We had the example where a rotating electric dipole stores angular momentum to the electromagnetic field. Did we need to use special relativity there? No. The only thing which matters is that we can use a rotating object as the source of waves. It could be a finger doing a circle on a drum skin as well.
The spin is the result of:
1. One can store angular momentum to a wave through a cyclic process.
2. That angular momentum is divided evenly to the energy quanta of the wave.
What happens in the derivation of the Dirac equation
We start from the Klein-Gordon equation and try to write it to an equivalent form where there are no second derivatives.
Since the Klein-Gordon equation is Lorentz covariant, there is a good chance that the equivalent form is, too.
We separate solutions which correspond to positive m and negative m. For that purpose, we use the matrix
1 0 0 0
0 1 0 0
0 0 -1 0
0 0 0 -1
as the coefficient of m Ψ. The "main components" of the negative energy solutions will appear in the lower components.
In the original Klein-Gordon equation, negative frequency solutions correspond to negative E. The Dirac spinor is just a new way to represent a basis of positive and negative frequency solutions of the Klein-Gordon equation.
The factorization of the Klein-Gordon operator, for some reason, also separates the spin_z up and spin_z down cases of solutions. We still need to analyze in more detail why that happens. The factorization has lots of symmetry. Maybe it is not that surprising that the separation happens.
A linearly oscillating source and spin 1/2
Since the Klein-Gordon equation is a general wave equation which allows a rotating object as the source, its quanta can have a non-zero spin.
Let us assume that the pair-producing source of the Klein-Gordon equation does linear back-and-forth movement. It is like a dipole radio transmitter. The energy which is pumped into the source contains no angular momentum.
In the case of photons, we needed to split the linear movement into two rotating dipoles to get the right model for quantum production. One dipole rotates clockwise and the other counter-clockwise when looked down from the positive z axis. Let us do that split in this case, too.
Let the frequency of the rotation of the source be f.
Because of the symmetry of negative and positive frequencies, the clockwise rotating source will produce both negative and positive frequency waves, and an equal amount.
Just as in the case of the photon, we can calculate that for work h f, the angular momentum stored in the field corresponds to spin 1. The energy and the angular momentum is evenly divided between the positive and negative frequency waves.
A quantum consists of the electron and the positron. The positive frequency wave corresponds to the electron. The energy of the electron is 1/2 h f and the spin_z is 1/2. The same for the positron.
The counter-clockwise rotating source produces electrons and positrons of spin_z -1/2.
Thus, the linearly oscillating source produces both electron and positron waves with spins up and down. What is the quantum of this whole process? It must be an electron-positron pair whose spins point at opposite directions, to ensure that the total angular momentum is zero.
Note that the rest mass of the quantum is 2 m_e, where m_e is the rest mass of the electron. If the momenta of the particles is zero, the quantum energy
h f = 2m_e c^2.
The source has 2X the frequency of the wave of the electron alone.
In deriving the spin 1/2, we did not use special relativity at all. Our assumptions were:
1. The source in the Klein-Gordon equation does a linear oscillation.
2. The correct way to model linear oscillation of a source is two rotating sources.
3. A quantum of the process is the electron-positron pair and its energy is h f, where f is the frequency of oscillation.
In the case of photons, we needed to split the linear movement into two rotating dipoles to get the right model for quantum production. One dipole rotates clockwise and the other counter-clockwise when looked down from the positive z axis. Let us do that split in this case, too.
Let the frequency of the rotation of the source be f.
Because of the symmetry of negative and positive frequencies, the clockwise rotating source will produce both negative and positive frequency waves, and an equal amount.
Just as in the case of the photon, we can calculate that for work h f, the angular momentum stored in the field corresponds to spin 1. The energy and the angular momentum is evenly divided between the positive and negative frequency waves.
A quantum consists of the electron and the positron. The positive frequency wave corresponds to the electron. The energy of the electron is 1/2 h f and the spin_z is 1/2. The same for the positron.
The counter-clockwise rotating source produces electrons and positrons of spin_z -1/2.
Thus, the linearly oscillating source produces both electron and positron waves with spins up and down. What is the quantum of this whole process? It must be an electron-positron pair whose spins point at opposite directions, to ensure that the total angular momentum is zero.
