Electron and graviton degrees of freedom

[unih]

Hi dear all
Please explain to a stupid dummy a very simple thing.
Take an a photon in 1+3 dimensions. How DOF it has? We all know that 2. How we calculate it?
a) 1) We have a spin 1 particle that should have 2s+1=3 spin state. So DOF=3.
2) We have 4 A_μ guys. One is out because of gauge invariance. So we have 3 left.
Does EOM gives us smth?
3) Electromagnetic tensor has 4×4 components. Its anti-symmetric so 4*3/2=6 components left. Bianci identities F_(αβ;γ)=0 gives us C₄³=4!/3!1!= 4 conditions (is it true? Or it is 6eq? Its the same as a symmetry conditions?), so we left with 2 DOF.
4) Electric and magnetic fields have three components each so we have 6. (4 Maxwell equations? ->2 )
What about photon in 1+D dim? Where and how zero photon mass get into play?

b) For graviton in 1+3.
1) h_αβ has 16 components. The matrix is symmetric so 4*5/2=10 components left. Bianci gives us h_[αβ;γ]=0 - 4 equations. The gauge conditions (i.e. coordinate transformations with non zero Jacobian) gives me 4 tranformation (1 for each coordinate). Left with 2 - Fine
2. In the 1+4 the same calculation give
5*6/2=15 (symetric matrix)
15-5=10 (because of gauge)
Bianci gives us how?
How DOF in the 5dim case?
Help!

 

[dextercioby]

It's all in the Hamiltonian formulation. The nr. of DOF is derived there by merely counting constraints.

 

[tom.stoer]

a) 1) We have a spin 1 particle that should have 2s+1=3 spin state. So DOF=3.

For massless particles the Poincare group rep. is not a vector rep.; it's not about spin but about the helicity and we are left with two helicity states

2) We have 4 A_μ guys. One is out because of gauge invariance. So we have 3 left.
Does EOM gives us smth?

It's most easily seen in the A°=0 gauge. A° is a Lagrangian multiplier, not a d.o.f. b/c there is no canonical conjugate momentum due to the missing ∂°A° term. So you eliminate one d.o.f. by A°=0 and a second one by the Gauß law constraint; 4-1-1 = 2.

What about photon in 1+D dim? Where and how zero photon mass get into play?

In one spatial dimension there is no dyn. d.o.f. for the el.-mag field, only a single zero mode with ∂₁A₁ = 0 which is solved by A₁ = const. survives; this corresponds to 2 components of A_μ and 2-1-1 = 0 as in 4 dim.