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International Journal of Quantum Chemistry

Spin‐free quantum electronic structure: Its second quantization and para‐Fermionics

Douglas J. Klein

Published Dec 15, 2009• 4 citations
Publisher

Abstract

Abstract First, the spin‐free formulation of electronic structure is reviewed within the standard first‐quantized framework, with special attention to its manner of correspondence to the standard antisymmetrized space‐spin (ASspin) formulation. Second, within this first‐quantized framework, note is made of some “discrepancies” in the correspondence, for example, regarding mismatches of symmetry‐related degeneracies. Third, a novel second‐quantized spin‐free formulation is introduced to overcome these “discrepancies,” and so achieve a one‐to‐one correspondence between ASspin wave functions and second‐quantized spin‐free wave functions. Yet further, this second‐quantized spin‐free description enables the identification of electrons as “para‐Fermions,” at least in the nonrelativistic limit of spin‐free interactions. From the point of view of elementary particle physics, a rather complete picture of the behavior of para‐Fermions emerges with electrons as a proxy, and so suggests that in fact para‐Fermions would be difficult to distinguish from ordinary Fermions, at least in this nonrelativistic limit or with but few such particles. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

DOI
10.1002/qua.22426

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arXiv:quant-ph/9502022•Physics Essays•Mar 1, 1998•6 citations
Abstract First, the spin‐free formulation of electronic structure is reviewed within the standard first‐quantized framework, with special attention to its manner of correspondence to the standard antisymmetrized space‐spin (ASspin) formulation. Second, within this first‐quantized framework, note is made of some “discrepancies” in the correspondence, for example, regarding mismatches of symmetry‐related degeneracies. Third, a novel second‐quantized spin‐free formulation is introduced to overcome these “discrepancies,” and so achieve a one‐to‐one correspondence between ASspin wave functions and second‐quantized spin‐free wave functions. Yet further, this second‐quantized spin‐free description enables the identification of electrons as “para‐Fermions,” at least in the nonrelativistic limit of spin‐free interactions. From the point of view of elementary particle physics, a rather complete picture of the behavior of para‐Fermions emerges with electrons as a proxy, and so suggests that in fact para‐Fermions would be difficult to distinguish from ordinary Fermions, at least in this nonrelativistic limit or with but few such particles. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focusses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti)ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body quasi-particle space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general. In fact, second quantization can be considered the first major cornerstone on which the theoretical framework of quantum field theory was built.
The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focuses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti) ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general.
Abstract The fermion unitary group formulation (UGF) of many‐body theory is based on the unitary group U(2n) where n is the number of freeon orbitals. This formulation, which conserves particle‐number but not spin, is isomorphic to the particle‐number‐conserving, second‐quantized formulation (SQF). In UGF we derive the familiar diagrammatic algorithm for matrix elements, M(Y) = (−1)H+L where H and L denote the numbers of hole lines and loops in the diagram D(Y) of M(Y). The unitary group derivation is considerably simpler than is the conventional, second‐quantized derivation that employs time‐dependence, Wick's theorem, normal‐order, and contractions. In neither fermion UGF nor SQF is spin conserved. We carry out in UGF the spin‐projection (symmetry adaptation to SU(2)) of the fermion vectors and obtain with a spin‐free Hamiltonian the same matrix elements as with the freeon UGF (part 24 of this series). The fermion unitary group formulation for a spin‐free Hamiltonian should be regarded as an alternate path to spin‐free quantum chemistry.
Abstract The representation of an N‐electron Schrödinger Hamiltonian on an orthonormal, spin‐free (freon) orbital product space is exactly modeled by a second‐degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the orbital product space with respect to the symmetric group yields Gel'fand states which provide base for irreducible represntations of the unitary group. These exist in closed form as, in consequence, does the representation of the model Hamiltonian in this same basis. We retain as physical only those states characterized by tableaux with no more than two columns for which the spin labelling is one‐half the difference in the lengths of the two columns. The unitary group formulation is equivalent to standard, number‐conserving, second‐quantized, many‐body theory.
Usually the only difference between relativistic quantization and standard one is that the Lagrangian of the system under consideration should be Lorentz invariant. The standard approaches are logically incomplete and produce solutions with unpleasant properties: negative-energy, superluminal propagation etc. We propose a two-projections scheme of (special) relativistic quantization. The first projection defines the quantization procedure (e.g. the Berezin-Toeplitz quantization). The second projection defines a casual structure of the relativistic system (e.g. the operator of multiplication by the characteristic function of the future cone). The two-projections quantization introduces in a natural way the existence of three types of relativistic particles (with 000, 12\frac{1}{2}21​, and 111 spins). Keywords: Quantization, relativity, spin, Dirac equation, Klein-Gordon equation, electron, Segal-Bargmann space, Berezin-Toeplitz quantization. AMSMSC Primary: 81P10, 83A05; Secondary: 81R30, 81S99, 81V45