L: Wave AND Particle Duality: Coherences AND polarizations

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.

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.

A free 2D spin in one of its states 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.

 

 

K. Filtering the 2D spin-cannot filter coherences

In the treatment here it is believed that before entering the field, the spin is a free particle and displays the √2 states. It is this spin that starts off in the superposed states of the two orthogonal axes, that depend upon the LHV, |±,r=q,f>n1=±1. Note here there are four pure states: two associated with n1=+1 and two with n1=-1, which cannot be simultaneously measured. We measure either n1=+1 states or n1=-1, but not both simultaneously. Half are averaged away when measured.

In the last entry I showed that the two dimensional spin cannot be fully characterized by measurement because of the Heisenberg Uncertainty Principle.  This results because when measured, half the states are randomized away.  In fact it was concluded that this 2D spin is capable of measurement along each orthogonal axis, but only one can be measured in an experiment.  This means that each such spin is theoretically capable of twice as many clicks as usual spin, even though only half can be detected.

Stated otherwise, the same number of events can be detected from usual spin as the 2D spin, but that number corresponds to only half the events that are possible from the 2D spin.

This point is important for resolving EPR.

Laboratory versus body fixed frame

In this entry I want to make only one small point and contrast the usual 3D Dirac equation and the new 2D equation. I write down both Dirac equations for a free particle spin in the two cases:

Which correspond to two different Dirac Algebras respectively:

The lower case labels, x,y,z denote the body fixed frame and the upper case labels, X, Y, Z denote the laboratory frame.  To see the difference, consider a Stern-Gerlach filter, 

On the Right Hand Side, the spins are displayed as in two states: up and down, |±>Z.  These are the states observed in experiment and are the usual states described nicely on the Bloch Sphere.  However just like the double slit experiment, these two states, as described by quantum mechanics, are found only after a large number of spins have passed the inhomogeneous magnetic field.  Passing one spin does not give the quantum state.

In other words, consistent with the way quantum mechanics is interpreted, we cannot predict if one spin will be deflected up or down.  Only after a statistically large number of spins have passed does the quantum result follow (called Malus’ Law).

Conclusion: the usual three dimensional Dirac equation leads to the usual point particle spin and, when filtered gives the observed states |±>. These quantum states are observed only after a statistical ensemble of spins pass the filter. Many different experiments are done with similarly prepared spins passing the same filter. These accumulate to give the usual observed probabilities of measuring spin up and spin down.

What happens to one spin?

In contrast, and a critical point, the 2D Dirac equation describes one spin that makes up the statistical ensemble. That is, the 2D equation describes a spin before it enters the filter.  Before filtering the free spin displays the √2 spin with four possible states.  As it reaches the filter, the spin deterministically is deflected up or down thereby showing that Nature is deterministic.  As the spin feels the deflection, one of the body fixed axes lines up with the laboratory field and the other precesses perpendicular to that axis and is randomized away as discussed in Blog J of this series.

In short, the 2D Dirac equation describes the state of a single spin before filtering and the usual 3D Dirac equation describes the statistical state that is measured after a large number of 2D spins have passed the filter.

It is that simple.

No spin density matrix for one spin

Now when a bunch of spins are polarized after being filtered, they are organized into pure state of |±>Z. These are polarized along the laboratory Z axis and in general, going back to U. Fano in the Reviews of Modern Physics in 1957, the way to handle a statistical ensemble of spins is with the spin density operator which is given by the well-known formula

Where Pz is the polarization of the ensemble of spins (how well do they point up or down).  If the state is pure up |+>Z, then Pz = 1, and if the state is pure down, |->Z, then Pz = -1.  Any other state is a superposition of the up and down states.  All this is so well known.  But notice, there are only two states.  The density operator ignores the quantum coherences (generally) because they phase randomize away for macroscopic observation. (just like we saw in blog J.)

What is the equivalent treatment for the spin before it enters the Stern-Gerlach filter?

Recall that Pz is the polarization of the ensemble.

Polarizations and coherences

In the treatment here it is believed that before entering the polarizaing field, the spin is a free particle and displays the √2 states. It is this spin that starts off in the superposed states of the two orthogonal axes, that depend upon the LHV, |±,r=q,f>n1=±1. Note here there are four pure states: two associated with n1=+1 and two with n1=-1, which cannot be simultaneously measured.  We measure either n1=+1 states or n1=-1, but not both simultaneously.  Half are averaged away when measured.

For this situation we want to display the quantum coherences, so we cannot use a density matrix since the quantum coherences will be lost.  Rather it is better to resolve the identity in the basis of the 2D spin.  See the figure, 

Here is depicted a filter set at angle “a” relative to the laboratory Z axis.  The 2D spin approaches the filter in one of its √2 states.  These states display both quantum polarizations, diagonal elements, and quantum coherences, off-diagonal elements, and resolving the identity in the 2D spin representation gives, 

The matrix elements are (after a lot of work) 

Where now pa± is the polarization of one spin (compare with the density matrix above.)

