scholarly journals THE SCHWINGER REPRESENTATION OF A GROUP: CONCEPT AND APPLICATIONS

2006 ◽  
Vol 18 (08) ◽  
pp. 887-912 ◽  
Author(s):  
S. CHATURVEDI ◽  
G. MARMO ◽  
N. MUKUNDA ◽  
R. SIMON ◽  
A. ZAMPINI
Keyword(s):  
Quantum Mechanics ◽  
Lie Group ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Direct Sum ◽  
Pure States ◽  
Group Concept ◽  
Set Up ◽  
Multiplicity Free

The concept of the Schwinger Representation of a finite or compact simple Lie group is set up as a multiplicity-free direct sum of all the unitary irreducible representations of the group. This is abstracted from the properties of the Schwinger oscillator construction for SU(2), and its relevance in several quantum mechanical contexts is highlighted. The Schwinger representations for SU(2), SO(3) and SU(n) for all n are constructed via specific carrier spaces and group actions. In the SU(2) case, connections to the oscillator construction and to Majorana's theorem on pure states for any spin are worked out. The role of the Schwinger Representation in setting up the Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group is brought out.

The Duflo non-commutative Fourier transform for the Lorentz group

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040011
Author(s):  
Giacomo Rosati
Keyword(s):  
Fourier Transform ◽  
Phase Space ◽  
Quantum System ◽  
Lie Group ◽  
Lorentz Group ◽  
Commutative Algebra ◽  
Cotangent Bundle ◽  
Irreducible Representations ◽  
Algebra Representation

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

scholarly journals Predicting crystal structures and properties of matter under extreme conditions via quantum mechanics: the pressure is on

2015 ◽  
Vol 17 (5) ◽  
pp. 2917-2934 ◽  
Author(s):  
Eva Zurek ◽  
Wojciech Grochala
Keyword(s):  
Quantum Mechanics ◽  
Crystal Structures ◽  
Quantum Mechanical ◽  
Extreme Conditions ◽  
Quantum Mechanical Calculations ◽  
Structures And Properties

The role of quantum mechanical calculations in understanding and predicting the behavior of matter at extreme pressures is discussed in this feature contribution.

scholarly journals A filtration on the ring of Laurent polynomials and representations of the general linear Lie algebra

2021 ◽  
Vol 32 (1) ◽  
pp. 9-32
Author(s):  
C. Choi ◽  
◽  
S. Kim ◽  
H. Seo ◽  
◽  
...  
Keyword(s):  
Lie Algebra ◽  
Irreducible Representations ◽  
General Linear ◽  
Laurent Polynomials ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Weight Multiplicity ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Multiplicity Free

We first present a filtration on the ring Ln of Laurent polynomials such that the direct sum decomposition of its associated graded ring grLn agrees with the direct sum decomposition of grLn, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in the associated graded ring grLn, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).

scholarly journals A Classical Interpretation of the Scrooge Distribution

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 619 ◽  
Author(s):  
William Wootters
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Distribution ◽  
Density Matrix ◽  
Pure State ◽  
Quantum Mechanical ◽  
Classical Probability ◽  
Pure States ◽  
Probability Simplex ◽  
Shed Light

The Scrooge distribution is a probability distribution over the set of pure states of a quantum system. Specifically, it is the distribution that, upon measurement, gives up the least information about the identity of the pure state compared with all other distributions that have the same density matrix. The Scrooge distribution has normally been regarded as a purely quantum mechanical concept with no natural classical interpretation. In this paper, we offer a classical interpretation of the Scrooge distribution viewed as a probability distribution over the probability simplex. We begin by considering a real-amplitude version of the Scrooge distribution for which we find that there is a non-trivial but natural classical interpretation. The transition to the complex-amplitude case requires a step that is not particularly natural but that may shed light on the relation between quantum mechanics and classical probability theory.

