## 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.

1. I have been busy the last week and not able to respond much. I am wondering what you think about any putative connection between quantions and twistors. It seems to me that quantions are in a way a sort of symplectic form of twisters.

Cheers LC

2. There is a deep indirect connection between quantions and twistors. The root cause is their origin in SO(2,4)~SU(2,2). The twistors represent the geometry and the quantions represent a (sub)algebra.

SO(2,4) is the group of conformal transformations in the conformal compactification of Minkowski space.

It is unclear at this time the role quantions will ultimately play in nature. In particular the problem is SU(3) and how it can come about (in quantionic QM, there is a way for SU(3) but there is a chirality problem). There were some papers about twistors and string theory, but string theory is at a different energy scale.

My personal opinion is that ultimately string theory is correct. The strongest piece of evidence is in the triplication of the elementary particle families. Each particle family is identical from the point of view of gauge symmetry or algebraic properties. Therefore there must be an additional ingredient needed to explain nature, and this cannot come straight from the canonical formalism like in loop quantum gravity.

1. My sense of string theory is not so much that I think “it is correct,” but rather that I think there is a stringy aspect to physics. The universe conserves quantum information according to error correction codes that restore entanglements. The code is E8 or more generally the Jordan J^3(O) that on a black hole horizon define a set of states. These are the Mathur “fuzzball” or net of strings on the black hole stretched horizon. So in order for what I am thinking to be correct the stringy picture of the world has to have some bearing as well.

I am thinking that the SO(4,2) ~ SU(2,2) being the one side of the double bundle system of twisters that this quantions is something involved with the CP^3 of projective twistor geometry. The quantions however, appear to have some sort of symplectomorphism structure to them. So the normal subgroup of SO(4,2) is extended by the inclusion of a symplectic group.

I am not sure if I am correct here, but I think this is the case that quantions are some sort of extension into symplectic structure. This could be important for understanding ADM based gravity theories, such as LQG. These appear to be a form of constraint system, such as NH = 0 classically or the Wheeler-DeWitt equation HΨ = 0, which selects the diffeomorphism system for the gravity sector.

Cheers LC

3. Lawrence,

About information conservation, I think I just solved the problem and I am writing two papers on it right now. From my point of view, I was looking to find a principle which separates abstract mathematics from reality. Initially I thought I had one, but it applied only to classical physics. Now I generalized it and is powerful enough to select classical and quantum mechanics, but reject hyperbolic quantum mechanics. I also developed the full math for hyperbolic QM as well and I can confidently state that it does not apply to the interior of black holes as it was speculated.

4. I spent some time over the Thanksgiving holiday thinking on how to succinctly express how I think this works. Of course this ended up being a lot of words for a blog entry, but I think it is few words given the nature of the subject.

There is a prospect of using the maximum entropy principle to sort this out. Information as I_n = log(ρ_{nn}) sums as entropy with S = -sum_n I_n ρ_{nn}. Entropy represents the number of ways microstates can be permuted so the macrostate remains the same. The black hole is then a macrostate that defines the upper bound of entropy for information that is packed into a bounding area given by the horizon. The symmetries or group structure for microstates of this system are not entirely known. However, the horizon is composed of units of area ~ Għ/c^3 that can hold a quibit. The event horizon is a null congruency that is mapped into the Fano projective plane. The projective plane is given by the sequence

F_4: 0 --- > B_4 --- > F_{52/16} --- > OP^2 --- > 0

Where B_4 = so(9) is the holographic group on the stretched horizon of Susskind and F_4 is the group for the 24 cell. F_4 is also the group system for the KS theorem. We then might have some sort of “game” that reshuffles qubits so the event horizon defines the maximum entropy in a region of space. This is then a game of qubits in sphere packings that define these groups.

There is a prospect for a type of Conway “game of life.” The J^3O system lies in a degenerate spacetime of 2 space plus time dimensions. The full 27 dimension of J^3O with the constraint on the trace is the 26 dimensions of the bosonic string. BTW, people tend to ignore the bosonic string, but for some formal reasons it is actually important. The three basis elements in J^3O are degenerate subspaces; they are one dimensional elements that under the splitting of the degeneracy are E8 spaces. The E8 group has as its discrete representation the Gosset polytope, which as a root space that is formally equivalent to the group E8 itself. This is a remarkable property of the exceptional E8, where the Weyl subgroup can construct the entire group. The roots of this polytope can represent states of the “elements” in the game.