Here “a” is the polarization vector, and “b” and “c” are orthogonal vectors which correspond to the representation (projection) of the 2D spin in the laboratory frame rather than the body fixed frame.

Can only detect polarizations

In an experiment only the diagonal elements can be detected.  The quantum coherences are lost.  However there is nothing special about the laboratory “Z” axis.  We could just have easily used the laboratory “X” axis.  If we make that change of representation, then all that happens is the quantum polarization and quantum coherences are flipped, to give (note “+” and ”-“ are interchanged) 

Now the tables have turned.  The polarization is the Z rep become coherences in the X rep and vice versa. In either rep, half the polarization is unavailable for measurement.  Each rep requires a different experimental set up.

This gives some food for thought.  The 2D spin completely agrees with the results of experiment.  However not everything can be measured in one experiment, so that only half the polarization available in the system can be detected.

It is the inability to detect the polarization from both axes of the spin that Bell’s inequalities appear to be violated.  This will be discussed later.

The main point here is that the set-up of the experiment limits our ability to measure all the polarization and what is missed is just the amount needed to resolve the EPR paradox and avoid entanglement.

J. Heisenberg Uncertainty and the 2D spin

Space is now no longer isotropic in the presence of a measuring probe, and so the √2 spin cannot form, nor can the mirror states. Since the spin is oriented some way, one axis is going to be closer to the applied field than the other. That one lines up while the other axis spins in the plane perpendicular to the applied field,

The 2D structured spin gives an example which provides a basic understanding of how Heisenberg Uncertainty Principle works. The mathematical basis for this Principle follows because observables in quantum mechanics sometimes do not commute.  Examples are position-momentum; energy-time; difference orthogonal components of the angular momentum vector. For spin, the Pauli spin operators do not commute.

Immediately it should jump out that the two components of the 2D spin are orthogonal and cannot simultaneously be measured. Each carries a magnetic moment and hence each has a spin operator which must be orthogonal. It follows from Heisenberg that only one of the two components can be measured in one experiment. One is always missed.

This point becomes important in analyzing EPR experiments. It also means that one cannot properly characterize spin by measuring it because the √2 spin states cannot be observed.

Here, my plan is to give a visualization of how only one of the two axes can be measured and the other one not.

Laboratory and Body Fixed Frame

In spectroscopy, it is common to define a body fixed frame for a molecule.  In that frame its structure is more easily expressed.  Then, using rotation matrices, the body frame is transformed to the laboratory frame, where experiments are done.  The same is done here for the 2D spin.  See the figure, 

On the left one 2D spin is show in its body fixed frame.  Each spin generally has a different frame, and all are related to the Lab frame by two angles.  Note also in the body frame, the two spin orientations are shown but only one exists at any instant. Here space is isotropic, so the √2 spin is shown bisecting the quadrants.

Let us now apply an external magnetic field along the laboratory Z axis.  Recall that a spin carries a magnetic moment that is so tiny that it is swamped by any external measuring field, so it lines up with that field,

Space is now no longer isotropic in the presence of a measuring probe, and so the √2 spin cannot form, nor can the mirror states. Since the spin is oriented some way, one axis is going to be closer to the applied field than the other.  That one lines up while the other axis spins in the plane perpendicular to the applied field,

 

Then it looks just like a usual spin ½ (shown in the middle above) with the same magnitude of magnetic moment, and is described by a single axis of quantization.  The 2D spin looks like a point particle when observed.

But it isn’t.  The application of a measuring probe makes it impossible to measure the √2 spin and the results are identical had the experiment assumed the usual point particle spin.

I know of only one quantum calculation of the Stern-Gerlach experiment, which the above depicts.  That is by Scully et al in 1969*.  I did the same calculation using the 2D spin and the results confirm the conclusion of Scully et al: that the linear momentum and the spin angular momentum couple so the particle is deflected in the direction of the applied field. The orthogonal states are averaged away.

*M. O. Scully, W. E. Lamb Jr., and A. Barut, On the Theory of the Stern-Gerlach Apparatus, Foundations of Physics,  1987, 17, 575.

Gedenken experiment

There is a mathematical “trick” that is often used to simplify things.  If one looks at an object which is spinning around some axis, it is possible to transform the equations so that they are all spinning at the same rate.  In other words we change coordinate frames to the rotating or spinning frame.  In that frame, the 2D spin as not spinning anymore and the two orthogonal components are frozen as in the figure below,

The orientation of the applied field is “a” which could very well be the laboratory Z axis.  The experimental result, which the 2D spin confirms, is that the Z component displays two pure states which are the usual spin ½ states of |±>Z. Experimentally, in a Stern-Gerlach experiment, the deflected spin leaves two spots on a photographic state, one for the “+” state and the other for the “-“ state.

The same results would be observed for the 2D spin.

