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Symmetry
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Physics based on Two-by-two Matrices
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Physics based on Two-by-two Matrices

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (28 February 2014) | Viewed by 48033

Special Issue Editor

Prof. Dr. Young S. Kim

E-Mail Website
Guest Editor
Center for Theoretical Physics, University of Maryland, College Park, MD, USA
Interests: physics of the Lorentz group; relativistic quantum mechanics; quantum optics; relativistic harmonic oscillators; internal space-time symmetries; Lorentz covariant quantum mechanics; physical consequences of Einstein’s E=mc2 ; combining the work of Wigner, Dirac, and Feynman
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Articles based on two-by-two matrices are invited. Your articles may contain original research results or a concise review based on your earlier publications. It is easier to read articles if written in the language of two-by-two matrices.

The two-by-two matrix is the mathematical instrument applicable to all branches of modern physics. If its determinant is one, this matrix has six independent parameters. It has three parameters if its elements are real. In addition to its role in developing theories in a given branch of physics, the same matrix formalism may be applicable to other areas of physics. It may thus be possible, using the same set of two-by-two matrices, to formulate new physical ideas based on what happens in a different branch of physics where the ideas are more firmly established.

It is generally assumed that the mathematics of this two-by-two matrix is well known. Get the eigenvalues by solving a quadratic equation, and then diagonalize the matrix by a rotation. This is not always possible. First of all, there are two-by-two matrixes that cannot be diagonalized. For some instances, the rotation alone is not enough for us to diagonalize the matrix. It is thus possible to gain a new insight to physics while dealing with these mathematical problems.

Prof. Dr. Young Suh Kim
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

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Keywords

  • physical applications of SU(2)
  • Sp(2)
  • SL(2,c)
  • GL(2,c)

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Published Papers (7 papers)

