Friday, April 13, 2012

Quantum Mechanics as Quantum Mathematics?

Quantum Mathematics (QM) suggests that the basic structures of Quantum Mechanics (QM) might reduce to fundamental mathematical and metamathematical structures, and that one even consider the possibility that Quantum Mechanics reduces to Quantum Mathematics with mathematician included or expressing it in a concice manner: QM=QM!

The notes below were stimulated by an observation raising a question about a possible connection between multiverse interpretation of quantum mechanics and quantum mathematics. The heuristic idea of multiverse interpretation is that quantum state repeatedly branches to quantum states which in turn branch again. The possible outcomes of the state function reduction would correspond to different branches of the multiverse so that one could save keep quantum mechanics deterministic if one can give a well-defined mathematical meaning to the branching. Could quantum mathematics allow to somehow realize the idea about repeated branching of the quantum universe? Or at least to identify some analog for it? The second question concerns the identification of the preferred state basis in which the branching occurs.

Quantum Mathematics briefly

Quantum Mathematics replaces numbers with Hilbert spaces and arithmetic operations + and × with direct sum ⊕ and tensor product ⊗.

  1. The original motivation comes from quantum TGD where direct sum and tensor product are naturally assigned with the two basic vertices analogous to stringy 3-vertex and 3-vertex of Feynman graph. This suggests that generalized Feynman graphs could be analogous to sequences of arithmetic operations allowing also co-operations of ⊕ and ⊗.

  2. One can assign to natural numbers, integers, rationals, algebraic numbers, transcendentals and their p-adic counterparts for various prime p Hilbert spaces with formal dimension given by the number in question. Typically the dimension of these Hilbert spaces in the ordinary sense is infinite. Von Neuman algebras known as hyper-finite factors of type II1 assume as a convention that the dimension of basic Hilbert space is one although it is infinite in the standard sense of the word. Therefore this Hilbert space has sub-spaces with dimension which can be any number in the unit interval. Now however also negative and even complex, quaternionic and octonionic values of Hilbert space dimension become possible.

  3. The decomposition to a direct sum matters unlike for abstract Hilbert space as it does also in the case of physical systems where the decomposition to a direct sum of representations of symmetries is standard procedure with deep physical significance. Therefore abstract Hilbert space is replaced with a more structured objects. For instance, the expansion ∑n xnpn of a p-adic number in powers of p defines decomposition of infinite-dimensional Hilbert space to a direct sum ⊕n xn⊗ pn of the tensor products xn⊗ pn. It seems that one must modify the notion of General Coordinate Invariance since number theoretic anatomy distinguishes between the representations of space-time point in various coordinates. The interpretation would be in terms of cognition. For instance, the representation of Neper number requires infinite number of pinary digits whereas finite integer requires onlya finite number of them so that at the level of cognitive representations general coordinate invariance is broken.

    Note that the number of elements of the state basis in pn factor is pn and m∈ {0,...,p-1} in the factor xn. Therefore the Hilbert space with dimension pn>xn is analogous to the Hilbert space of a large effectively classical system entangled with the microscopic system characterized by xn. p-Adicity of this Hilbert space in this example is for the purpose of simplicity but raises the question whether the state function reduction is directly related to cognition.

  4. On can generalize the concept of real numbers, the notions of manifold, matrix group, etc... by replacing points with Hilbert spaces. For instance, the point (x1,..,xn) of En is replaced with Cartesian product of corresponding Hilbert spaces. What is of utmost importance for the idea about possible connection with the multiverse idea is that also this process can be also repeated indefinitely. This process is analogous to a repeated second quantization since intuitively the replacement means replacing Hilbert space with Hilbert space of wave functions in Hilbert space. The finite dimension and its continuity as function of space-time point must mean that there are strong constraints on these wave functions. What does this decomposition to a direct sum mean at the level of states? Does one have super-selection rules stating that quantum inteference is possible only inside the direct summands?

  5. Could one find a number theoretical counterpart for state function reduction and preparation and unitary time evolution? Could zero energy ontology have a formulation at the level of the number theory as earlier experience with infinite primes suggest? The proposal was that zero energy states correspond to ratios of infinite integers which as real numbers reduce to real unit. Could zero energy states correspond to states in the tensor product of Hilbert spaces for which formal dimensions are inverses of each other so that the total space has dimension 1?

