L: Wave AND Particle Duality: Coherences AND polarizations
The wave-particle duality concept is at the heart of the foundations of quantum mechanics. Here I want to show that the 2D spin gives a clear visualization that the wave-particle question comes down to how we look at spins. That is, the wave or particle nature depends upon how we set up the experiment.
This is old hat, of course, going back to the Einstein-Bohr debates, but the 2D spin gives a different perspective.
The question comes down to this: “wave or particle”, or, “wave and particle”. The most widely accepted version is the “or” group which includes the Copenhagen Interpretation (CI). But this is what bothered EPR, who claimed that quantum mechanics does not live up to a physical theory because QM cannot describe both position and momentum, and likewise two orthogonal axes of angular momentum. Recall that the 2D spin has two orthogonal angular momentum axes and so both cannot be measured simultaneously: in other words the 2D spin must satisfy the Heisenberg Uncertainty relations.
EPR make perfect sense except for their assumption (reasonable to them and me) that interactions between particles drops off with some inverse power law, and so are local interactions. On this point they were repudiated because of Bell’s Inequalities (BI) upon which the notion of non-locality rests. Non-locality is equivalent to the presence of those quantum channels that stretch over light-years and still remain intact between an EPR pair. Quantum channels are supposed to provide the conduits that allow particles to remain entangled after they have separated. (They make no physical sense to me.)
Filter spins: we must choose a direction to measure.
One of my objectives is, of course, to show that the violation of BI does not require entanglement and so restores locality to Nature. Using the 2D spin, the extra quantum correlations exists because each spin has magnitude √2 larger than when measured.
More on this later. Here I want to talk about the wave-particle duality because in the case of the 2D spin it acts as both a particle and a wave. Below I show again the √2 spin approaching a filter polarized in the “a” direction.
In Blog K of this series the states, which the polarizer resolves, are the usual spin states of |±>Z. Hence to summarize Blog K, the 2D spin basis can be used to resolve the observed states. Two representations are given. The first is when the polarizer is measured relative to the laboratory Z axis,
And the second is relative to the laboratory X axis,
The matrix elements are
Here “a” is the polarization vector. Both vectors “b” and “c” depend upon the LHV which orient the structured spin in the lab frame. These two vectors are orthogonal and so have magnitude √2.
Changing reps; changing experiments
The difference between the two representations, the laboratory Z rep and the laboratory X rep, simply means that as we change the representation, we flip between p+and p- , and this flips the experimental alignment.
Notice that what the flipping does is change the way we look at the spin: in the 31 quadrant, (b+c) or the 3-1 quadrant (b-c). If one is diagonal in the above matrices, the other is not, and vice versa.
Look at the Z rep above first. This means we are doing an experiment that is set up with the laboratory which defines an axis Z. In that rep the diagonal elements have particle nature and the off-diagonal elements have wave nature. Since the experiment is set up to measure only particle nature, the wave nature, although present, cannot be detected. Hence we have both wave and particle nature simultaneously.
But there is nothing sacred about the Z rep, and the second matrix above has changed to the X rep. Note again that the plus and minus are flipped. What was “particle” nature in the Z rep has become “wave” nature in the X rep, and vice versa.
However, who needs a representation? Well we do but Nature does not. Hence for a free electron drifting through space, it exists in one of two pure states and these bisect the quadrants as seen above.
Wave AND particle
I think this sheds light on things that have been said since the beginnings of quantum mechanics. In particular the concept that underpins the CI, which is: we set up an experiment (say Z rep) then we measure the observable compatible with that experiment, but not the (non-commuting) complementary observables. QM cannot describe the complementary observables which is why EPR said QM is incomplete.
The 2D spin shows that the experiment forces the observables to have either particle or wave nature, but disagrees with CI which states the unmeasurable wave nature does not exist. It does exist in this work, and to retrieve it one simply does what the CI suggests, set up a different experiment.
Now perhaps those who know BI can see where this is leading. That 2D spin carries polarization along two axes and each axis contributes √2 correlation to the CHSH form of BI. Hence this spin, I will show, accounts for all the correlation, 2√2, that violates BI. This definitely goes contrary to Bell’s theorem, and as a result, non-locality is history.
My conclusion is that the 2D spin gives a clear answer to the wave-particle duality. I discussed the origin of the Heisenberg Uncertainty Principle in Blog J, and the wave-particle duality is just another way of looking at it. I hope I have shown that the wave and particle nature (or the polarizations and coherences in the two matrices above) provide evidence that my model gives a view to Heisenberg Uncertainty which is physically reasonable.