Counting events

Note in the figure in the spinning frame that the two orthogonal components are frozen along directions “a’ and “d”.  We have already stated that the “a” component lies along the laboratory frame and gives the two states |±>a. Experimenters refer to the detection of different states as “channels”.  That is, for this experiment, there are two channels in the “a” direction and every time a “+” state is measured, it is added to the “+” channel bin.  A bin is simply a place (a computer file) which gives the number of events that occur in that channel.  Likewise there is a “-“ channel and bin.  Creating an inhomogeneous magnetic field and setting up the channels and bins is the job of the experimentalist.  He counts “clicks” which are the responses of events from detectors.  This is the experimental set up.

Now let us look at the “d” direction.  In the spinning frame it looks the same as the “a” direction, so what the experimentalist must do to detect along the “d” direction is to build the same sort of apparatus but which is spinning at the same speed as the spin is spinning.  Although this is an impossible task, we can think about it and perform a gedenken experiment.

If it were possible to build an apparatus like the above, then we could measure simultaneously along both the “a” and the  ”d” axes.  Then we would have four distinct channels and four distinct bins.  Two bins would count clicks along the laboratory axis and two bins would count clicks in the spinning frame.

This means that the 2D spin displays twice as many clicks as does the usual spin which has only one axis.

In reality, knowing that spins spin at megahertz frequencies, it seems an impossible task to be able to create such an apparatus (although I am quick to add that experimentalists are extremely creative and may be able to pull something off like this somehow.)

Conclusion

The conclusion is that although four clicks are theoretically possible for the 2D spin, only two can ever be measured.  Coming back to the Heisenberg Uncertainty Principle, it is clear why it is impossible to simultaneously measure along two orthogonal axes of spin quantization: we cannot build such an apparatus.

This makes a difference for the EPR coincidence experiments because in a given run, N coincidences are detected.  However there are actually 2N coincidences possible but only half can ever be detected in a given experiment.

The reality is that it is unlikely that such an apparatus can be built and so those spinning orthogonal components simply phase randomize and are not detected.

This also underlines something that is quite unexpected.  Physics is an experimental science.  However for spin, if it turns out the 2D version is viable, then there are some states that can never be observed.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I. What does a Singlet State look like?

When I was a graduate student, and studying quantum mechanics, I came across a statement by Heisenberg which impressed me.  We have no trouble visualizing the macroscopic world.  It is our common environment and when someone throws a ball to you, you need no Newtonian mechanics to catch it.  If you had to catch an electron, well you have no idea without some help.  That is because an electron is part of the microscopic world, which is impossible for us to visualize without knowing the equations.  This is where Heisenberg came in and said that the only way we can actually visualize the microscopic world is through understanding the equations.

It is through the equations, and only the equations, that we can form a mental picture of microscopic processes.  Such mental images are very useful.

As Heisenberg said, we develop “visualizability” of the microscopic world through following the logic of nature which for us is mathematics.

The usual singlet state

In blog H. of this series, I used the usual depiction the singlet state of, say, the hydrogen molecule like:

This certainly gives us a visualization of the singlet state showing the two electron having opposite spin states of up and down thereby cancel out their magnetic moments.  On the other hand, as all chemists know, the electron charge on each electron pushes them apart, while the opposite magnetic moments pull them together.  A lot of chemistry is based upon this.

Recall from blog H that these two spins in a singlet state are entangled.

If that singlet state is somehow separated, like in the coincident photon experiments, it is assumed that the two spins remain entangled no matter how far apart:

These are the so-called “quantum or EPR channels” which are supposed to exist but no-one can explain. These “quantum channels” are the quantum version of “classical channels” (like a phone line), which are said to be necessary for our understanding of things like “quantum teleportation” as well as playing a major role in most of quantum information theory.

That is, the two spins are believed to retain the singlet state whether they are close together, locally like in a bond, or if the have separated to the other ends of the universe, and are non-local.

Wrong! There is no physical basis that I can see that justifies this view. (help me out here if you can.)

Since EPR believed in locality, I am sure that they would turn in their graves if they knew these non-existent quantum channels were named after them.

The 2D spin.

The structured 2D spin that I have been talking about leads to a very different visualization of the singlet,

In this view, the two electron charges repel and are just balanced by the attraction of the magnetic poles.  Recall that there is no magnetic moment of magnitude √2µ here because that state can only exist for free electrons. Singlet states, and higher order states, have interacting electrons. The structured spin gives a physically reasonable representation of the singlet.  At least it makes sense to me and fits together nicely.

Consider what happens if this were an electron and a positron.  Then the charge on one of them would change from negative to positive and the two charges and the magnetic moments would both be attractive. This leads to a visualization of electron-positron annihilation into energy.

What happens when the singlet above is separated?  As soon as the two electrons are far enough away from each other, and in an isotropic environment, the √2 state will appear.  Hence the separation will look like this:

There is no entanglement.  Note also that the two spins have the same orientation but opposite angular momentum.  That is the Local Hidden Variables are the same for the two.  This is the 2D spin representation of an EPR pair. These two spins move in opposite directions (conservation of linear momentum) towards filters where they can be detected, and the correlation between them can be measured.