Unitary process and state function reduction in ZEO

The minimal view about unitary process and state function reduction is provided by ZEO.

  1. Zero energy states correspond to a superposition of pairs of positive and negative energy states. The M-matrix defining the entanglement coefficients is product of Hermitian square root of density matrix and unitary S-matrix, and various M-matrices are orthogonal and form rows of a unitary U-matrix. Quantum theory is square root of thermodynamics. This is true even at single particle level. The square root of the density matrix could be also interpreted in terms of finite measurement resolution.

  2. It is natural to assume that zero energy states have well-defined single particle quantum numbers at the either end of CD as in particle physics experiment. This means that state preparation has taken place and the prepared end represents the initial state of a physical event. Since either end of CD can be in question, both arrows of geometric time identifiable as the Minkowski time defined by the tips of CD are possible.

  3. The simplest identification of the U-matrix is as the unitary U-matrix relating to each other the state basis for which M-matrices correspond to prepared states at two opposite ends of CD. Let us assume that the preparation has taken place at the "lower" end, the initial state. State function reduction for the final state means that one measures the single particle observables for the "upper" end of CD. This necessarily induces the loss of this property at the "lower" end. Next preparation in turn induces localization in the "lower" end. One has a kind of time flip-flop and the breaking of time reversal invariance would be absolutely essential for the non-triviality of the process.
The basic idea of Quantum Mathematics is that M-matrix is characterized by Feynman diagrams representing sequences of arithmetic operations and their co-arithmetic counterparts. The latter ones give rise to a superposition of pairs of direct summands (factors of tensor product) giving rise to same direct sum (tensor product). This vision would reduce quantum physics to generalized number theory. Universe would be calculating and the consciousness of the mathematician would be in the quantum jumps performing the state function reductions to which preparations reduce.

Note that direct sum, tensor product, and the counterpart of second quantization for Hilbert spaces in the proposed sense would be quantum mathematics counterpart for set theoretic operations, Cartesian product and formation of the power set in set theory.

ZEO, state function reduction, unitary process, and quantum mathematics

State function reduction acts in a tensor product of Hilbert spaces. In the p-adic context to be discussed n the following xn⊗ pn is the natural candidate for this tensor product. One can assign a density matrix to a given entangled state of this system and calculate the Shannon entropy. One can also assign to it a number theoretical entropy if entanglement coefficients are rationals or even algebraic numbers, and this entropy can be negative. One can apply Negentropy Maximization Principle to identify the preferred states basis as eigenstates of the density matrix. For negentropic entanglement the quantum jump does not destroy the entanglement.

Could the state function reduction take place separately for each subspace xn⊗ pn in the direct sum ⊕n xn⊗ pn so that one would have quantum parallel state function reductions? This is an old proposal motivated by the many-sheeted space-time. The direct summands in this case would correspond to the contributions to the states localizable at various space-time sheets assigned to different powers of p defing a scale hierarhcy. The powers pn would be associated with zero modes by the previous argument so that the assumption about independent reduction would reflect the super-selection rule for zero modes. Also different values of p-adic prime are present and tensor product between them is possible if the entanglement coefficients are rationals or even algebraics. In the formulation using adeles the needed generalization could be formulated in a straightforward manner.

How can one select the entangled states in the summands xn⊗ pn? Is there some unique choice? How do unitary process and state function reduction relate to this choice? Could the dynamics of Quantum Mathematics be a structural analog for a sequence of state function reductions taking place at the opposite ends of CD with unitary matrix U relating the state basis for which single particle states have well defined quantum numbers either at the upper or lower end of CD? Could the unitary process and state function reduction be identified solely from the requirement that zero energy states correspond to tensor products Hilbert spaces, which correspond to inverses of each other as numbers? Could the extension of arithmetics to include co-arithmetics make the dynamics in question unique?

What multiverse branching could mean?