The irreducible representation of the E8 was worked out in 2007, and it was an enormous undertaking. The problem was solved by factoring the Vogon polynomial representation of the group. This produced the set of possible representation of E8. To describe the new result requires one more level of abstraction. The ways that E8 manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E8. The new result is a complete list of these representations of E8, and a precise description of the relations between them, all encoded in a matrix with 205,263,363,600 entries. This amounts to 60Gb of data, which is 50 time larger than the human genome. We might consider a situation where the elements in a game of life interact with each other according to some irrep of E8. Of course this appears massively complex. It was enough to compute these irreps, albeit with computer technology that by today's standards is probably a bit old. It is another to imagine some sort of dynamics from this.

5. What dynamics would this be? It would be sort of maximal entropy calculation. The interrelationships between these elements would be driven by a potential (of sorts) that pushes the system towards maximum entropy. This would then construct the set of entanglements (connections etc) which give the solution, or the space of solutions. The entropy function(al) is what defines the maps between a pure vacuum |0> in a Hilbert space H_M and the pure state of radiation or bosons |i> for a black hole of mass M to a BH of mass M – E_n with the Hilbert space ⊕_nH_{M-E_n}. Here ⊕_n means sum_n. This space of state has a dimension given by the integer partition of N. The density of string states is related to this as well. The integer partition is given by the holomorphic functions derived from the Dedekind η-function. This will then govern the operators that give the unitary or meromorphic map from |i>}0> --- > |j>n> with

U(t) |i>|0> = sum_{n,j}^iC_{n,j}^i|j>|n>.

The details of the operators C_{n,j}^i is not known, but I conjecture that they are some sort of ergodic matrices formed from an ensemble space of irreps of the E8. The “entropy”of these operators is given on a coarse grained level by

sum_j = C^†_nC_m ~= w_nδ_{mn}, w_n = e^{-E_nβ}

and a finer grained information on these operators would be given by a distribution over the set of irreps of the E8. What is interesting is to ask whether the ergodic structure of these matrices defines this distribution according to an integer partition function.

The growth and decay of a black hole is probably closely related to or not different from any type of quantum transition. The process involves a Hamiltonian for the black hole, the radiation emitted as some interaction Hamiltonian that couples the two H = H_{bh} + H_r + H_{int}. The interaction Hamiltonian is the generator for a unitary operator U(t') = exp(i∫^t’ H_{int}dt, where t’ is defined to be at the page time, or approximately about the time the BH entanglement is entirely with the emitted radiation. The states for the black hole plus radiation in the early phase are |i>|0> that is mapped into |j>|n> here |i> and |j> are in H_{bh}and |0> and |n> are the microstates of the black hole and radiation respectively. The black hole evolves from M to M - E_n and there exists a vast number of ways to arrange the Hilbert space for the microstate to give rise to this macrostate. BH of mass M – E_n with the Hilbert space ⊕_nH_{M-E_n}. Here ⊕_n means sum_n. This space of state has a dimension given by the integer partition of N. The standard thermodynamics variables are at play with the density matrix

ρ = sum_nw_n|n><n| w_n = e^{-E_nβ}/Z

with Z = sum_ne^{-E_nβ} and β = 8πM ~ 1/T_{hawking}.

Here dim(H_{M-E_n}) = E_n = Ne^{-E_nβ}. The entropy of the states in the black hole and radiation is given by their Hilbert spaces H_b and H_r with

dim(H_b⊗H_r) = sum_n dim(H_{M-E_n}) = NZ.

This unfortunately exceeds the Bekenstein bound for the black hole entropy.

6. Now construct operators that give the unitary or meromorphic map from |i>}0> --- > |j>n> with

U(t) |i>|0> = sum_{n,j}^iC_{n,j}^i|j>|n>.

The details of the operators C_{n,j}^i is not known, but I conjecture that they are some sort of ergodic matrices with random phases formed from an ensemble space of irreps of the E8. The “entropy”of these operators is given on a coarse grained level by

sum_j = C^†_nC_m ~= w_nδ_{mn}, w_n = e^{-E_nβ}.

The matrix C_{n,j}^i = describes the evolution of a black hole in the state i to the state j with the emission of n particles. The matrix is further defined as C_n = where the particle or radiation states are Boulware vacua states. We have the counter form

U|i>|0> = sum_n|i>|n>

Unitarity U†U = 1 implies

sum_{n,j}C^i_{n,j}C*^k_{n,j} = δ^{ij}, sum_nC^†C_n = 1_M

The density matrix for the black hole before the Page time t' is

ρ = U|i>|0><0| = sum_j = = w_nδ_{mn}.

