Generalized Uncertainty Relation in the Non-commutative Quantum Mechanics

In this paper the non-commutative quantum mechanics (NCQM) with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar _{eff}}{2}\) is discussed. Four each uncertainty relation, wave functions saturating each uncertainty relation are explicitly constructed. The unitary operators relating the non-commutative position and momentum operators to the commutative position and momentum operators are also investigated. We also discuss the uncertainty relation related to the harmonic oscillator.     

Probing Uncertainty Relations in Non-Commutative Space

In this paper, we compute uncertainty relations for non-commutative space and obtain a better lower bound than the standard one obtained from Heisenberg’s uncertainty relation. We also derive the reverse uncertainty relation for product and sum of uncertainties of two incompatible variables for one linear and another non-linear model of the harmonic oscillator. The non-linear model in non-commutating space yields two different expressions for Schrödinger and Heisenberg uncertainty relation. This distinction does not arise in commutative space, and even in the linear model of non-commutative space.

     

A Minimal Framework for Non-Commutative Quantum Mechanics

Deformation quantisation is applied to ordinary Quantum Mechanics by introducing the star product in a configuration space combining a Riemannian structure with a Poisson one. A Hilbert space compatible with such a configuration space is designed. The dynamics is expressed by a Hermitian Hamiltonian containing a scalar potential and a one-form potential. As a simple illustration, it is shown how a particular type of non-commutativity of the star product is interpretable as generating the Zeeman effect of ordinary Quantum Mechanics.     

Preparation and Measurement: Two Independent Sources of Uncertainty in Quantum Mechanics

In the Copenhagen interpretation the Heisenberg inequality QP/2 is interpreted as the mathematical expression of the concept of complementarity, quantifying the mutual disturbance necessarily taking place in a simultaneous or joint measurement of incompatible observables. This interpretation was criticized a long time ago and has recently been challenged in an experimental way. These criticisms can be substantiated by using the generalized formalism of positive operator-valued measures, from which an inequality, different from the Heisenberg inequality, can be derived, precisely illustrating the Copenhagen concept of complementarity. The different roles of preparation and measurement in creating uncertainty in quantum mechanics are discussed.     

Generalized Quantum Mechanics for Separated Systems

We give seven necessary physical conditions on a property lattice for to describe two quantum systems when they are separated.     

On Quantum Fluctuations Relations with Generalized Energy Measurements

Quantum fluctuations relations are typically derived with projective measurements of energy at the beginning and the end of the protocol. Though projective measurements are easy to treat theoretically, they may be difficult to implement in experiments. We show that recent results on the force protocol with generalized measurements remain valid with more general forms of quantum evolution. In the case considered, states of an open quantum system are transformed according to an arbitrary trace-preserving completely positive map. The energy measurements used in the protocol are prescribed as a superposition of projective ones. Such measurements can be parameterized by a continuous variable substituted into acceptance functions of the apparatuses. The role of limited measurement precision is also clarified here. As is known, the standard Jarzynski equality and the Tasaki–Crooks theorem remain valid for the case of unital quantum channels. Using characteristic functions, we recast standard relations for the scenario with generalized measurements. With respect to the Jarzynski equality, generalized measurements lead to a negative shift in observed values of the difference of free energies. A deviation of the actual quantum channel from unital ones may also generate some shift with an arbitrary sign.     

Generalized Uncertainty Principle and Quantum Electrodynamics

In the present work the role that a generalized uncertainty principle could play in the quantization of the electromagnetic field is analyzed. It will be shown that we may speak of a Fock space, a result that implies that the concept of photon is properly defined. Nevertheless, in this new context the creation and annihilation operators become a function of the new term that modifies the Heisenberg algebra, and hence the Hamiltonian is not anymore diagonal in the occupation number representation. Additionally, we show the changes that the energy expectation value suffers as result of the presence of an extra term in the uncertainty principle. The existence of a deformed dispersion relation is also proved.     

