Reciprocal Relativity of Noninertial Frames and the Quaplectic Group

Low, Stephen G.

doi:10.1007/s10701-006-9051-2

Cited by 19 publications

(34 citation statements)

References 6 publications

(10 reference statements)

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“…One may introduce a similar invariant space-time-momentum-energy line element under quaplectic transformations which assumes Born reciprocity as an accepted group of intrinsic symmetry. If such quaplectic group is considered to be fundamental, then it would imply a maximum rate of change of momentum and appear as a new fundamental constant [26]. Here, however, we confine our discussion in the light of Born-Green reciprocity and its validity is the heart of our analysis.…”

Section: The Principle Of Reciprocity: the Born-green Formalismmentioning

confidence: 98%

“…• Yet, constructing a general theory defining Born reciprocity as an intrinsic symmetry [26], continued as an interesting field of research even in modern times by adapting group theoretical methods [27][28][29][30][31] which are employed to study a group, namely the quaplectic group. With the advent of renormalization theory and Higg's mechanism in the last seventy years [32][33][34][35], both in theory as well as in experiments, it has also been suggested that Born's reciprocity may be the underlying physical reason for the T-duality symmetry in string theory [25,35,36], and may be of relevance in developing a quantum geometry [37,38].…”

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confidence: 99%

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Born-Kothari Condensation for Fermions

Ghosh

2017

Entropy

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Abstract:In the spirit of Bose-Einstein condensation, we present a detailed account of the statistical description of the condensation phenomena for a Fermi-Dirac gas following the works of Born and Kothari. For bosons, while the condensed phase below a certain critical temperature, permits macroscopic occupation at the lowest energy single particle state, for fermions, due to Pauli exclusion principle, the condensed phase occurs only in the form of a single occupancy dense modes at the highest energy state. In spite of these rudimentary differences, our recent findings [Ghosh and Ray, 2017] identify the foregoing phenomenon as condensation-like coherence among fermions in an analogous way to Bose-Einstein condensate which is collectively described by a coherent matter wave. To reach the above conclusion, we employ the close relationship between the statistical methods of bosonic and fermionic fields pioneered by Cahill and Glauber. In addition to our previous results, we described in this mini-review that the highest momentum (energy) for individual fermions, prerequisite for the condensation process, can be specified in terms of the natural length and energy scales of the problem. The existence of such condensed phases, which are of obvious significance in the context of elementary particles, have also been scrutinized.

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Section: The Principle Of Reciprocity: the Born-green Formalismmentioning

confidence: 98%

mentioning

confidence: 99%

Born-Kothari Condensation for Fermions

Ghosh

2017

Entropy

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“…In this form, it is clear that the global symmetry group of the system, namely the so-called quaplectic group [10][11][12], is indeed isomorphic to…”

Section: The Complex Parametrizationmentioning

confidence: 99%

“…An alternative realization of the Weyl-Heisenberg group is provided by the following (2D + 2) × (2D + 2) real matrices [12],…”

Section: Appendixmentioning

confidence: 99%

“…The idea of reciprocity has found resonance with various attempts to generalize the framework for the fundamental interactions-for example, in the context of string and M-theory [5,6], in the guise of bi-crossproduct algebras and physics at the Planck scale [7], in 'two-time' formulations [8], or in ad hoc 'noncommutative geometry' extensions of perturbative field theory [9]. Born-Green reciprocity can be viewed [10][11][12][13] as an alternative paradigm for generalized wave equations, which specify unitary irreducible representations of the full symmetry group, in the same way that relativistic wave equations establish unitary irreducible representations of the Poincaré group in four dimensions. It can be argued [10][11][12][13] that the appropriate invariance group is the so-called quaplectic group Q(3, 1) ∼ = U(3, 1) H (4), or more generally in D spacetime dimensions, the group Q(D − 1, 1) ∼ = U(D − 1, 1) H (D) of reciprocal relativity, the semi-direct product of the pseudo-unitary group of linear transformations between x µ and p µ which preserve both the extended metric d 2 and the symplectic form, with the WeylHeisenberg group.…”

Section: Introductionmentioning

confidence: 99%

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World-line quantization of a reciprocally invariant system

Govaerts

Jarvis

Morgan

et al. 2007

J. Phys. A: Math. Theor.

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We present the world-line quantization of a system invariant under the symmetries of reciprocal relativity (pseudo-unitary transformations on 'phasespace coordinates ' (x µ (τ ), p µ (τ )) which preserve the Minkowski metric and the symplectic form, and global shifts in these coordinates, together with coordinate-dependent transformations of an additional compact phase coordinate, θ(τ )). The action is that of free motion over the corresponding Weyl-Heisenberg group. Imposition of the first class constraint, the generator of local time reparametrizations, on physical states enforces identification of the world-line cosmological constant with a fixed value of the quadratic Casimir of the quaplectic symmetry group Q(D − 1, 1) ∼ = U(D − 1, 1) H (D), the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberg group (the central extension of the global translation group, with central extension associated with the phase variable θ(τ )). The spacetime spectrum of physical states is identified. Even though for an appropriate range of values the restriction enforced by the cosmological constant projects out negative norm states from the physical gauge invariant spectrum, leaving over spin zero states 5 Fellow of the Stellenbosch Institute for Advanced Study (STIAS), Stellenbosch, Republic of South Africa, http://academic.sun.ac.za/stias/. 6 On sabbatical leave from

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Born’s Reciprocal Gravity in Curved Phase-Spaces and the Cosmological Constant

Castro

2012

Found Phys

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The main features of how to build a Born's Reciprocal Gravitational theory in curved phase-spaces are developed. By recurring to the nonlinear connection formalism of Finsler geometry a generalized gravitational action in the 8D cotangent space (curved phase space) can be constructed involving sums of 5 distinct types of torsion squared terms and 2 distinct curvature scalars R, S which are associated with the curvature in the horizontal and vertical spaces, respectively. A Kaluza-Klein-like approach to the construction of the curvature of the 8D cotangent space and based on the (torsionless) Levi-Civita connection is provided that yields the observed value of the cosmological constant and the Brans-Dicke-Jordan Gravity action in 4D as two special cases. It is found that the geometry of the momentum space can be linked to the observed value of the cosmological constant when the curvature in momentum space is very large, namely the small size of P is of the order of (1/R Hubble ). Finally we develop a Born's reciprocal complex gravitational theory as a local gauge theory in 8D of the deformed Quaplectic group that is given by the semi-direct product of U(1, 3) with the deformed (noncommutative) Weyl-Heisenberg group involving four noncommutative coordinates and momenta. The metric is complex with symmetric real components and antisymmetric imaginary ones. An action in 8D involving 2 curvature scalars and torsion squared terms is presented.

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Reciprocal Relativity of Noninertial Frames and the Quaplectic Group

Cited by 19 publications

References 6 publications

Born-Kothari Condensation for Fermions

Born-Kothari Condensation for Fermions

World-line quantization of a reciprocally invariant system

Born’s Reciprocal Gravity in Curved Phase-Spaces and the Cosmological Constant

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