Note that the rest mass of the quantum is 2 m_e, where m_e is the rest mass of the electron. If the momenta of the particles is zero, the quantum energy
h f = 2m_e c^2.
The source has 2X the frequency of the wave of the electron alone.
In deriving the spin 1/2, we did not use special relativity at all. Our assumptions were:
1. The source in the Klein-Gordon equation does a linear oscillation.
2. The correct way to model linear oscillation of a source is two rotating sources.
3. A quantum of the process is the electron-positron pair and its energy is h f, where f is the frequency of oscillation.
How does the Dirac equation know that the electron spin is 1/2?
In the preceding section we showed that a linearly oscillating source produces waves with both positive and negative frequencies, and both the spin up and down.
How does the Dirac equation "know" that the spin of the electron is 1/2?
It may be that the Dirac equation describes correctly a half of the Klein-Gordon wave which we produced with the linearly oscillating source. We need to investigate this in more detail.
Conjecture 1. A spatially circularly polarized wave solution of the Klein-Gordon equation stores h-bar/2 of angular momentum per m of "rest mass energy", if m is non-zero. A linearly polarized wave must be broken down to circularly polarized components to reveal the "hidden" angular momentum.
A rotating source which radiates electron-positron pairs
A basic principle of quantum mechamics is: If we have a rotating source, its angular momentum is quantized in amounts of h-bar. It can radiate a quantum whose energy is h f, where f is the frequency of rotation.
Let us work with the Klein-Gordon wave equation. We assume that it can handle the type of source we have.
We conjecture that if h f is greater or equal to twice the electron rest mass, the source can radiate pairs. Each particle in the pair carries h-bar/2 of angular momentum.
In practice, the produced electron and positron have opposite spins. We conjecture that the source in such a case is doing linear oscillation.
The annihilation of a pair usually produces two photons of opposite spin. If we invert time, we produce a pair. It looks sensible that two photons with opposite spins make a linearly oscillating source to the electron-positron field. The source may through some mechanism be the strong electric field.
The quantum of the Klein-Gordon electron-positron field is the pair electron-positron. If we have an arbitrary wave, we can separate it to positive and negative frequency components to know the electron wave component and the positron wave component. Here the positive and negative frequencies correspond to the sign of E in a basis wave
exp(-i (E t - p x)).
The "rotation" which differentiates the electron and the positron happens in an abstract complex space.
The spin is rotation which happens in the familiar 3-dimensional space. The rotating source stores angular momentum to the Klein-Gordon field.
We conjecture that the Dirac equation separates the Klein-Gordon waves which have positive versus negative angular momentum when looked down from the positive z axis.
That is, if the waves were born from a source which rotates clockwise or counter-clockwise, the Dirac equation separates them to different components. If we look at the wave very far away, it looks almost like a plane wave. But the big difference is that the wave contains stored angular momentum. The wave can be approximated by a plane wave plus information about the angular momentum. That is probably the origin of the Dirac spinor. The basis waves of the Dirac equation are like plane waves which contain a little more information in the spinor.
The stored angular momentum, that is, the spin, is present already in the familiar Klein-Gordon waves. The Dirac equation just pinpoints it.
In practice, the produced electron and positron have opposite spins. We conjecture that the source in such a case is doing linear oscillation.
The annihilation of a pair usually produces two photons of opposite spin. If we invert time, we produce a pair. It looks sensible that two photons with opposite spins make a linearly oscillating source to the electron-positron field. The source may through some mechanism be the strong electric field.
The quantum of the Klein-Gordon electron-positron field is the pair electron-positron. If we have an arbitrary wave, we can separate it to positive and negative frequency components to know the electron wave component and the positron wave component. Here the positive and negative frequencies correspond to the sign of E in a basis wave
exp(-i (E t - p x)).
The "rotation" which differentiates the electron and the positron happens in an abstract complex space.
The spin is rotation which happens in the familiar 3-dimensional space. The rotating source stores angular momentum to the Klein-Gordon field.
We conjecture that the Dirac equation separates the Klein-Gordon waves which have positive versus negative angular momentum when looked down from the positive z axis.
