Date: 04/30/2003 at 04:59:34 From: Dr. Narinder Pushkarna Subject: Tensor What is a polarized tensor and how can we prove that it has rank two?
Date: 05/07/2003 at 21:26:29 From: Doctor Nitrogen Subject: Re: Tensor Hi, Dr. Pushkarna: A "tensor" is a mathematical quantity that transforms according to certain rules. One use for it is that it helps one make calculations independent of charts (coordinate systems). The "rank" of a tensor has to do with how many indices appear in the tensor symbol. As for the definition of a "polarized tensor," I could find no mathematical definition and so had to consult with a particle physicist I have the honor of knowing to some extent. According to this physicist, in particle physics, a "polarized vector" and a "polarized tensor" have to do with a radiating beam of particles, such as a beam of deuterium ions. In particle physics, certain particles have "mixed spin states," and in certain conditions these particles, when inside a beam of particles such as might be formed in a supercollider like the one at Fermilab or Brookhaven, remain invariant under either a 360-degree (polarized vector) or 180-degree (polarized tensor) rotation. I realize this is a very light glossing over of a very extensive physics subject, so you might use the searcher Google (www.google.com) to look for the keywords tensor polarized beam You will find several more links dealing with the topic of "polarized tensor." If you would also like to know more about mathematical tensors, I recommend the following books: (1) General Relativity and Cosmology, G.C. McVittie, University of Illinois Press, 1965, chapter 1, pages 10-38. (2) Gravitation and Inertia, Ignacio Ciufolini, J. A. Wheeler, Princeton University Press, 1995, ISBN #0-691-03323-4, Mathematical Appendix, pages 404-436. (3) A Brief on Tensor Analysis, Second Edition, James G. Simmonds, Springer-Verlag, 1991, ISBN #0-387-94088-X. (4) Tensor Spaces and Exterior Algebra, Takeo Yokonuma, Mathematical Monographs, American Mathematical Society, 1992, ISBN #0-8218-4564-0. Reference (3) above gives physical applications for tensors in Classical Mechanics. Reference (4) is quite mathematically rigorous. In addition, I have received a more mathematically clarifying definition from a quantum gravity theorist of a "polarized tensor." The explanation however involves some knowledge of differential p- forms. Loosely defined mathematically by the theoretical physicist I mentioned, a polarized tensor is a differential p-form expression involving a rank k tensor F_(i_1)......(i_k): F = [F_(i_1)......(i_k)]dx^i_1 /\ ......../\ dx^i_k, where " /\ " denotes the wedge product for the differential form, and for which, if the tensor is "projected" in a certain direction, then 'something varies' in another direction perpendicular to that component which is projected. Example: Suppose k = 2. Then F_(i_1)......(i_k) = F_(i_1)(i_2) is a rank 2 tensor. According to the quantum gravity theorist I consulted, this tensor could "project" as F_(i_1)(i_2) -----> V_(i_1) = ( F_(i_1)(i_2) )n_i_2. Please note that the arrow above does not particularly represent some kind of "homomorphic or other map"; I am just using it to indicate projection is taking place for the rank 2 tensor F_(i_1)(i_2). To make this look more clear, you can replace i_1 with i and i_2 with j, then sum i, j as, in turn, 1, 2, and use for the rank 2 tensor F_ij instead of F_(i_1)(i_2). I used indices (i_1),..........,(i_k), for a more generalized tensor of rank k. If it is the case that ( F_(i_1)(1) )n_1 =/= ( F_(i_1)(2) )n_2, or, alternatively, using the indices i, j instead, ( F_(i)(1) )n_1 =/= ( F_(i)(2) )n_2, then this rank 2 tensor would be polarized. I hope this helped answer the questions you had concerning your mathematics problem. - Doctor Nitrogen, The Math Forum http://mathforum.org/dr.math/
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