Could QM allow to identify a mathematical counterpart for the branching of quantum states to quantum states corresponding to preferred basis? Could one can imagine that a superposition of states ∑ cnΨn in a direct summand xn⊗ pn is replaced by a state for which Ψn belong to different direct summands and that branching to non-intefering sub-universes is induced by the proposed super-selection rule or perhaps even induces state function reduction? These two options seem to be equivalent experimentally. Could this decoherence process perhaps correspond to the replacement of the original Hilbert space characterized by number x with a new Hilbert space corresponding to number y inducing the splitting of xn⊗ pn? Could the interpretation of finite integers xn and pn as p-adic numbers p1≠ p induce the decoherence?

This kind of situation is encountered also in symmetry breaking. The irreducible representation of a symmetry group reduces to a direct sum of representations of a sub-group and one has in practice super-selection rule: one does not talk about superpositions of photon and Z0. In quantum measurement the classical external fields indeed induce symmetry breaking by giving different energies for the components of the state. In the case of the factor xn⊗ pn the entanglement coefficients define the density matrix characterizing the preferred state basis. It would seem that the process of branching decomposes this state space to a direct sum 1-D state spaces associated with the eigenstates of the density matrix. In symmetry breaking superposition principle holds true and instead of quantum superposition for different orientations of "Higgs field" or magnetic field a localization selecting single orientation of the "Higgs field" takes place.

Could state function reduction be analogous process? Could non-quantum fluctuating zero modes of WCW metric apper as analogs of "Higgs fields". In this picture quantum superposition of states with different values of zero modes would not be possible, and state function reduction might take place only for entanglement between zero modes and non-zero modes.

The replacement of a point of Hilbert space with Hilbert space as a second quantization

The fractal character of the Quantum Mathematics is what makes it a good candidate for understanding the self-referentiality of consciousness. The replacement of the Hilbert space with the direct sum of Hilbert spaces defined by its points would be the basic step and could be repeated endlessly corresponding to a hierarchy of statements about statements or hierarchy of nth order logics. The construction of infinite primes leads to a similar structure.

What about the step leading to a deeper level in hierarchy and involving the replacement of each point of Hilbert space with Hilbert space characterizing it number theoretically? What could it correspond at the level of states?

  1. Suppose that state function reduction selects one point for each Hilbert space xn⊗ pn. The key step is to replace this direct sum of points of these Hilbert spaces with direct sum of Hilbert spaces defined by the points of these Hilbert spaces. After this one would select point from this very big Hilbert space. Could this point be in some sense the image of the Hilbert space state at previous level? Should one imbed Hilbert space xn⊗ pn isometrically to the Hilbert space defined by the preferred state xn⊗ pn so that one would have a realization of holography: part would represent the whole at the new level. It seems that there is a canonical manner to achieve this. The interpretation as the analog of second quantization suggest the identification of the imbedding map as the identification of the many particle states of previous level as single particle states of the new level.

  2. Could topological condensation be the counterpart of this process in many-sheeted spacetime of TGD? The states of previous level would be assigned to the space-time sheets topologically condensed to a larger space-time sheet representing the new level and the many-particle states of previous level would be the elementary particles of the new level.

  3. If this vision is correct, second quantization performed by theoreticians would not be a mere theoretical operation but a fundamental physical process necessary for cognition! The above proposed unitary imbedding would imbed the states of the previous level as single particle states to the new level. It would seem that the process of second quantization, which is indeed very much like self-reference, is completely independent from state function reduction and unitary process. This picture would conform with the fact that in TGD Universe the theory about the Universe is the Universe and mathematician is in the quantum jumps between different solutions of this theory.
Returning to the motivating question: it seems that the endless branching of the states in multiverse interpretation cannot correspond to a repeated second quantization but could have interpretation as a decoherence identifiable as delocalization in zero modes. If state function is allowed, it corresponds to a localization in zero modes analogous to Higgs mechanism. The Quantum Mathematics realization for a repeated second quantization would represent a genuinely new kind of process which does not reduce to anything already known.


At 6:00 PM, Anonymous Orwin O'Dowd said...

As reported on RESONAANCES, the Higgs signal at CERN has an unexpected profile and location in parameter space. The parameter concerned, c, appears to reflect in the sign of the coupling to the Yukawa potential, as noted in comments. In the underlying algebra, its a question of cosets.

What was always something of a vortex now resembles rather a thermal turbulence, with the symmetry-breaking perhaps reflecting particle specific heat, conjugate to mass. Per length, that's acceleration complementary to mass!!

