scholarly journals DUALITY AND THE EQUIVALENCE PRINCIPLE OF QUANTUM MECHANICS

2001 ◽  
Vol 16 (23) ◽  
pp. 3853-3865 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Equivalence Principle ◽  
Heisenberg Algebra ◽  
Explicit Construction ◽  
Quantum Mechanical ◽  
Quantum Theories ◽  
Quantum Numbers ◽  
Starting Point ◽  
Topological Limit ◽  
Classical Phase Space

Following a suggestion by Vafa, we present a quantum-mechanical model for S duality symmetries observed in the quantum theories of fields, strings and branes. Our formalism may be understood as the topological limit of Berezin's metric quantization of the upper half-plane H, in that the metric dependence of Berezin's method has been removed. Being metric-free, our prescription makes no use of global quantum numbers. Quantum numbers arise only locally, after the choice of a local vacuum to expand around. Our approach may be regarded as a manifestly nonperturbative formulation of quantum mechanics, in that we take no classical phase space and no Poisson brackets as a starting point. Position and momentum operators satisfying the Heisenberg algebra are defined and their spectra are analysed. We provide an explicit construction of the Hilbert space of states. The latter carries no representation of SL (2,R), due to the lifting of the metric dependence. Instead, the reparametrization invariance of H under SL (2,R) induces a natural SL (2,R) action on the quantum-mechanical operators that implements S duality. We also link our approach with the equivalence principle of quantum mechanics recently formulated by Faraggi–Matone.

scholarly journals QUANTUM-MECHANICAL DUALITIES ON THE TORUS

2004 ◽  
Vol 19 (23) ◽  
pp. 1733-1744 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Differentiable Structure ◽  
Recent Developments ◽  
The Creation ◽  
Classical Phase Space ◽  
Creation Operators

On classical phase spaces admitting just one complex-differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e. independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-differentiable structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.

Constructing deterministic models for quantum mechanical systems

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040007
Author(s):  
Gerard ’t Hooft
Keyword(s):  
Quantum Mechanics ◽  
Essential Element ◽  
Quantum Mechanical ◽  
Interpretation Of Quantum Mechanics ◽  
Final States ◽  
Deterministic Models ◽  
Quantum Theories ◽  
Initial States ◽  
Quantum Mechanical Systems ◽  
Entire Theory

A sharper formulation is presented for an interpretation of quantum mechanics advocated by the author. We claim that only those quantum theories should be considered for which an ontological basis can be constructed. In terms of this basis, the entire theory can be considered as being deterministic. An example is illustrated: massless, noninteracting fermions are ontological. Subsequently, as an essential element of the deterministic interpretation, we put forward conservation laws concerning the ontological nature of a variable, and the uncertainties concerning the realization of states. Quantum mechanics can then be treated as a device that combines statistics with mechanical, deterministic laws, such that uncertainties are passed on from initial states to final states.

scholarly journals On the Relativistic Quantum Mechanics of a Particle in Space with Minimal Length

2012 ◽  
Vol 57 (9) ◽  
pp. 942
Author(s):  
Ch.M. Scherbakov
Keyword(s):  
Quantum Mechanics ◽  
Commutation Relation ◽  
Relativistic Quantum ◽  
Heisenberg Algebra ◽  
Minimal Length ◽  
Group Symmetry ◽  
Relativistic Quantum Mechanics ◽  
Noncommutative Space ◽  
Quantum Mechanical

A noncommutative space and the deformed Heisenberg algebra [X,P] = iħ{1 – βP2}1/2 are investigated. The quantum mechanical structures underlying this commutation relation are studied. The rotational group symmetry is discussed in detail.

A Principal of Linear Covariance for Quantum Mechanics and Its Consequences Taking one Beyond Symmetry

1997 ◽  
Vol 52 (1-2) ◽  
pp. 46-48
Author(s):  
Oktay Sinanoğlu
Keyword(s):  
Quantum Mechanics ◽  
Superposition Principle ◽  
Point Group ◽  
Equivalence Classes ◽  
Quantum Mechanical ◽  
Quantum Numbers ◽  
Unitary Transformations ◽  
Linear Covariance ◽  
Group Symmetries

Abstract A Principle of Linear Covariance is stated which follows from the "superposition principle" of quantum mechanics. Accordingly, quantum mechanical equations should be written in linearly covariant form which makes them look the same under non-unitary as well as unitary transformations. The principle leads to a non-unitary classification of all molecules (and clusters and solids) into distinct equivalence classes giving hitherto unknown relations between isomeric molecules. One also gets kinetic and thermic selection rules for chemical reactions. All these are independent of, and far more general than any unitary or point group symmetries. The invariants found for each class of molecules or clusters allow qualitative electronic deductions and are more generally applicable than symmetry based quantum numbers.