Now the whole point is it is well established (incorrectly in my view) that without entanglement, one can never account for the violation of Bell’s Inequalities: the basis upon which non-locality rests.  Therefore if I can show, and I will in later blogs, that this model carries all the quantum correlation without entanglement, then the 2D spin serves as a counter example which disproves Bell’s Theorem.

Summary

In the next few blogs, I will show that a product state formed from the 2D spin can, in fact, account for all the correlation between an EPR pair that violates Bell’s Inequalities. No entanglement necessary.

It also contains some unexpected, yet reasonable results.  First, indeed it can be shown that the correlation is the same as if the spins were entangled because the larger magnetic moment makes for more correlation.  However, note that the two axes of quantization are orthogonal.  Since each of those two axes carry magnetic moments which must also be orthogonal, their operators do not commute.  Hence only the correlation from one axis (either one) can be detected, but not from the two simultaneously.  To do so would violate the Heisenberg Uncertainty principle.  Just like position and momentum, two complementary experiments are needed in order to measure along both axes of a 2D spin.

What this means is that only half the correlation can be obtained from a singlet in one experiment.  This means that EPR experiments actually satisfy Bell’s inequalities even though it carries all the correlation!

The two complementary experiments on the 2D spin would measure all the correlation, but the results from either are identical to those from usual spin. Naysayers will point to this and say “so you have no way to know if the 2D spin is a better choice than usual spin.” Indeed what is measured is a spin with a single axis even though it has two.

However when you look into it, the 2D spin leads to no disagreements with EPR experiment, but does require a new interpretation of them.

Which spin is correct? “The proof of the pudding is in the eating”, and some of it comes together as I will show.

H. Entanglement or Disentanglement

I have said that the √2 spin only exists when space is isotropic. However when interacting, say with other particles, although the √2 magnetic moment is destroyed, none-the-less the 2D structure remains and this actually removes entanglement from quantum theory!!

The 2D structured spin leads to the removal of all entangled states in quantum mechanics.  Read on, but first the perspective.

The point about this series is that I have found a Dirac algebra and Dirac Equation which gives a completely different physical interpretation of spin but rests on the same mathematical basis as usual spin. Since measurement cannot distinguish between the two, there is a choice.  Fundamentally the choice is between reality and anti-reality. It is also a choice between disentanglement and entanglement.

Since the 2D spin changes into usual spin upon measurement, it seems reasonable that the 2D spin is more fundamental than the usual formulation.

Usual spin is described by the Pauli spin operators and the identity, (I, σ1, σ2, σ3) with a permanent magnetic moment along any direction (say pointing to the surface of a Bloch sphere). The new structured spin is described by (I, σ1, iσ2, σ3) has permanent magnetic moments along the 1 and 3 axes, and these add, giving a total magnitude for the magnetic moment which is √2 larger than can be measured.

What consequences does a larger magnetic moment have on properties that depend upon spin? Think of the clouds of free electrons that are found throughout the universe. Any calculation that involves free spin needs to be examined again.

Bound electrons and entanglement

I have said that the √2  spin only exists when space is isotropic.  However when interacting, say with other particles, although the √2  magnetic moment is destroyed, none-the-less the 2D structure remains and this actually removes entanglement from quantum theory!!

So what is entanglement in a nutshell?

“Entanglement” was coined in 1936 by Schrodinger who said of it,

I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.

Think of a singlet state, the smallest entangled state.  We see this in the hydrogen molecule in high school chemistry classes,

The two electron spins line up, north-south and south-north to give zero angular momentum.  Since these electrons are close enough to interact, the singlet state is a local state.

Although two electrons are indistinguishable in Nature, we mortals need labels to keep track of things, so we make them distinguishable,

But if we write a product state, then upon permutation the wave function does not change sign:

However if the state is entangled, then permutation of 1 and 2 is odd, as it must be for fermions,

Equation (2)

The reason this state is entangled is that it cannot be written as a product state.  The two cannot be separated into a product state. It is “an undivided whole”. But is it?

Now the difficulty with entangled particles is most people believe that when a singlet  is separated (e.g.as is done in  EPR coincidence experiments), the entangled states remain entangled over any distance.  No one knows how. To me it makes no sense.  It requires us to accept instantaneous action-at-a-distance.

However if the 2D spin is used, then entanglement disappears from the theory and all entangled states can be written as a sum of product states.  First, the figure below is how a singlet might look if formed from the 2D spin,

The repulsion of the charge is just balanced by the attraction of the magnetic moments. The two spins share the same LHV (Local Hidden Variables) which also means they share the same body fixed coordinate frame.

Coherency

Usual spin is defined by a density operator which for pure states looks like (U. Fano 1957 Rev. Mod. Physics)

Which simply says a single spin has the usual states of |±1><±1|.  Of course this is what is observed.

