M. Calixto

Algebraic quantization, good operators and fractional quantum numbers

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 ofgood (andbad) operators. The analysis of “constrained” Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for quantum operators without classical analogue and 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 “anomalous” operators, non-equivalent quantization and the latter, fractional quantum numbers. These provide the rationale behind flux quantization in superconducting rings and Fractional Quantum Hall Effect, respectively.

Vacuum Radiation and Symmetry Breaking in Conformally Invariant Quantum Field Theory

Abstract:The underlying reasons for the difficulty of unitarily implementing the whole conformal group SO(4,2) in a massless Quantum Field Theory (QFT) on Minkowski space are investigated in this paper. Firstly, we demonstrate that the singular action of the subgroup of special conformal transformations (SCT), on the standard Minkowski space

Entropic uncertainty and the quantum phase transition in the Dicke model

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Extended MacMahon–Schwingerʼs Master Theorem and conformal wavelets in complex Minkowski space

, cannot be primarily associated with the vacuum radiation problems, the reason being more profound and related to the dynamical breakdown of part of the conformal symmetry (the SCT subgroup, to be more precise) when representations of null mass are selected inside the representations of the whole conformal group. Then we show how the vacuum of the massless QFT radiates under the action of SCT (usually interpreted as transitions to a uniformly accelerated frame) and we calculate exactly the spectrum of the outgoing particles, which proves to be a generalization of the Planckian one, this recovered as a given limit.

Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U(2,2) and Born's Reciprocity

We show that the description of the quantum phase transition in terms of the entropic uncertainty relation turns out to be more suitable than in terms of the standard variance-based uncertainty relation. The entropic uncertainty relation detects the quantum phase transition in the Dicke model and it provides a correct description of the quantum fluctuations or quantum uncertainty of the system.

The electromagnetic and Proca fields revisited: A unified quantization

Abstract We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D 4 = SO ( 4 , 2 ) / ( SO ( 4 ) × SO ( 2 ) ) of the conformal group SO ( 4 , 2 ) (locally isomorphic to SU ( 2 , 2 ) ) in 1 + 3 dimensions. The manifold D 4 can be mapped one-to-one onto the future tube domain C + 4 of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L h 2 ( D 4 , d ν λ ) and L h 2 ( C + 4 , d ν ˜ λ ) of square integrable holomorphic functions with scale dimension λ and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a λ-extension of Schwingerʼs Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L h 2 ( D 4 , d ν λ ) . SMT is related to MacMahonʼs Master Theorem (MMT) and an extension of both in terms of Louckʼs SU ( N ) solid harmonics is also provided for completeness. Convergence conditions are also studied.

Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

We revise the use of eight-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwingers Master theorem is commented in connection with closure relations. An adaptation of the Borns Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the eight-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.

Group approach to quantization of Yang-Mills theories: a cohomological origin of mass

We all wish to thank J. Guerrero for valuable discussions. M. C. and M. N. are grateful to the Spanish M.E.C. for a F.P.I. and postdoctoral F.P.U. grant, respectively, and M. N. to the Imperial College for its hospitality.

Group quantization on configuration space: Gauge symmetries and linear fields

Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk

Phase space analysis of first-, second- and third-order quantum phase transitions in the Lipkin-Meshkov-Glick model

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