scholarly journals CHIRAL QUANTIZATION ON A GROUP MANIFOLD

1994 ◽  
Vol 09 (23) ◽  
pp. 4149-4168 ◽  
Author(s):  
ZBIGNIEW HASIEWICZ ◽  
PRZEMYSLAW SIEMION ◽  
WALTER TROOST
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Geometric Quantization ◽  
Quadratic Relation ◽  
Classical Analysis ◽  
Quantum Sphere ◽  
Exchange Algebra ◽  
Group Manifold ◽  
Left And Right ◽  
The One

The phase space of a particle on a group manifold can be split into left and right sectors, in close analogy with the chiral sectors in Wess–Zumino–Witten models. We perform a classical analysis of the sectors, and geometric quantization in the case of SU(2). The quadratic relation, classically identifying SU(2) as the sphere S3, is replaced quantum-mechanically by a similar condition on noncommutative operators ("quantum sphere"). The resulting quantum exchange algebra of the chiral group variables is quartic, not quadratic. The fusion of the sectors leads to a Hilbert space that is subtly different from the one obtained through a more direct (unsplit) quantization.

scholarly journals Covariant phase space and soft factorization in non-Abelian gauge theories

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Temple He ◽  
Prahar Mitra
Keyword(s):  
Phase Space ◽  
Ward Identity ◽  
Gauge Theories ◽  
Minkowski Spacetime ◽  
Soft Gluon ◽  
Careful Study ◽  
Abelian Gauge ◽  
Gauge Sector ◽  
Vacuum States ◽  
The One

Abstract We perform a careful study of the infrared sector of massless non-abelian gauge theories in four-dimensional Minkowski spacetime using the covariant phase space formalism, taking into account the boundary contributions arising from the gauge sector of the theory. Upon quantization, we show that the boundary contributions lead to an infinite degeneracy of the vacua. The Hilbert space of the vacuum sector is not only shown to be remarkably simple, but also universal. We derive a Ward identity that relates the n-point amplitude between two generic in- and out-vacuum states to the one computed in standard QFT. In addition, we demonstrate that the familiar single soft gluon theorem and multiple consecutive soft gluon theorem are consequences of the Ward identity.

scholarly journals Hamiltonian charges in the asymptotically de Sitter spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Maciej Kolanowski ◽  
Jerzy Lewandowski
Keyword(s):  
Phase Space ◽  
Initial Data ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Physical Meaning ◽  
De Sitter ◽  
Killing Vectors ◽  
Asymptotically Flat ◽  
Angular Momenta ◽  
The One

Abstract We generalize a notion of ‘conserved’ charges given by Wald and Zoupas to the asymptotically de Sitter spacetimes. Surprisingly, our construction is less ambiguous than the one encountered in the asymptotically flat context. An expansion around exact solutions possessing Killing vectors provides their physical meaning. In particular, we discuss a question of how to define energy and angular momenta of gravitational waves propagating on Kottler and Carter backgrounds. We show that obtained expressions have a correct limit as Λ → 0. We also comment on the relation between this approach and the one based on the canonical phase space of initial data at ℐ+.

A quantization of the electromagnetic field

1983 ◽  
Vol 61 (8) ◽  
pp. 1172-1183
Author(s):  
Anton Z. Capri ◽  
Gebhard Grübl ◽  
Randy Kobes
Keyword(s):  
Hilbert Space ◽  
Electromagnetic Field ◽  
Gauge Invariance ◽  
Free Field ◽  
External Source ◽  
Field Level ◽  
Manifest Covariance ◽  
The One ◽  
Physical Spaces

Quantization of the electromagnetic field in a class of covariant gauges is performed on a positive metric Hilbert space. Although losing manifest covariance, we find at the free field level the existence of two physical spaces where Poincaré transformations are implemented unitarily. This gives rise to two different physical interpretations of the theory. Unitarity of the S operator for an interaction with an external source then forces one to postulate that a restricted gauge invariance must hold. This singles out one interpretation, the one where two transverse photons are physical.

