Spin

[Lizwi]

I am failing to understand spin. 

I believe any object when rotated 360 degrees will return it faces to their original position. 

What does spin 1/2 and spin 3/2 mean?

[joigus]

When you rotate an object, the way it behaves is not necessarily the way ordinary things rotate.

The particular model an object follows when rotating is called "representation of the rotation group".

A spin 1/2 particle, when rotating, carries what's called a spinor representation of O(3). There can be several intuitive ways to picture this, but in the end it's an abstract property. The one I like best is this Balinese traditional dance. Sorry I wasn't able to find a more dexterous dancer:

Spin 3/2 (Rarita-Schwinger) particles are believed not to be fundamental. In fact, there are arguments in QFT for particles of spin other than 0, 1/2, 1 or 2 not to exist.

A hypothetical spin 3/2 particle would go back to its original state if rotated an angle of 4pi/3.

[swansont]

QM is a bit weird 

Spin 1/2 means the particle has a component of angular momentum (typically taken as the z-component) of 1/2 h-bar 

[Markus Hanke]

7 hours ago, Lizwi said:

I believe any object when rotated 360 degrees will return it faces to their original position. 

The rotation isn’t in 3D Euclidean space, but in the 4D Minkowski spacetime of Special Relativity. The geometry of that spacetime is hyperbolic, so the situation is more complex than what can be easily visualised. Note also - and that is important - that spin is not a function of spacetime coordinates, so visualising it as some kind of rotation about itself is highly misleading. Rather, the rotation involved is one of the wavefunction about a hyperbolic angle in spacetime - in other words, a Lorentz transformation between inertial frames.

I think a better way to understand spin is to take it to signify what kind of mathematical object the quantum mechanical wavefunction of the entity in question needs to be, in order for it to be compatible with both the laws of quantum mechanics and Special Relativity. Spin-0 means we are dealing with a scalar, spin-½ means it is a Dirac spinor (bispinor), spin-1 means it is a vector, and spin-2 means it is a rank-2 tensor. Of course all these object types are closely related, in that they are all representations of the Lorentz group - that group which captures the symmetries of spacetime. So spin is at its heart a relativistic phenomenon, and an expression of symmetry.