Frame Transformations for Fermions (original) (raw)

Ju l 2 01 9 Quantum versus classical angular momentum

Angular momentum in classical mechanics is given by a vector. The plane perpendicular to this vector, in accordance to central field theory, determines the space in which particle motion takes place. No such simple picture exists in quantum mechanics. States of a particle in a central field are proportional to spherical harmonics which do not define any plane of motion. In the first part of the paper we discuss the angular distribution of particle position and compare it to the classical probabilistic approach. In the second part, the matter of addition of angular momenta is discussed. In classical mechanics this means addition of vectors while in quantum mechanics ClebschâȂŞGordan coefficients have to be used. We have found classical approximations to quantum coefficients and the limit of their applicability. This analysis gives a basis for the so called "vector addition model" used in some elementary textbooks on atomic physics. It can help to understand better the addition of angular momenta in quantum mechanics.

Rigid particle and its spin revisited

Foundations of Physics 37 (2007) 40-79, 2007

The arguments by Pandres that the double valued spherical harmonics provide a basis for the irreducible spinor representation of the three dimensional rotation group are further developed and justified. The usual arguments against the inadmissibility of such functions, concerning hermiticity, orthogonality, behavior under rotations, etc., are all shown to be related to the unsuitable choice of functions representing the states with opposite projections of angular momentum. By a correct choice of functions and definition of inner product those difficulties do not occur. And yet the orbital angular momentum in the ordinary configuration space can have integer eigenvalues only, for the reason which have roots in the nature of quantum mechanics in such space. The situation is different in the velocity space of the rigid particle, whose action contains a term with the extrinsic curvature.

Chapter 3 Quantum Theory in Space-time I-frames

Why do we need space-time descriptions? Well, quantum states are basic elements of abstract Hilbert space and as such they do not have space-time localization. Abstract quantum mechanics provides a general framework that would cover all possibilities a system may show up; it is a foundational construct but one does not get hints to construct actual models for most operators. The only thing one can require in abstract space is existence that is ensured by endowing the framework with a definite mathematical structure. But the world of laboratory experiments imposes space-time frameworks.

A reconstruction of quantum theory for spinning particles

arXiv (Cornell University), 2022

As part of a probabilistic reconstruction of quantum theory (QT), we show that spin is not a purely quantum mechanical phenomenon, as has long been assumed. Rather, this phenomenon occurs before the transition to QT takes place, namely in the area of the quasi-classical (here better quasiquantum) theory. This borderland between classical physics and QT can be reached within the framework of our reconstruction by the replacement p → M (q, t), where p is the momentum variable of the particle and M (q, t) is the momentum field in configuration space. The occurrence of spin, and its special value 1/2 , is a consequence of the fact that M (q, t) must have exactly three independent components M k (q, t) for a single particle because of the three-dimensionality of space. In the Schrödinger equation for a "particle with spin zero", the momentum field is usually represented as a gradient of a single function S. This implies dependencies between the components M k (q, t) for which no explanation exists. In reality, M (q, t) needs to be represented by three functions, two of which are rotational degrees of freedom. The latter are responsible for the existence of spin. All massive structureless particles in nature must therefore be spin-one-half particles, simply because they have to be described by 4 real fields, one of which has the physical meaning of a probability density, while the other three are required to represent the momentum field in three-dimensional space. We derive the Pauli-Schrödinger equation, the correct value g = 2 of the gyromagnetic ratio, the classical limit of the Pauli-Schrödinger equation, and clarify some other open questions in the borderland between classical physics and QT.

Vector parametrization, partial angular momenta and unusual commutation relations in physics

Chemical Physics, 2003

When studying an N -particle system by means of N À 1 vectors i.e., by means of a vector parametrization, one unavoidably comes across several angular momenta: not only the total angular momentum of the system but also the various partial angular momenta corresponding to the motion of the various vectors. All these momenta can, in addition, be referred to a variety of reference frames. The use of vector parametrizations and partial angular momenta in physics greatly simplifies the classical as well as quantum expressions of the kinetic energy. The present paper is devoted to a detailed and rigorous study of the partial angular momenta and the various commutation relations they satisfy, in particular the unusual commutation relations whose origin is traced back to the very structure of the coordinate changes used to define the Body-Fixed (BF) frames. The direct quantization of the classical expressions of the kinetic energy obtained in the context of various vector parametrizations is also given in detail. It turns out to be an efficient extension of well-known quantization procedures to the case where supernumerary quasi-momenta are used. As an illustration, the case of a four-particle system is treated in detail for a particular choice of the BF frames. Finally, the analogies between the classical and quantum approaches are emphasized.

Spin–fermion mappings for even Hamiltonian operators

Journal of Physics A: Mathematical and General, 2005

We revisit the Jordan-Wigner transformation, showing that -rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators-it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan-Wigner's result. It provides direct mapping of local quantum spin problems into local fermionic problems (and viceversa), under the (rather physical) requirement that the latter are described by Hamiltonian's which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermions models and spin 1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground state energies. PACS numbers: 2003 PACS number(s): 75.10.Jm,05.30.-d,03.65.Fd,71.10.-w

Rotating frames and gauge invariance in two-dimensional many-body quantum systems

Journal of Physics A: Mathematical and General, 2003

We study the quantization of many-body systems in two dimensions in rotating coordinate frames using a gauge invariant formulation of the dynamics. We consider reference frames defined by linear and quadratic gauge conditions. In both cases we discuss their Gribov ambiguities and commutator algebra. We construct the momentum operators, inner-product and Hamiltonian in both types of gauges, for systems with and without translation invariance. The analogy with the quantization of QED in noncovariant gauges is emphasized. Our results are applied to quasi-rigid systems in the Eckart frame.

A dynamic systems approach to fermions and their relation to spins

EPJ Quantum Technology, 2014

The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. It just requires knowing drift and control Hamiltonians of an experimental setup. Then one can easily determine all the states that can be reached from any given initial state. Likewise all the quantum operations that can be simulated with a given setup can be identified. Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.

The Schrödinger eigenfunctions for the half-integral spins

Physica A-statistical Mechanics and Its Applications - PHYSICA A, 1999

In this paper we are interested in approaching the problem of the spin from a different point of view. We will show that the spin is neither basically relativistic nor quantum mechanical but reflects just a symmetry property related to the Lie algebra to which it is associated – a Lie algebra that may also be associated with the classical Poisson bracket. The classical approach will be compared with the usual quantum one to stress their formal similarities. With this “classical” representation of the spin by means of phase-space functions we proceed to the usual quantization procedure to derive a Schrödinger equation for the half-integral spin. We then solve this equation to obtain the half-integral spin eigenfunctions. The connection between this approach and that using the Heisenberg matrix calculus will also be worked out.

O ct 2 00 3 Rotational invariance and the spin-statistics theorem

2003

In this article, the rotational invariance of entangled quantum states is investigated as a possible cause of the Pauli exclusion principle. First, it is shown that a certain class of rotationally invariant states can only occur in pairs. This is referred to as the coupling principle. This in turn suggests a natural classification of quantum systems into those containing coupled states and those that do not. Surprisingly, it would seem that Fermi-Dirac statistics follows as a consequence of this coupling while the Bose-Einstein follows by breaking it. In section 5, the above approach is related to Pauli's original spin-statistics theorem and finally in the last two sections, a theoretical justification, based on Clebsch-Gordan coefficients and the experimental evidence respectively, is presented.