The 2D spin, in contrast, carries not only the diagonal states, but also off-diagonal quantum coherence. These depend upon how we represent the spin. Above, and below, we use the “Z” representation.  Above, it is the laboratory frame, Z, and below it is the body frame of the spin, z. This means that the contributions from the “x” axis are off-diagonal, and appear as a coherences. Coherences cannot be directly measured. Here are two states out of the possible eight which you can contrast with the above equations,

This spin not only shows the diagonal polaraizations, but also coherences.

Of course it is easy to switch from the “z” rep. to the “x” rep in which case the diagonal terms become off-diagonal coherences, and the off-diagonal terms become diagonal polarizations.  That process just flips the z and x reps showing that there is really no difference in physical interpretation between the two reps.  However although experiments are possible in either representation, only one can be measured at a time. This is fundamentally because the two axes do not commute and Heisenberg’s Uncertainty Principle comes into play.

Disentangling the Singlet state

Simply calculate the matrix representation of the singlet by substituting the single spin states.  The matrix representation of the singlet in the usual states of equation 2,

Now let’s take the states of two single 2D spins, (s1+++,  s2+-+, … etc). and note these are coherent states like above.  The two spins that form a product must be in the opposite quadrants (to have opposite angular momentum). But there are four quadrants and therefore four ways the two spins can form a product state.  So summing all four contributions gives,

It is easy to plug the Pauli spin matrices into the above and add up all the products to give the expression for the entangled singlet (see end).  In other words, the 2D spin shows that there is no entanglement.  The singlet state can be written as a sum of product states.  This is not possible using usual spin.  The 2D spin, however, carries the quantum coherences which just do the trick to allow the states to disentangle.

It might be useful to connect this with the 2D Dirac equation discussed in earlier blogs of this series. Recall that mirror states were formulated, and these are seen in the above equation: each line of that equation has the last subscript as “+” on the first product and “-” on the second. Same for others.  Each state operator in the above product is described by a non-hermitian and not-Lorentz invariant equation.  If this were solved, then and the products taken, the hermitian singlet state, which is Lorentz invariant, would result.

A final note of speculation is based on the notion that each line of the above singlet disentanglement corresponds to a possible orientation of the two spins, but those two spins can be in only one quadrant at any instance.  Therefore, upon separation, it is suggested here that the product state from only one of the four lines exists, and this, when separated, forms the single 2D spin in free space which is described by the 2D Dirac equation.

It does not stop here.  The three entangled triplet states (the other Bell states) are similarly decomposed into products, and so on for all entangled states composed of any number of particles. In short, it is easy to modify the Clebsch-Gordan series that couples angular momentum to include the 2D spin. That series contains no entanglement.

This is one of the papers I got up on the quantum archives before the moderators blacklisted me for my ideas.  Paper is called Separation of Bell States.

In summary, the 2D spin plays a role for all spin interactions and removes entanglement from quantum theory.

Entanglement is a property of quantum mechanics, but not of Nature.

Here is the simple disentangled proof:

 

G. Structured 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.

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

Figure 1. The usual spin of an electron as observed. The spin aligns with the external magnetic field giving a point particle with intrinsic angular momentum.

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.

Figure 2. The two orientations of the 2D spin in its body fixed frame. The spin has a magnetic moment directed along the x and z axes forming the resultants that bisect the quadrants.

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.

Figure 3. Application of a measuring probe destroys the isotropy of space and therefore also the mirror and parity states. One physical axis lines up with the field and the other precesses in the perpendicular plane, thereby randomizing half the available polarization.

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.

Summary

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.

 

F. My Black Swan

The book called The Black Swan by Nassim Nicholas Taleb finds many events are unpredictable and occur suddenly, and therefore have a large impact on our lives and thinking.  All swans in Europe were white, leading to the paradigm that all swans are white. However the discovery of one black swan in Australia,  changes all this leading to a new paradigm about swans.  Black Swan events in history, geopolitics and science often occur ; they are unpredictable, they have a large impact, and they are usually only explained after the observation.

Basically Taleb believes that our lack of knowledge is as important as our knowledge. If we know too much, then we make predictions which are usually a continuation of our present history.  We really cannot predict: the stock market, world wars, epidemics, down to a sudden illness or a personal crisis: all are black swans.  They can indicate either positive of negative events, but in all cases it is their suddenness that takes us by surprise and only then can we start to deal with the impact.

Structured Spin Black Swan

I believe that my 2D structured spin falls into the category of a Black Swan event. Other events in physics are Black Swans: the world is not flat; the Earth is not the center of the universe. Then we have the ultraviolet catastrophe of Black body radiation, explained by Planck, which ushered in quantum theory. There is the impact of relativity, then Black Holes and the Big Bang. These events, and others, shaped our view of Nature and the present paradigm which contends that quantum mechanics (or quantum field theory) is the most basic description of Nature.

