scholarly journals QUANTUM MECHANICS AS A SPONTANEOUSLY BROKEN GAUGE THEORY ON A U(1) GERBE

2008 ◽  
Vol 05 (02) ◽  
pp. 233-252 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Gauge Symmetry ◽  
Configuration Space ◽  
Fibre Bundle ◽  
Higgs Mechanism ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Classical Action ◽  
Yang Mills

Any quantum-mechanical system possesses a U(1) gerbe naturally defined on configuration space. Acting on Feynman's kernel exp(iS/ħ), this U(1) symmetry allows one to arbitrarily pick the origin for the classical action S, on a point-by-point basis on configuration space. This is equivalent to the statement that quantum mechanics is a U(1) gauge theory. Unlike Yang–Mills theories, however, the geometry of this gauge symmetry is not given by a fibre bundle, but rather by a gerbe. Since this gauge symmetry is spontaneously broken, an analogue of the Higgs mechanism must be present. We prove that a Heisenberg-like noncommutativity for the space coordinates is responsible for the breaking. This allows to interpret the noncommutativity of space coordinates as a Higgs mechanism on the quantum-mechanical U(1) gerbe.

scholarly journals Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

2018 ◽  
Vol 4 (1) ◽  
pp. 47-55
Author(s):  
Timothy Brian Huber
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Mechanical System ◽  
Schrodinger Equation ◽  
Supersymmetric Quantum Mechanics ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Intertwining Operator

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

scholarly journals A NOTE ON THE QUANTUM-MECHANICAL RICCI FLOW

2009 ◽  
Vol 24 (27) ◽  
pp. 4999-5006
Author(s):  
JOSÉ M. ISIDRO ◽  
J. L. G. SANTANDER ◽  
P. FERNÁNDEZ DE CÓRDOBA
Keyword(s):  
Quantum Mechanics ◽  
Configuration Space ◽  
Ricci Flow ◽  
Riemannian Metric ◽  
Quantum Mechanical ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Flow Equations ◽  
Emergent Quantum Mechanics

We obtain Schrödinger quantum mechanics from Perelman's functional and from the Ricci-flow equations of a conformally flat Riemannian metric on a closed two-dimensional configuration space. We explore links with the recently discussed emergent quantum mechanics.

scholarly journals BMN gauge theory as a quantum mechanical system

2003 ◽  
Vol 558 (3-4) ◽  
pp. 229-237 ◽  
Author(s):  
N. Beisert ◽  
C. Kristjansen ◽  
J. Plefka ◽  
M. Staudacher
Keyword(s):  
Gauge Theory ◽  
Mechanical System ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Probability in quantum mechanics

1978 ◽  
Vol 10 (4) ◽  
pp. 725-729 ◽  
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

Probability in quantum mechanics

1978 ◽  
Vol 10 (04) ◽  
pp. 725-729
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

scholarly journals GRAVITY AND YANG–MILLS THEORY

2010 ◽  
Vol 19 (14) ◽  
pp. 2379-2384 ◽  
Author(s):  
SUDARSHAN ANANTH
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Mechanical ◽  
Yang Mills ◽  
Mechanical Description ◽  
Quantum Mechanical Description ◽  
Theory Of Gravity

Three of the four forces of Nature are described by quantum Yang–Mills theories with remarkable precision. The fourth force, gravity, is described classically by the Einstein–Hilbert theory. There appears to be an inherent incompatibility between quantum mechanics and the Einstein–Hilbert theory which prevents us from developing a consistent quantum theory of gravity. The Einstein–Hilbert theory is therefore believed to differ greatly from Yang–Mills theory (which does have a sensible quantum mechanical description). It is therefore very surprising that these two theories actually share close perturbative ties. This essay focuses on these ties between Yang–Mills theory and the Einstein–Hilbert theory. We discuss the origin of these ties and their implications for a quantum theory of gravity.

scholarly journals MASS DEFORMATIONS OF SUPER YANG–MILLS THEORIES IN D = 2 + 1, AND SUPER-MEMBRANES: A NOTE

2009 ◽  
Vol 24 (03) ◽  
pp. 193-211 ◽  
Author(s):  
ABHISHEK AGARWAL
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theories ◽  
Mass Term ◽  
Quantum Mechanical ◽  
Explicit Formulas ◽  
Yang Mills ◽  
Mechanical Models ◽  
Mass Terms ◽  
Matter Fields ◽  
Matrix Quantum

Mass deformations of supersymmetric Yang–Mills theories in three spacetime dimensions are considered. The gluons of the theories are made massive by the inclusion of a nonlocal gauge and Poincaré invariant mass term due to Alexanian and Nair, while the matter fields are given standard Gaussian mass-terms. It is shown that the dimensional reduction of such mass-deformed gauge theories defined on R3 or R × T2 produces matrix quantum mechanics with massive spectra. In particular, all known massive matrix quantum mechanical models obtained by the deformations of dimensional reductions of minimal super Yang–Mills theories in diverse dimensions are shown also to arise from the dimensional reductions of appropriate massive Yang–Mills theories in three spacetime dimensions. Explicit formulas for the gauge theory actions are provided.

scholarly journals A glance beyond the quantum model

2009 ◽  
Vol 466 (2115) ◽  
pp. 881-890 ◽  
Author(s):  
Miguel Navascués ◽  
Harald Wunderlich
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Physical Theory ◽  
Quantum Mechanical ◽  
Quantum Model ◽  
Microscopic Structure ◽  
Quantum Mechanical System ◽  
Classical Physics ◽  
Quantum Limits ◽  
The Universe

One of the most important problems in physics is to reconcile quantum mechanics with general relativity, and some authors have suggested that this may be realized at the expense of having to drop the quantum formalism in favour of a more general theory. Here, we propose a mechanism to make general claims on the microscopic structure of the Universe by postulating that any post-quantum theory should recover classical physics in the macroscopic limit. We use this mechanism to bound the strength of correlations between distant observers in any physical theory. Although several quantum limits are recovered, such as the set of two-point quantum correlators, our results suggest that there exist plausible microscopic theories of Nature that predict correlations impossible to reproduce in any quantum mechanical system.

CONFORMAL PARASUPERSYMMETRY IN QUANTUM MECHANICS

1989 ◽  
Vol 04 (26) ◽  
pp. 2519-2529 ◽  
Author(s):  
STEPHANE DURAND ◽  
LUC VINET
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Symmetry Algebra ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
One Dimensional ◽  
Complete Determination ◽  
Symmetry Generators

Conformal parasupersymmetry of order 2 is exemplified using a one-dimensional quantum mechanical system. Symmetry generators are seen to realize trilinear structure relations. The relevant representations of this novel symmetry algebra are constructed and shown to allow for a complete determination of the energy spectrum and wave functions of the system.

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