Generalized Fierz Identities and the Superselection Rule for Geometric Multispinors Presenter: William M. Pezzaglia Jr. (wpezzag@clifford.org)

Department of Physics, Santa Clara University, Santa Clara, CA 95053, USA

Presentation at: Second Max Born Symposium, Wroclaw, Poland, Sept. 24-27, 1992. Presentation at: San Francisco State University, November 30, 1992

Abstract The inverse problem, to reconstruct the general multivector wave function from the observable quadratic densities, is solved for 3D geometric algebra. It is found that operators which are applied to the right side of the wave function must be considered, and the standard Fierz identities do not necessarily hold except in restricted situations, corresponding to the spin-isospin superselection rule. The Greider idempotent and Hestenes quaterionic spinors are included as extreme cases of a single superselection parameter.

This talk was summarized in the submission to the proceedings. Preprint version available at: gr-qc/9211018

• Up to index of other talks at: http://www.clifford.org/~wpezzag/talks.html • This URL: http://www.clifford.org/wpezzag/talk/92wroclaw/index.html

Updated: 2001Dec13

Index to Transparencies

1. Introduction (Title Page)

II. Algebra of Standard Pauli Spinor Densities • A. Spinor Densities • B. Fierz Identity • C. Inverse Theorem

III. Algebra of Geometric Multispinors • A. Multispinor Densities

o 1. Multivector Density o 2. Isospin Density o 3. Bilateral Density

• B. General Fierz Identities o 1. Form of Identities o 2. Some Results o 3. Raw Identities

• C. Interpretation of R_jk o 1. Isometry Transformations o 2. Induced Change in R_jk o 3. Parameterize R_jk

IV. Inverse Theorem and Superselection Rule • A. Inverse Theorem

o 1. Reconstruction Equation o 2. Simplest Case o 3. Three other choices

• B. Special Cases o 1. Unitary/Quaternionic (Hestenes) o 2. Singular/Idempotent (Greider) o 3. Introduce Isospin

• C. SuperSelection Parameter o 1. Superposition Invalid o 2. Parameter Definition o 3. Generalized Parameterization

V. Extensions/Summary • A. Extensions to 4D Dirac • B. Standard Dirac Bispinor Algebra • C. Multivector Dirac Theory