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 Analysis on Complete Set of Fock States with Explicit Wavefunctions for the Covariant Harmonic Oscillator Problem

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 39
Author(s):  
Suzana Bedić ◽  
Otto Kong
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Fock Space ◽  
Minkowski Spacetime ◽  
Lorentz Symmetry ◽  
Inner Product ◽  
Covariant Quantum ◽  
Fock States ◽  
Complete Set ◽  
State Wavefunctions

The earlier treatments of the Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems. The complete set of Fock states is obtained, together with the corresponding explicit wavefunctions and their inner product integrals free from any divergence problem and with Lorentz symmetry fully maintained without additional constraints imposed. By a simple choice of the pseudo-unitary representation of the underlying symmetry group, motivated from the perspective of the Minkowski spacetime as a representation for the Lorentz group, we obtain the natural non-unitary Fock space picture commonly considered, although not formulated and presented in the careful details given here. From a direct derivation of the appropriate basis state wavefunctions of the finite-dimensional irreducible representations of the Lorentz symmetry, the relation between the latter and the Fock state wavefunctions is also explicitly shown. Moreover, the full picture, including the states with a non-positive norm, may give a consistent physics picture as a version of Lorentz covariant quantum mechanics. The probability interpretation for the usual von Neumann measurements is not a problem, as all wavefunctions restricted to a definite value for the `time’ variable are just like those of the usual time independent quantum mechanics. A further understanding from a perspective of the dynamics from the symplectic geometry of the phase space is shortly discussed.

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 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.

One-dimensional quantum mechanics with Dunkl derivative

2019 ◽  
Vol 34 (24) ◽  
pp. 1950190
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Scalar Product ◽  
Momentum Operator ◽  
Inner Product ◽  
One Dimensional ◽  
Coordinate Representation ◽  
The One ◽  
Infinite Wall

In this paper, we consider the quantum mechanics with Dunkl derivative. We use the Dunkl derivative to obtain the coordinate representation of the momentum operator and Hamiltonian. We introduce the scalar product to find that the momentum is Hermitian under this inner product. We study the one-dimensional box problem (the spin-less particle with mass m confined to the one-dimensional infinite wall). Finally, we discuss the harmonic oscillator problem.

scholarly journals A scale-covariant quantum spacetime

2014 ◽  
Vol 26 (04) ◽  
pp. 1450006 ◽  
Author(s):  
Claudio Perini ◽  
Gabriele Nunzio Tornetta
Keyword(s):  
Minkowski Spacetime ◽  
C Algebra ◽  
Covariant Quantum ◽  
Covariant Model ◽  
Dilation Group ◽  
Free Scalar ◽  
Neutral Field ◽  
Noncommutative Spacetime ◽  
Scalar Neutral Field ◽  
Quantum Spacetime

A noncommutative spacetime admitting dilation symmetry was briefly mentioned in the seminal work [8] of Doplicher, Fredenhagen and Roberts. In this paper, we explicitly construct the model in detail and carry out an indepth analysis. The C*-algebra that describes this quantum spacetime is determined, and it is shown that it admits an action by *-automorphisms of the dilation group, along with the expected Poincaré covariance. In order to study the main physical properties of this scale-covariant model, a free scalar neutral field is introduced as an investigation tool. Our key results are then the loss of locality and the irreducibility, or triviality, of special field algebras associated with regions of the ordinary Minkowski spacetime. It turns out, in the conclusions, that this analysis allows also to argue on viable ways of constructing a full conformally covariant model for quantum spacetime.

scholarly journals Beyond Causal Explanation: Einstein’s Principle Not Reichenbach’s

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 114
Author(s):  
Michael Silberstein ◽  
William Mark Stuckey ◽  
Timothy McDevitt
Keyword(s):  
Quantum Mechanics ◽  
Quantum Information ◽  
Causal Explanation ◽  
Causal Structure ◽  
Minkowski Spacetime ◽  
Physical Principle ◽  
Causal Principle ◽  
Complete Rejection ◽  
Epr Correlations ◽  
Tsirelson Bound

Our account provides a local, realist and fully non-causal principle explanation for EPR correlations, contextuality, no-signalling, and the Tsirelson bound. Indeed, the account herein is fully consistent with the causal structure of Minkowski spacetime. We argue that retrocausal accounts of quantum mechanics are problematic precisely because they do not fully transcend the assumption that causal or constructive explanation must always be fundamental. Unlike retrocausal accounts, our principle explanation is a complete rejection of Reichenbach’s Principle. Furthermore, we will argue that the basis for our principle account of quantum mechanics is the physical principle sought by quantum information theorists for their reconstructions of quantum mechanics. Finally, we explain why our account is both fully realist and psi-epistemic.

QUANTUM SUPERSPACE, q-EXTENDED SUPERSYMMETRY AND PARASUPERSYMMETRIC QUANTUM MECHANICS

1993 ◽  
Vol 08 (28) ◽  
pp. 2657-2670 ◽  
Author(s):  
K. N. ILINSKI ◽  
V. M. UZDIN
Keyword(s):  
Quantum Mechanics ◽  
Extended Supersymmetry ◽  
Harmonic Oscillator ◽  
Unit Circle ◽  
Canonical Quantization ◽  
Matrix Representations ◽  
Dimensional Matrix ◽  
Continuous Parameter ◽  
Superspace Formalism

We describe q-deformation of the extended supersymmetry and construct q-extended supersymmetric Hamiltonian. For this purpose we formulate q-superspace formalism and construct q-supertransformation group. On this basis q-extended supersymmetric Lagrangian is built. The canonical quantization of this system is considered. The connection with multi-dimensional matrix representations of the parasupersymmetric quantum mechanics is discussed and q-extended supersymmetric harmonic oscillator is considered as a simplest example of the described constructions. We show that extended supersymmetric Hamiltonians obey not only extended SUSY but also the whole family of symmetries (q-extended supersymmetry) which is parametrized by continuous parameter q on the unit circle.

scholarly journals Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

10.53733/90 ◽  
2021 ◽  
Vol 52 ◽  
pp. 109-143
Author(s):  
Astrid An Huef ◽  
Marcelo Laca ◽  
Iain Raeburn
Keyword(s):  
Regular Representation ◽  
Crossed Product ◽  
Ideal Structure ◽  
Toeplitz Algebra ◽  
Natural Numbers ◽  
Kms States ◽  
C Algebra ◽  
Affine Semigroup ◽  
Natural Dynamics ◽  
The Right

We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

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