Spin!
In our quest for a local and realistic description of quantum mechanics we have developed a full concrete model incorporating many aspects of quantum behavior (quantization, uncertainty principle, wave-particle duality, Born rule, momentum entanglement, etc.), which we have been presenting in previous posts.
However, a further class of genuine quantum processes requires a description of an additional particle property, which is intrinsic spin. I have recently added spin in my model, as reported in my most recent ArXiv publication. With this post, we are going to discuss how spin can emerge from a more fundamental local-realistic mechanism.
In addition to source momentum, particles have an intrinsic property that is a rational number comprised between -1 and +1, which we shall call "source spin" and denote as s0. In addition to momentum polarization, they also have another vector quantity, that is, three rational numbers {μ0d} such that
We shall call this property "spin polarization". While s0 remains constant, polarization is prone to change at each time the particle experiences a magnetic field. We shall represent such a field with three numbers B = {βd}BM, where BM is the magnitude of the field and such that the sum of the βd squared is one. Clearly, β represents the unit vector along which the physical field is directed.
The evolution of the polarization follows the rule
 "\mu_d[n+1]=\mu_d[n]-\gamma\mu_M B_M (\mu_{d+1}[n]\beta_{d+2}[n]-\mu_{d+2}[n]\beta_{d+1}[n])")
where n is the iteration number, μM is the polarization magnitude, γ represents the gyromagnetic ratio, the indexes must be taken modulo three, and μd[n0]=μ0d. We note that, if the βd are constant (constant orientation of the magnetic field), the sum of the squares of the polarizations are constant.
We further introduce the quantity
that is clearly constant if the field is constant. This quantity clearly corresponds to the cosine of the angle between the polarization and the field orientations.
Finally, we define the "spin" as a binary quantity (for spin 1/2 particles, we shall see in a future post how that changes for higher spin particles) s ∈ {-1,1} that is determined by the fundamental rule
 "s[n]= \mathrm{sign}(s_0+M[n])")
A simple check allows to affirm that the expected value of spin is the quantity M. For this reason, we shall denote this quantity as "spin propensity", in analogy to the momentum propensity already introduced in previous posts.
The main effect of spin is to parametrize magnetic forces, which arise in the presence of an inhomogeneous magnetic field (field variable in space). Assuming a constant field orientation β and a constant field inhomogeneity orientation ν, the magnetic force is directed along ν and is easily found to be
In addition to these rules, we introduce an External Reset, similar to that affecting span: when an external force is experienced, including a magnetic force, the polarizations 'jump' to the value
Consequently, the spin propensity jumps to the current value of spin, as easily shown by applying its definition above.
That's almost all we need. A further rule ("spin flip") will be presented in a future post. With these rules we will be able to represent typical quantum behavior such as spin quantization in Stern-Gerlach apparatuses or alike, and even spin entanglement. We will discuss these topics in future posts.
However, a further class of genuine quantum processes requires a description of an additional particle property, which is intrinsic spin. I have recently added spin in my model, as reported in my most recent ArXiv publication. With this post, we are going to discuss how spin can emerge from a more fundamental local-realistic mechanism.
In addition to source momentum, particles have an intrinsic property that is a rational number comprised between -1 and +1, which we shall call "source spin" and denote as s0. In addition to momentum polarization, they also have another vector quantity, that is, three rational numbers {μ0d} such that
We shall call this property "spin polarization". While s0 remains constant, polarization is prone to change at each time the particle experiences a magnetic field. We shall represent such a field with three numbers B = {βd}BM, where BM is the magnitude of the field and such that the sum of the βd squared is one. Clearly, β represents the unit vector along which the physical field is directed.
The evolution of the polarization follows the rule
where n is the iteration number, μM is the polarization magnitude, γ represents the gyromagnetic ratio, the indexes must be taken modulo three, and μd[n0]=μ0d. We note that, if the βd are constant (constant orientation of the magnetic field), the sum of the squares of the polarizations are constant.
We further introduce the quantity
that is clearly constant if the field is constant. This quantity clearly corresponds to the cosine of the angle between the polarization and the field orientations.
Finally, we define the "spin" as a binary quantity (for spin 1/2 particles, we shall see in a future post how that changes for higher spin particles) s ∈ {-1,1} that is determined by the fundamental rule
A simple check allows to affirm that the expected value of spin is the quantity M. For this reason, we shall denote this quantity as "spin propensity", in analogy to the momentum propensity already introduced in previous posts.
The main effect of spin is to parametrize magnetic forces, which arise in the presence of an inhomogeneous magnetic field (field variable in space). Assuming a constant field orientation β and a constant field inhomogeneity orientation ν, the magnetic force is directed along ν and is easily found to be
In addition to these rules, we introduce an External Reset, similar to that affecting span: when an external force is experienced, including a magnetic force, the polarizations 'jump' to the value
Consequently, the spin propensity jumps to the current value of spin, as easily shown by applying its definition above.
That's almost all we need. A further rule ("spin flip") will be presented in a future post. With these rules we will be able to represent typical quantum behavior such as spin quantization in Stern-Gerlach apparatuses or alike, and even spin entanglement. We will discuss these topics in future posts.
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