What is the number system of quantum mechanics?

Inherently relativistic quantum mechanics

It was a cold 2007 January morning during rush our in L’enfant Plaza metro station in Washington DC when one of the best violinist virtuoso in the world, Joshua Bell

played on a Stradivarius violin some of the best classical pieces for about an hour. Do you think his masterpiece performance drew a crowd? The lottery kiosk nearby was attracting much more attention.

In the same 2007 year, Emile Grgin published his Structural Unification of Quantum Mechanics and Relativity book. Still to this day people buy into the “lottery” idea that quantum mechanics is solely about information and Born’s rule.

But maybe Grgin’s results were “crackpot”. That was the reaction of the archive when http://arxiv.org/abs/1204.3562 was reclassified from the quantum section to the general section dedicated to “laymen’s fantasies”.

So let’s prove quantionic quantum mechanics is the real deal and not some crazy idea. First quantions are the simplest non-trivial type I von Neumann algebra. All linear algebras have matrix and vector representations which come in pairs: left algebra and a column vector, and right algebra and a row vector.

For example, a complex number z = a + ib can be represented as:

a   -b

b  a

and

a

b

OR

a  b

-b a

and

a  b

It is a simple theorem that for linear associative algebras the left and right matrix representations commute. For quantions, the left and column representations are:

q1   q3   0    0

q2   q4   0    0

0     0    q1  q3

0     0    q2  q4

and

q1

q2

q3

q4

and the right and row representations are:

q1   0    q3   0

0    q1    0    q3

q2   0    q4   0

0    q2    0   q4

and

q1 q2 q3 q4

Both representations play a major role in the physics. For now, since in the 4x4 matrix algebra representation the left L and a right R quantions have non-overlapping parts, the von-Neumann double commutant theorem (http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem) holds: L^{``} = L In turn, this puts the corresponding quantionic Hilbert module theory on a solid foundation.

To bring the discussion on a more known ground, here is the 1-to-1 mapping between quantions and spinors:

q1                                 -Psi2

q2        <----->sqrt(2)     Psi3^*

q3                                   Psi1

q4                                   Psi4^*

Psi1                                 q3

Psi2    <----->1/sqrt(2)   -q1

Psi3                                 q2^*

Psi4                                 q4^*

Dirac’s current

j^mu = Psi^{dagger} gamma^0 gamma^mu Psi

is the same as:

j^mu = (q^* q)^mu

Dirac’s equation in quantionic formalism reads:

D |q) = i m Gamma^1 |q^* )

With D a right derivation quantion.

The Klein-Gordon equation in quantionic formalism reads:

(D’Alembert + m^2) |q) = 0

with |q) the column quantion.

Dirac went from Klein Gordon’s equation to a linear equation by talking the square root of the d’Alembertian:

(gamma^mu partial_mu) * (gamma^mu partial_mu) =   D’Alembert

quantions offer another decomposition of D’Alembert’s operator:

D^sharp D = D’Alembert

With D a derivarion right quantion and the sharp operation the parity-reversed transformation P of a quantion (quantions have the discrete CPT = I symmetry).

Quantionic time evolution is best understood in the gauge theory sense, but to wrap the standard quantum mechanics description, recall this from Hardy’s paper:

|mn> = 1/sqrt(2) (|m> + |n>)

|MN> = 1/sqrt(2) (|m> + i|n>)

and that there were

½ N(N-1) projectors of the form |mn><mn|

Because quantionic quantum mechanics has a non-commutative number system the corresponding most general projectors here is of the form:

|mn>P<mn|

|MN>P<MN|

with P^2=P and P a unit quantion. We know p+ = (1+sigma)/2 and  p- = (1-sigma)/2, are spin projector operators where sigma are the Pauli matrices.

From here it is not hard to show that are 4 linear independent unit quantionic projectors P (one for identity and 3 for each of the Pauli matrices) corresponding to the spin degree of freedom, or to the 4 spinor components. Hardy’s formula then reads:

K = 4N^2 for N quantions

An instructive exercise for the reader is to investigate why quaternionic quantum mechanics does not have projectors of this type (I’ll give the answer in the next post).

Next time I’ll finish the quantionic quantum mechanics presentations from the gauge theory point of view.

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