Description of Spin and Statistics in Nonrelativistic Quantum Theories Based on Local Currents

Abstract

We continue the investigation of the preceding paper into the irreducible representations of local nonrelativistic current algebras. Here, we concentrate on the new features which arise in classifying the representations of the current algebra with regard to particle statistics, and in including internal variables such as spin.

The anyon fluid in the Bogoliubov approximation

“…We emphasize that this algebra is identical to that satisfied by p and j'. Thus the two sets of operators {p, ji) and {p, J ' ) lead to different representations of the same current algebra [17]. That these representations are unitarily inequivalent is easily seen from the spectrum of the angular momentum operator, which is different in the two cases:…”

Particle spin from representations of the diffeomorphism group

“…Next, we observe that within each SU(2)-in variant subspace of the original representation, we have a new representation of the algebra of Eq. ( 1), augmented by spin density operators [1,4], and a corresponding representation of the local current group associated with this algebra [3]. We rewrite Eq.…”

“…The corresponding infinite-dimensional Lie group is a semidirect product e^Λjf, where £f is Schwartz' space of test functions under addition, and Jf is a group of diffeomorphisms of 1R 3 under composition [1][2][3]. To describe non-relativistic particles with spin, however, it was thought necessary to introduce at the outset additional operators Σ(h) = jΣ(x) h(x)dx, where Σ(x) is the spin density [4], The Lie group generated by the Σ-operators is the local SU(2) current group [5][6][7][8][9].…”

The Hamiltonian and generating functional for a nonrelativistic local current algebra

“…In Quant um Mecha nics the free Hamil tonia n for Boson s and Fermi ons is forma lly the same; H = -ti 42 · How eve r, the domai ns are diffe rent; symm etric func tions for Boson s and antis ymmet ric funct ions for Fermi ons. As a resul t the Free Bose Hamil tonia n and the Free Fermi Hamil tonia n are diffe rent opera tors (13) with disti nct spect ra (5) Hope fully there will be a syste matic metho d for dete rmin ing A(x, p) for 47 e a given poten tial. Eq.…”