scholarly journals On Analog of Fourier Transform in Interior of the Light Cone

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Tatyana Shtepina
Keyword(s):  
Fourier Transform ◽  
Lorentz Group ◽  
Light Cone ◽  
Inverse Transform ◽  
Irreducible Components ◽  
One To One

We introduce an analog of Fourier transformFhρin interior of light cone that commutes with the action of the Lorentz group. We describe some properties ofFhρ, namely, its action on pseudoradial functions and functions being products of pseudoradial function and space hyperbolic harmonics. We prove thatFhρ-transform gives a one-to-one correspondence on each of the irreducible components of quasiregular representation. We calculate the inverse transform too.

The Duflo non-commutative Fourier transform for the Lorentz group

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040011
Author(s):  
Giacomo Rosati
Keyword(s):  
Fourier Transform ◽  
Phase Space ◽  
Quantum System ◽  
Lie Group ◽  
Lorentz Group ◽  
Commutative Algebra ◽  
Cotangent Bundle ◽  
Irreducible Representations ◽  
Algebra Representation

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

FOURIER SINE AND COSINE TRANSFORMS ON BOEHMIAN SPACES

2013 ◽  
Vol 06 (01) ◽  
pp. 1350005 ◽  
Author(s):  
R. Roopkumar ◽  
E. R. Negrin ◽  
C. Ganesan
Keyword(s):  
Fourier Transform ◽  
Cosine Transform ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Fourier Cosine Transform ◽  
One To One ◽  
The Fourier Transform

We construct suitable Boehmian spaces which are properly larger than [Formula: see text] and we extend the Fourier sine transform and the Fourier cosine transform more than one way. We prove that the extended Fourier sine and cosine transforms have expected properties like linear, continuous, one-to-one and onto from one Boehmian space onto another Boehmian space. We also establish that the well known connection among the Fourier transform, Fourier sine transform and Fourier cosine transform in the context of Boehmians. Finally, we compare the relations among the different Boehmian spaces discussed in this paper.

The generalized uncertainty principle from the doubly special relativity: Algebraic approach, Ramsauer effect and delta potential

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950052 ◽  
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi ◽  
Min Su Kang ◽  
Hee Hun Sung ◽  
Jinyeob Maeng ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Quantum Mechanics ◽  
Plane Wave ◽  
Generalized Uncertainty Principle ◽  
Algebraic Approach ◽  
Delta Potential ◽  
Doubly Special Relativity ◽  
Inverse Transform ◽  
Ramsauer Effect ◽  
Gaussian Packet

In this paper, we consider the DSR-GUP which was first proposed by two of us [W. S. Chung and H. Hassanabadi, Phys. Lett. B 785, 127 (2018)]. Here, we introduce the deformed derivative corresponding to DSR-GUP to discuss the quantum mechanics based on DSR-GUP in an algebraical way. We use the deformed Fourier transform and its inverse transform to discuss the plane wave in [Formula: see text]-space and Gaussian packet. We also extend our discussion to the (1 + 1)-dimensional case. As examples, we discuss the Ramsauer–Townsend effect and delta potential.

scholarly journals Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

Universe ◽  
2019 ◽  
Vol 5 (8) ◽  
pp. 184 ◽  
Author(s):  
Victor Miguel Banda Guzmán ◽  
Mariana Kirchbach
Keyword(s):  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Dirac Spinor ◽  
Product Spaces ◽  
Group Representations ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Irreducible Components ◽  
Carrier Space ◽  
Lorentz Group Representations

The momentum-independent Casimir operators of the homogeneous spin-Lorentz group are employed in the construction of covariant projector operators, which can decompose anyone of its reducible finite-dimensional representation spaces into irreducible components. One of the benefits from such operators is that any one of the finite-dimensional carrier spaces of the Lorentz group representations can be equipped with Lorentz vector indices because any such space can be embedded in a Lorentz tensor of a properly-designed rank and then be unambiguously found by a projector. In particular, all the carrier spaces of the single-spin-valued Lorentz group representations, which so far have been described as 2 ( 2 j + 1 ) column vectors, can now be described in terms of Lorentz tensors for bosons or Lorentz tensors with the Dirac spinor component, for fermions. This approach facilitates the construct of covariant interactions of high spins with external fields in so far as they can be obtained by simple contractions of the relevant S O ( 1 , 3 ) indices. Examples of Lorentz group projector operators for spins varying from 1 / 2 –2 and belonging to distinct product spaces are explicitly worked out. The decomposition of multiple-spin-valued product spaces into irreducible sectors suggests that not only the highest spin, but all the spins contained in an irreducible carrier space could correspond to physical degrees of freedom.

Spectral Analysis on Upper Light Cone in R3 and the Radon Transform

1988 ◽  
Vol 40 (6) ◽  
pp. 1458-1481 ◽  
Author(s):  
Antoni Wawrzyñczyk
Keyword(s):  
Spectral Analysis ◽  
Symmetric Space ◽  
Lorentz Group ◽  
Radon Transform ◽  
Light Cone ◽  
General Procedure ◽  
Natural Action ◽  
3 Dimensional ◽  
Generalized Radon Transform ◽  
Image Position

The upper light cone L in R3 is a homogeneous space of the 3-dimensional Lorentz group G. It may be identified with the space of horocycles in the upper hyperboloide H which is the symmetric space associated to G. There exists a duality between H and L (see [5] p. 144) and a general procedure leads to a generalized Radon transform:and the dual Radon transformThese operations commute with the natural action of the group G.

scholarly journals A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Simple Proof ◽  
Linear Canonical Transform ◽  
Quaternion Fourier Transform ◽  
Inverse Transform ◽  
Fundamental Relationship ◽  
Quaternion Linear Canonical Transform

We provide a short and simple proof of an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by considering the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). We show how this relation allows us to derive the inverse transform and Parseval and Plancherel formulas associated with the QLCT. Some other properties of the QLCT are also studied.

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