# Particle pointing away or from a charge?

1. Jun 4, 2010

### hankaaron

Can spin be thought of as a particle "pointing away or from a charge? For example, an electron's mass is mostly pointing toward the positively charged nucleus and away from another electron or negative charge.

You can also think of the electron as being tilted off-axis, in the same way the earth is tilted forward/away from the sun.

Regards,
hankaaron.

2. Jun 4, 2010

### f95toli

Re: Spin

No.
It is a bit unfortunate that the word "spin" was chosen for this property. Although it is relevant for e.g. calculating the angular momentum etc it has nothing to do with anything "spinning" in the usual sense of the word, there is no simple geometrical intepretation of what it is; there is certainly nothing pointing in different directions.

I prefer to think of it as a fundamental property of the particle, just as e.g. charge. It just "is".

3. Jun 4, 2010

### tom.stoer

Re: Spin

Nevertheless spin is related to the rotational symmetry and can be derived from the structure of spacetime = from the Lorentz group ~ SO(3,1).

4. Jun 7, 2010

### zonde

Re: Spin

Does this relation to rotational symmetry means that spin can be described with the help of pseudovector?

If that's the case then it is a bit unclear why it can't be described as intrinsic magnetic dipole without reference to rotation as magnetic field too is described with pseudovector.

5. Jun 7, 2010

### tom.stoer

Re: Spin

Spin is just spin :-)

Some remarks concerning group and representation theory:
SO(3) is the rotational symmetry leaving a sphere invariant; this is the symmetry respected by 3-space. Constructing the SO(3) Lie group one finds that it can be generated by "infinitesimal rotations", namely an so(3) Lie algebra.

Analysing so(3) further one finds that the the SU(2) Lie group has the same algebra, that means that SU(2) rotations is a 2-dim. complex space (which is clearly different from a 3-dim. real space) are generated by the same (!) algebra, i.e. so(3) = su(2).

Something similar works in 4-dim. space-time. The symmetry group is SO(3,1), its algebra so(3,1) = su(2) + su(2). That's the reason why spin follows from the 4.-dim. space-time.

From SO(3,1) one knows scalars, vectors and higher tensors, (e.g. the el.-mag. field strength tensor which is a tensor field of rank 2). From su(2) one can derive additional spinor representations which are different from vectors and tensors.

One derives these representations from the algebras; the representations are labelled by the spin j; the correspondence is
j = 0: scalar (e.g. pions)
j = 1: vector (e.g. photon)
j = 2 : 2-tensor
...

j = 1/2 (e.g. electron)
j = 3/2
...

Spin is different from vectors and tensors. One strange property of a spin 1/2 object is that rotation it by 360° degrees is not the identity, but "minus the identity". In order to create the identity one must rotate a spin 1/2 object by 2*360° = 720° degrees!

6. Jun 17, 2010

### cbd1

Re: Spin

By my understanding, an elementary particle is a point particle. This would mean that there is nothing really to "spin", i.e. have angular momentum.

So, elementary particle "spin" is really just a number involved with the pauli exclusion principle and more of a categorizing number added to it than a physical description of a particle actually spinning.

Any more experienced/real physicists to comment on the validity of this?

7. Jun 18, 2010

### tom.stoer

Re: Spin

I agree that there is nothing which is really spinning. But I would not use the term "point particle"; it's a quantum state carrying a certain (algebraic) representation of a symmetry group

8. Jun 18, 2010

### haushofer

Re: Spin

Why not use the phrase "point particle"? Experiments up till now indicate that electrons, quarks etc. are just that, right?

To laymen I always explain "spin" as just an intrinsic property of a particle which labels it, just like mass. Which is group-theoretically sound.

9. Jun 18, 2010

### tom.stoer

Re: Spin

Point-like particles sounds like a classical concept whcih is misleading in quantum field theory.

10. Jun 18, 2010

### peteratcam

Re: Spin

This has bugged me for ever - just because two groups have the same algebra doesn't mean they are the same group. SO(3,1) is what is often called the Lorentz group - it does not have spinor representations! A different group, SL(2,C) has the same Lie algebra as the Lorentz group, and does have spinor representations.

I don't like the way that spin is apparently 'derived' from looking at the representation theory of the Lorentz group, cos it's not there. If someone could put some wise words into explaining the logical deductions around spin I would be massively grateful - does the universe favour double covers, is that the message?

11. Jun 18, 2010

### tom.stoer

Re: Spin

Correct.

I did not say that. I said that spin is derived from the symmetry of spacetime; and this is in a certain sense $$SU(2)\otimes SU(2)$$, not only $$SO(3,1)$$.

One can define a $$sl(2,C)$$ matrix

$$X = x_\mu T^\mu$$

based on the $$sl(2,C)$$ generators $$T^a$$ and the identity $$T^0 = id$$. The 4-dim. length is

$$x_\mu x^\mu = det X$$

The symmetry of this expression is captured by two matrices:

$$det\,X \to det\,X^\prime = det\,(AXB^\dagger)$$.

And as there are two matrices already at the space-time level there is $$SU(2) \otimes SU(2)$$ symmetry.

The reason why we are not familiar with half-integer representation is that macroscopic objects always belong to integer representations; so the underlying symmetry is there, but only at the microscopic level.

I have no idea why nature decided to be $$SU(2)\otimes SU(2)$$ symmetric instead of $$SO(3,1)$$ symmetric.