Let us first assume that the scalar field in the Klein-Gordon equation is real.
A Klein-Gordon wave in a drum skin is longitudinal, in a sense
We can generate a circularly polarized electromagnetic wave by rotating an electric dipole. The helicity of the photon is then 1 or -1. Can we do something similar with a Klein-Gordon wave?
The Klein-Gordon field is much like a drum skin (though three-dimensional). We can press the drum skin with our finger. The pressing finger is a "charge" of the field.
^
/
● metal sphere
/
v movement
^
|
● finger
|
v movement
If we move our finger back and forth on the skin, we generate a wave into the skin. If we put a metal sphere on the drum skin at some distance from the finger, the sphere will roll back and forth. The sphere is an antenna which absorbs a part of the dipole wave.
We can easily create waves where the back and forth movement of the sphere is to the direction of the wave source. But making a wave where the metal sphere moves normally to the direction of the wave source seems to be hard, or impossible.
radiated
energy
● spot 1
destructive
interference
● spot 2
We can approximate the wave by imagining that a finger alternately presses the spots 1 and 2 in the diagram. The wave is formed by the interference of the simple circular waves originating from the spots.
Normal to the line spot 1 - spot 2, the waves have a 180 degree phase shift, and almost total destructive interference.
Most of the energy is radiated upward or downward.
The drum skin wave is longitudinal in the sense that the metal sphere moves back and forth to the direction of the wave source. This is a major difference from electromagnetic waves which are always transverse.
Waves in a solid
Sound waves in a solid can be either longitudinal pressure waves or transverse shear waves.
Now we see that scalar field waves correspond to pressure waves. Transverse waves may be similar to electromagnetic waves.
Suppose that we disturb a solid by moving a point inside it along a circulat route. We input energy and angular momentum in both the transverse waves and the longitudinal waves.
Suppose that have a small device which harvests energy from vibrations of the solid. We believe that the device can absorb the entire energy and angular momentum in a single quantum of the transverse wave. The classical wave corresponding to a single quantum "collapses" and is absorbed entirely.
The collapse is not a classical process but purely a quantum process.
Question. Can a small device absorb the entire pressure wave, too? That is, can it absorb the whole energy and angular momentum in the scalar wave?
If the answer to the question is yes, then the quantum of a scalar wave can carry spin angular momentum. Its spin is not 0.
We have not found a theoretical argument which would resolve the question. Maybe it is possible to find the answer empirically?
Conclusions
It is easy to create a circularly polarized electromagnetic wave. Such a wave will move a freely floating electric charge along a circular path. Transverse waves can be used in this way to carry "spin" angular momentum into a small absorbing device, the charge.
Waves in a scalar field are longitudinal. They will move a charge back and forth.
The above are results for a classical field.
However, the collapse of the entire field energy to a small absorbing device is a quantum process.
It is not clear to us what happens in the collapse in a scalar field. Does the entire angular momentum get absorbed into the device? Does it make the device to spin or does it input linear momentum to the device so that it gains "orbital angular momentum" with respect to the emitting system.
We should analyze both the momentum and the angular momentum in the field. If the system which disturbs the field is symmetric, then, classically, the field will only receive angular momentum, not momentum at all relative to the laboratory frame. If the field is entirely absorbed by a device, that device cannot receive orbital angular momentum because then it would also receive momentum!
However it is possible that the field does receive momentum from the emitting device if the field contains just one quantum. It is like throwing a grain of sand from a carousel. In that case, the absorbing device will catch the grain. But if the absorbing device is not on the path of the grain, how can it catch it?
In the next blog post we will analyze the momentum aspect in the process.
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