ON THE TAXONOMY OF FLATTENED MOEBIUS STRIPS

2012 ◽  
Vol 21 (01) ◽  
pp. 1250004 ◽  
Author(s):  
J. S. AVRIN
Keyword(s):  
Quantum Mechanics ◽  
Inner Product ◽  
Permutation Groups ◽  
Structural Factors ◽  
Planar Extension ◽  
Convolution Process ◽  
Fusion Concept ◽  
Related Application

The taxonomy of flattened Moebius strips (FMS) is reexamined in order to systematize the basis for its development. An FMS is broadly characterized by its twist and its direction of traverse. All values of twist can be realized by combining elementary FMS configurations in a process called fusion but the result is degenerate; a multiplicity of configurations can exist with the same value of twist. The development of degeneracy is discussed in terms of several structural factors and two principles, conservation of twist and continuity of traverse. The principles implicate a corresponding pair of constructs, a process of symbolic convolution, and the inner product of symbolic vectors. Combining constructs and structural factors leads to a systematically developed taxonomy in terms of twist categories assembled from permutation groups. Taxonomical structure is also graphically revealed by the geometry of an expository edifice that validates the convolution process while displaying the products of fusion. A formulation that combines some of the algebraic precepts of Quantum Mechanics with the primitive combinatorics and degeneracies inherent to the FMS genus is developed. The potential for further investigation and application is also discussed. An appendix outlines the planar extension of the fusion concept and another summarizes a related application of convolution.

scholarly journals CRYPTO-HERMITIAN APPROACH TO THE KLEIN–GORDON EQUATION

2017 ◽  
Vol 57 (6) ◽  
pp. 462 ◽  
Author(s):  
Iveta Semoradova
Keyword(s):  
Quantum Mechanics ◽  
Gordon Equation ◽  
Inner Product ◽  
Indefinite Inner Product ◽  
Probability Interpretation ◽  
Klein Gordon ◽  
Common Problems ◽  
Klein Gordon Equation

We explore the Klein-Gordon equation in the framework of crypto-Hermitian quantum mechanics. Solutions to common problems with probability interpretation and indefinite inner product of the Klein-Gordon equation are proposed.

scholarly journals Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 471 ◽  
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Hermitian Operator ◽  
Quantum Systems ◽  
Time Dependent ◽  
Inner Product ◽  
Geometric Framework ◽  
Heisenberg Picture ◽  
Metric Operator ◽  
Recent Developments

A non-Hermitian operator H defined in a Hilbert space with inner product ⟨ · | · ⟩ may serve as the Hamiltonian for a unitary quantum system if it is η -pseudo-Hermitian for a metric operator (positive-definite automorphism) η . The latter defines the inner product ⟨ · | η · ⟩ of the physical Hilbert space H η of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.

scholarly journals A ‘Dysonization’ scheme for identifying quasi-particles using non-Hermitian quantum mechanics

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120056 ◽  
Author(s):  
Katherine Jones-Smith
Keyword(s):  
Quantum Mechanics ◽  
High Temperature ◽  
High Temperature Superconductivity ◽  
Spin Waves ◽  
Low Energy ◽  
Inner Product ◽  
Inner Products ◽  
Weakly Interacting

Dyson analysed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of the non-Hermitian quantum mechanics formalism at our disposal when considering Dyson’s work, both technically and contextually. Here, we recast Dyson’s work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly ‘Hermitian’. Then we extend his scheme to doped anti-ferromagnets described by the t – J model, with hopes of shedding light on the physics of high-temperature superconductivity.

scholarly journals The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

2018 ◽  
Vol 474 (2217) ◽  
pp. 20180264 ◽  
Author(s):  
David Krejčiřík ◽  
Vladimir Lotoreichik ◽  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Adjoint Operator ◽  
Inner Product ◽  
Sufficient Condition ◽  
Lagrange Equations ◽  
Entire Class ◽  
Inner Products ◽  
Metric Operator ◽  
A Charge

We propose a unique way to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimizing a ‘Hilbert–Schmidt distance’ to the original inner product among the entire class of admissible inner products. We prove that either the minimizer exists and is unique or it does not exist at all. In the former case, we derive a system of Euler–Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supported by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.

scholarly journals PT -symmetric quantum state discrimination

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120160 ◽  
Author(s):  
Carl M. Bender ◽  
Dorje C. Brody ◽  
João Caldeira ◽  
Uwe Günther ◽  
Bernhard K. Meister ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Quantum State ◽  
Success Probability ◽  
Inner Product ◽  
Perfect State ◽  
Quantum State Discrimination ◽  
Single Measurement ◽  
State Discrimination ◽  
Symmetric Quantum ◽  
Ordinary Quantum

The objective of this paper is to explain and elucidate the formalism of quantum mechanics by applying it to a well-known problem in conventional Hermitian quantum mechanics, namely the problem of state discrimination. Suppose that a system is known to be in one of two quantum states, | ψ 1 〉 or | ψ 2 〉. If these states are not orthogonal, then the requirement of unitarity forbids the possibility of discriminating between these two states with one measurement; that is, determining with one measurement what state the system is in. In conventional quantum mechanics, there is a strategy in which successful state discrimination can be achieved with a single measurement but only with a success probability p that is less than unity. In this paper, the state-discrimination problem is examined in the context of quantum mechanics and the approach is based on the fact that a non-Hermitian -symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states. It is shown that it is always possible to choose this inner product so that the two states | ψ 1 〉 and | ψ 2 〉 are orthogonal. Using quantum mechanics, one cannot achieve a better state discrimination than in ordinary quantum mechanics, but one can instead perform a simulated quantum state discrimination, in which with a single measurement a perfect state discrimination is realized with probability  p .

scholarly journals Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 22
Author(s):  
Suzana Bedić ◽  
Otto C. W. Kong ◽  
Hock King Ting
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Regular Representation ◽  
Minkowski Spacetime ◽  
Inner Product ◽  
Minkowski Metric ◽  
C Algebra ◽  
Covariant Quantum ◽  
Metric Operator ◽  
Variable Domains

We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.

scholarly journals Anti-PT Transformations and Complex PT-Symmetric Superpartners

2021 ◽  
Author(s):  
Sehban Kartal ◽  
Taha Koohrokhi ◽  
Ali Mohammadi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Inner Product ◽  
Quantum Mechanical ◽  
Probabilistic Interpretation ◽  
Quantum Mechanical System ◽  
Complex Conjugation ◽  
New Formalism ◽  
Shape Invariant ◽  
Symmetric Quantum

Abstract A quantum mechanical system with unbroken super-and parity-time (PT)-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex PT-symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a PT-symmetric quantum theory is correlated with the calculation of a new linear operator called the C operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-PT (A PT) conjugation to redefine a new version of the inner product without any additional considerations. This PT-supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.

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