The Higgs field then resembles a vacuum temperature. I'm interested in new uncertainties like this: also mass-frequency and length^2 hypothesized for the gravitational noise experiment.

At 6:21 PM, Blogger Santeri Satama said...

John Baez has had similar ideas towards Hilbert spaces from Category theoretical point of view, with which you are probably allready familiar with, but just in case:

Is Baez' approach subsumed by your Quantum Math, complementary or in contradiction?


On a more personal note about the role of holography in mathematical (self-)consciousness (in a very mathematically limited spaces like my mind) I remember developing an obsession about triangular bipyramid made of two tetrahedra when I first came in touch with TGD, and now I assume that the obsession was about extremely simplified version of CD. Does Quantum Math also explain and predict also holographic mathematical relations like idea of complex 8D CD as most simple triangular bipyramid? There was also strong philosophical connotation to Heraclitus, who considered fire - symbolized by tetrahedron in Plato's works - the most fundamental element. During my obsession I also reinvented the tetrahedral numbers, which for me was proof enough of Platonic anamnesis. Looking now again, wikipedia tells that
- infinite sum of tetrahedral reciprokes is 3/2 (distance of the real part Riemann hypothesis from the -1 of the unit circle)
- Watermolecule is (almost) a tetrahedron
- The parity of tetrahedral numbers follows the repeating pattern odd-even-even-even. - which brings to mind the Minkowskian symmetry break +t,-x,-y,-z
- number 5 that was important in ATP-ADP etc comes up couple ways: T5 = T4 + T3 + T2 + T1; and there are only five numbers that are both 3D tetrahedral and 2D triangular

That said, it's allways good to remember the original meaning of the word 'symbol' - part that reminds of the whole. :)

At 9:23 PM, Anonymous ◘Fractality◘ said...


"Universe would be calculating and the consciousness of the mathematician would be in the quantum jumps performing the state function reductions to which preparations reduce."

And for the non-mathematicians?



At 9:49 PM, Anonymous said...

To Orwin:

If Higgs has "unexpected profile and location in parameter space" it is not Higgs at all since the coupling of standard model Higgs to fermions are fixed. Fermio-phobic or top-phobic Higgs is reported to provide a better fit. The coupling to photon pairs is too large by factor of 3 for large transverse momenta and quite generally too large by a factor of 2. The production of W pairs is too low.

One can also consider SUSY Higgs but also this leads to difficulties according to the article that I commented. And standard SUSY is almost completely eliminated. The remaining hope is that these are still statistical fluctuations and it turns out to be Higgs after all.

I am skeptic. M_89 pion like, its super-partner consisting of squark pair, or its "satellite" analogous to those discovered in case of ordinary pion and also some other mesons and baryons could be in question and correspond to infrared Regge trajectory with energy scale of 20 GeV. TGD allows also genuine elementary particle behaving as scalars as finally became clear but they are definitely not Higgs like states.

Ceasing to talk about Higgs will be a difficult after these years of God particle hype for tax payers. There is also the very unpleasant possibility that the genuine signals for new physics have been intentionally eliminated: to my best understanding only those anomalous events consistent with the mainstream proposes are accepted. This is horribly stupid and might become a monument for the legendary arrogance of particle physicists.

Also the plans for the next accelerator assume that Higgs will be found and that it will be used to high precision measurements related to Higgs and main stream SUSY. The situation is very very difficult politically and I believe that the interpretation as Higgs will not be given up easily.

At 10:11 PM, Anonymous said...

To Santeri:

I know something about the work of Baez at the level of ideas. The notion of n-category or I think n-object involves also a hierarchy analogous to that of statements about statements. At the level of technical detail I am unable to say anything. The inspiration for Quantum Math does not come from n-categories which are too far from physics for me but from generalized Feynman graphs and the construction of infinite primes where one repeatedly second quantizes. Probably the "repeated something" implies a lot of structural similarities.

It would be interesting to look whether n-objects could be understood in terms of second quantization like procedure.

TGD and ZEO lead to surprisingly detailed analogies with mathematical consciousness. Representation of Boolean algebra in terms of fermion Fock states emerges naturally. Zero energy statements have interpretation as superpositions of Boolean statements A-->B. Sequences of bits correspond to Boolean statements with infinite number of bits forming continuum in 2-adic topology and this generalizes to p-adic case. Infinite primes allow interpretation as hierarchy of statements about... or as a hierarchy of n:th order logics. Quantum mathematics has analogous interpretation.

