Helicity states degrees of freedom

In summary, the number of helicity states for a massless graviton in d dimensions can be determined by the formula d(d-3)/2, but for a particle of spin s=2, it has 2s+1=5 states. The additional 4 helicity states come from the number of independent components of an (p+1)-antisymmetric tensor in (d-2) transverse dimensions. This can be seen in Tomas Ortin's book "Gravity and Strings" on page 295.
  • #1
Neitrino
137
0
Hello,

1)Why does the number of helicity states depend on space-time dimension ?

Masselss graviton in 4 dimensional space has 2 helicity states ( 2 degrees of freedom). In 5 dimensional space it has (still massless in 5 D) 5 helicity states (5 degrees of freedom) ...

In 6 dimesional space according to formula d(d-3)/2 it has 9 helicity states ... but beeing a particle of spin s=2 it has 2s+1=5 states so where from are the additional 4 helicity states...

thanks
 
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  • #2
Neitrino said:
Hello,

1)Why does the number of helicity states depend on space-time dimension ?

Masselss graviton in 4 dimensional space has 2 helicity states ( 2 degrees of freedom). In 5 dimensional space it has (still massless in 5 D) 5 helicity states (5 degrees of freedom) ...

In 6 dimesional space according to formula d(d-3)/2 it has 9 helicity states ... but beeing a particle of spin s=2 it has 2s+1=5 states so where from are the additional 4 helicity states...

thanks

If you recall that in 4d for the photon only 2 transverse modes which can be picturized to be the left and right circularly polarized. For d-dimensions there are d-2 transverse directions and so according to me the number of degrees of freedom i.e. the number of helicities ought to be 2 * (no. of polarizations mixing two transverse directions) = 2* (d-2)(d-3)/2= (d-2)(d-3). Can you refer me the text/paper from where you quoted the helicities = d*(d-3)/2 formula.
 
  • #3
Tomas Ortin gravity and strings page 295
 
  • #4
Neitrino said:
Tomas Ortin gravity and strings page 295

Ah I see that now. Thats just the number of independent components (polarizations) of an (p+1)-antisymmetric tensor in (d-2) transverse dimensions i.e. (d-2) C _(p+1) = (d-2)!/(p+1)!(d-p-3)! while the massive case you can have longitudinal modes so you count the number of components of an antisymmetric tensor with (p+1) indices in (d-1) spatial directions i.e. (d-1)C_(p+1).
 

1. What are helicity states degrees of freedom?

Helicity states degrees of freedom refer to the two possible spin orientations of a particle along its direction of motion, either parallel or antiparallel. In quantum mechanics, these two states are represented by the eigenstates of the helicity operator, which measures the projection of the particle's spin angular momentum onto its direction of motion.

2. How are helicity states degrees of freedom different from spin states?

Helicity states degrees of freedom are a subset of spin states. While spin states refer to the overall spin angular momentum of a particle, helicity states specifically refer to the spin orientation along the direction of motion. For particles with spin, the helicity states can be different from the overall spin states due to the relativistic effects of their motion.

3. What is the importance of helicity states degrees of freedom?

Helicity states degrees of freedom play a crucial role in understanding the behavior of elementary particles. They are conserved in many interactions, making them useful in describing and predicting the outcomes of particle collisions. Additionally, the helicity states of particles can affect their decay rates, leading to differences in the particles' lifetimes.

4. Can a particle have both helicity states at the same time?

No, a particle can only have one helicity state at a time. This is because the helicity operator is a projection operator, meaning it can only measure the projection of the spin onto the direction of motion. Therefore, a particle with a definite spin orientation along its direction of motion cannot simultaneously have the opposite orientation.

5. Are helicity states degrees of freedom always conserved?

No, helicity states degrees of freedom are not always conserved. While they are conserved in many interactions, there are instances where helicity can change, such as in weak interactions or in the presence of a strong magnetic field. However, the total angular momentum, which includes both spin and orbital angular momentum, is always conserved.

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