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Research

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1054 KiB  
Article
Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere
by Sibel Başkal, Young S. Kim and Marilyn E. Noz
Symmetry 2014, 6(3), 473-515; https://doi.org/10.3390/sym6030473 - 25 Jun 2014
Cited by 6 | Viewed by 8193
Abstract
The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group Sp(2). It is shown that this oscillator system contains the essential features of Wigner’s little groups dictating the [...] Read more.
The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group Sp(2). It is shown that this oscillator system contains the essential features of Wigner’s little groups dictating the internal space-time symmetries of particles in the Lorentz-covariant world. The little groups are the subgroups of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. It is shown that the damping modes of the oscillator correspond to the little groups for massive and imaginary-mass particles respectively. When the system makes the transition from the oscillation to damping mode, it corresponds to the little group for massless particles. Rotations around the momentum leave the four-momentum invariant. This degree of freedom extends the Sp(2) symmetry to that of SL(2, c) corresponding to the Lorentz group applicable to the four-dimensional Minkowski space. The Poincaré sphere contains the SL(2, c) symmetry. In addition, it has a non-Lorentzian parameter allowing us to reduce the mass continuously to zero. It is thus possible to construct the little group for massless particles from that of the massive particle by reducing its mass to zero. Spin-1/2 particles and spin-1 particles are discussed in detail. Full article
(This article belongs to the Special Issue Physics based on Two-by-two Matrices)
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258 KiB  
Article
Invisibility and PT Symmetry: A Simple Geometrical Viewpoint
by Luis L. Sánchez-Soto and Juan J. Monzón
Symmetry 2014, 6(2), 396-408; https://doi.org/10.3390/sym6020396 - 22 May 2014
Cited by 12 | Viewed by 6242
Abstract
We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry [...] Read more.
We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Möbius transformations, as parallel displacements, a geometric action that has no Euclidean analogy. Full article
(This article belongs to the Special Issue Physics based on Two-by-two Matrices)
330 KiB  
Article
Closed-Form Expressions for the Matrix Exponential
by F. De Zela
Symmetry 2014, 6(2), 329-344; https://doi.org/10.3390/sym6020329 - 29 Apr 2014
Cited by 5 | Viewed by 9519
Abstract
We discuss a method to obtain closed-form expressions of f(A), where f is an analytic function and A a square, diagonalizable matrix. The method exploits the Cayley–Hamilton theorem and has been previously reported using tools that are perhaps not sufficiently appealing to [...] Read more.
We discuss a method to obtain closed-form expressions of f(A), where f is an analytic function and A a square, diagonalizable matrix. The method exploits the Cayley–Hamilton theorem and has been previously reported using tools that are perhaps not sufficiently appealing to physicists. Here, we derive the results on which the method is based by using tools most commonly employed by physicists. We show the advantages of the method in comparison with standard approaches, especially when dealing with the exponential of low-dimensional matrices. In contrast to other approaches that require, e.g., solving differential equations, the present method only requires the construction of the inverse of the Vandermonde matrix. We show the advantages of the method by applying it to different cases, mostly restricting the calculational effort to the handling of two-by-two matrices. Full article
(This article belongs to the Special Issue Physics based on Two-by-two Matrices)
179 KiB  
Article
Dynamical Relation between Quantum Squeezing and Entanglement in Coupled Harmonic Oscillator System
by Lock Yue Chew and Ning Ning Chung
Symmetry 2014, 6(2), 295-307; https://doi.org/10.3390/sym6020295 - 23 Apr 2014
Cited by 5 | Viewed by 6769
Abstract
In this paper, we investigate into the numerical and analytical relationship between the dynamically generated quadrature squeezing and entanglement within a coupled harmonic oscillator system. The dynamical relation between these two quantum features is observed to vary monotically, such that an enhancement in [...] Read more.
In this paper, we investigate into the numerical and analytical relationship between the dynamically generated quadrature squeezing and entanglement within a coupled harmonic oscillator system. The dynamical relation between these two quantum features is observed to vary monotically, such that an enhancement in entanglement is attained at a fixed squeezing for a larger coupling constant. Surprisingly, the maximum attainable values of these two quantum entities are found to consistently equal to the squeezing and entanglement of the system ground state. In addition, we demonstrate that the inclusion of a small anharmonic perturbation has the effect of modifying the squeezing versus entanglement relation into a nonunique form and also extending the maximum squeezing to a value beyond the system ground state. Full article
(This article belongs to the Special Issue Physics based on Two-by-two Matrices)
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198 KiB  
Article
Spacetime Metrics from Gauge Potentials
by Ettore Minguzzi
Symmetry 2014, 6(2), 164-170; https://doi.org/10.3390/sym6020164 - 27 Mar 2014
Viewed by 5210
Abstract
I present an approach to gravity in which the spacetime metric is constructed from a non-Abelian gauge potential with values in the Lie algebra of the group U(2) (or the Lie algebra of quaternions). If the curvature of this potential vanishes, the [...] Read more.
I present an approach to gravity in which the spacetime metric is constructed from a non-Abelian gauge potential with values in the Lie algebra of the group U(2) (or the Lie algebra of quaternions). If the curvature of this potential vanishes, the metric reduces to a canonical curved background form reminiscent of the Friedmann S3 cosmological metric. Full article
(This article belongs to the Special Issue Physics based on Two-by-two Matrices)
229 KiB  
Article
Pseudo Hermitian Interactions in the Dirac Equation
by Orlando Panella and Pinaki Roy
Symmetry 2014, 6(1), 103-110; https://doi.org/10.3390/sym6010103 - 17 Mar 2014
Cited by 4 | Viewed by 5646
Abstract
We consider a (2 + 1)-dimensional massless Dirac equation in the presence of complex vector potentials. It is shown that such vector potentials (leading to complex magnetic fields) can produce bound states, and the Dirac Hamiltonians are η-pseudo Hermitian. Some examples have been [...] Read more.
We consider a (2 + 1)-dimensional massless Dirac equation in the presence of complex vector potentials. It is shown that such vector potentials (leading to complex magnetic fields) can produce bound states, and the Dirac Hamiltonians are η-pseudo Hermitian. Some examples have been explicitly worked out. Full article
(This article belongs to the Special Issue Physics based on Two-by-two Matrices)

Review

Jump to: Research

335 KiB  
Review
Quantum Local Symmetry of the D-Dimensional Non-Linear Sigma Model: A Functional Approach
by Andrea Quadri
Symmetry 2014, 6(2), 234-255; https://doi.org/10.3390/sym6020234 - 17 Apr 2014
Cited by 1 | Viewed by 5643
Abstract
We summarize recent progress on the symmetric subtraction of the Non-Linear Sigma Model in D dimensions, based on the validity of a certain Local Functional Equation (LFE) encoding the invariance of the SU(2) Haar measure under local left transformations. The deformation of the [...] Read more.
We summarize recent progress on the symmetric subtraction of the Non-Linear Sigma Model in D dimensions, based on the validity of a certain Local Functional Equation (LFE) encoding the invariance of the SU(2) Haar measure under local left transformations. The deformation of the classical non-linearly realized symmetry at the quantum level is analyzed by cohomological tools. It is shown that all the divergences of the one-particle irreducible (1-PI) amplitudes (both on-shell and off-shell) can be classified according to the solutions of the LFE. Applications to the non-linearly realized Yang-Mills theory and to the electroweak theory, which is directly relevant to the model-independent analysis of LHC data, are briefly addressed. Full article
(This article belongs to the Special Issue Physics based on Two-by-two Matrices)
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