Search results for "quantization"

Multi-Resolution Analysis and Fractional Quantum Hall Effect: an Equivalence Result

2001

In this paper we prove that any multi-resolution analysis of $\Lc^2(\R)$ produces, for some values of the filling factor, a single-electron wave function of the lowest Landau level (LLL) which, together with its (magnetic) translated, gives rise to an orthonormal set in the LLL. We also give the inverse construction. Moreover, we extend this procedure to the higher Landau levels and we discuss the analogies and the differences between this procedure and the one previously proposed by J.-P. Antoine and the author.

Extracting three-body observables from finite-volume quantities

2015

Scattering and transition amplitudes with three-hadron final states play an important role in nuclear and particle physics. However, predicting such quantities using numerical Lattice QCD is very difficult, in part because of the effects of Euclidean time and finite volume. In this review we highlight recent formal developments that work towards overcoming these issues. We organize the presentation into three parts: large volume expansions, non-relativistic nonperturbative analyses, and nonperturbative studies based in relativistic field theory. In the first part we discuss results for ground state energies and matrix elements given by expanding in inverse box length, $1/L$. We describe com…

Relating the finite-volume spectrum and the two-and-three-particle S matrix for relativistic systems of identical scalar particles

2017

Working in relativistic quantum field theory, we derive the quantization condition satisfied by coupled two- and three-particle systems of identical scalar particles confined to a cubic spatial volume with periodicity $L$. This gives the relation between the finite-volume spectrum and the infinite-volume $\textbf 2 \to \textbf 2$, $\textbf 2 \to \textbf 3$ and $\textbf 3 \to \textbf 3$ scattering amplitudes for such theories. The result holds for relativistic systems composed of scalar particles with nonzero mass $m$, whose center of mass energy lies below the four-particle threshold, and for which the two-particle $K$ matrix has no singularities below the three-particle threshold. The quan…

Perturbative results for two and three particle threshold energies in finite volume

2015

We calculate the energy of the state closest to threshold for two and three identical, spinless particles confined to a cubic spatial volume with periodic boundary conditions and with zero total momentum in the finite-volume frame. The calculation is performed in relativistic quantum field theory with particles coupled via a $\lambda \phi^4$ interaction, and we work through order $\lambda^3$. The energy shifts begin at ${\cal O}(1/L^3)$, and we keep subleading terms proportional to $1/L^4$, $1/L^5$ and $1/L^6$. These terms allow a non-trivial check of the results obtained from quantization conditions that hold for arbitrary interactions, namely that of L\"uscher for two particles and our re…

Linear response theory: many-body formulation

2013

Quantization as a consequence of the group law

1982

A method of gemetric quantization which solely makes use of the structure of the symmetry group of the dynamical system is proposed; the classical limit is discussed along similar lines. The method is applied to two examples, the free particle and the harmonic oscillator.

The pinch technique at two loops

1999

It is shown that the fundamental properties of gauge-independence, gauge-invariance, unitarity, and analyticity of the $S$-matrix lead to the unambiguous generalization of the pinch technique algorithm to two loops.

Two- and Three-Pion Finite-Volume Spectra at Maximal Isospin from Lattice QCD.

2019

We present the three-pion spectrum with maximal isospin in a finite volume determined from lattice QCD, including excited states in addition to the ground states across various irreducible representations at zero and nonzero total momentum. The required correlation functions, from which the spectrum is extracted, are computed using a newly implemented algorithm which speeds up the computation by more than an order of magnitude. On a subset of the data we extract a nonzero value of the three-pion threshold scattering amplitude using the $1/L$ expansion of the three-particle quantization condition, which consistently describes all states at zero total momentum. The finite-volume spectrum is p…

The Quantum Scalar Field in Spherically Symmetric Loop Quantum Gravity

2013

We consider the quantization of a spherically symmetric gravitational system coupled to a massless scalar field within the loop quantum gravity framework. Our results rely on the uniform discretizations method developed during the last years. We minimize the associated discrete “master constraint” using a trial state whose gravitational part is peaked around the classical Schwarzschild solution.

Algebraic Quantization, Good Operators and Fractional Quantum Numbers

1995

The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the ``failure" of the Ehrenfest theorem is clarified in terms of the already defined notion of {\it good} (and {\it bad}) operators. The analysis of ``constrained" Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric Quantization. This study is illustrated with the examples of the free particle on the circumference and the charged particle in a homogeneous magnetic field on the torus, both examples featuring ``anomal…