Spin of the photon derived from classical physics plus the Planck-Einstein relation
Let us study the classical limit of polarized photons. The limit of many coherent photons has to make sense classically.
For a photon, the spin is equivalent to the circular polarization of the electromagnetic wave.
Suppose that we have a rotating electric dipole. It will emit electromagnetic radiation which carries away angular momentum. Actually, all the energy that the dipole radiates is originally mechanical rotation energy. We may have a motor which makes the dipole to rotate, and the mechanical energy is emitted as electromagnetic waves.
electromagnetic
waves
\ | /
\ | /
rotating electric ______
dipole
e- --------- e+
|
| \
__|__ \
| |
motor
The electromagnetic waves thus contain angular momentum. When a photon is absorbed by an antenna, the angular momentum moves to the antenna and tries to make the antenna to rotate to the same direction as the dipole which sent the wave.
Above we have a classical interpretation for the photon spin. Linearly polarized waves do not carry angular momentum, but quantum mechanics states that each photon still has spin 1 or -1.
If the length of the rotating dipole is 2r, and the motor has to exert a force F to one end of the dipole, then the power of the motor is
P = E / t = F ω r,
where ω is the angular velocity. The motor transfers angular momentum to the field at the rate
L / t = F r.
The ratio E / L is ω.
According to the Planck-Einstein relation, a photon has the energy
E = h f.
We know empirically that its angular momentum is
L = h / (2π).
The ratio E / L is 2π f, which is equal to ω. We were able to derive the spin = 1 of the photon from classical physics plus the Planck-Einstein relation. Our result generally implies that if we can pump energy to a field with a rotating object, then the quantum of the field has the spin 1 or -1.
https://en.wikipedia.org/wiki/Vortex
A question is, if a circularly polarized wave in some sense is a vortex, too.
Why is the photon spin +1 or -1, why not 0? That is, why is a single absorbed energy quantum of the field always circularly polarized and not linearly polarized?
In rowing, an oar which is moved linearly in water will produce two vortices spinning to opposite directions. Could the circular polarization of photons have an analogous origin? The vortices made by an oar can be interpreted as a "particle" and an "antiparticle". For a photon of spin +1, the antiparticle is a photon of spin -1.
A linear dipole radio transmitter
<---> <--->
e- ------------- e+
Suppose that we have a horizontal electric dipole which we alternately make longer and shorter. It will emit a linearly polarized electromagnetic wave. The dipole is a radio transmitter.
One can certainly extract angular momentum from a linearly polarized wave, too. Just put another dipole vertically in the field. The dipole is an antenna:
e-
|
|
e+
The alternating horizontal electric field of our horizontal radio transmitter will alternately turn the vertical dipole clockwise and counter-clockwise.
If we assume that the the vertical dipole can absorb a quantum of energy, E = h f, from the field, then that quantum also carried angular momentum.
What about a quantum which carries no angular momentum? If our radio antenna is horizontal, then it might absorb a quantum E = h f which had no angular momentum. Do we have any argument against the existence of such spin = 0 quanta?
The radio transmitter cannot produce all its energy output in spin = 0 quanta, because then the vertical antenna would not turn. But can it emit at least some of its output in spin = 0 quanta? Classically, the vertical antenna can receive energy and angular momentum from any linearly polarized radio wave, even if the wave is extremely weak. Suppose that our radio transmitter produces some spin = 0 quanta. If we can filter all spin = 1 and spin = -1 quanta out, what is the remaining wave? It cannot be a classical wave because a classical wave can always be absorbed by our vertical antenna. This is a heuristic argument against the existence of spin = 0 photons. They would contradict the classical theory of electromagnetism.
Longitudinal electromagnetic waves
If we put a test charge in line with our horizontal antenna:
test charge e- e- ----------- e+
then some of the energy that we feed in our radio transmitter will go to moving the test charge. The static electric fields of the charges in the antenna will move it. We could say that the test charge is receiving "longitudinal" electromagnetic waves. The intensity of such waves falls down very rapidly when the distance from the transmitter grows. That is why they normally are not called waves at all, but just an electric force. Transverse electromagnetic waves can carry energy over vast distances.
What is the spin of a hypothetical quantum of longitudinal waves? We can extract angular momentum from longitudinal waves, too. If there exists a quantum of longitudinal waves. it cannot have the spin = 0.
What is the difference between a linearly polarized wave and a vortex?
