There are ~10 times in the above where I tried to use bracket notation. I didn’t notice that bracket notation gets hidden… I will try to recreate the missing parts using () instead of angled brackets.

1-3: I know that if one calculates (a|a) for a quaternion series, the result will always be a positive real number. If one has a quaternion series operator O, then (a|O|a) will be a real number and all three imaginary terms will be zero. I was able to show that (1|U|1) = (1, 0, 0, 0).

4. It turns out that (+|V|+) = (1/2, 1/2, 0, 0).

5. What I decided to do was calculate (0|V|0) = (0, 1, 0, 0).

6-7. This leads to (0|Vi|0) = (1, 0, 0, 0). Combine these so: (1|U|1)(0|Vi|0) = (1, 0, 0, 0). Nice.

8. When I multiplied jk and U, took that and calculated (1|jkU|1), the number was a pure j.

9. (1|jjkU|1) = (1, 0, 0, 0)

My bad for not noticing.

]]>It is cold here in Massachusetts, but I was not able to warm myself by the claimed flaming garbage fire discussed in the main blog. I was able to spot a few issues in the calculation with the state |1> for Alice, |+> for Bob, U/V stuff.

I recently rolled my own tools for doing quaternion series quantum mechanics calculations. Just last week I took my first mini-class on quantum computation during MIT’s IAP. The subject of the CHSH inequality came up which provided a perfect test for my software. All I had to do was set j=k=0, and if I was doing things right, I had to get the anti-correlation of -2 2^(1/2) out of the process. The code was up to the task. Great, and boring too since the CHSH inequality has been proven long ago for complex numbers.

It then became possible to extend the CHSH inequality to quaternions, to go from (1 i, 0, 0) to (n i, m j, p k) where n, m, and p can take whatever real values. The only thing that changed was a normalization factor. That change is important at two levels. First it is so simple a computer can do it. Second it means that quaternion series quantum mechanics is not exactly complex-valued quantum mechanics. If anyone wants to see the details of the work, it is on a GitHub Jupyter notebook, to avoid filters, url of bit dot ly slash vp dash CHSH.

That was my warm up exercise I did before trying to figure out the calculations presented in the main blog.

My approach was simple… I know that if one calculates a for a quaternion series, the result will always be a positive real number. If one has a quaternion series operator O, then a will be a real number and all three imaginary terms will be zero. I was able to show that = (1, 0, 0, 0). This is a good thing. In the details of the calculation, it is also not too interesting since only real values go into it, no imaginaries take the stage.

It turns out that = (1/2, 1/2, 0, 0). This means that V for the state |+> is not of interest because it is not an operator, so cannot be observed.

Can we ask a different question that is of interest? What I decided to do was calculate = (0, 1, 0, 0). This too fails, but can be fixed.

Vi =

| -i 0 |

| 0 1 |

This leads to = (1, 0, 0, 0). Combine these so: = (1, 0, 0, 0). Nice.

Some might object that Vi does not look like what they expect for a Hermitian matrix. The operator Vi most definitely is not a Hermitian operator. It is unclear at this moment if Vi would be considered part of work by Carl Bender on non-Hermitian quantum mechanics, or something novel, the transition from one imaginary number to three. While you may judge a complex Hermitian matrix by its trace which must be real, you cannot do so for the considerably more diverse quaternion series. The trace rule still exists for quaternion series if and only if two of the three imaginaries are zero. What happens in the other cases? I have no idea, this is too new to know. Fun.

What about the suggested rotation? We can predict it won’t work: quaternion series are very fussy about direction, so if this was chosen without calculations, the odds are great it is a little off. That said, if the pattern holds, it will be simple to fix.

The pattern holds. I decided to rotate both U and V with the rotation matrix I chose to call jk. When I multiplied jk and U, took that and calculated , the number was a pure j. That can be fixed with multiplication by -j. Here is the result:

jjkU =

| 1 -i |

| -k j |

= (1, 0, 0, 0)

The same story applies to Vi, including the -j. The rotation does nothing to alter the odds. So I formally disagree that Alice can communicate a bit to Bob. A Jupyter notebook is available on GitHub at bit dot ly slash vp dash why dash complex.

**The Big Picture: non-locality is space-like separated reflections of information**

A little more effort is required to implement quaternion series quantum mechanics. There are normalization factors and more care to determine where things have to point in 3D space. The relatively simple rules for Hermitian matrices have to be expanded in ways that currently are not clear. Why bother when the calculations will generate the same probabilities?

There is a physical interpretation to wave functions and their conjugates. Treat a wave function as a series of events in space-time. The conjugate of that series of events will flip the signs of the 3D spatial terms while keeping the time part identical. This has some similarities to a regular mirror which flips the sign of one of the three spatial directions. A mirror reflection always feels distant and unreachable. Mirrors involve photons bouncing around which is not what is going on in quantum mechanics. Instead, this is a mathematical reflection that always necessarily must be done for observations in quantum mechanics. One takes the information one has – one state being say (1, 2, 3, 4) – and calculates its conjugate, (1, -2, -3, -4). These two pieces of information are necessarily space-like separated because the times are the same but the spatial locations are opposite.

I have been living with this idea for two months. I think it raises more questions than it answers, but it does start to answer a question that has been around since the 1920s: what physical reason is there that requires quantum mechanics to be non-local? Non-locality is due to 3D spatial reflections required so that has a least lower bound of zero. Or for a non-technical audience: physicists have to use mirrors to get the math right of almost empty space. Mirrors are odd when used all the time.

]]>Amplitudes are complex numbers, not just imaginary ones. But more to the point, real numbers don’t “exist in the real world” any more than complex ones do! They’re both mathematical constructions. The question addressed in this post was just: why does the latter mathematical construction and not the former accurately describe this part of physics?

]]>Nevertheless, the hand-wavy argument does seem correct: the dynamics of the measurement must be expressible as some set of terms within the global Hamiltonian, and so they must respect the dimension pairing that block-diagonalizes said Hamiltonian.

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