What is the number system of quantum mechanics?                   

Quantionic quantum mechanics

So far we have seen that quantum mechanics can be expressed over real numbers, complex numbers, and quaternions. Physically, quantum mechanics over reals and quaternions do not lead to new predictions. Also this short list implies that quantum mechanics is about Born rule and information. We shall see that this is not the complete story.

Quantionic quantum mechanics is a direct counter example to arbitrarily restricting the number system to division algebras. Quantionic quantum mechanics was discovered by Emile Grgin (or Gergin) – I am proud to state that my “Grgin number” is one. Working in Peter Bergmann’s group, Grgin joined forces with Aage Petersen, Bohr’s personal assistant and investigated a line of thought from Bohr about the correspondence principle. This resulted in the composability principle and a dual quantum-classical mechanics framework (pre C* algebra) better than Segal’s quantum algebraic formalism. This work happened in early 70s and was forgotten after Grgin left academia and went to work for the industry. Recently Grgin retired and restarting working in this area resulting in quantionic quantum mechanics – a towering achievement which unfortunately is not well known.

Probably the best way to think quantionic quantum mechanics is terms of Darwin’s evolution: it is a “missing link” between regular quantum mechanics and gauge theory. When talking about quantions, one can either take the point of view of standard quantum mechanics, or the point of view of field theory. Its physical content is identical with Dirac’s theory of the electron, but its formalism is most illuminating.

So let start the story from the trusted Born’s rule. This implies that quantum mechanics is only about information. But is it? All experiments are done in space-time and events require 4 coordinate numbers (x,y,x,t) to be located. In turn Lorentz transformations and special relativity teaches us about 4-vectors, so in a relativistic quantum mechanics it is natural to think not of probabilities, but of probabilities currents. The starting point of quantionic quantum mechanics is a generalization of Born’s rule to probabilities currents (called the “Zovko interpretation”-after the person who discovered it).

Complex numbers have two norms, let’s call them A for algebraic and M for metric. In matrix representation a complex number z = a+ib correspond to a 2x2 matrix:

 a   b

-b  a

The algebraic norm A is defined as: A(z) = z^{dagger} z:

 a  -b     a  b   =  (a^2 + b^2)  1  0

 b   a    -b  a                           0  1

The metric norm is the determinant: M(z) = det (z) = a^2 + b^2

The value is the same, but the meaning is very different. Quantionic quantum mechanics aims to lift this degeneracy and have two fully distinct norms.

Quantions are based on the SL(2, C) ~ SO(3,1) isomorphism and are defined as follows:

q1  q3   0   0

q2  q4   0   0

0    0   q1  q3

0    0   q2  q4

with q1, q2, q3, q4 complex numbers. The multiplication rule for the algebra of quantions (q1, q2, q3, q4) follows from the matrix multiplication rule.

OK, this is a bit dry, and to put it in physics context, for quantions M correspond to relativity or with picking a particular frame of reference, and A corresponds to quntum mechanics and the inner product.

For a quantion Q:

A(Q) = Q^dagger Q :  a future oriented 4-vector

M(Q) = det(Q) : a complex number

The reason for the block diagonal zero elements have to do with gauge theory and quantion-spinor correspondence, and to simplify the idea, we can talk for now about a reduced quantion:

q1  q3

q2  q4

A hermitean reduced quation:

 r   z

z*  s

corresponds to a 4-vector: (p0, p1, p2, p3) in the Minkowski space as follows:

p0 + p3        p1+ i p2

p1 – i p2       p0 – p3

More important, A(Q) = Q^dagger Q is a future oriented 4-vector. (recall from prior posts that spin factors are realizations of the Jordan algebra)

The fundamental theorem of quantionic algebra is that the algebraic and metric norm commute: AM(Q) = MA(Q)

Pictorially this can be represented as:

        q   ----------------A------------------->A(q) = future oriented 4-vector

         |                                                       / |         on Minkowski space

         |                                                    /    |

p -------------------A------------------>A(p) = null vector

         |                                                  |       |                                                    

|        M                                               M     |

|        |                                                  |       M

M     |                                                  |        |

|       z   ---------------A----------------------->x  

|                                                           |    /    

|                                                           |  /

0 ------------------A---------------------> 0

with p a quantion of determinant zero

The 0-x axis represents the good old-fashion Born rule: the experimental predictions are positive probabilities.

A(q) inside the Minkowski cone A(p) represents Zovko interpretation as probability current.

But is quantionic quantum mechanics nothing but complex quantum mechanics with relativistic symmetry? No, the story is subtle (and this is where spin enters the picture).

In standard complex (non-relativistic) quantum mechanics, a state omega of a quantum mechanical system is a linear functional omega on the space of observables A with the property:

Omega (A^dagger A) >= 0 for all observables in A

Omega (I) = 1

The Hilbert space A has the inner product:

<A, B> = Tr (A^dagger B)

In quantionic quantum mechanics the inner product does not involve the trace. Here:

<A, B> = M (A^dagger B) = determinant (A^dagger B)

In turn this demands the existence of spin.

If I can coin a word, in complex quantum mechanics one talks of qubits, while in quantionic quantum mechanics one talks of z-bits (z from Zovko).

In quantionic quantum mechanics superposition is based on quantions. Recall Adler’s requirements for a modulus function N:

(1)        N(0) = 0

(2)        N(phi) > 0 if phi is not 0

(3)        N(r phi) = |r| N(phi)

(4)        N(phi_1 + phi_2) <= N(phi_1) + N(phi_2)

where N = AM in this case.

Now (2) has to be relaxed: there are quantions phi for which N(phi) = 0 and also (4) is not true in general.

The fact that (4) does not hold is not a problem because it does not hold in general for Hilbert modules either which generalize C* algebras for vector bundles (field theory). Quantionic quantum mechanics can be understood as electroweak gauge theory!!!

Naively, the relaxation of condition (2) can be treated with the tools of C* algebra in the usual GNS construction:

Here singular means that the sesquilinear form may fail to satisfy the non-degeneracy property of inner product. By the Cauchy–Schwarz inequality, the degenerate elements,x in A satisfying ρ(x* x)= 0, form a vector subspace I of A. By a C*-algebraic argument, one can show that I is a left ideal of A. The quotient space of the A by the vector subspace I is an inner product space. The Cauchy completion of A/I in the quotient norm is a Hilbert space H.

In quantionic quantum mechanics this quotient space is empty! This is another clue that there is something fundamentally different at play here and that the Jordan spin factor algebras, although embeddable in the nxn matrix case is a beast of its own.

From the zero determinant quantions, it is clear that quantionic algebra is a non-division associative algebra!!!

And why do we need division in quantum mechanics? It is silly to arbitrarily insist on a mathematical property just for our convenience. In what quantum mechanics book or paper have you seen any direct division by wavefunctions?

The beauty of quantionic quantum mechanics is that nothing needs to be postulated except Zovko’s interpretation. In subsequent posts we’ll see how Dirac equation follows naturally, we’ll investigate the spinor-quantionic correspondence, we’ll give a counter example for Hardy’s formula: K = N^r (for quantionic quantum mechanics the formula is K = 4 N^r), and we’ll present quantionic quantum mechanics from the gauge theory point of view.

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