Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

Moretti, Valter; Oppio, Marco

doi:10.1142/s0129055x19500132

Cited by 8 publications

(4 citation statements)

References 35 publications

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“…We shall not address here the problem of the apparent absence of physical systems described in real Hilbert spaces [St60,StGu61,MoOp17] and the possibility (or impossibility) of quaternionic formulations [FJSS62,Ad95,Gan17,MoOp17b]. Instead, we restrict attention to a celebrated result, provided by Gleason's theorem [Gl57], regarding the notion of quantum state.…”

Section: Gleason's Theorem and Troubles With The Quaternionic Formulamentioning

confidence: 99%

The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

Moretti

Oppio

2018

Ann. Henri Poincaré

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Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Solér's theorem. Quantum states are here pictured in terms of σ-additive probability measures over the non-Boolean lattice of orthogonal projectors of the considered Hilbert space. Gleason's theorem proves that, if the Hilbert space is either real or complex and some technical hypothes are true, then these measures are one-to-one with standard density matrices used by physicists recovering and motivating the familiar notion of state. The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, the formulation of this extension [Va68] is partially mathematically incorrect due to some peculiarities of the notion of trace in quaternionic Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is devoted to present Gleason-Varadarajan's theorem into a technically correct and physically meaningful form valid for the three types of Hilbert spaces. In particular, we prove that only the real part of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

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Section: Gleason's Theorem and Troubles With The Quaternionic Formulamentioning

confidence: 99%

The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

Moretti

Oppio

2018

Ann. Henri Poincaré

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“…In other words, the lattice structure of propositions in quantum physics does not suggest the Hilbert space to be complex. More recently, Moretti and Oppio [3] gave stronger motivation for the Hilbert space to be complex which rests on the symmetries of elementary relativistic systems.…”

Section: Introductionmentioning

confidence: 99%

Quantum description of angles in the plane

2022

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The real plane with its set of orientations or angles in [0, π) is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral quantization, linear polarisation of light as a quantum measurement, interpretation of entanglement leading to the violation of Bell inequalities, and spin one-half coherent states viewed as two entangled angles.

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“…It is worth recalling that the theory of slice regular functions has significant applications in various areas of mathematics, as quaternionic functional calculus (see e.g. [12,20,21,30]), twistor theory (see e.g. [4,5,14]), operator semigroup theory (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Slice-by-slice and global smoothness of slice regular and polyanalytic functions

Ghiloni¹

2020

Preprint

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The concept of slice regular function over the real algebra H of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let Ω be an open subset of H, which intersects R and is invariant under rotations of H around R. A function f : Ω → H is slice regular if it is of class C 1 and, for all complex planes C I spanned by 1 and a quaternionic imaginary unit I, the restriction f I of f to Ω I = Ω ∩ C I satisfies the Cauchy-Riemann equations associated to I, i.e., ∂ I f I = 0 on Ω I , whereWe define global slice polyanalytic functions of order n as the functions f : Ω → H, which admit a decomposition of the form f (x) = n−1 h=0 x h f h (x) for some slice regular functions f 0 , . . . , f n−1 . Global slice polyanalytic functions of any order n are slice polyanalytic of the same order n. The converse is not true: for each n ≥ 2, we give examples of slice polyanalytic functions of order n, which are not global.The aim of this paper is to study the continuity and the differential regularity of slice regular and global slice polyanalytic functions viewed as solutions of the slice-by-slice differential equations ∂ n I f I = 0 on Ω I and as solutions of their global version ϑ n f = 0 on Ω \ R.Our quaternionic results extend to the monogenic case.

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Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

Cited by 8 publications

References 35 publications

The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

Quantum description of angles in the plane

Slice-by-slice and global smoothness of slice regular and polyanalytic functions

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