Erratum: The abstract complex Lorentz transformation group with real metric. I. Special relativity formalism to deal with the holomorphic automorphism group of the unit ball in any complex Hilbert space [J. Math. Phys. 35, 1408–1426 (1994)]

1994 ◽  
Vol 35 (7) ◽  
pp. 3770-3770 ◽  
Author(s):  
Abraham A. Ungar
Keyword(s):  
Hilbert Space ◽  
Automorphism Group ◽  
Unit Ball ◽  
Special Relativity ◽  
Lorentz Transformation ◽  
Transformation Group ◽  
Complex Hilbert Space ◽  
Holomorphic Automorphism ◽  
Holomorphic Automorphism Group ◽  
Math Phys

scholarly journals Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

2021 ◽  
pp. 2150083
Author(s):  
Vikramjeet Singh Chandel
Keyword(s):  
Automorphism Group ◽  
Unit Ball ◽  
Complex Plane ◽  
Unit Disc ◽  
Interpolation Problem ◽  
Necessary Conditions ◽  
Holomorphic Automorphism ◽  
Spectral Unit Ball ◽  
Holomorphic Automorphism Group

In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows — generalizing a result of Ransford–White for the spectral unit ball — that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider [Formula: see text]-point and [Formula: see text]-point Pick–Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

scholarly journals A Spacetime Symmetry Approach to Relativistic Quantum Multi-Particle Entanglement

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1259 ◽  
Author(s):  
Abraham A. Ungar
Keyword(s):  
Special Relativity ◽  
Symmetry Group ◽  
Lorentz Transformation ◽  
Lorentz Group ◽  
Transformation Group ◽  
Relativity Theory ◽  
Symmetry Violation ◽  
3 Dimensional ◽  
Dimensional Spacetime ◽  
The Common

A Lorentz transformation group SO(m, n) of signature (m, n), m, n ∈ N, in m time and n space dimensions, is the group of pseudo-rotations of a pseudo-Euclidean space of signature (m, n). Accordingly, the Lorentz group SO(1, 3) is the common Lorentz transformation group from which special relativity theory stems. It is widely acknowledged that special relativity and quantum theories are at odds. In particular, it is known that entangled particles involve Lorentz symmetry violation. We, therefore, review studies that led to the discovery that the Lorentz group SO(m, n) forms the symmetry group by which a multi-particle system of m entangled n-dimensional particles can be understood in an extended sense of relativistic settings. Consequently, we enrich special relativity by incorporating the Lorentz transformation groups of signature (m, 3) for all m ≥ 2. The resulting enriched special relativity provides the common symmetry group SO(1, 3) of the (1 + 3)-dimensional spacetime of individual particles, along with the symmetry group SO(m, 3) of the (m + 3)-dimensional spacetime of multi-particle systems of m entangled 3-dimensional particles, for all m ≥ 2. A unified parametrization of the Lorentz groups SO(m, n) for all m, n ∈ N, shakes down the underlying matrix algebra into elegant and transparent results. The special case when (m, n) = (1, 3) is supported experimentally by special relativity. It is hoped that this review article will stimulate the search for experimental support when (m, n) = (m, 3) for all m ≥ 2.

Geometry on the Unit Ball of a Complex Hilbert Space

1978 ◽  
Vol 30 (01) ◽  
pp. 22-31 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Hyperbolic Geometry ◽  
Complex Hilbert Space ◽  
Open Unit ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Open Unit Ball ◽  
Similar Role ◽  
Necessary And Sufficient

Furnishing the open unit ball of a complex Hilbert space with the Carathéodory-differential metric, we construct a model which plays a similar role as that of the Poincaré model for the hyperbolic geometry. In this note we study the question whether or not through a point in the model not lying on a given line there exists a unique perpendicular, and give a necessary and sufficient condition for the existence of a unique perpendicular. This enables us to divide a triangle into two right triangles. Many trigonometric identities in a general triangle are easy consequences of various identities which hold on a right triangle.

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