Quantum Superposition and Entanglement

2020 ◽  
pp. 154-171
Author(s):  
M. Suhail Zubairy
Keyword(s):  
Quantum Mechanics ◽  
Quantum Teleportation ◽  
Entangled State ◽  
Quantum Mechanical ◽  
The Novel ◽  
Quantum Superposition ◽  
Multiple Objects ◽  
Mechanical Description ◽  
Quantum Mechanical Description ◽  
Set Up

In this chapter, the notion of quantum superposition of states is introduced through the example of a polarized photon. This brings out the novel feature that the state of the system depends on how the experiment is set up. The paradoxical consequences of quantum superposition, such as a cat can be simultaneously dead and alive, are also discussed. This is the essence of the famous Schrödinger’s cat paradox. This description motivates another important consequence of quantum mechanical description of the multiple objects, namely, their ability to exist in an entangled state. The properties of the two objects can remain entangled no matter how far away they are from each other and thus have the ability to influence each other. After discussing these aspects of quantum mechanics, the application of quantum entanglement to novel phenomena of quantum teleportation and quantum swapping are presented.

scholarly journals Causality and Modelling in the Sciences: Introduction

Disputatio ◽  
2017 ◽  
Vol 9 (47) ◽  
pp. 423-427
Author(s):  
María Jiménez-Buedo ◽  
Federica Russo
Keyword(s):  
Climate Change ◽  
Quantum Mechanics ◽  
System Biology ◽  
Special Issue ◽  
Interdisciplinary Dialogue ◽  
Set Up ◽  
Economics And Biology ◽  
Scientific Fields

Abstract The advantage of examining causality from the perspective of modelling is thus that it puts us naturally closer to the practice of the sciences. This means being able to set up an interdisciplinary dialogue that contrasts and compares modelling practices in different fields, say economics and biology, medicine and statistics, climate change and physics. It also means that it helps philosophers looking for questions that go beyond the narrow ‘what-is-causality’ or ‘what-are-relata’ and thus puts causality right at the centre of a complex crossroad: epistemology/methodology, metaphysics, politics/ethics. This special issue collects nine papers that touch upon various scientific fields, from system biology to medicine to quantum mechanics to economics, and different questions, from explanation and prediction to the role of both true and false assumptions in modelling.

scholarly journals RICCI FLOW, QUANTUM MECHANICS AND GRAVITY

2009 ◽  
Vol 06 (03) ◽  
pp. 505-512
Author(s):  
JOSÉ M. ISIDRO ◽  
J. L. G. SANTANDER ◽  
P. FERNÁNDEZ DE CÓRDOBA
Keyword(s):  
Quantum Mechanics ◽  
Lie Group ◽  
Ricci Flow ◽  
Mechanical Model ◽  
Quantum Dynamics ◽  
Deterministic Model ◽  
Quantum Mechanical ◽  
Deterministic System ◽  
Canonical Transformations ◽  
The Given

It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantization, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space ℂℙd-1 of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base ℂℙd-1. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.

A reconnaissance of tropical resources during Revolutionary years: the role of the Paris Museum d'Histoire Naturelle

1991 ◽  
Vol 18 (3) ◽  
pp. 333-362 ◽  
Author(s):  
MADELEINE LY-TIO-FANE
Keyword(s):  
South Africa ◽  
Natural History ◽  
Scientific Research ◽  
Extensive Literature ◽  
War Of Independence ◽  
Leading Role ◽  
Scientific Principles ◽  
Set Up ◽  
Joseph Ii

SUMMARY The recent extensive literature on exploration and the resulting scientific advances has failed to highlight the contribution of Austrian enterprise to the study of natural history. The leading role of Joseph II among the neutral powers which assumed the carrying trade of the belligerents during the American War of Independence, furthered the development of collections for the Schönbrunn Park and Gardens which had been set up on scientific principles by his parents. On the conclusion of peace, Joseph entrusted to Professor Maerter a world-encompassing mission in the course of which the Chief Gardener Franz Boos and his assistant Georg Scholl travelled to South Africa to collect plants and animals. Boos pursued the mission to Isle de France and Bourbon (Mauritius and Reunion), conveyed by the then unknown Nicolas Baudin. He worked at the Jardin du Roi, Pamplemousses, with Nicolas Cere, or at Palma with Joseph Francois Charpentier de Cossigny. The linkage of Austrian and French horticultural expertise created a situation fraught with opportunities which were to lead Baudin to the forefront of exploration and scientific research as the century closed in the upheaval of the Revolutionary Wars.

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>

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