Tensors and Spinors Defined

Date: 09/30/1999 at 03:42:50
From: David Eisenstein
Subject: Tensors and Spinors

What are tensors and spinors? Can you explain with a small number of 
visual examples?

Date: 09/30/1999 at 06:52:27
From: Doctor Mitteldorf
Subject: Re: Tensors and Spinors

Dear David,

That's a tall order. Tensors and spinors have little in common, aside 
from the fact that they're both rectangular arrays of numbers.

If you can visualize a vector, it's a little further stretch to 
visualize a tensor. The easiest example to imagine is the 
compressibility tensor for a crystal. Think of the pressure on a 
crystal as a DIRECTIONAL force: it may be squeezed harder in the 
east-west direction than the north-south direction; and think of the 
crystal's response as directional as well: it may be easy to compress 
in the north-south direction, and more resistant in the east-west 
direction; furthermore, while you squeeze in the north-south 
direction, the crystal may expand outward in the east-west direction 
because it's "oozing out the sides."

All this means that the compressibility of a tensor can be 
directional; to fully describe the compressibility, you need to list 
every direction in which the compressing force can be applied, and 
every direction in which the crystal can shrink (or grow!) in 
response. Hence you have 9 numbers that behave very similarly to the 
dyadic product of two vectors. The dyadic product is all 9 
combinations of products of the 3 components of one vector with the 3 
components of another. A dyadic product transforms as a tensor.

That's a credible "visual" description of a tensor; but spinors are a 
lot harder. You have to begin with some appreciation of quantum 
mechanics, in all its strangeness. Spinors only have meaning, as far 
as I know, in the context of quantum mechanics. In classical 
(pre-quantum) mechanics, an electron has a position and a velocity in 
every moment. In quantum mechanics, the electron has an "amplitude" 
for appearing at every point in infinite space. The amplitude can be 
converted into a probability that the electron will be found there, 
but it also carries additional significance that describes how one 
electron interacts with another. You have this very strange phenomenon 
that if you have one electron with an amplitude for being at point 
(x,y,z) and another with an amplitude for being at the same point, 
it's possible that the two amplitudes add up, and then there's not 
twice as much probability but 4 times as much probability that one of 
the two electrons will be found there; or, if the amplitudes work 
against each other, it's possible that they cancel out and there's 
zero probability of finding an electron at that spot. These amplitudes 
behave like waves in that regard: if the crest of one wave falls on 
the trough of another wave, the two can cancel out locally; 
conversely, two crests appearing together have 4 times the energy of 
either one separately...

But I digress. What's a spinor? When Wolfgang Pauli (about 1930) was 
trying to figure out how to account for the spin of an electron, how 
to include it in the quantum mechanics for the position and motion of 
the electron, he found that it worked to express an amplitude that the 
spin of the electron pointed north, and another amplitude that the 
spin pointed south. The two amplitudes listed together form a spinor. 
 
This may seem ho-hum, until you think about a few reasons it's 
strange. First: what happened to east and west? Why isn't there an 
amplitude for the spin to point east? Or up or down? In classical 
mechanics, spin is a vector: it has an x, a y and a z component. Why 
in quantum mechanics would it have only 2 components? And why would 
they be for +x and -x rather than for x and y and z? This is the 
strange world of QM. Once you know the x and -x components of an 
electron's spin, you know all that can be said about it. You can't 
predict what the measurement of its y and z spin will be, and if you 
could, you would lose all knowledge of what its x spin was.

Most elementary particles are like this: they just have two-component 
spinors. But a few have 3-component spinors, sometimes notated as 
"spin-up, spin-zero, and spin-down". The 2-component spinors are just 
"spin-up and spin-down."  

This is hardly visual and hardly intuitive, but quantum mechanics is a 
strange world, even for physicists who become adept at following its 
rules.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/