What is the direction of angular momentum vector of a photon?

In summary, the spin angular momentum of a photon is either parallel or antiparallel to its momentum due to its zero mass. The little group, which is the subgroup that leaves a particle's 4-momentum invariant, is SO(3) for massive particles and ISO(2) for massless particles. The 4-momentum is important because it is tangent to the particle's worldline. The little group is relevant because it helps define the particle's angular momentum.
  • #1
jartsa
1,576
137
Let's say momentum vector of a photon points to direction D. What are the possible directions that this photon's angular momentum vector can point to?

The spin angular momentum is the angular momentum that I'm interested of.
 
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  • #2
A photon's intrinsic angular momentum ("spin") is either parallel or antiparallel to its momentum. This is a consequence of its mass being zero.
 
  • #3
jtbell said:
A photon's intrinsic angular momentum ("spin") is either parallel or antiparallel to its momentum. This is a consequence of its mass being zero.
How do you reconcile this with the fact that angular momentum operators along perpendicular directions do not commute, and thus satisfy an uncertainty relation?
 
  • #4
This is incorrect. The members of the Lorentz group that define a particle's angular momentum form the "little group", i.e. the subgroup that leaves its 4-momentum invariant. For a particle with mass the little group is SO(3), and as you say the three generators of this group do not commute. For a massless particle the little group is the Euclidean group ISO(2), i.e. rotations and translations in two dimensions. One generator of this group is the rotation with axis along the direction of motion. The eigenvalues of this operator are Sz, the particle's helicity, namely ±1 for a photon. The other two generators are null rotations in the two transverse directions. They commute with each other, and for photons their effect is simply a gauge transformation.
 
  • #5
Bill_K said:
The members of the Lorentz group that define a particle's angular momentum form the "little group", i.e. the subgroup that leaves its 4-momentum invariant.
Why exactly are the symmetry operations that leave the 4-momentum invariant the ones used to define angular momentum? I thought that the whole reason why angular momentum is connected with rotation is that in nonrelativistic classical mechanics angular momentum is the Noether charge of SO(3)-invariance, so in nonrelativistic quantum mechanics the angular momentum operators are identified with the generators of SO(3). Where does the "little group" come into play in all this? Why would it be of paramount importance to keep the 4-momentum invariant under the exponential of the angular momentum operators?
 
  • #6
Particles in 3-space are points. The symmetries that leave a point invariant are SO(3).

Particles in 4-dimensional spacetime are lines. Massive particles are represented by timelike lines. The symmetries that leave a timelike line invariant are SO(3). This is the "little group" of a massive particle.

Massless particles are represented by null lines. The symmetries that leave a null line invariant are ISO(2). This is the "little group" of a massless particle.

The reason we care about the 4-momentum is because this vector is tangent to the particle's worldline.

If it were possible to have a particle that appears for one instant and then disappears, so that it is pointlike in spacetime, then its angular momentum would be in a representation of the full SO(3,1).
 

1. What is angular momentum vector of a photon?

The angular momentum vector of a photon is a physical quantity that describes the rotational motion of a photon. It is a mathematical vector that has both magnitude and direction.

2. Does a photon have angular momentum?

Yes, a photon does have angular momentum. This is because it has both energy and momentum, which are related to each other through the concept of angular momentum.

3. In which direction does a photon's angular momentum point?

The direction of a photon's angular momentum points in the direction of its propagation. This means that it is perpendicular to the direction of the photon's movement.

4. How is the direction of a photon's angular momentum determined?

The direction of a photon's angular momentum is determined by the direction of its electric and magnetic fields. These fields are perpendicular to each other, and their cross product gives the direction of the angular momentum vector.

5. Can the direction of a photon's angular momentum change?

Yes, the direction of a photon's angular momentum can change. This can happen when the photon interacts with other particles or fields, such as through scattering or absorption. In these cases, the photon's angular momentum can be transferred to the interacting particle or field.

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