How do we define the spin or the helicity of a particle?
One way to define it is that a particle whose spin is n can carry
-n, -n + 1, ..., n - 1, n
times ħ of angular momentum L relative to some axis.
A circularly polarized photon carries either -ħ or ħ of angular momentum relative to its polarization axis. A linearly polarized photon carries zero angular momentum.
Is there any difference in how much angular momentum can a single photon or a single graviton carry?
Apparently, no.
A lopsided quadrupole
center
of mass
M R r m
●----------×-----------------------------•
|
v ω
The center of mass is marked with ×. The distance of M from it is R and the distance of m from it is r. The angular velocity is ω.
An example of such a system is the hydrogen atom where M is the proton and m is the electron. We may imagine that the electron has a pretty high principal quantum number n and orbits the proton relatively far from it.
The system clearly sends out gravitational waves. That can cause the system to decay to a lower energy state. In the case of the hydrogen atom, the system loses a certain quantum of energy E and a certain angular momentum L relative to the center of mass of the system.
Let us use the Bohr model for a hydrogen atom. The small mass m decays to a state where the principal quantum number n is one less. We assume that n > 0 is large. Let
f = ω / (2 π).
Then the energy loss of the system is
E = h f,
and the loss of the orbital angular momentum of m is
ħ = h / (2 π),
where h is Planck's constant.
The kinetic energy and the angular momentum of the large mass M is negligible in the system.
Let us add to M and m electric charges Q and q of the same sign, such that
Q / q = M / m.
Then the system sends out an electromagnetic quadrupole wave which is analogous to the gravitational wave.
A symmetric quadrupole
M M
●----------×----------●
|
v ω
Let us have a quadrupole where M and m are equal. Then the system is essentially two dipoles, both M, rotating around their center. A jump to a lower energy quantum state requires that both masses M lose one ħ of angular momentum L around the center.
The graviton in this case is assumed to carry
2 ħ
of angular momentum away.
Note that since the system is symmetric, the field returns to its original state already after a rotation through an angle π. One may then imagine that the frequency is 2 f, and the quantum of energy is
2 h f.
That is, the imagined quantum is two dipole quanta "glued" together.
We did not refer to any special property of gravity. The quadrupole could as well be electric where the two charges have the same sign.
Can we "split" a quadrupole quantum?
What does it mean that a quadrupole quantum has the energy 2 h f and the angular momentum 2 ħ?
Can we split it into two halves?
Let us imagine that inside a classical quadrupole wave there is a classical dipole antenna which is tuned to the frequency f. Obviously, the antenna cannot absorb much energy from the quadrupole wave.
However, if we tune the dipole to the frequency 2 f, then it obviously can absorb energy in units of 2 h f, and angular momentum in units of ħ. However, we cannot really say that it "splits" a quadrupole quantum. Rather, the dipole absorbs a full quantum and then emits a low-energy quantum with the excess ħ of angular momentum.
We may imagine an antenna with a "gearbox" such that it converts a frequency 2 f to a frequency f, and stores the harvested energy to an oscillator whose frequency is f. Such a system might be able to "split" quadrupole quanta.
We can generate pure dipole gravitational waves
The best known gravitational waves come from mergers of black holes. They are quadrupole waves.
However, it is easy to generate dipole waves, too. Simply construct a dipole wave in empty space, and let masses M absorb the energy in that wave. Run time backwards. Now you have masses M sending a pure dipole wave.
What literature says about the spin 2
Lubos Motl (2013) writes that a photon's angular momentum in the z direction, jz, cannot be zero because of "gauge symmetries". This is a strange claim. If we have an electric charge oscillating in the x direction at a frequency f, it should be able to decay to a lower state and give up an energy quantum h f. It will not give up any angular momentum relative to the z axis because it does not have any. That is, we should have a linearly polarized photon where the polarization is to the x direction.
Motl writes that because of diffeomorphisms, a graviton only can have
jz = +- 2 ħ.
Again, this is strange because we can, e.g., have two masses M attached with a rubber band, and oscillating linearly to the x direction. They should be able to radiate gravitons which have jz zero.
Conclusions
In this blog we have speculated that the photon is not really the property of the electromagnetic wave, but describes the decay of an antenna to a lower energy state. That is, the quantum is determined by the antenna which sends it.
We did not find any fundamental difference between electromagnetism and gravity. It makes sense to claim that both the photon and the graviton have the same spin, or helicity, 1.
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