Photons, helicity and energy and angular momentum conservation
z
^
|
|
| <----- + ----->
charge moves
back and forth
According to the link, the system creates sums of two helicity states
J = |+1 ħ ⟩
+
| -1 ħ ⟩
where J is the angular momentum to the direction of the photon momentum.
z
^
|
|
| + -------- -
rotating dipole
We may also create photons by rotating a dipole from the middle. Let the axis of the rotation be to the positive or negative z direction. For a single photon the helicity is
J = +-1 ħ.
Is any photon that moves to the direction of the z axis an eigenstate of the helicity?
A photon moving at the right angle to the z axis probably is the sum |+1 ħ ⟩ + | -1 ħ ⟩.
But now we face a dilemma: if we have a rotating antenna which absorbs a photon, how can we simultaneously conserve energy and angular momentum?
z
^
| + ------ -
| absorbing
| + ------ -
| emitting
Suppose that we have an emitting rotating dipole whose axis points to the z direction, and an absorbing dipole with the axis to the z direction. The receiving dipole is placed at an arbitrary direction from the emitting one.
The dipoles rotate at approximately the same rate.
If a photon jumps between the dipoles, it carries the angular momentum +-ħ and the energy hf. How can we explain this transfer if the helicity of the photon is parallel/antiparallel to its momentum?
A possible solution: if we treat the electromagnetic field as a classical field, then the emitting dipole certainly can speed up the rotation of the absorbing dipole. Now, if a transition conserves energy and angular momentum, then it is allowed. In this description we did not refer to a photon at all, and neither to its helicity.
If the axis of the absorbing dipole is not parallel to the emitting dipole, then a simple transition is not possible. Several photons must be absorbed and emitted. Classically this corresponds to waves scattered from the absorbing dipole.
Can a scalar field exist at all: is it possible that the quantum always has the spin 0?
Question. Is there any field where J could be zero always? If we disturb the field with a rotating movement, is it possible that the angular momentum is carried away by J = 0 particles?
^
|
• particle
^
|
●------------● rotating "antenna"
|
v
• particle
|
v
<----- r----->
We use a motor to rotate the antenna. The axis of the rotation points out from the screen.
If J = 0, then the angular momentum must be carried by particles which depart (tangentially) from some radius r from the center of the rotating movement.
We could say that the particles carry away "orbital angular momentum" rather than spin angular momentum.
The model with two emitted particles is ugly.
Let us try another model.
z
^
| ● ------ ●
| absorbing
| ● ------ ●
| emitting
Let us assume that we have an emitting rotating system and an absorbing rotating system, like we had for the electromagnetic field.
The systems interact through a classical field. The rotation can be transmitted between the systems through this field.
Suppose that the emitting system loses a quantum of energy and angular momentum, and that quantum shows up in the absorbing system.
We say that a quantum of the classical field carried energy and angular momentum. The quantum had a "spin" that differed from zero.
In this model, the "spin" of the quantum can never be zero. It can always carry angular momentum from one rotating system to another rotating system.
Conjecture. The spin of all bosons differs from zero. The Higgs boson cannot have the spin zero.
Question. Fermions have half-integer spins. Consequently, the spin cannot be zero. Can we find an explanation to this?
Conclusions
What is the spin of a boson? The spin question is intertwined to the fundamental question: what is the quantum of a classical field? It is the "quantization" problem of fields.
In this blog we have claimed that we must work with classical fields and only "quantize" the start and end states of a process. We are not allowed to assume that intermediate phases in the process consist of well-defined quanta. This is in the spirit of quantum mechanics: do not assume "hidden variables" in intermediate states.
Thus, the question of what is the spin of a boson, is somewhat vague. A boson only exists as an intermediate state. However, we conjecture that all bosons can carry angular momentum. The "spin" of a boson cannot be 0.
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