We have an instinctive grasp of what is a vortex in water and what is a wave. What exactly is the difference? If we have a short block of floating wood, we can make it rotate with a vortex, but with a wave it will just turn a little one way and then move back.
We might imagine the rotating dipole transmitter forming a huge vortex which fills the whole space and makes all dipole antennas rotate the same way. Or we may imagine that the transmitter send "mini-vortices", photons, which are absorbed by the antennas.
The linear dipole transmitter is classically quite close to two rotating transmitters which rotate to opposite directions. We might say that its linear wave is formed by two superimposed vortices.
Do the quanta of all force fields carry angular momentum?
An oar makes vortexes to water. The photon has a non-zero spin. Is there any way to transfer energy in any force field with quanta which have spin = 0?
What about classical physics? Is there a way to transfer energy over a distance in a way where no device can extract angular momentum from the energy transfer? In water, waves as well as vortexes and turbulence involve changes in angular momentum, at least locally.
If we have a perfect linear flow of water, then no floating object can extract angular momentum from it, but neither can it extract any energy. The same holds for a homogeneous gravitational field.
What about longitudinal sound waves? We can extract angular momentum by attaching a light plate to a heavy object. The plate will move with the sound waves while the heavy object stays still.
In classical physics, does transferring energy through a periodic motion allow extraction of angular momentum in all possible cases?
Why do the quanta of the electron field carry angular momentum?
In classical physics, particles do not need to carry angular momentum. We can throw grains of sand, and there is no need for the grains to rotate.
What exactly happens in pair production? We support the Feynman hypothesis that a pair production is like an electron scattering from a high-energy photon, where a space coordinate has been flipped with the time coordinate.
Is there a reason why a scattering electron must carry the spin 1/2?
In scattering, in a Feynman diagram, the electron first absorbs the photon and later emits it. Suppose that the spin of the photon is 1. If the electron has the spin -1/2 before the absorption, then its spin after the absorption is 1/2, until it emits the photon, and then its spin is again -1/2.
How does an electron actually scatter from a photon or a rotating dipole field? Is the magnetic interaction much stronger than the electric interaction?
We had the classical model where the electron flies at the speed of light in a circle of radius 2 * 10^-13 m. That explains its angular momentum. The magnetic moment is double of this classical model.
A 511 keV photon is born from an electric dipole where an electron and a positron circle their center of mass at the speed of light, and the length of the orbit is one Compton wavelength 2 * 10^-12 m. The radius is then 3 * 10^-13 m.
The magnetic fields of the dipole and the electron which comes close to the dipole are of the same order of magnitude. The magnetic potential reaches 511 keV at a distance of the order 10^-13 m, while the electric potential would require a distance of the order 10^-15 m.
The rotating tube model of the electron-positron pair explains spin 1/2
Classically, the energy density of the electric field is E^2, where E is the electric field strength. Can we interpret that the electric field is a material object which can move and rotate? Sure. We can measure the magnetic component at different places and determine if the electric field is moving or rotating.
The uncertainty principle in quantum mechanics states that if an object can rotate, then its angular momentum in the z direction is some n * h-bar, where n can also be 0. We cannot determine the angular momentum components to the x, y, z directions simultaneously, which means that the object always has rotational uncertainty. One cannot put an extended object in a non-rotating state.
Thus, the electric field of the electron has to rotate. But why is its spin 1/2 or -1/2?
What is the wavelength of a complex object in quantum mechanics? We can give the wavelength of an atom or even a molecule, though it contains also very light parts like electrons which individually would have a wavelength much longer than the atom as a whole? Quantum mechanics seems to include the principle:
Principle of combination 1. If we have a complex object of mass m whose parts are bound to each other, "entangled" in some sense, then we can treat it as a single wave of a pointlike object of mass m. In periodic motion, there has to be an integer number of wavelengths of the wave of the combined object. Destructive interference prohibits other periodic orbits.
Each individual part in the complex object contains the information of the location and the momentum of the whole complex object. But what about parts which are far away and contain information of the object?
