Spin

Usually physical encyclopedias write the following about spin in quantum mechanics. In quantum mechanics spin is a fundamental property of elementary particles, atomic nuclei and hadrons. Spin direction is treated as an important intrinsic degree of freedom. Spins have important theoretical implications and practical applications. The electron spin is the key to the Pauli exclusion principle and to the understanding of the periodic system of chemical elements. Spin-orbit coupling leads to the fine structure of atomic spectra. Electron spins play an important role in magnetism. The photon spin is associated with the polarization of light. For instance, all the electrons have spin s = 1/2. The photon spin is thought to be 1.

There have been a number of approaches to understanding the phenomenon of spin (see, reviews, i.e. in papers [1,2]).

Submicroscopic mechanics points to the fact that a moving canonical particle periodically changes its characteristics, i.e. its basic parameters, such as velocity $\upsilon$, mass $m$ and charge $e$ are not constant but oscillate on the length of the spatial period $\lambda$, which has been identified with the particle's de Broglie wavelength. These parameters have their maximum value in points $\lambda \cdot l$ of the particle's path (where $l=0,1,2,3,...$) and zero in points $\lambda/2 \cdot l$ of the particle's path (where $l=1,3,5,7,...$). Parameters which supplement the mass and charge, namely the tension (or the rugosity) of space $\xi$ and the magnetic charge $g$ respectively, oscillate in counter-phase: from zero in points $\lambda \cdot l$ of the particle's path (where $l=0,1,2,3,...$) to the maximum value in points $\lambda/2 \cdot l$ of the particle's path (where $l=1,3,5,7,...$).

In submicroscopic mechanics we can introduce one more parameter, an intrinsic motion of a particle, which can easily be mapped to the notion of spin in quantum mechanics. Namely, a pulsation of the volume of a particulate cell can be associated with the manifestation of spin as used in quantum mechanics [1]. In the real physical space in the section of $\lambda$ the moving particle changes its shape from the initial strain bean-like shape in points $\lambda \cdot l$ of the particle's path (where $l=0,1,2,3,...$) to the non-strain spherical shape in points $\lambda/2 \cdot l$ (where $l=1,3,5,7,...$). This intrinsic motion may have the same kinetic energy as the kinetic energy of the particle's translational movement, $m\upsilon^2/2$.

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The mentioned intrinsic motion, pulsation, features the quantum equation

(1)
\begin{align} \Bigl({\hat{\vec\Pi}}^{\ 2}_{\alpha}/2m - e_{\alpha}\varepsilon_{\alpha} \Bigr) \chi_{\alpha}=0. \end{align}

Allowing that the induction $\cal B$ is aligned solely with the $O^e$ axis, instead of equation (1) we can write

(2)
\begin{align} \bigl({\hat\Pi_{\alpha 1}}^2/2m + {\hat\Pi_{\alpha 2}}^2/2m - e_{\alpha}\varepsilon_{\alpha} \bigr) \chi_{\alpha}=0. \end{align}

where the operators satisfy the equation

(3)
\begin{align} [\hat\Pi_{\alpha 1},\hat\Pi_{\alpha 2}]_{\_}=ie\hbar{\cal B}. \end{align}

With allowance for the function $e_{\alpha} = \pm 1$, we can get the solution for the eigenfunction $\chi_{\alpha}$ and the expression for the eigenfunctions $\varepsilon _{\uparrow (\downarrow)}$ in the representation of the so-called operator of the spin projection onto the axis $Oe^3$:

(4)
\begin{align} \varepsilon_{\uparrow (\downarrow)}= \frac {e \cal B} {m} {\hat S}_3, \end{align}

where, as is known, the eigenvalues of this operator

(5)
\begin{align} S_{\uparrow (\downarrow) 3}=\pm\hbar/2. \end{align}

Correction (4) to the particle energy which is due to the intrinsic degree of freedom, should be inserted into the total particle spectrum renormalising the eigenvalue $E$ to the value $E + e {\cal B} S_ {\uparrow (\downarrow) 3}/m$. The renormalised equation goes into the Pauli equation. They differ only in the mass quantity: in submicroscopic mechanics the relativistic mass of the particle $m$ enters the Schrödinger equation with the spin component, which includes the interaction of spin with an applied magnetic field. But only the mass at rest $m_0$ appears in the conventional Pauli equation. However, if the terms proportional to $v_0 /c$ are accounted for, the Schrödinger equation with the spin component derived in the framework of submicroscopic mechanics and the Pauli equation coincide.

In the submicroscopic approach, the Dirac equation is derived from the Hamiltonian that includes the intrinsic motion of the particle in question (a new term ${ c^{\kern 1pt 2} \ {\vec {\pi}}}^{\ 2}$$_{\uparrow (\downarrow)}$ that has not been taken into account so far), i.e.

(6)
\begin{align} H_{\uparrow (\downarrow)}^{\rm part. tot}= \sqrt{ c^2\vec p^{\ 2} +c^2\vec\pi^{\ 2}_{\uparrow (\downarrow)} + m_0^2c^4}. \end{align}

The Hamiltonian (6) includes additional terms associated with two possible projections of intrinsic pulsations of particles. Therefore, if we decompose the square root in expression (6), which has a matrix form, we must obtain the equation in a matrix form too. This is the inner reason why the Dirac equation should possess matrix components associated with the particle spin.

Particles that have an integral spin are particles combined of simple particles with half-integer spin.

Since the photon is not a particle, but a quasi-particle of real physical space, we cannot write down any quantum equation for it in the form similar to equation (1). That is why the notion of spin cannot be applied to this field particle which is an excitation of space. In the case of the photon we should rather use the term "integer-valued polarization" $\pm 1.$

Bibliography

[ 1] V. Krasnoholovets, On the nature of spin, inertia and gravity of a moving canonical particle, Indian Journal of Theoretical Physics 48, no. 2, pp. 97-132 (2000) (also arXiv.org e-print archive, http://arXiv.org/abs/quant-ph/0103110).

[2] V. Krasnoholovets, Submicroscopic deterministic quantum mechanics, International Journal of Computing Anticipatory Systems 11, pp. 164-179 (2002) (also http://arXiv.org/abs/quant-ph/0109012).

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