Understanding Spinors: Not Your Typical Tensor

[jcsd]

What mathematically speaking is a spinor? Why isn't it a tensor? I didn't find the mathworld definition very useful at all as it describes it as a complex column vector which really tells me nothing especially as we usually think of such an object as a tensor!

 

[zefram_c]

A spinor and a tensor are differentiated mathematically by the way they transform under a Lorentz transformation, and by what they mean physically.

Spinor - See http://particle.phys.uvic.ca/~blokland/phys506a/lec10.pdf slide 4 for how a spinor transforms across reference frames.

Tensor - See http://farside.ph.utexas.edu/teaching/jk1/lectures/node10.html for how a tensor transforms.

Note that the transformation for a spinor cannot be put in tensor form, so they are different objects. Physically, spinors arise as solutions to the Dirac equation and describe fermions. I *think* tensors describe fields with integer spin (ie the EM field), but I don't know that for certain.

Edit: shouldn't this go in either Linear Algebra or Quantum physics?

 

[jcsd]

So basically they ARE vectors BUT they exist in a different vector space (a two dimensional complex space as opposed to a four dimensional real space which makes sense in some perverse way)?

So if that is true the obvious quetsion is what is the space that spinors 'live' in meant to represent?

 

[zefram_c]

Pretty much. Also spinors have the so-called associated 'dual spinors' and generally a calculation will involve both a spinor and its dual (eg. the coupling of the electron spinor to the EM vector potential contains the spinor, the EM potential and the dual spinor).

Two other potentially useful links:

http://www.compsoc.net/~fedja/twistors/node3.html
http://encyclopedia.thefreedictionary.com/Spinor

 

[jcsd]

aha that wiki definition is much more enlightening then the mathworld definitnion.