scholarly journals Pathways to relativistic curved momentum spaces: de Sitter case study

2016 ◽  
Vol 25 (02) ◽  
pp. 1650027 ◽  
Author(s):  
Giovanni Amelino-Camelia ◽  
Giulia Gubitosi ◽  
Giovanni Palmisano
Keyword(s):  
Momentum Space ◽  
Hopf Algebras ◽  
Affine Connection ◽  
Planck Scale ◽  
Test Case ◽  
De Sitter ◽  
Group Manifold ◽  
The One ◽  
Natural Test

Several arguments suggest that the Planck scale could be the characteristic scale of curvature of momentum space. As other recent studies, we assume that the metric of momentum space determines the condition of on-shellness while the momentum space affine connection governs the form of the law of composition of momenta. We show that the possible choices of laws of composition of momenta are more numerous than the possible choices of affine connection on a momentum space. This motivates us to propose a new prescription for associating an affine connection to momentum composition, which we compare to the one most used in the recent literature. We find that the two prescriptions lead to the same picture of the so-called [Formula: see text]-momentum space, with de Sitter (dS) metric and [Formula: see text]-Poincaré connection. We then show that in the case of “proper dS momentum space”, with the dS metric and its Levi–Civita connection, the two prescriptions are inequivalent. Our novel prescription leads to a picture of proper dS momentum space which is DSR-relativistic and is characterized by a commutative law of composition of momenta, a possibility for which no explicit curved momentum space picture had been previously found. This momentum space can serve as laboratory for the exploration of the properties of DSR-relativistic theories which are not connected to group-manifold momentum spaces and Hopf algebras, and is a natural test case for the study of momentum spaces with commutative, and yet deformed, laws of composition of momenta.

scholarly journals Market of Stocks During Crisis Looks Like a Flock of Birds

2020 ◽  
Author(s):  
Bahar Afsharizand ◽  
Pooya H. Chaghoei ◽  
A A. Kordbacheh ◽  
A Trufanov ◽  
G.Reza Jafari
Keyword(s):  
Phase Space ◽  
Collective Behavior ◽  
Unique Point ◽  
Property A ◽  
Vicsek Model ◽  
Cooperative Effects ◽  
Cooperative Movement ◽  
Vector Velocity ◽  
Work Cooperative ◽  
The One

According to its inner property, a crisis in the financial market can be considered as a collective behavior phenomenon. Through the prism of collective behavior, the crisis does not happen if the companies are independent of each other. In this work, cooperative movement processes in a stock market are investigated in a manner similar to that Vicsek first described collective behavior for self-propelled entities. To this end, a phase space is defined as the one in which the return of volume of transactions versus return of price is represented with each share in each day corresponding to a unique point in the space. The findings of the observation show that during times of crisis, the phase space is limited with the vector velocity of shares in the same direction. In contrast, on a regular day, the phase space is entirely accessible, with vector velocity aligned randomly. Moreover, in line with the Vicsek model, an order parameter is introduced, which evaluates the cooperative effects for the shares so that the higher the value of this parameter, the stronger the collective behavior of the shares.

One-Dimensional Harmonic Oscillator in Quantum Phase Space

2011 ◽  
Vol 110-116 ◽  
pp. 3750-3754
Author(s):  
Jun Lu ◽  
Xue Mei Wang ◽  
Ping Wu
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Quantum Phase ◽  
Space Representation ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Matrix Mechanics ◽  
The Matrix ◽  
Quantum Wave ◽  
The One

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

scholarly journals Does the HM mechanism work in string theory?

