Let us put a covering space around the full circle, such that in the covering space points are identified when their angular distance is 720 degrees. In a normal circle, points are identified after 360 degrees.
In the covering space, the electron can orbit in the normal way such that its wave function does a full cycle during a round of the orbit.
The covering space is like a coil which consists of two loops. The covering space is mapped non-bijectively to the 360 degree circle in the natural way, such that every point in the circle is an image of 2 points in the covering space.
The rotating electron lives in the covering space. An external magnet lives in the normal 1+3D space.
The electron will see the magnet double: the magnet appears at two points in the covering space. It will also see the magnetic field lines double. This might explain why the interaction of the rotating electron with the external magnetic field is 2X of what one would expect - its gyromagnetic ratio g is 2.
The question is what is physics like in the covering space. Do external interactions that depend on rotation have a double strength?
It is not at all clear that the physics in the covering space should produce results that differ from the ordinary 1+3D space. We can certainly build physics which agree 100% with the ordinary space.
Quantum mechanically, the strange angular momentum 1/2 h-bar is the outstanding feature of the electron. Therefore, it probably is responsible for the equally strange gyromagnetic ratio 2.
The Dirac equation, and probably the Pauli equation, too, are aware of the angular momentum as well as the gyromagnetic ratio. The equations do not give us an intuitive model of the rotation and the associated waves. We have been trying to sketch a model, but cannot yet connect it to the equations.
https://arxiv.org/abs/1207.5752
David Delphenich in his arxiv.org paper writes about the history of the Pauli and Dirac equations. There were efforts to find an intuitive model for the electron spin 1/2, but no one has succeeded in it.
We in this blog claim that the existence of a spin is a necessary consequence of a quantized field if the source can feed angular momentum into the field. The challenges are linking this idea to the known equations, and showing the intuitive reason for g = 2. The idea of a covering space is not too intuitive, and even less intuitive is the claim that the magnetic field is seen as having a double strength by an electron living in the covering space. We did not find a plausible explanation.
The fact that in the Pauli equation, the whole kinetic term, including the linear momentum p, is subject to the Pauli matrices, suggests that the "correct" momentum space is the space of 2 x 2 Hermitian matrices. Rather than being a vector
(1, 0, 0) * p_x
+ (0, 1, 0) * p_y
+ (0, 0, 1) * p_z,
the momentum is represented as a 2 x 2 matrix
σ_1 * p_x
+ σ_2 * p_y
+ σ_3 * p_z,
where the sum matrix operates on a two-component wave function
(Ψ_+,
Ψ_-).
The Pauli equation is a natural mathematical generalization of the kinetic term to a 2-component wave function. Mathematically, the Pauli equation is "intuitive" in this sense. But physically, it is hard to comprehend.
We explained in the previous blog post that if we put angular momentum in a field, then the wave function collapse cannot be modeled with a simple one-component wave function of a point particle. The angular momentum is a global property of the field and cannot be formulated in a hamiltonian which only looks at a single point in spacetime of a one-component wave function.
It is then natural to ask how we should add more "degrees of freedom" to the wave function, so that it describes the angular momentum, too. The Pauli equation shows that adding another component to the wave function does the trick. But why does it work?
It is not at all clear that the physics in the covering space should produce results that differ from the ordinary 1+3D space. We can certainly build physics which agree 100% with the ordinary space.
Quantum mechanically, the strange angular momentum 1/2 h-bar is the outstanding feature of the electron. Therefore, it probably is responsible for the equally strange gyromagnetic ratio 2.
The Dirac equation, and probably the Pauli equation, too, are aware of the angular momentum as well as the gyromagnetic ratio. The equations do not give us an intuitive model of the rotation and the associated waves. We have been trying to sketch a model, but cannot yet connect it to the equations.
https://arxiv.org/abs/1207.5752
David Delphenich in his arxiv.org paper writes about the history of the Pauli and Dirac equations. There were efforts to find an intuitive model for the electron spin 1/2, but no one has succeeded in it.
We in this blog claim that the existence of a spin is a necessary consequence of a quantized field if the source can feed angular momentum into the field. The challenges are linking this idea to the known equations, and showing the intuitive reason for g = 2. The idea of a covering space is not too intuitive, and even less intuitive is the claim that the magnetic field is seen as having a double strength by an electron living in the covering space. We did not find a plausible explanation.
The Pauli equation revisited
The fact that in the Pauli equation, the whole kinetic term, including the linear momentum p, is subject to the Pauli matrices, suggests that the "correct" momentum space is the space of 2 x 2 Hermitian matrices. Rather than being a vector
(1, 0, 0) * p_x
+ (0, 1, 0) * p_y
+ (0, 0, 1) * p_z,
the momentum is represented as a 2 x 2 matrix
σ_1 * p_x
+ σ_2 * p_y
+ σ_3 * p_z,
where the sum matrix operates on a two-component wave function
(Ψ_+,
Ψ_-).
The Pauli equation is a natural mathematical generalization of the kinetic term to a 2-component wave function. Mathematically, the Pauli equation is "intuitive" in this sense. But physically, it is hard to comprehend.
We explained in the previous blog post that if we put angular momentum in a field, then the wave function collapse cannot be modeled with a simple one-component wave function of a point particle. The angular momentum is a global property of the field and cannot be formulated in a hamiltonian which only looks at a single point in spacetime of a one-component wave function.
It is then natural to ask how we should add more "degrees of freedom" to the wave function, so that it describes the angular momentum, too. The Pauli equation shows that adding another component to the wave function does the trick. But why does it work?
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