Polarized Tensors

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/