Hyperbolic Hilbert space

Xuegang, Yu

doi:10.1007/bf03042009

Cited by 10 publications

(11 citation statements)

References 3 publications

(4 reference statements)

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“…The hyperbolic numbers can be used also for the description of the orbital angular momentum, which will be shown in this section (compare with Xuegang [22]). A spacelike relativistic vector x µ can be parametrized in relativistic spherical coordinates as…”

Section: Orbital Angular Momentum and Single Particle Potentialsmentioning

confidence: 99%

“…Considerations of a non-relativistic hyperbolic Hilbert space with respect to the Born formula have been given by Kocik [18], Khrennikov [19,20], and Rochon and Tremblay [21]. Xuegang et al investigated the Dirac wave equation, Clifford algebraic spinors, a hyperbolic Hilbert space, and the hyperbolic spherical harmonics in hyperbolic spherical polar coordinates [22,23,24]. The hypercomplex numbers, and the geometries generated by these numbers, have been investigated by Catoni et al [25].…”

Section: Introductionmentioning

confidence: 99%

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Relativistic quantum physics with hyperbolic numbers

Ulrych¹

2005

Physics Letters B

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A representation of the quadratic Dirac equation and the Maxwell equations in terms of the three-dimensional universal complex Clifford algebraC3,0 is given. The investigation considers a subset of the full algebra, which is isomorphic to the Baylis algebra. The approach is based on the two Casimir operators of the Poincaré group, the mass operator and the spin operator, which is related to the Pauli-Lubanski vector. The extension to spherical symmetries is discussed briefly. The structural difference to the Baylis algebra appears in the shape of the hyperbolic unit, which plays an integral part in this formalism.

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Section: Orbital Angular Momentum and Single Particle Potentialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Relativistic quantum physics with hyperbolic numbers

Ulrych¹

2005

Physics Letters B

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“…We note that our definition of the hyperbolic scalar product is different from the definitions given in [8,9] and [19].…”

Section: Definitionmentioning

confidence: 95%

Bicomplex Quantum Mechanics: II. The Hilbert Space

Rochon

Tremblay

2006

AACA

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Using the bicomplex numbers T ∼ = Cl C (1, 0) ∼ = Cl C (0, 1) which is a commutative ring with zero divisors defined by2 = 1 and i 1 i 2 = j = i 2 i 1 , we construct hyperbolic and bicomplex Hilbert spaces. Linear functionals and dual spaces are considered on these spaces and properties of linear operators are obtained; in particular it is established that the eigenvalues of a bicomplex self-adjoint operator are in the set of hyperbolic numbers.

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“…Taking hyperbolic Minkowski space as base space, we can abstract a kind hyperbolic Hilbert space [5].If let discrete structure of Minkowski space corresponds to n-dimensional Hilbert phase space and every discrete phase cell corresponds to a microscopic foreign body's quantum state, Minkowski space's discrete structure can be relate to quantum foreign body's microscopic state. In the discrete structure, take k=1 for any phase cell, (6) can be written as: X 2 − X 1 = ∆X = θ. Obviously,X 1 and X 2 are space-time points in C region and have particle character, θ is a space-time point in Ξ region and have wave character.…”

Section: The Causality Of Physical Event In Minkowski Spacementioning

confidence: 99%