I would say that Bell’s Inequities and his theorem (that any Local Hidden Variable Theory be non-local) are also Black Swans, because the existence of quantum correlation leads to the violation of Bell’s Inequalities.  This has been interpreted, (incorrectly in my view) that entanglement must persists when entangled states are separated.  Hence Nature is to have non-local interactions-how?

Before Bell, ( say in the mid 1960’s) everyone believed Nature to be local, although not deterministic.  Today non-locality, or instantaneous-action-at-a-distance, is accepted by most Physicists, and the experimental evidence rests upon one type of experiment (photon-coincidence).

Clearly the non-locality Black Swan is mind-boggling because no-one understands it.

But I believe it to be wrong.  This does not mean that I believe Bell’s mathematical proof of his inequalities is wrong, although doubters exist.  I also completely believe the experimental results from photon coincidence experiments.  I do not believe, however, that the interpretation of the experimental results is correct.  My spin gives a local and realistic view of Nature. I think choosing that makes more sense that accepting the absurd.

Consequences of the Black Swan of structured spin.

If my 2D spin were accepted, and people started to study it, I am certain many more problems in physics, unknown to me, could be resolved.  What impact is there to Nature if the magnetic moment of an undisturbed electron is √2 larger than can be measured?  Certainly I have used this successfully to account for the violation of Bell’s Inequalities without entanglement–that it gives a resolution of the EPR paradox. Similarly I believe that quantum information theory (computing, teleportation and quantum cryptography), are not moving along very well.  I believe that emphasis should be placed on controlling the LHV and not entanglement.  But there are many more problems that exist: the double slit experiment, dark matter, parity breaking in beta decay and prediction of mirror states;, etc.

If we suppose the standard model is correct, then I have a problem. The standard model is composed of 16 particles )[excluding the Higgg’s boson), and they all have spin of either 1/2 or 1.  All these particles are considered to be point particles.  Hey, even the Black Hole that exploded to form our universe is treated as a singularity in the theory.

You can imagine the opposition I receive when I say that these 1/2 spin particles  (all 12 of them in the standard model) are not point particles, but have a 2D structure.  Well, it seems that in the initial Big Bang, what  happened in the first 10-37 seconds is not understood.  Maybe in this early stage, the two axes of my structured spin 1/2 were formed as a condensate to produce those 12 spins. However to suggest that spin has structure, means that the Standard Model needs to be looked at some more.

Another impact of a 2D spin is that it means that Quantum Theory is not the most fundamental description of Nature.  Physics is and experimental science, and if the 2D spin exists, then it must lie deeper than quantum mechanics. For me, although measurement is critical for us to obtain knowledge, measurement is simply another interaction in Nature.  Nature does not care if we measure, so it seems to me there is nothing fundamentally wrong in assuming there are structures below our ability to measure.

Entanglement, said Schrodinger in 1936, is not a difference between classical and quantum mechanics, but the difference.  I agree but go further and state that entanglement is a property of quantum mechanics, but not of Nature.

Local Reality

Finally, of all the other points I can make, the structured 2D spin restores locality and determinism to Nature and this makes me very happy, and I believe would be accepted, especially if many other problems in Physics can be rationalized using the structured spin.

That is why I see my Black Swan as paradigm changing.  For me it is a no-brainer to do away with non-locality and indeterminism, but then others, who have not gone through my objective developments, like in parts A to E in this series, will simply accept the status quo and in most cases, usual spin causes no problems.

Still it would be nice to get our understanding of Nature right.

I will add other blogs later in which I will show some aspects of the Coincidence Photon experiments, and how the structured spin leads to two simultaneous coincidences, although only one can be detected.  This means that half the EPR correlation cannot be detected and, big surprise, Bell’s Inequalities are no longer violated.

 

 

E. 2D Dirac Equation and Spin

One therefore has a choice. Accept usual spin that leads to entangled states and a non-local and indeterministic foundation of Nature. Alternately, you can choose the 2D structured spin which gives both a local and realistic view of Nature. Experimentally, the two cannot be distinguished and so the treatment here is not inconsistent with any experimental results.

 

In my previous blog, D. A  Dirac equation for mirror states, it was shown that the two dimensional Dirac algebra leads to mirror states, ψ±

This can be re-written by combining the two mirror states,

Upon reflection of the 1 and 3 axes the mirror states are interchanged

See the figure,

It follows that the above superpostion gives odd and even parity states,

All this, however, is destroyed by the application of a vector field needed for measurement.

I have a more prosaic view of mirrors states and am not quite able to grasp the notion that there exists a whole mirror world all around us which we cannot see. Rather I just see one state being described in a RH coordinate system and its mirror image in a LH one. From my understanding, Nature does not need a coordinate system, but we do.  Therefore the state, and its mirror image are identical to Nature.