Quantum Fisher Information and Uncertainty Relations

It is well known that the Cramér–Rao inequality places a lower bound for quantum Fisher information in terms of the variance of any quantum measurement. We establish an upper bound for quantum Fisher information of a parameterized family of density operators in terms of the variance of the generator. These two bounds together yield a generalization of the Heisenberg uncertainty relations from statistical estimation perspective.     

Quantum Thermodynamic Uncertainty Relations,Generalized Current Fluctuations and Nonequilibrium Fluctuation–Dissipation Inequalities

Thermodynamic uncertainty relations (TURs) represent one of the few broad-based and fundamental relations in our toolbox for tackling the thermodynamics of nonequilibrium systems. One form of TUR quantifies the minimal energetic cost of achieving a certain precision in determining a nonequilibrium current. In this initial stage of our research program, our goal is to provide the quantum theoretical basis of TURs using microphysics models of linear open quantum systems where it is possible to obtain exact solutions. In paper [Dong et al., Entropy 2022, 24, 870], we show how TURs are rooted in the quantum uncertainty principles and the fluctuation–dissipation inequalities (FDI) under fully nonequilibrium conditions. In this paper, we shift our attention from the quantum basis to the thermal manifests. Using a microscopic model for the bath’s spectral density in quantum Brownian motion studies, we formulate a “thermal” FDI in the quantum nonequilibrium dynamics which is valid at high temperatures. This brings the quantum TURs we derive here to the classical domain and can thus be compared with some popular forms of TURs. In the thermal-energy-dominated regimes, our FDIs provide better estimates on the uncertainty of thermodynamic quantities. Our treatment includes full back-action from the environment onto the system. As a concrete example of the generalized current, we examine the energy flux or power entering the Brownian particle and find an exact expression of the corresponding current–current correlations. In so doing, we show that the statistical properties of the bath and the causality of the system+bath interaction both enter into the TURs obeyed by the thermodynamic quantities.     

Energy-Time Uncertainty Relation within the Framework of Axiomatics of Quantum Mechanics

The present paper is devoted to an interpretation of energy-time uncertainty relations for potentials that are constant in time and have different spatial profiles (constant and Gaussian potentials) within the framework of approaches based on axiomatics of quantum theory and nonstationary perturbation theory. The results obtained indicate that the energy-time relation has no strictly axiomatic meaning determined by the Heisenberg uncertainty relations.     

Quantum General Relativity,Torsion and Uncertainty Relations

It is shown that in gravitational theories with torsion one is led to commutation rules corresponding to Landau-Peierls type uncertainty relations.     

Velocity Operators and Time-Energy Relations in Relativistic Quantum Mechanics

According to both Dirac's and Kemmer's relativistic quantum theories, the eigenvalues of the velocity operator are +c and –c. This false result is avoided if certain alternative particle coordinates are adopted. Another advantage is that the new coordinates occur in additional constants of the motion. These are sui generis angular momenta obtained by taking the vector product of the nonstandard coordinates with the linear momentum. An additional virtue of the new velocity operator is that, like in classical mechanics, it is proportional to the linear momentum. Besides, the zeroth component of the new set of coordinates does not commute with the hamiltonian, which results in a genuine indeterminacy relation between time and energy.     

Generalized Uncertainty Principle and the Quantum Entropy of Rotating Black Hole

The entropy of rotating Kerr-Newman-Kasuya black hole due to massive charged fields （bosons and fermions） is calculated by using the new equation of state density motivated by the generalized uncertainty relation. The result shows the entropy does not depend on the mass and the charge but the parameter A, the area A and the spin of the fields. Moreover, an improved approximation is provided to calculate the scalar entropy.     

Theory of Dynamical Systems and the Relations Between Classical and Quantum Mechanics

We give a review of some works where it is shown that certain quantum-like features are exhibited by classical systems. Two kinds of problems are considered. The first one concerns the specific heat of crystals (the so called Fermi–Pasta–Ulam problem), where a glassy behavior is observed, and the energy distribution is found to be of Planck-like type. The second kind of problems concerns the self-interaction of a charged particle with the electromagnetic field, where an analog of the tunnel effect is proven to exist, and moreover some nonlocal effects are exhibited, leading to a natural hidden variable theory which violates Bell's inequalities.