scholarly journals GRAVITOMAGNETIC MOMENTS AND DYNAMICS OF DIRAC (SPIN ½) FERMIONS IN FLAT SPACE–TIME MAXWELLIAN GRAVITY

2004 ◽  
Vol 19 (25) ◽  
pp. 4207-4229 ◽  
Author(s):  
HARIHAR BEHERA ◽  
P. C. NAIK
Keyword(s):  
Quantum Mechanics ◽  
Equivalence Principle ◽  
Relativistic Quantum ◽  
Flat Space ◽  
Space Time ◽  
Relativistic Quantum Mechanics ◽  
Quantum Mechanical ◽  
Gravitational Effects ◽  
Spin Particle ◽  
Dirac Spin

The gravitational effects in the relativistic quantum mechanics are investigated in a relativistically derived version of Heaviside's speculative gravity (in flat space–time) named here as "Maxwellian gravity." The standard Dirac's approach to the intrinsic spin in the fields of Maxwellian gravity yields the gravitomagnetic moment of a Dirac (spin ½) particle exactly equal to its intrinsic spin. Violation of the Equivalence Principle (both at classical and quantum-mechanical level) in the relativistic domain has also been reported in this work.

Modelling Flexible Protein-Ligand Binding in p38α MAP Kinase using the QUBE Force Field

2019 ◽  
Author(s):  
Joshua Horton ◽  
Alice Allen ◽  
Daniel Cole
Keyword(s):  
Quantum Mechanics ◽  
Force Field ◽  
Map Kinase ◽  
Binding Free Energy ◽  
Quantum Mechanical ◽  
Enhanced Sampling ◽  
Relative Binding ◽  
Stringent Test ◽  
Flexible Protein ◽  
P38α Map Kinase

<div><div><div><p>The quantum mechanical bespoke (QUBE) force field is used to retrospectively calculate the relative binding free energy of a series of 17 flexible inhibitors of p38α MAP kinase. The size and flexibility of the chosen molecules represent a stringent test of the derivation of force field parameters from quantum mechanics, and enhanced sampling is required to reduce the dependence of the results on the starting structure. Competitive accuracy with a widely-used biological force field is achieved, indicating that quantum mechanics derived force fields are approaching the accuracy required to provide guidance in prospective drug discovery campaigns.</p></div></div></div>

Quantum Theory

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Principle ◽  
Quantum Spin ◽  
Quantum States ◽  
Wave Functions ◽  
Symbolic Representations ◽  
Quantum Numbers ◽  
The Subject ◽  
Heisenberg's Uncertainty Principle

The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being representations of quantum states as well as a source of wave functions. The quantum box and quantum spin kets are specified, as are the quantum numbers that identify them. Operators are introduced and defined in part as the symbolic representations of observable quantities such as position, momentum and quantum spin. Eigenvalues and eigenkets are defined and discussed, with the former identified as the possible outcomes of a measurement. Bras, the counterpart to kets, are introduced as the means of forming probability amplitudes from kets. Products of operators are examined, as is their role underpinning Heisenberg’s Uncertainty Principle. A variety of symbol manipulations are presented. How measurements are believed to collapse linear superpositions to one term of the sum is explored.

scholarly journals IMPLICATIONS OF A QUANTUM MECHANICAL TREATMENT OF THE UNIVERSE

1998 ◽  
Vol 13 (05) ◽  
pp. 347-351 ◽  
Author(s):  
MURAT ÖZER
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Early Universe ◽  
Mechanical Treatment ◽  
Correspondence Principle ◽  
Scale Factor ◽  
Quantum Mechanical ◽  
Quantum Mechanical Treatment ◽  
The Standard Model ◽  
The Universe

We attempt to treat the very early Universe according to quantum mechanics. Identifying the scale factor of the Universe with the width of the wave packet associated with it, we show that there cannot be an initial singularity and that the Universe expands. Invoking the correspondence principle, we obtain the scale factor of the Universe and demonstrate that the causality problem of the standard model is solved.

scholarly journals Exceptional nonrelativistic effective field theories with enhanced symmetries

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Tomáš Brauner
Keyword(s):  
Effective Field Theories ◽  
Explicit Construction ◽  
Effective Field ◽  
Field Theories ◽  
Relativistic Case ◽  
Starting Point ◽  
Higher Power ◽  
Mere Presence ◽  
Spatial Rotations

Abstract We initiate the classification of nonrelativistic effective field theories (EFTs) for Nambu-Goldstone (NG) bosons, possessing a set of redundant, coordinate-dependent symmetries. Similarly to the relativistic case, such EFTs are natural candidates for “exceptional” theories, whose scattering amplitudes feature an enhanced soft limit, that is, scale with a higher power of momentum at long wavelengths than expected based on the mere presence of Adler’s zero. The starting point of our framework is the assumption of invariance under spacetime translations and spatial rotations. The setup is nevertheless general enough to accommodate a variety of nontrivial kinematical algebras, including the Poincaré, Galilei (or Bargmann) and Carroll algebras. Our main result is an explicit construction of the nonrelativistic versions of two infinite classes of exceptional theories: the multi-Galileon and the multi-flavor Dirac-Born-Infeld (DBI) theories. In both cases, we uncover novel Wess-Zumino terms, not present in their relativistic counterparts, realizing nontrivially the shift symmetries acting on the NG fields. We demonstrate how the symmetries of the Galileon and DBI theories can be made compatible with a nonrelativistic, quadratic dispersion relation of (some of) the NG modes.

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