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  • 1
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    Symmetry, Vol. 5, Pages 271-286: A New Route to the Majorana Equation (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-09-27
    Description: In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby, use is made of the complex conjugation operator and Pauli spin matrices only. The eigenfunctions of the two-component complex Majorana equation are also calculated. The associated quantum fields are found to describe particles and antiparticles, which have opposite mean helicities and are not their own antiparticles, but correspond to two independent degrees of freedom. The four-component real Dirac equation in its Majorana representation is shown to be the natural outcome of the two-component complex Majorana equation. Both types of equations come in two forms, which correspond to the irreducible left- and right-chiral representations of the Lorentz group.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 2
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    Symmetry, Vol. 5, Pages 344-354: Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-12-10
    Description: Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the Brillouin zone. In this paper, we will illustrate this property by considering two-dimensional (2D)-hexagonal lattices characterized by a possible two-fold degenerate band at the K points with a linear dispersion (Dirac points). By combining scanning tunneling spectroscopy and angle-resolved photoemission, we study the electronic properties of a similar system: the Ag/Cu(111) interface reconstruction characterized by a hexagonal superlattice, and we show that the gap opening at the K points of the Brillouin zone of the reconstructed cell is due to the symmetry breaking of the wave vector group.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 3
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    Symmetry, Vol. 5, Pages 253-270: Supersymmetric Version of the Euler System and Its Invariant Solutions (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-07-13
    Description: In this paper, we formulate a supersymmetric extension of the Euler system of equations. We compute a superalgebra of Lie symmetries of the supersymmetric system. Next, we classify the one-dimensional subalgebras of this superalgebra into 49 equivalence conjugation classes. For some of the subalgebras, the invariants have a non-standard structure. For nine selected subalgebras, we use the symmetry reduction method to find invariants, orbits and reduced systems. Through the solutions of these reduced systems, we obtain solutions of the supersymmetric Euler system. The obtained solutions include bumps, kinks, multiple wave solutions and solutions expressed in terms of arbitrary functions.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 4
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    Symmetry, Vol. 4, Pages 379-426: Symmetry-Adapted Fourier Series for the Wallpaper Groups (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-03-09
    Description: Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between the Fourier coefficients. Visualising the equivalencies of Fourier coefficients by means of discrete 2D maps reveals how direct-space symmetry is transformed into coefficient-space symmetry. Explicit expressions are given for the Fourier series and Fourier coefficient maps of both real and complex functions, readily applicable to the description of the properties of 2D materials like graphene or boron-nitride.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 5
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    Symmetry, Vol. 4, Pages 225-250: Classical Knot Theory (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-03-09
    Description: This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 6
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    Symmetry, Vol. 4, Pages 39-115: Knots on a Torus: A Model of the Elementary Particles (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-03-09
    Description: Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, mass, charge, CPT invariance and more. There are no quarks to provide fractional charge, no gluons to sequester them within nucleons and no “colors” to avoid violating Pauli’s principle. Nor do we require the importation of an enigmatic Higgs boson to confer mass upon the particles of our world. All the requisite attributes emerge simply (and relativistically invariant) as a result of particle conformation and occupation in and of spacetime itself, a spacetime endowed with the imprimature of general relativity. Also emerging are some novel tools for systemizing the particle taxonomy as governed by the gauge group SU(2) and the details of particle degeneracy as well as connections to Hopf algebra, Dirac theory, string theory, topological quantum field theory and dark matter. One exception: it is found necessary to invoke the munificent geometry of the icosahedron in order to provide, as per the group “flavor” SU(3), a scaffold upon which to organize the well-known three generations—no more, no less—of the particle family tree.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 7
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    Symmetry, Vol. 4, Pages 626-643: Dirac Matrices and Feynman’s Rest of the Universe (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-03-09
    Description: There are two sets of four-by-four matrices introduced by Dirac. The first set consists of fifteen Majorana matrices derivable from his four γ matrices. These fifteen matrices can also serve as the generators of the group SL(4, r). The second set consists of ten generators of the Sp(4) group which Dirac derived from two coupled harmonic oscillators. It is shown possible to extend the symmetry of Sp(4) to that of SL(4, r) if the area of the phase space of one of the oscillators is allowed to become smaller without a lower limit. While there are no restrictions on the size of phase space in classical mechanics, Feynman’s rest of the universe makes this Sp(4)-to-SL(4, r) transition possible. The ten generators are for the world where quantum mechanics is valid. The remaining five generators belong to the rest of the universe. It is noted that the groups SL(4, r) and Sp(4) are locally isomorphic to the Lorentz groups O(3, 3) and O(3, 2) respectively. This allows us to interpret Feynman’s rest of the universe in terms of space-time symmetry.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 8
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    Symmetry, Vol. 4, Pages 566-580: On the Notions of Symmetry and Aperiodicity for Delone Sets (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-03-09
    Description: Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 9
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    Symmetry, Vol. 5, Pages 1-46: Taylor–Socolar Hexagonal Tilings as Model Sets (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-03-09
    Description: The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice Q in the plane. If XQ denotes the set of all Taylor–Socolar tilings with centers on Q, then XQ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of Q.The Q-adic completion Q of Q is a natural factor of XQ and the natural mapping XQ → Q is bijective except at a dense set of points of measure 0 in /Q. We show that XQ consists of three LI classes under translation. Two of these LI classes are very small, namely countable Q-orbits in XQ. The other is a minimal dynamical system, which maps surjectively to /Q and which is variously 2 : 1, 6 : 1, and 12 : 1 at the singular points. We further develop the formula of what determines the parity of the tiles of a tiling in terms of the coordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull XQ; more precisely the two hulls are mutually locally derivable.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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  • 10
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    Symmetry, Vol. 4, Pages 452-473: Supersymmetric Quantum Mechanics and Solvable Models (2013)
    MDPI Publishing
    In: Symmetry
    Details
    Publication Date: 2013-03-09
    Description: We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of ħ-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ħ.
    Electronic ISSN: 2073-8994
    Topics: Mathematics , Physics
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