Re: Massive Fermions in Four Spatial Dimensions

On May 15, 1:15 pm, Dr Tim <timrobin...@xxxxxxxxxxxxxxx> wrote:
We know that massive 1/2 spin particles in three spatial dimensions
have two spin eigenstates, known as "up" and "down".
How many eigenstates would here be if there were four spatial
dimensions?
How many in "n" spatial dimensions?

The number of spatial dimensions doesn't matter once you state the
particle's spin:

When you say the particle has spin 1/2, it means the state of the
particle is a vector(*) in the doublet representation SU(2) -- this is
the representation with dimension 2. If you had a spin 1 particle for
example, it would have a triplet state -- the representation with
dimension 3.

A representation, as you recall, is a vector space + some matrices,
and the dimension of the representation is the dimension of the vector
space. The dimension of the vector space determines how many
eigenstates there are, because the eigenstates form the basis of the
vector space. That is why spin 1/2 particles have "up" and "down" --
these form the basis of a vector space of dimension 2. (or rather they
form *a* basis).

So! You see this entire discussion has nothing to do with the number
of spatial dimensions. The only thing that matters is the spin, and
some group-theoretic considerations.

(*) actually a spinor

.

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