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### 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
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.

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
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.

mathematics problem.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Analysis
College Physics

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