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This troubles me: We are talking about time and space being equivalent, but still only consider Spin in the $x$, $y$ or $z$-direction. What's Spin in time dimension? Is it distinction between particles and antiparticles?

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Boost transformation correspond to rotation of time into space; so, in principle, their generators can be taken as analog of spin in time direction. However generators of boosts satisfy very different algebra from the spin algebra. This is because space and time aren't fully "equivalent" as can be seen from the signature of the metric (-1,1,1,1). –  user10001 Nov 12 '13 at 14:33
Good question. Is is possible to devise a theoretical experiment demonstrating rotation into and out of time dimension? –  ja72 Nov 12 '13 at 16:12
@ja72 rotation into and out of time dimension would be just to increase an object's speed. –  jinawee Nov 12 '13 at 17:25
Great so it would be an observable effect then. –  ja72 Nov 12 '13 at 17:33
@ja72, the object becomes shorter. In a sense, it becomes longer in the time direction because its head and tail cross our $t=\mathrm{const}$ hypersurface for different values of proper time. (Put synchronized clocks on the front and rear of a moving train, and they will appear out of synch with a stationary observer) –  lionelbrits Nov 20 '13 at 17:48

3 Answers 3

It is not true that "we" consider spin only in spatial directions, except "we" means something like "undergraduate students", maybe. Instead relativistic physics is all controled by spacetime spinors, namely by representatins of the double cover of the Poincaré Lie group of spacetime translations, rotations and boosts.

Maybe the best way to get an intuitive feeling for what a timelike spinor "means", physically, is to see how two such spinors combine into a twistor and how that endodes momentum and chirality of massless particles.

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If you want to do ordinary non-relativistic quantum mechanics with electrons without having to put spin in "by hand", you would start with the Dirac equation $(i\gamma^\mu \partial_\mu - m) \psi = 0$ and use this to derive the Schrödinger equation as the $|\mathbf{p}|^2 \ll m^2$ limit. The field $\psi$ is a 4-component spinor, although the number 4 is misleading (it would also have 4 components in 5 spacetime dimensions).

In any event, if you go to the rest frame of the electron, i.e., $\mathbf{p}=0$, then there are 4 solutions to the Dirac equation: $\uparrow, \downarrow$ with energy $E=m$ (electrons with spin up, down respectively) and $\uparrow, \downarrow$ with energy $E=-m$ (positrons).

For $\mathbf{p} \ne 0$, the 4 solutions become coupled, and instead of spin, it is better to talk about helicity or chirality.

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The thing with spin is that it does not quite have a physical interpretation, it is rather a mathematical formulation for the intrinsic angular momentum of a particle. Spin corresponds to an orthogonal translation of a Lorentz Group (a lie group), which one can usually represent in terms of spinors. Spinors have very interesting, and complex, properties, which are the reason as to why their behavior is usually in time dimensions become so hard to understand.

Let me know if you want a further explanation on the subject. Hope it helped!

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