboard Cite

A new approach to quantum theory is proposed in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the discrete case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is shown here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. This simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. The interpretation inferred from the proposed foundation here may be called a general epistemic interpretation of quantum theory. A special case of this interpretation is QBism; it also has a relationship to several other interpretations.

show abstract

Why and whence the Hilbert space in quantum theory?

Cited by 2 publications

References 24 publications

An alternative foundation of quantum theory

An alternative foundation of quantum theory

An Alternative Foundation of Quantum Theory

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