If an electron and a positron are always born as a pair and the positron always preserves the information of the electron rotational state, then in rotation, we have to treat the combined object the electron & the positron as one. A full rotation of the combined object consists of a full rotation of both the electron and the positron. If we just look at the electron wave function, it looks like there was just a 1/2 wavelength in the rotation. This is the origin of the strange spin value 1/2.
rotating tube
e- ============= e+
But does the positron preserve information of the rotational state of its brother electron? What if we use a magnetic field to turn the positron? One may imagine that the rotational angle is a hidden variable in both the electron and the positron. Maybe turning the positron does not affect the angle but rather the orientation of the rotational axis? Maybe the electron and positron are the ends of a flexible rotating tube. One can turn the ends to any direction, but the low-level rotation of the tube stays the same.
e- ========= e+ e- ========= e+
B
annihilation
What if we annihilate the positron with an electron B? If we accept the Feynman interpretation that annihilation means scattering backward in time, then the annihilation really means gluing together the two tubes. Our electron will still have a brother positron. In the diagram, the two electrons and two positrons are just one particle bouncing back and forth in time.
What if we speed up the electron, so that its time slows down relative to the laboratory frame, consequently its rotation slows down, and the positron loses the knowledge of the electron rotation angle?
Problem 2. When does the Principle of combination 1 lose its validity if time in part of the system is slowed down by increasing the velocity of that part? How can we cancel the entanglement of two quantum mechanical systems?
Why is the electron spin 1/2 or -1/2, and not 1 or 0? In the hydrogen atom, the electron may have the orbital angular momentum 0. That is probably because then the electric field outside the atom is zero. One cannot know if the atom is rotating by measuring the magnetic component of the electric field.
The electron total spin cannot be 0, because we can measure the rotational state of its electric field. But can the z component of the spin be 0? Then we would know that the spin axis is in the x,y plane. That is probably prohibited by an uncertainty principle.
An electron spin 1 or -1 is possible in our rotating tube model. Maybe it has not been observed because it would be stable only under a magnetic field of 10^8 tesla or more.
We did not need to use special relativity at all in deriving the electron spin 1/2. It is enough to assume that the electric field can be a rotating object, and that the rotation of the electron and its brother positron forms a "bound state" or entangled state where the (hidden) rotation angle of both particles is perfectly correlated.
Why does the spin 1/2 follow from the Dirac equation? Dirac does not need to assume that the electron is entangled with a positron.
Why is the g factor in the electron magnetic moment 2? We calculated in an earlier post that if the electron goes in a circle at the speed c * sqrt(3) / 2, then the gyromagnetic ratio is 2, according to special relativity. Is there a reason why the electric field of the electron should rotate just at that speed?
rotating tube
e- ============= e+
But does the positron preserve information of the rotational state of its brother electron? What if we use a magnetic field to turn the positron? One may imagine that the rotational angle is a hidden variable in both the electron and the positron. Maybe turning the positron does not affect the angle but rather the orientation of the rotational axis? Maybe the electron and positron are the ends of a flexible rotating tube. One can turn the ends to any direction, but the low-level rotation of the tube stays the same.
e- ========= e+ e- ========= e+
B
annihilation
What if we annihilate the positron with an electron B? If we accept the Feynman interpretation that annihilation means scattering backward in time, then the annihilation really means gluing together the two tubes. Our electron will still have a brother positron. In the diagram, the two electrons and two positrons are just one particle bouncing back and forth in time.
What if we speed up the electron, so that its time slows down relative to the laboratory frame, consequently its rotation slows down, and the positron loses the knowledge of the electron rotation angle?
Problem 2. When does the Principle of combination 1 lose its validity if time in part of the system is slowed down by increasing the velocity of that part? How can we cancel the entanglement of two quantum mechanical systems?
Why is the electron spin 1/2 or -1/2, and not 1 or 0? In the hydrogen atom, the electron may have the orbital angular momentum 0. That is probably because then the electric field outside the atom is zero. One cannot know if the atom is rotating by measuring the magnetic component of the electric field.
The electron total spin cannot be 0, because we can measure the rotational state of its electric field. But can the z component of the spin be 0? Then we would know that the spin axis is in the x,y plane. That is probably prohibited by an uncertainty principle.
An electron spin 1 or -1 is possible in our rotating tube model. Maybe it has not been observed because it would be stable only under a magnetic field of 10^8 tesla or more.
We did not need to use special relativity at all in deriving the electron spin 1/2. It is enough to assume that the electric field can be a rotating object, and that the rotation of the electron and its brother positron forms a "bound state" or entangled state where the (hidden) rotation angle of both particles is perfectly correlated.