That is, if the waves were born from a source which rotates clockwise or counter-clockwise, the Dirac equation separates them to different components. If we look at the wave very far away, it looks almost like a plane wave. But the big difference is that the wave contains stored angular momentum. The wave can be approximated by a plane wave plus information about the angular momentum. That is probably the origin of the Dirac spinor. The basis waves of the Dirac equation are like plane waves which contain a little more information in the spinor.
The stored angular momentum, that is, the spin, is present already in the familiar Klein-Gordon waves. The Dirac equation just pinpoints it.
The gyromagnetic ratio g = 2
Our model should be able to explain why the magnetic moment of the electron is 2X of the most straightforward orbiting particle model.
If the electron is a point particle, and it can be described with plane waves of type
exp(-i (E t - p x)),
then it is hard to build a model where the gyromagnetic ratio is anything else than 1. If the "mass charge" and the electric charge are bound in the same way into the point particle, then the angular momentum and the magnetic moment should have the same ratio in all kinds of movement, whether it is translation or rotation.
When a pair is produced, there first is an electron present and a positron with an opposite spin present. If a particle B interacts with the pair, it will measure that the magnetic field is 2X of what one would expect from a lone electron.
But according to the Feynman interpretation, the electron and the positron are just one and the same particle. The particle B interacts with two copies of the same electron.
Suppose then that the positron has flown away. If B comes to interact with the electron, does it now interact just with one copy of the electron? If we are allowed to switch the time and space dimensions, then B should interact both with the "arriving" electron and the "departing" electron. This may be the reason why the magnetic moment of the electron is twice of what one would expect.
In Feynman diagrams, the pair production and photon scattering are drawn in a symmetric fashion:
^
| time
photon e-
\ /
\ /
\/ emission
|
| absorption
/\
/ \
/ e-
photon
e+ e-
\ /
\ /
\______/ pair production
/ \
/ \
photon photon
Why is the angular momentum just the spin 1/2 then? If we measure the angular momentum, should we measure both the arriving electron and the departing electron?
When we measure the electric charge of the electron, why is there no doubling if the model above is right?
How can the Dirac equation "know" about the mechanism we sketched above? It correctly predicts the magnetic moment.
The gyromagnetic ratio 2 seems to require some internal degree of freedom for the electron. A point particle would have it 1. The Dirac equation, for an unknown reason, conjures up the right internal degree of freedom. The four components of the Dirac spinor are interdependent, but allow some freedom.
We have not yet found an intuitive reason for the value 2, but let us keep trying.
If the electron is a point particle, and it can be described with plane waves of type
exp(-i (E t - p x)),
then it is hard to build a model where the gyromagnetic ratio is anything else than 1. If the "mass charge" and the electric charge are bound in the same way into the point particle, then the angular momentum and the magnetic moment should have the same ratio in all kinds of movement, whether it is translation or rotation.
When a pair is produced, there first is an electron present and a positron with an opposite spin present. If a particle B interacts with the pair, it will measure that the magnetic field is 2X of what one would expect from a lone electron.
But according to the Feynman interpretation, the electron and the positron are just one and the same particle. The particle B interacts with two copies of the same electron.
Suppose then that the positron has flown away. If B comes to interact with the electron, does it now interact just with one copy of the electron? If we are allowed to switch the time and space dimensions, then B should interact both with the "arriving" electron and the "departing" electron. This may be the reason why the magnetic moment of the electron is twice of what one would expect.
In Feynman diagrams, the pair production and photon scattering are drawn in a symmetric fashion:
^
| time
photon e-
\ /
\ /
\/ emission
|
| absorption
/\
/ \
/ e-
photon
e+ e-
\ /
\ /
\______/ pair production
/ \
/ \
photon photon
Why is the angular momentum just the spin 1/2 then? If we measure the angular momentum, should we measure both the arriving electron and the departing electron?
When we measure the electric charge of the electron, why is there no doubling if the model above is right?
How can the Dirac equation "know" about the mechanism we sketched above? It correctly predicts the magnetic moment.
The gyromagnetic ratio 2 seems to require some internal degree of freedom for the electron. A point particle would have it 1. The Dirac equation, for an unknown reason, conjures up the right internal degree of freedom. The four components of the Dirac spinor are interdependent, but allow some freedom.
We have not yet found an intuitive reason for the value 2, but let us keep trying.
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