Many-sheeted space-time would allow a representation for the hierarchy of infinite primes in terms of hierarchy of space-time sheets and maybe also for Quantum Mathematics. Finite measurement/cognitive representation is of course essential: this would lead to discretization also at the level of Hilbert spaces. A fascinating question is whether quantum measurement theory in TGD sense could have interpretation in terms of quantum mathematics as cautiously suggested above.

In particular, the idea about mathematical necessity of state function reduction as something induced by something analogous to Higgs mechanism/spontaneous magnetization, and having direct counterpart at the level of pinary series representation of p-adic number translated to direct sum composition of Hilbert space such that different powers of p correspond to super-selection sectors is very attractive. This super-selection rule would be extremely natural since in p-adic topology sets consisting of points with norm p^-n are disjoint for different values of n. In the measurement entanglement the cognition of conscious observer would be represented by p-adic Hilbert space and state function reduction would be induced by cognition for mathematical consistency. Von Neumann said something akin to this but without p-adics.

At 10:14 PM, Anonymous said...

To Fractality:

This statement sounded so good, so poetic!;-). I had the feeling that I am saying too much unless we accept that also mathematicians have emotions and feelings and sensory perceptions. Maybe we can agree about this;-).

At 9:39 PM, Anonymous ◘Fractality◘ said...


I consider myself to be a poet at times so I am by no means surprised. ;)

In TGD consciousness framework, is the "analog I" made possible by the magnetic body of Earth? Interesting to note that humans haven't always had this "analog I" which would be explained in TGD by the emergence of the next "onion layer" of dark matter/magnetic body?


At 8:31 PM, Anonymous said...

To Fractality:

Maybe the layers of personal magnetic body integrate the sensory input to single self representation. I have even half-seriously considered the possibility that the magnetic body could at the level of topology and shape mimic the anatomy of brain and even entire biological body.

Persinger's findings relating to God helmet in particular experience of fusing with God might be understood as emergence of new layer to magnetic body. The motivation for TGD inspired theory of consciousness was just this kind of long lasting experience.

See .
Manic-depressive cycle might also relate to periodic dynamics in which some upper layer of magnetic body is lost and regained. There are interesting connections also with bicamerality claiming that ancient people experienced of being in continual contact with God and schizophrenics are the modern bicamerals. What could be the articulation for this in terms of magnetic body? See .

The illusions such as moving train illusion could be due to the motion of magnetic body. Also OBEs (out-of-body experiences) could have similar origin. See .

At 9:32 PM, Anonymous ◘Fractality◘ said...


Yes, the ancients experienced "Gods" when the magnetic bodies of the planets in our solar system interacted with the magnetic body of Earth and our own magnetic bodies. The plasma displays associated with these interactions also added psychological effects ;)Perhaps these psychological effects were due to the plasmoids as simple life forms mediating entanglement/sharing mental images as you have written about. Has it been resolved yet whether each individual possesses magnetic body or if each individual emerges from and uses the magnetic body of Earth?

The illusions such as moving train illusion could be due to the motion of magnetic body.

Same principle applies to standing on the edge of a cliff? I've experienced this.

I thought that magnetic body is fixed in regards to physical body. How would one feel the motion of magnetic body if it is fixed?


At 10:40 PM, Blogger Ulla said...

Manic-depressive cycle might also relate to periodic dynamics in which some upper layer of magnetic body is lost and regained. There are interesting connections also with bicamerality claiming that ancient people experienced of being in continual contact with God and schizophrenics are the modern bicamerals. What could be the articulation for this in terms of magnetic body?

These things follow a dopamine gradient balance and shifts. It seems that dopamine is one molecule that can expand the coherent phase in our bodies, possibly aided by endorphins and acetylcholin. But this yet doesn't explain OBE nor minds. The detatchment of the magnetic body from the material body is problematic and can only mean that we have a stronger entanglement than our physical body?

