The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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 ( 
to look for the keywords

   tensor polarized beam

You will find several more links dealing with the topic of "polarized 

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-

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,

          " /\ " 

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 

Example: Suppose k = 2. Then


                = F_(i_1)(i_2)

is a rank 2 tensor. According to the quantum gravity theorist I 
consulted, this tensor could "project" as


                -----> 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


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 

instead of
                         F_(i_1)(i_2). I used indices


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 
Associated Topics:
College Analysis
College Physics

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.