2013 ◽  
Vol 91 (1) ◽  
pp. 75-80
Author(s):  
Alireza Sepehri ◽  
Somayyeh Shoorvazi ◽  
Mohammad Ebrahim Zomorrodian
Keyword(s):  
Hilbert Space ◽  
Black Hole ◽  
String Theory ◽  
Correspondence Principle ◽  
Black Hole Entropy ◽  
Closed String ◽  
Centre Of Mass ◽  
Final State ◽  
Information Transformation ◽  
Left And Right

The correspondence principle offers a unique opportunity to test the Horowitz and Maldacena mechanism at the correspondence point “the centre of mass energies around (Ms/(gs)2)”. First by using the Horowitz and Maldacena proposal, the black hole final state for closed strings is studied and the entropy of these states is calculated. Then, to consider the closed string states, a copy of the original Hilbert space is constructed with a set of creation–annihilation operators that have the same commutation properties as the original ones. The total Hilbert space is the tensor product of the two spaces Hright ⊗ Hleft, where in this case Hleft/right denote the physical quantum state space of the closed string. It is shown that closed string states can be represented by a maximally entangled two-mode squeezed state of the left and right spaces of closed string. Also, the entropy for these string states is calculated. It is found that black hole entropy matches the closed string entropy at transition point. This means that our result is consistent with correspondence principle and thus HM mechanism in string theory works. Consequently the unitarity of the black hole in string theory can be reconciled. However Gottesman and Preskill point out that, in this scenario, departures from unitarity can arise due to interactions between the collapsing body and the infalling Hawking radiation inside the event horizon and information can be lost. By extending the Gottesman and Preskill method to string theory, the amount of information transformation from the matter to the state of outgoing Hagedorn radiation for closed strings is obtained. It is observed that information is lost for closed strings.

The uniform exponential stability of a class of linear differential–difference equations in a Hilbert space

1981 ◽  
Vol 89 (3-4) ◽  
pp. 201-215 ◽  
Author(s):  
Richard Datko
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Exponential Stability ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Exponential Stability ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

WIGNER THEORY OF THE NAMBU STRING: (II) THE CLOSED STRING

1992 ◽  
Vol 07 (17) ◽  
pp. 4107-4148 ◽  
Author(s):  
F. COLOMO ◽  
L. LUSANNA
Keyword(s):  
Minkowski Space ◽  
Canonical Basis ◽  
Open String ◽  
Closed String ◽  
Zero Modes ◽  
Left And Right ◽  
The One

A set of relative variables for the closed string with P2>0 is found, which has Wigner covariance properties. They allow one to obtain global Lorentz-invariant abelianizations of the constraints, like for the open string, and then global Lorentz-invariant canonically conjugated gauge variables are found. But now there are two extra zero modes in the constraints and in the gauge variables, related to the gauge arbitrariness of the origin σ0 of the circle σ∈(−n, π) embedded in Minkowski space, σ↦xμ(σ). By means of the multitemporal approach a noncanonical redundant set of Dirac observables for the left and right modes is defined; they transform as spin-1 Wigner vectors and satisfy constraints of the same kind as in σ models. The quantization is not made, because a canonical basis of observables is still lacking, but the program to be followed to find them is just the same as the one delineated for the case of the open string.

The direct and inverse problem in Newtonian scattering

1991 ◽  
Vol 118 (1-2) ◽  
pp. 119-131 ◽  
Author(s):  
M. A. Astaburuaga ◽  
Claudio Fernández ◽  
Víctor H. Cortés
Keyword(s):  
Inverse Problem ◽  
Phase Space ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Dimensional Case ◽  
Classical Particle ◽  
Scattering Operator ◽  
One Dimensional ◽  
The One

SynopsisIn this paper we study the direct and inverse scattering problem on the phase space for a classical particle moving under the influence of a conservative force. We provide a formula for the scattering operator in the one-dimensional case and we settle the properties of the potential that can be deduced from it. We also study the question of recovering the shape of the barriers which can be seen from −∞ and ∞. An example is given showing that these barriers are not uniquely determined by the scattering operator.

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