The Dirac equation obtained from the 2D Dirac algebra is non-hermitian and not Lorentz invariant, so cannot have solutions that are physical states. They describe a state, and its mirror image.  You cannot have one without the other, so they occur together. However simply take the sum and difference of the first equation to obtain the following two equations in terms of states of definite parity,

Now we come to a critical part.  The two mirror states are shown in the figure and if you add them, the even parity state is independent of the  “2” axis, and if you subtract them, then the odd parity state is independent of the “1” and “3” axes.  Looking at the second equation above, clearly the first operator does not depend on the 2 axis and the second operator does not depend upon the 1 and 3 axes.  Hence the second equation has two terms that vanish.

2D Dirac and Phase Time Equations

This leaves the first equation, and since the even and odd parity states are orthogonal, the two separate into two independent equations.  One describes a constant precession,

which gives the rotational or phase time.  The second equation is what I am after, which is the two dimensional Dirac Equation,

This equation is both hermitian and Lorentz invariant.

It is important to keep in mind that this is one spin oriented relative to some coordinate frame by angles r=(Θ,Φ) (the body-fixed frame which are Local Hidden Variables LHV) and refers to this particular spin.  In general every spin is oriented differently, and so the spin operator above differs for each spin.

Two Lorentz Invariants

There are two Lorentz invariants, One bisects the 3,1 plane and the other bisects the 3,-1  plane (i.e the even and odd quadrants of the body fixed frame),

We now have another LHV.  In addition to the two angles that orient the structured spin in 3D space, r=(Θ,Φ), the integer, n1 =±1 indicates which quadrant the spin is oriented. See the figure,

The two possible spin orientations in the spin's body fixed frame, either in the even or odd quadrants.

 

Since the 1 and 3 axes are assumed to have a magnetic moment of μ the magnitude of the two Lorentz invariants is √2μ. That means that when this spin is not observed, it has a magnitude of √2 larger than observed. This must have some interesting consequences.

Spin Operator

It is straight forward to find the spin states from the above Pauli spin operator,  These states depend upon the LHV (r and and n1 )

which is the same form as the usual Pauli spin operator, and so it is possible to take over the usual treatment of spin and define the spin operator,

Finally the magnetic moment is root 2 larger than is observed, and it lies along the bisectors of the body fixed frame in the 13 plane.

Summing up

The purpose of this blog is to show that using a different Dirac Algebra, a 2D Dirac equation is obtained and the spin from this bisects the quadrants of the spin’s body fixed frame. The treatment is as mathematically sound as the usual treatment of spin, and therefore puts the structured spin on the same firm footing as usual spin 1/2.

One therefore has a choice.  Accept usual spin that leads to entangled states and a non-local and indeterministic foundation of Nature. Alternately, you can choose the 2D structured spin which gives both a local and realistic view of Nature. Experimentally, the two cannot be distinguished and so the treatment here is not inconsistent with any experimental results.

 

D. A Dirac equation for mirror states

That is, these two states are reflections of each other, see the figure, The operation of reflection via P13 changes one state into its mirror image. This is exactly the property sort by Yang and Lee to solve the fact that parity is not conserved for the electro-weak force. Using their example, if cobalt atoms undergo beta decay, and you watch it in a mirror, then the magnetic moments are not reflected, and so parity is violated.

It is shown here that the different Dirac Algebra leads to a different Dirac Equation and this diplays mirror states which combine to give states of odd and even Parity.

I suppose those who do look at this blog wonder what I am up to because who’s interested in these equations?  Well I am not trying to reach the general public right now, but on the other hand the Dirac equation and spin are something all undergraduate physics students study.  I have felt for a long time a different look into the foundations of physics is needed.  Things have not moved much in areas where entanglement is used as a resource, although papers abound.  I believe people are beginning to think there must be some way out of the problems with physics, from dark matter down to our failure to describe the double slit experiment. We must be missing something.

Then of course there is the absurdity of non-locality, my pet peeve.

I believe the way out of many of these problems is to accept that spin has a 2D structure rather than being a point particle.  Both structured and point particle spin rest on the same firm mathematical foundations and offer a choice between the status quo, and something that answers a lot of questions, at least for me.  So I thought I would write down those ideas in the simplest way, yet detailed enough so that those who have the background can follow.

A different Dirac equation

In the last entry, Part C, “A different Dirac Algebra”, I noted that the small change in the gamma matrix γ2 of replacing the Pauli spin operator σ2 by iσ2 makes a big difference.  First it changes one of the three spatial variables from our usual 3D real space to 2D space.  (I am tempted to say “Welcome to Flatland”).  That spatial variable changes into a new time variable, a rotational time, which is frequency or phase.

This new time is analogous to usual linear time but for angular motion.  So just as different inertial frames have different linear times, so the flat plane of 2D spin can rotate at different relative frequencies.  Spin is a different sort of matter from mass with momentum, having angular momentum rather than linear momentum.  There are two types of angular momentum, orbital (like moon moving around the Earth) and spin (like a spinning top). It is not surprising that angular time arises for structured particles.

Usual spin is considered to a point particle, so angular time cannot exist.