Why does the spin 1/2 follow from the Dirac equation? Dirac does not need to assume that the electron is entangled with a positron.
Why is the g factor in the electron magnetic moment 2? We calculated in an earlier post that if the electron goes in a circle at the speed c * sqrt(3) / 2, then the gyromagnetic ratio is 2, according to special relativity. Is there a reason why the electric field of the electron should rotate just at that speed?
A spinor is a "square root" of a vector?
The spin 1 or -1 of a photon is explained in literature from the fact that the 4-dimensional current density, the four-current is a vector.
When a vector is rotated 360 degrees, it is again identical with the original vector. In contrast, the Dirac spinor has to be "rotated", or more precisely, has to move on a Möbius strip a path of 720 degrees to return to the original value. In a sense, a spinor is a square root of a vector.
The rotation of an ordinary vector can be understood as moving it along a straight strip which is obtained when a ring is cut from a circular disk by removing the center.
The Möbius strip is obtained by cutting the ring open, flipping one end 180 degrees, and gluing together.
Rotation of a single particle can be understood as moving a vector along a straight ring for 360 degrees.
Our rotating tube model describes two particles rotating in unison. The rotation of the whole system contains one full cycle of the wave function. If we just look at one end of the tube, maybe we can describe its behavior with a spinor? The combination of two spinors should in some sense be equivalent to a vector.
If we have two particles rotating strictly in unison, at the same location or separated by some distance, can we describe one particle as a spinor, and the combination of the two spinors is an ordinary vector which describes the rotation of the combined system?
Since we can change the direction of the electron spin and its brother positron spin independently, the model is more complicated than what we get by just gluing two particles together. That gives rise to the spinor.
The combined system the electron & the positron would have a rotational state on a 1X sphere, but when the system is split into two particles, each of those particles has to be described on the 2X sphere with the modified topology.
A spinor is a vector in a 2X sphere where we identify opposite points?
If the electron wave only does 1/2 wavelengths in one round, we have the problem of destructive interference. But we get rid of the interference by making the orbit 2X in spatial size. The electron can orbit on the surface of such a sphere.
Then we identify each point in the sphere with the opposite point, to get a new topology. Can we project the 2X sphere to a normal 1X sphere in the space, such that identified points are mapped to the same point in the 1X sphere? The mapping is certainly possible if we want to map a 2X circle to a 1X circle. What about the 2-dimensional case?
If we have a point on the 2X sphere, with a latitude c degrees and a longitude d degrees, where would we map it on the 1X sphere?
If we map (c, d) to (2c, 2d), does that work? Probably yes.
If the rotational state of the electron lives on the 2X sphere, and it is projected to the 1X sphere in our universe, then the properties of spinors might find an explanation.
The combined system the electron & the positron would have a rotational state on a 1X sphere, but when the system is split into two particles, each of those particles has to be described on the 2X sphere with the modified topology.
The photon can decay into an elecron-positron pair?
Our rotating dipole model of the first section in this blog post suggests that the photon is a small pointlike rotating electric dipole of spin 1. It is like a miniature copy of the rotating dipole which produced the electromagnetic field.
The rotating dipole of the photon can split, decaying into the negative charge part and the positive charge part. Since these new particles start from the same spacetime point, they have to carry away the angular momentum in their own spin, which is 1/2. This would be the origin of a produced pair.
The photon wave is (at least roughly) described by the Klein-Gordon equation. We need to find out why a "square root" of the Klein-Gordon equation describes the ends of the dipole once they manage to break free. The square root is the Dirac equation.
Water waves are molecules in a vertical circular motion, vortices are horizontal circular motion
There is an intriguing analogue between water and the electromagnetic field.
A wave in water is a temporary "hill" or "valley". But a vortex is a semi-permanent valley. The electron may be analogous with a vortex because it is a permanent valley in the electric field.
If the surface of water is sloping, then a vortex may tend to move upward. That would be analogous to electric attraction.
The positron might be an "anti-vortex" which is a permanent hill. There are no anti-vortices in water, though.
When powerful waves in water meet head on, they probably do not produce vortices. Waves hitting an obstacle may produce vortices.
The vortex analogy might explain why there are permanent charges in the electromagnetic field.
The Poynting vector, 1/μ_0 * E × B, does make circles around the electron. If there is energy circling the electron, there has to be a centripetal force which keeps it in the orbit. We need to check what is the Poynting vector. The energy of the electric field outside the classical radius, 3 * 10^-15 m, is the electron mass 511 keV. The energy of the magnetic field is probably much more.