Oscillations in emotions and cognitions can be partly at least, explained by stochastic resonence (open systems). Today I feel this can be a way to explain how meridians and 'chi' work. Also our window-effect, 'free will', strong love, spiritual people? This path must be explored.

Think, it is quite possible to explain the topologic, fractal and hierachic body and mind - using TGD :) and mainstream. WAU.

At 10:51 PM, Blogger Ulla said...

How would one feel the motion of magnetic body?

It feels fabuolous, like waves of sweet love that open up your body and take away all bad feelings, all pain, wash you from inside out :) Makes you shine like a white light sometimes. This is why it is called 'enlightment'. Sometimes it can even be seen from outside, some claim (the glory). Once experienced you will never forget it :)

Our left brain wants an explanation, and so we call it God?

At 11:33 PM, Anonymous said...

To Fractality:

I have interpreted the experience of experience like the nasty feeling in stomach when you are near the cliff from the accelerated motion of magnetic body with respect to biological body. Imagined motion would be real motion of the magnetic body relative to biological body. Same experience results when biological body is falling but do not test this conjecture;-). Quite generally, motor imagination could involve motor actions of magnetic body.

I have also talked a lot about plasmoids assigned to magnetic flux quanta. It would be interesting to talk everything again by bringing in negentropic entanglement and hierarchy of Planck constants.

At 11:43 PM, Anonymous said...

To Ulla:

I tend to see dopamine as one particular information molecule whose basic purpose is to create connections in the web of magnetic flux tubes associated with biological body. All these molecules attach to receptors and this process would generate a negentropic bond between the sender of the molecule and its receiver.

In quantum computer picture this is nothing but addressing and the process building these connections would be analogous to the running of a computer program which at basic level creates connections between address for input variables, for sub-programs defining fundamental functions , and for output variables. The output of quantum computer program could be also emotion/ emotional expression as the analog of printing or picture at computer screen. Or motor action!

An interesting question is whether similar connections exist between dark matter magnetic body and biological body. Receptors are very complex molecules: could they really exist as dark matter at distance of order Earth from Earth's surface?

Or could the attachment to receptor be replaced by the analog of password defined as a temporal pattern of say electromagnetic field and perhaps obeying genetic code as Gariaev's findings about remote gene expression suggest?

At 11:52 PM, Anonymous said...

To Ulla:

God as collective consciousness - say assignable to the magnetic body of a community or even Mother Gaia - would allow to take a different attitude to religion instead of the usual yes or no. We would create the God to a high degree since we represent its mental images. God would directly reflect our level of development.

The God of old testament was a monster and psychopath as the story of Job, which is still told without questioning the ethics of this fellow, demonstrates. Completely comparable to Hitler or Stalin.

Accepting the new interpretation this would have a very simple explanation: this God was indeed a primitive caveman. Our challenge is to create a more civilized God instead of repeating old religious dogmas.

At 12:12 PM, Blogger Ulla said...

Ye, I agree with you completely. You know I have many times felt like Job who the Gods played childish games with :)

Pain would in this scenario be only a decoherent subsystem, like cancer? The challenge would be to create a sufficiently large coherent field that could again bind the body systems together? Maybe the Ormus can do that? Or dopamine. Interesting that psychosis sometimes makes pain vanish. They get too much dopamine?

In the stochastic resonance model, that doesn't in itself explain anything in detail, noise is equivalent with dopamine in creating a bigger wave and nerve pulses/receptor answers? Endorphine would be a subsystem to dopamine etc. This is quite cool, and agree with what meridians tell us.

The resonance is the interference of two waves, but the paticle/position is the same, just compressed/enlarged, showing blue/redshift? Interestingly DM has also been suggested to express blueshift :) E=mc2 tells us the blue phase is mc (not quantized)? This large expansion is not yet known? Vanishing matter would be explained in this way. Missing matter at LHC yet?

What kind of math does these express? Not big enough to be p-adics, so it must be thermodynamic potenses. They have found out it forms squares. This means that your NMP can be used?

But this is nothing special for life?

At 1:23 PM, Blogger Ulla said...

"Conclusion: Our findings support the notion that information processing is not only dependent on signal strength but also on a certain amount of basic noise, reflecting the overall energy state of the brain."

At 1:28 PM, Blogger Ulla said... Stochastic Hierarchical Systems: Excitable Dynamics


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