Algebraic Fermion Bosonization (original) (raw)
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Bosonization of Non-Relativistic Fermions and W-Infinity Algebra
Modern Physics Letters A, 1992
We discuss the bosonization of non-relativistic fermions in one-space dimension in terms of bilocal operators which are naturally related to the generators of W-infinity algebra. The resulting system is analogous to the problem of a spin in a magnetic field for the group W-infinity. The new dynamical variables turn out to be W-infinity group elements valued in the coset W-infinity/H where H is a Cartan subalgebra. A classical action with an H gauge invariance is presented. This action is three-dimensional. It turns out to be similar to the action that describes the color degrees of freedom of a Yang–Mills particle in a fixed external field. We also discuss the relation of this action with the one recently arrived at in the Euclidean continuation of the theory using different coordinates.
Algebraic Bosonization: The Study of the Heisenberg and Calogero–Sutherland Models
International Journal of Modern Physics A, 1997
We propose an approach to treat (1 + 1)-dimensional fermionic systems based on the idea of algebraic bosonization. This amounts to decompose the elementary low-lying excitations around the Fermi surface in terms of basic building blocks which carry a representation of the W 1+∞ × W 1+∞ algebra, which is the dynamical symmetry of the Fermi quantum incompressible fluid. This symmetry simply expresses the local particle-number current conservation at the Fermi surface. The general approach is illustrated in detail in two examples: the Heisenberg and Calogero-Sutherland models, which allow for a comparison with the exact Bethe Ansatz solution.
Chiral Charged Fermions, One-Dimensional Quantum Field Theory and Vertex Algebras
2000
We give an explicit L 2-representation of chiral charged fermions using the Hardy-Lebesgue octant decomposition. In the " pure" case such a representation was already used by M. Sato in holonomic field theory. We study both "pure" and " mixed" cases. In the compact case we rigorously define unsmeared chiral charged fermion operators inside the unit circle. Using chiral fermions we orient our findings towards a functional analytic study of vertex algebras as one dimensional quantum field theory.
Introduction to Algebraic and Constructive Quantum Field Theory
Physics Today, 1993
Wiener transform 41 1.8. The structure of r and wave-particle duality 1,9. Implications of wave-particle duality 1.10. Characterization of the free boson field 62 1.11. The complex wave representation 1.12. Analytic features of the complex wave representation 70 2, The Free Fermion Field 2, I. Clifford systems 2.2. Existence of the free fennion field 80 2.3. The real wave representation • 2.4. The complex wave representation 3. Properties of the Free Fields 3,1. lmroduction 3.2. The exponential laws 3.3. Irreducibility 3.4. Representation of the orthogonal group by measurepreserving transfonnations 100 3.5. Bosonic quantization of symplectic dynamics 3.6. Fermionic quantization of orthogonal dynamics 5. C•-Algebraic Quantization 5.1. Introduction 5.2. Weyl algebras over a linear symplectic space 5.3. Regular statcs of me gcneral boson field 5.4. Thc representalion-independem Clifford algebra 5.5. Lexicon: The distribution of occupation numbers 4. Absolute Continuity and Unitary Implementabllity 4.1. Introduction 4.2. Equivalence of distributions 4.3. Quasi-invariant distributions and Weyl systems 4.4. Ergodicity and irreducibility ofWeyl pairs 4.5. Infinite products of Hilbert spaces 4.6. Affine transforms of the isononnal distribution 4.7. Implememabilily of orthogonal transformations on the fermion field
Physical Review D, 1996
The technique of extended dualization developed in this paper is used to bosonize quantized fermion systems in arbitrary dimension D in the low energy regime. In its original (minimal) form, dualization is restricted to models wherein it is possible to define a dynamical quantized conserved charge. We generalize the usual dualization prescription to include systems with dynamical non-conserved quantum currents.
Fermion-boson interactions and quantum algebras
Physical Review C, 2002
Quantum Algebras (q-algebras) are used to describe interactions between fermions and bosons. Particularly, the concept of a suq(2) dynamical symmetry is invoked in order to reproduce the ground state properties of systems of fermions and bosons interacting via schematic forces. The structure of the proposed suq(2) Hamiltonians, and the meaning of the corresponding deformation parameters, are discussed. PACS numbers: 21.60.-n; 21.60.Fw; 02.20.Uw. I. INTRODUCTION Group-theoretical methods have contributed significantly to the study of the nuclear quantum-many-body problem. Classical examples are the Lipkin model [1], the Elliot SU(3) model [2], the various realizations of Arima and Iachello Interacting Boson Model (IBM) [3], the Schütte and Da Providencia model [4], and the bi-fermion algebraic model of Geyer et al. [5], among other important contributions. We refer the reader to the review article of Klein and Marshalek [6], for a comprehensive presentation of the problem.
ALGEBRAIC QUANTUM FIELD THEORY
Philosophy of Physics, 2007
Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by S. Doplicher, R. Haag, and J. E. Roberts (DHR); and we give an alternative proof of Doplicher and Roberts' reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to Roberts and the abstract duality theorem for symmetric tensor * -categories, a self-contained proof of which is given in the appendix. * Forthcoming in Handbook of the Philosophy of Physics, edited by Jeremy Butterfield and John Earman. HH wishes to thank: Michael Müger for teaching him about the Doplicher-Roberts Theorem; the editors for their helpful feedback and patience; and David Baker, Tracy Lupher, and David Malament for corrections. MM wishes to thank Julien Bichon for a critical reading of the appendix and useful comments.
Fermions in the lowest Landau level. Bosonization, W [infinity] algebra, droplets, chiral bosons
Physics Letters B, 1992
We present field theoretical descriptions of massless (2+1) dimensional nonrelativistic fermions in an external magnetic field, in terms of a fermionic and bosonic second quantized language. An infinite dimensional algebra, W ∞ , appears as the algebra of unitary transformations which preserve the lowest Landau level condition and the particle number. In the droplet approximation it reduces to the algebra of area-preserving diffeomorphisms, which is responsible for the existence of a universal chiral boson Lagrangian independent of the electrostatic potential. We argue that the bosonic droplet approximation is the strong magnetic field limit of the fermionic theory. The relation to the c = 1 string model is discussed.