If we model the electron as the current loop whose radius is 3 * 10^-13 m, then close to the loop, the magnetic field is of the order 10^8 tesla and the electric field is 10^17 V/m.
The Poynting vector is
10^6 * 10^8 * 10^17 = 10^31 W/m^2.
Does this make sense? The electron moves at almost the speed of light. The energy transfer in the loop has the power
m c^2 c / C = m c^3 / C
= 10^-30 * 27 * 10^24 * 10^12 / 3 W
= 10^7 W,
where C is the loop circumference.
If the area through which the Poynting vector moves is in the ballpark (3 * 10^-13 m)^2, the Poynting power is 10^6 W. The figures look reasonable.
The centripetal force to keep the electron in the loop is
m v^2 / r = 10^-30 * 10^17 * 10^13 / 3 N
= 0.3 N.
The electric repulsion between two electrons at the distance 3 * 10^-13 m is
k e^2 / r^2
= 10^10 * 3 * 10^-38 * 10^25 N
= 3 * 10^-3 N.
The centripetal force which we need is roughly 1,000X the electric force.
If we assume that a water whirlpool has the velocity of the water constant on the surface, then the depth of the vortex will follow the 1/r law which is like the Coulomb potential of electric force.
A simple calculation shows that the kinetic energy in a water whirlpool is roughly the same as the potential energy in the valley which the vortex makes in water.
A vortex in water is always a valley. In a hypothetical water where the centrifugal acceleration has the sign flipped to the normal case, a vortex would be a hill in water.
http://www.worksofvalerychalidze.com/uploads/1/0/2/8/102863812/1126751print1a.pdf
The Soviet dissident Valery Chalidze (1938 - 2018) developed some kind of a vortex hypothesis for electromagnetism. His model is quite complicated and probably does not help us much. Fluid models and vortices in them were researched in the 19th century, before elementary particles were discovered. Apparently, no one was able to create a satisfactory fluid model of electromagnetism.
The Poynting vector, 1/μ_0 * E × B, does make circles around the electron. If there is energy circling the electron, there has to be a centripetal force which keeps it in the orbit. We need to check what is the Poynting vector. The energy of the electric field outside the classical radius, 3 * 10^-15 m, is the electron mass 511 keV. The energy of the magnetic field is probably much more.
If we model the electron as the current loop whose radius is 3 * 10^-13 m, then close to the loop, the magnetic field is of the order 10^8 tesla and the electric field is 10^17 V/m.
The Poynting vector is
10^6 * 10^8 * 10^17 = 10^31 W/m^2.
Does this make sense? The electron moves at almost the speed of light. The energy transfer in the loop has the power
m c^2 c / C = m c^3 / C
= 10^-30 * 27 * 10^24 * 10^12 / 3 W
= 10^7 W,
where C is the loop circumference.
If the area through which the Poynting vector moves is in the ballpark (3 * 10^-13 m)^2, the Poynting power is 10^6 W. The figures look reasonable.
The centripetal force to keep the electron in the loop is
m v^2 / r = 10^-30 * 10^17 * 10^13 / 3 N
= 0.3 N.
The electric repulsion between two electrons at the distance 3 * 10^-13 m is
k e^2 / r^2
= 10^10 * 3 * 10^-38 * 10^25 N
= 3 * 10^-3 N.
The centripetal force which we need is roughly 1,000X the electric force.
If we assume that a water whirlpool has the velocity of the water constant on the surface, then the depth of the vortex will follow the 1/r law which is like the Coulomb potential of electric force.
A simple calculation shows that the kinetic energy in a water whirlpool is roughly the same as the potential energy in the valley which the vortex makes in water.
A vortex in water is always a valley. In a hypothetical water where the centrifugal acceleration has the sign flipped to the normal case, a vortex would be a hill in water.
http://www.worksofvalerychalidze.com/uploads/1/0/2/8/102863812/1126751print1a.pdf
The Soviet dissident Valery Chalidze (1938 - 2018) developed some kind of a vortex hypothesis for electromagnetism. His model is quite complicated and probably does not help us much. Fluid models and vortices in them were researched in the 19th century, before elementary particles were discovered. Apparently, no one was able to create a satisfactory fluid model of electromagnetism.
No comments:
Post a Comment