The invisible side of quantum spin

There is a realm of states that lies below our ability to measure. The limit of measurement is set by the Heisenberg Uncertainty Principle which states complementary observables, like position and momentum, cannot be measured simultaneously; likewise for components of the Pauli spin vector operator. Fundamental to measurement is to use a probe and it destroys the isotropy of space. Since the quantum spin states discussed here only exist in isotropy, they cannot be measured.

When Stern and Gerlach did their experiments on silver atoms, the deflection of particles up or down revealed two states defined relative to an axis of quantization, usually the laboratory Z axis. Theoretically quantum spin naturally falls out from the Dirac field. In both cases a probe must be used. The picture that emerges of quantum spin is the well-known point particle having a single axis of quantization, defined by the orientation of the probe field. What happens when the probe is removed?

I think that everyone simply believes that the spin remains the same. It just continues to be polarized in the direction it last was seen and remains a point particle. Spinors are, after all, a distinct and unique property of matter that exist throughout the universe. Quantum spin is everywhere and of fundamental importance, could we have got it wrong? (I hear a resounding “No”.)

There is another view which is radically different and yet fits all the experimental data. It also does away with entanglement and non-locality and leads to a local realistic violation of Bell’s inequalities. In this entry I want to visualize spin in an isotropic environment.

I will assume quantum spin has two axes, σ_z and σ_x and these carry magnetic moments of equal magnitude. There is no experimental way of proving this choice is incorrect. This is because in the presence of a probe, one axis randomizes while the other is polarized. The axis that is more aligned with the probe will nutate and “line up” with it. The other axis will precess in the plane perpendicular to the probe and average away any correlation associated with it, (see Figure 1). In this situation the 2D spin becomes the usual point particle, but when the probe is removed, it falls back into its 2D structure, sitting somewhere in space and not interacting with anything.

Figure 1

Since two axes means a quantum spin has structure, then it must have its own body fixed coordinate system, called (x, y, z). Hence every 2D spin is generally randomly oriented relative to each other and to some arbitrary laboratory frame, denoted (X,Y,Z). Since we are below measurement, then the angles that relate each spin to a common frame must be Local Hidden Variables, LHV, and are the two polar angles, θ,φ.  So far we can depict the 2D spin as in figure 2.

Figure 2. The two axes of quantization of a spin 1/2.

Since the two axes are indistinguishable, their labels are interchangeable. This leads to another departure from the way quantum mechanics is applied. Usually quantum mechanics measures a statistical ensemble of particles described by a density operator. In contrast every spin in that ensemble is 2D and its individual state can be expressed as, (click equation to enlarge)

To compare, the usual density operator is written for a pure state, polarized in the laboratory Z direction

Besides body and laboratory frames being different, the significant difference between the two is the presence of off-diagonal components. These contain all the information about the x axis. In the case of Eq.(1.1) the eigenstates are

and we have of course,

Figure 3. The state that can never be seen: the superposition of two indistinguishable axes of quantization.

The usual reason for neglecting the off-diagonal terms in Eq.(1.2) is decoherence. If they do exist, then due to random phases and interference, the information that could exist in the off-diagonal terms cannot be detected macroscopically (that is at the level of a quantum ensemble). In contrast the states from Eq.(1.1) are
and have eigenvalues of
These states are superpositions of the x and z states within one quantum spin. This means a single spin can interfere with itself which has implications for the double slit experiment, for one.  These new states, Eq.(1.5) cannot exist in the presence of a probe. Nonetheless there is no reason to reject them just because we cannot measure them. Figure 2 is now changed in Figure 3 to depict these new states, with magnitude of √2. see Eq.(1.6).

There is, however, more to this because there is nothing special about the first quadrant shown in Figures 2 and 3. The spin can equally well be in the other quadrants, and this is depicted in Figure 4.

This means that we have to introduce two more LHV to indicate which quadrant the 2D spin occupies. These are integers, (n_z,n_x=±1,±1). The states now come from

and these other LHV must label the states, |±,√2>_nz,nx. Symmetry shows, however, that the quadrants for the two even are identical, and differ from the odd by a sign change.

A single 2D spin has two states that are superpositions of its two axes as seen from Eq.(1.5). However this is where superposition ends because a single particle cannot be in two places at the same time. Hence at any instant it can only occupy one quadrant. Since there is no advantage of one quadrant over another, rather than superposition, a single spin might resonates between different quadrants.

Figure 4. A effective magnetic monopole: the quadrants of the body fixed frame of a 2D spin showing the superposed states.

This leads to the view that when a probe is removed, the usual point particle spin becomes a 2D structured particle that resonates with a net resultant angular momentum of zero: an effective magnetic monopole. When the probe is turned on, the spin sits up.

Is this view of sub quantum spin correct? To me it comes down to which is weirder: To accept non-local entanglement and a point particle spin with intrinsic angular momentum; or a local realistic structured spin that has two axes of quantization and obviates non-locality.