A different Dirac Equation

As soon as we have a Dirac Algebra, we can immediately write down the Dirac Equation.  For the different Dirac Algebra of

the different Dirac equation has two forms

 

This equation is neither hermitian nor Lorentz invariant* which is usually bad news for any equation, but let’s move on and see where this leads.

The reason it has two forms lies in the fact that space is locally completely isotropic for an isolated particle, say an electron in interstellar space. When a spin interacts with something, like when we try to measure it, then space becomes anisotropic (we turn on a magnetic field (which is a vector) to measure its magnetic moment).

If space is isotropic, then the labels 1 and 3 are indistinguishable and can be permuted without changing the physical meaning of the equation. Therefore if 1 and 3 are interchanged, the 2 term changes sign.

Let P13 be the operator that permutes these two labels, then

And we get the two forms above for the different Dirac equation.

Mirror States

In fact the operator P13 is a reflection operator.

Applied to the Dirac equation above it gives

 

That is, these two states are reflections of each other, see the figure, The operation of reflection via P13 changes one state into its mirror image.  This is exactly the property sort by Yang and Lee to solve the fact that parity is not conserved for the electro-weak force.  Using their example, if cobalt atoms undergo beta decay, and you watch it in a mirror, then the magnetic moments are not reflected, and so parity is violated.

This does not make much sense but is an experimental reality.

However when beta decay occurs and we do not observe it, then parity is conserved.

Parity States

From the above different Dirac equation, the two mirror states exist simultaneously. Therefore we can define states as sums and differences of the two,

These states have definite parity when reflected,

And so parity is conserved in this different Dirac equation, but destroyed when we try to measure those states.

The next step is to change that non-hermitian, non- Lorentz invariant Dirac equation from mirror states to states of definite parity, .  When this is done, the resulting equation is both hermitian and Lorentz invariant.

And that makes good physical sense.

Summary

Mirror states were postulated to resolve the parity breaking of the electro-weak force and from the different Dirac algebra, they appear naturally.  Since there is no external field, the magnetic moments (that bisect the quadrants of the 13 plane), are reflected faithfully.  In contrast, in the presence of a magnetic field, the 1 and 3 components cannot be permuted because each axis is generally oriented differently with respect to the field, and are no longer indistinguishable.  Hence the mirror states cannot exist when observed, and parity is broken.

I will discuss this different Dirac equation in the next entry.

(*)  A non-hermitian equation means that things we observe can be complex (z=x±iy), rather than real (z=x).  An equation that is not Lorentz invariant means that physics at different places in the Universe is different.

C. A Different Dirac Algebra

But two times? The first is the usual linear time that differs in different inertial frames. The second is a rotational time which rotates in the plane of the 2D flat space. This is a phase time or a frequency and accounts for the different relative rotations of 2D objects in different inertial frames.

The Usual Dirac Algebra

Paul Dirac’s genius was to realize that the Klein-Gordon equation, which conserves energy and mass, would perhaps show more structure if it were a first order differential equation in space-time rather than the second order Klein-Gordon equation. This led to spin and anti-matter for starters. The trick he used required that a set of four matrices (called the gamma matrices) had to anti-commute, (i and j take values of 0, 1, 2, 3)

Where I4 is a 4×4 identity matrix and the metric tensor given by,

This means that there is one time variable, the +1, and three spatial variables, the -1.  Good old space-time again.

It is not necessary to go into the properties of the gamma-matrices.  They are well known. However their representation in terms of the Pauli spin matrices is important for what is to follow, so here they are,

There are two spins here: one for matter and the other for antimatter.  From the Dirac equation, spin emerges as a Lorentz invariant defined by the operators {I2123}.

A Different Dirac Algebra

At the end of the first blog (A) in this series, I said that there is a different way to define the gamma matrices for a spin ½.  That is one of the spatial matrices, say γ2, is replaced by (i is the imaginary number)

Why not?  The matrices still anti-commute, so by definition they form a Dirac algebra. It has a different metric tensor,

which means that there are two time variables and two spatial variables.  That is certainly different.  It represents a flat structure with only two spatial dimensions, maybe an anyon?

But two times?  The first is the usual linear time that differs in different inertial frames.  The second is a rotational time which rotates in the plane of the 2D flat space.  This is a phase time or a frequency and accounts for the different relative rotations of 2D objects in different inertial frames.

So what does this different Dirac algebra lead to?  First spin is now defined by {I21,23} .  Second, (opps), the corresponding Dirac equation is neither hermitian nor Lorentz Invariant. It turns out that is not an issue because the solutions lead to Mirror Matterand both mirror states must exist simultaneously: a reflection must have an object

Those Mirror states lead to a 2-dimensional Dirac equation. There is fun with parity too.

At the level of a spin ½ there are only two possibilities which make physical sense {I2,σ1,σ2,σ3} and {I2,σ1,2,σ3}. Adding more imaginary numbers gets rid of space and that is no good.  You might say adding one imaginary number also does not make sense, but it does. I’ll show that next time.