Normal ordering of the su(1, 1) ladder operators for the quasi-number states of the Morse oscillator

2020 ◽  
Vol 384 (19) ◽  
pp. 126493
Author(s):  
Xuanhao Chang ◽  
Sergey V. Krasnoshchekov ◽  
Vladimir I. Pupyshev ◽  
Dmitry V. Millionshchikov
Keyword(s):  
Morse Oscillator ◽  
Ladder Operators ◽  
Normal Ordering

scholarly journals Nonrelativistic near-BPS corners of $$ \mathcal{N} $$ = 4 super-Yang-Mills with SU(1, 1) symmetry

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Stefano Baiguera ◽  
Troels Harmark ◽  
Nico Wintergerst
Keyword(s):  
Classical Limit ◽  
Particle Number ◽  
Matrix Theory ◽  
Companion Paper ◽  
Positive Definite ◽  
Normal Ordering ◽  
Yang Mills ◽  
Three Sphere ◽  
Dilatation Operator ◽  
Definite Expressions

Abstract We consider limits of $$ \mathcal{N} $$ N = 4 super Yang-Mills (SYM) theory that approach BPS bounds and for which an SU(1,1) structure is preserved. The resulting near-BPS theories become non-relativistic, with a U(1) symmetry emerging in the limit that implies the conservation of particle number. They are obtained by reducing $$ \mathcal{N} $$ N = 4 SYM on a three-sphere and subsequently integrating out fields that become non-dynamical as the bounds are approached. Upon quantization, and taking into account normal-ordering, they are consistent with taking the appropriate limits of the dilatation operator directly, thereby corresponding to Spin Matrix theories, found previously in the literature. In the particular case of the SU(1,1—1) near-BPS/Spin Matrix theory, we find a superfield formulation that applies to the full interacting theory. Moreover, for all the theories we find tantalizingly simple semi-local formulations as theories living on a circle. Finally, we find positive-definite expressions for the interactions in the classical limit for all the theories, which can be used to explore their strong coupling limits. This paper will have a companion paper in which we explore BPS bounds for which a SU(2,1) structure is preserved.

scholarly journals Hermite Functions and Fourier Series

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 853
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano del Olmo
Keyword(s):  
Fourier Transform ◽  
Hermite Functions ◽  
Quantum Oscillator ◽  
Linear Combinations ◽  
Ladder Operators ◽  
Orthonormal Bases ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
The One ◽  
Square Integrable Functions

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

scholarly journals Canonical brackets of a toy model for the Hodge theory without its canonical conjugate momenta

2015 ◽  
Vol 30 (20) ◽  
pp. 1550115 ◽  
Author(s):  
D. Shukla ◽  
T. Bhanja ◽  
R. P. Malik
Keyword(s):  
Present Method ◽  
Hodge Theory ◽  
Rigid Rotor ◽  
Normal Ordering ◽  
Creation And Annihilation Operators ◽  
Basic Concepts ◽  
Definition Of ◽  
Continuous Symmetries ◽  
Internal Symmetries ◽  
Theory Lead

We consider the toy model of a rigid rotor as an example of the Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism and show that the internal symmetries of this theory lead to the derivation of canonical brackets amongst the creation and annihilation operators of the dynamical variables where the definition of the canonical conjugate momenta is not required. We invoke only the spin-statistics theorem, normal ordering and basic concepts of continuous symmetries (and their generators) to derive the canonical brackets for the model of a one [Formula: see text]-dimensional (1D) rigid rotor without using the definition of the canonical conjugate momenta anywhere. Our present method of derivation of the basic brackets is conjectured to be true for a class of theories that provide a set of tractable physical examples for the Hodge theory.

The development of a new Morse-oscillator based rotation-vibration Hamiltonian for H3+

1985 ◽  
Vol 112 (1) ◽  
pp. 183-202 ◽  
Author(s):  
V. Špirko ◽  
Per Jensen ◽  
P.R. Bunker ◽  
A. Čejchan
Keyword(s):  
Morse Oscillator

W-ALGEBRA REALIZATIONS AND W-GRAVITY ANOMALIES

1991 ◽  
Vol 06 (32) ◽  
pp. 2995-3003 ◽  
Author(s):  
C. M. HULL ◽  
L. PALACIOS
Keyword(s):  
Path Integral ◽  
Current Algebra ◽  
Gravity Anomalies ◽  
Normal Ordering ◽  
Path Integral Quantization ◽  
Transformation Rules ◽  
Higher Derivative ◽  
Quantum Current ◽  
Boson Model ◽  
Background Charge

The coupling of scalars fields to chiral W3 gravity is reviewed. In general the quantum current algebra generated by the spin-two and three currents does not close when the "natural" regularization (corresponding to the normal ordering with respect to the modes of ∂ϕi) is used, and the non-closure reflects matter-dependent anomalies in the path integral quantization. We consider the most general modification of the current, involving higher derivative "background charge" terms, and find the conditions for them to form a closed algebra in the "natural" regularization. These conditions can be satisfied only for the two-boson model. In that case, it is possible to cancel all the matter-dependent anomalies by adding finite local counter terms to the action and modifying the transformation rules of the fields.

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