Tensors and Spinors DefinedDate: 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/ |
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