The unitarity condition is in the trace a statement of the probabilities of the system. The weights of these probabilities are given by the Boltzmann factors w_n = e^{-E_nβ}/Z. This can be expressed as the matrix product

sum_jC^†i_{j,n}C^i_{j,m} ~= w_nδ_{mn}.

The diagonal elements with n = m contains Ne^{-E_nβ} terms. The unitarity condition gives 1/N times this and the result is then e^{-E_nβ}. Off diagonal portions are equally numerous, but the phases are considered to be random. This system is to a degree of approximation considered to be similar to spontaneous emission. A detailed analysis of the phases is more complex, but that is a part of what we may be able to do for a set of coherent histories. This term is then equal in number, but the Poisson result for the statistical deviation of the phases gives a result that is ~ 1/sqrt{N} the diagonal result and may be ignored in a semi-classical limit.

We may now compare the new and old black hole. The new black hole has the BH and radiation R entropy equal S_b = S_r and the joint entropy is zero with

I_{br} = S_b + S_r - S_{br}

with

S{br} = 0, S_b = S_r = log(Z) + βE-bar, I_{br} = 2S_b

for E-bar = sum_n E_n e^{-E_nβ}.

The radiation is maximally entangled with the black hole and carries the thermal entropy. The mutual information is maximal and the joint entropy is zero, which is what changes over time as the black hole ages.

The older black hole is time evolved by the matrix element U so that

ρ(t’) = U(t’)ρ(0)U^†(t’)

where the initial density matrix is given by the Boulware and BH states as ρ(0) = (1/N)1_M|0><0|. The time evolved density matrix is then

ρ(t’) = (1/N)U^†(t’)1_M|0><0|U(t’) = (1/N)sum_{jk}C^i_{j,n}C*^i_{k,m} = (1/N)C_nC*_m

The joint entropy is S_{br} = log N in contrast to log(NZ) which is in line with the Bekenstein bound. The entropies are then

S_b = log(N) - βE-bar, S_r = log(N) - βE-bar, S_{br} = log(N), I_{br} = log(Z)

which is the state of the old black hole.

7. This is where quantum error correction codes enter the game. We set the black hole up as existing in the 2+1 spacetime of the degenerate J^3O. This precludes the existence of gravitons, other than possible physics of gauge bosons that have emergent graviton physics in colorless entanglements. We may then take this and extend it to four dimensions later on. The system is also “upgraded” with a quantum error correction code. The code we us is E8 itself. The Steiner system S(8, 5, 3) is a QECC that measures the Hamming distance for error that occur. The code operates on a code space H_c and has a set of basis elements |i_c>. The dimension of the code space is dim H_c << N, which insures that errors which occur in the system are corrected with fidelity. However, as errors are corrected the coding space grows. By this it is meant that the number of nonvacuum states increases. The code space holds transfered information so that unitarity is conserved. While the code space is “virtual” the entropy associated with it is physically real and contained in the black hole. Once the entropy grows to N the Bekenstein bound is no longer applicable. This is a manifestation of the fire wall problem.

I have spent considerable time on this today. I will continue with this if there is interest. The proposal is to exploit the E6 subgroup of the E8 and its connection to the twistor space CP^3. The “quantum” part of this is the E6, and the remainder of the E8 is considered to be ergodic matrices with random phases. In this way we can consider the overlap elements such as C^†i_{j,n}C^i_{j,m} and jC^†i_{j,n}C^i_{k,n}. Note here that we have in the first equal elements on the H_b and in the second equal elements with H_r. With this we can evaluate twistor space elements of the BCFW recursion for elementary quantum corrections.

Sorry for using up so much bandwidth

Cheers LC

8. Lawrence,

This was an impressive tour de force on your end. Do you want to write one or a series of guest posts at this blog to present it in detail?

Recently I uploaded http://arxiv.org/abs/1311.6461 and I'll discuss it in subsequent posts.

9. I could do that. I may though wait until the winter break starts. I also have to review a paper for a journal, and the paper is mathematically pretty stiff. I have been reading a paper by Nima Arkani-Hamed with Cachazo and others. The paper is on the BCFW method for computing quantum amplitudes. This involves twistor theory and the quantions seem in some way to be related to twistors.

Cheers LC