G. Structured Spin
Same mathematical basis
By changing the Dirac algebra from
a different formulation of spin emerges which rests on the same firm mathematical basis as usual spin. The new algebra has two time variables and two spatial variables. The spatial variables give the 2D Dirac equation and finds the new spin operators as Lorentz Invariants. Besides the usual linear time, the new time is quite different, being a rotational or phase time. Since spin now has structure, it can precess relative to spins in different inertial frames. Hence it plays the same role for angular momentum that linear time plays for linear momentum.
This means that we are presented with a choice: is spin that which we observe, or is it a 2D structured particle which, when measured, is indistinguishable from the former?
In this entry I will describe the 2D spin and how it changes our understanding of Nature.
Brief summary of Non-locality and persistent entanglement.
Quantum mechanics is considered to be the most fundamental theory of Nature. There are many interpretations and the most accepted of these is the Copenhagen primarily due to Nils Bohr, which rests on complementarity: incompatible observables cannot simultaneously exist, like position and momentum. Devise an experiment to measure one, and the other is not detected. One acts like a particle and the other a wave. Einstein, Podolsky and Rosen (EPR) famously disagreed with Bohr and showed that a physical theory must account for all elements of physical reality, thus they asserted quantum theory is incomplete.
Thirty years later, in 1964, Bell proposed his inequalities and theorem that essentially establishes non-local connectivity between separated particles. This repudiates EPR and establishes the persistence of entanglement after particles separate (non-locality) as a resource and property of Nature.
However a search on the Internet using “quantum weirdness” establishes that no-one can give a rational explanation for non-locality. If you accept non-locality, you are an anti-realist. If you do not, then you are a realist. Today the vast majority of physicists are anti-realists.
A point particle spin
I assume that readers have a good understanding of usual spin ½ so I do not have to discuss it: a point particle of intrinsic angular momentum displays two states |↑> and |↓>, spin up and spin down in the laboratory frame of reference, see Figure 1.
The reason these states are defined in the laboratory frame is because that is where experiments are performed. This requires turning on a probe which interacts with the spins and filters them into one of their two states. Recall that quantum mechanics cannot predict which state will be found. Rather all that can be known is the probability that a spin will be in one of its two states before filtering. Only after a statistically large number of spins have passed do the results agree with the predictions of quantum mechanics. This is the statistical interpretation. A large number of spins make up a quantum ensemble. Quantum mechanical spin states are ensembles of many spins, not individual spin states.
The 2D spin
Note that the 2D spin has two orthogonal axes of spin quantization, and these do not commute. Hence axes 1 and 3 (x and z) are incompatible which means the two cannot be measured simultaneously. This is why the 2D spin looks the same as usual spin when measured. The 2D spin gives a clear example of the Heisenberg Uncertainty relations.
The main difference suggested here is that the ensembles are made up of individual spins which have a 2D structure rather than being point particles. Once again, this is not postulated, but follows from the new Dirac algebra. Since they have structure, the spins can be conveniently viewed in their individual body fixed frames of reference, see Figure 2.
In the body fixed frame (x,y,z), a spin has two possible orientations, which bisect the (x,z)-plane in the even and odd quadrants. The unit vectors are given by,
Since the body fixed frame is related to the laboratory frame by a rotation by θ,φ and its orientation within each body fixed frame is given by n1=±1, the parameters (θ,φ,n1) are Local Hidden Variables (LHV). Averaging over those LHV is the same as ensemble averages, and must retrieve the quantum mechanical results. In a later blog, I will show this to be true by computer simulation.
If each axis, (x,z), carries a magnetic moment of magnitude µ, then the magnitude of the magnetic moment that bisects the quadrants is √2 µ. One question to ask is what are the consequences of this larger magnetic moment? The reader might have ideas.
Anisotropy and measurement
As seen in Figure 3, the application of an external field destroys the 2D spin. Recall that the mirror states and the states of definite parity (Part E of this series), depend upon the indistinguishably of the 1 and 3 labels (the x and z axes) within the 2D Dirac equation, and this requires that space be isotropic.
If the field is oriented between 0 and 45 degrees in the even quadrant, then the z axis lines up with the field and the x axis precesses in the plane perpendicular to the field and averages away, see Figure 3. Between 45 and 90 degrees, the x axis lines up and the z axis averages away.
Hence one important result is that the 2D spin is deterministic. We know from its orientation before it is filtered whether it is in the up or down state.
However the axis that precesses are quantum coherences and these are phase randomized away and make no contribution to measurement. This means the act of measurement destroys the polarization associated with the axis that precesses. Upon measurement, one must accept that only one axis can be measured so that any experiment can only detect half the spin polarization present in the system. This is simply a manifestation of the Heisenberg Uncertainty relations: the two spin axes of the 2D spin carry angular momentum which do not commute.
2D spin states
When not measured, however, 2D spin generally has differently oriented spin operator for every spin in the ensemble. Whereas the usual spin states observed are either up or down states, each 2D spin has a spin operator oriented along either one of the two bisecting directions. Rather than the usual two pure states from usual spin, the 2D spin displays four pure states: two for each orientation in its body frame, (Figure 2 pure states along each of the directions nn1).
The Pauli spin operator associated with the 2D spin is mathematically the same as usual spin. There are two Lorentz invariants of the 2D Dirac equation,
It is easy to find the eigenstates for this operator which depend on the LHV,
The states are given by
which are super-positions of the x and the z axes in the body fixed frame (not the usual laboratory frame). None-the-less they have the same usual representations as usual spin,
but once again in the body frame.
The usual approach to structured particles is to transform the states from the body fixed frame into the laboratory frame, where experiments are done. This is obtained by a simple rotation by angles θ,φ . Averaging over these angles for a specific quadrant must give the ensemble averaged result from quantum mechanics.
Although the mathematical basis for both usual and 2D spin is equivalent, two very different views of Nature emerge. The choice between the two spins will be made on the ability of one to resolve problems, and which is more physically appealing. In the following blogs, I will show that entanglement is not needed to account for the violation of Bell’s inequalities. I believe that this spin will shed light on the Double Slit experiment and perhaps other problems unknown to me.
The 2D spin does not only exist in an isotropic environment. Higher states that are entangled in quantum mechanics are not entangled if the 2D spin states are used. The singlet can be written as a sum of products. Therefore entanglement is not a property of Nature, but it is a valid approximation and a useful property of quantum mechanics.
Adopting 2D spin and a local realistic view of Nature is unlikely to interfere with the current success of quantum mechanics for most problems. It does, however, shift the emphasis. For example in quantum information theory, controlling entanglement should be replaced by controlling the LHV.