**6. Spin**

It is possible to construct the row or column vectors whose components are different functions of types (2) and (3), or, respectively, of types (4) and (5). Each component of these vectors is associated with a different network of groups of capacitors in parallel or, respectively, of inductances in parallel, afferent to a single actuator. If the relativistic covariance rules are applied to these vectors, they become spinors. We then have the Pauli spinors in the non-relativistic limit and the Dirac spinors in the general relativistic case. It thus becomes possible, in principle, to extend the present electrical analogy to include the spin of elementary particles and their internal degrees of freedom such as isospin or strangeness. Here, we limit ourselves to an observation on the spin.

A generic wave function of spin 12 takes, in the Pauli algebra, the following form [18,19]:

$$\begin{array}{rclcrcl}\psi(\stackrel{\rightarrow}{\mathbf{x}},t)&=&a(\stackrel{\rightarrow}{\mathbf{x}},t)\,\psi\_{z}^{+}&+&\beta(\stackrel{\rightarrow}{\mathbf{x}},t)\,\psi\_{z}^{-}&\ &|aa\ast|\ &+&|\beta\beta\ast|\ &=&1\ .\end{array}$$

where →*x* = (*<sup>x</sup>*, *y*, *z*) is the spatial position, and the spinors *ψ*<sup>+</sup>*z* and *ψ*<sup>−</sup>*z* represent fully polarized beams along the *z* axis, corresponding to the two eigenvalues of the spin projection along that axis. The meaning of (7) in the present representation is that a single actuator (particle) exchanges charge with two distinct networks, each of which is associated with one of the two projections, and keeps them both open. The (complex) coefficients *α* and *β* measure the relative intensity of the exchange with the relevant network, in the terms seen in Section 3. The rotation of the frame of reference of an angle *χ* around a spatial axis →*n* (with →*n* = 1) changes coefficients *α* and *β* according to the following law [18,20]:

$$\psi \to \left. \epsilon^{i\frac{\chi}{2}(\stackrel{\rightarrow}{n} \cdot \stackrel{\rightarrow}{\sigma^{\cdot}})} \psi \right. \\ = \left[ I \cos \frac{\chi}{2} + i \left( \stackrel{\rightarrow}{n} \cdot \stackrel{\rightarrow}{\sigma^{\cdot}} \right) \sin \frac{\chi}{2} \right] \psi \tag{8}$$

For example, if *α*(<sup>→</sup>*x* , *t*) = 1 and *β*(<sup>→</sup>*x* , *t*) = 0, a rotation around the *z* axis gives *α*(<sup>→</sup>*x* , *t*) = *ei χ*2 , *β*(<sup>→</sup>*x* , *t*) = 0. This means that the exchange is sensible to rotations. It must be borne in mind that a variation in axes is always associated with a modification of the experimental situation, such as fixing a direction to a magnetic field or reorienting a polarizer. This modification involves a variation in the charge exchange between the particle and the two networks. The directional dependence of the exchange represents, in a certain sense, a directional structure of the particle itself. However, this structure, or the internal "direction" of the particle, has nothing to do with the rotation of a solid object in three-dimensional space. This is consistent with the non-classical nature of the spin [21].
