Friday, August 17, 2012

Emergent Braided Matter of Quantum Geometry

Ulla gave in the comment section of previous posting a link to an article by Bilson-Thompson , Hackett , Kauffman, and Wan with title "Emergent Braided Matter of Quantum Geometry". The article summarizes the recent state of an attempt to replace space-time continuum with discrete structure involving braids. The article satisfies high technical standards - note that one of the authors is Louis Kauffman, a leading knot theorist. The mathematician inside me is however skeptic for several reasons.

  1. Continuous symmetries - in particular Lorentz and Poincare invariance are lost. One must however get continuous space-time, which we use to organize our observational data and this requires ad hoc assumptions to get "long wave length limit" of the theory.

    The notion of space-time dimension becomes questionable. In algebraic topology continuous space is replaced with a web of simplexes of various dimensions embedded into the space. The simplexes with maximum number of vertices define the dimension of manifold. Genuine discretization would not allow the imbedding and one would lose all information about manifold. Everything would reduce to combinatorics and trying to get continuous space-time from mere combinatorics is like squaring of circle.

    The notion of distance realized in terms of Riemann geometry is fundamental for (quantum) physics. Without it one has just topological quantum field theory. Braids indeed appear naturally in topological quantum theory. The notion of metric must be introduced in ad hoc manner if space-time is discretized.

  2. The proposed approach identifies particles as 3-braids (this does not work and sin network is essential for the proposed generalization). The introduction of braiding is in conflict with the original idea about discreteness at fundamental level. Braiding requires the imbedding of the spin network into some continuous space. Continuous imbedding space would of course make possible to introduce also the notion of length and also "long wave length limit" could be more than a trick of magician. The continuous space-time seems to pop up irresistibly even in these noble attempts to get rid of it!

  3. The identification of braid invariants with standard model quantum numbers might look like an innocent operation. Braid invariants are however discrete topological invariants whereas standard model quantum numbers are group theoretical invariants. The latter ones are much more refined requiring continuum topology, differential structure, and Riemann metric. These quantum numbers are always with respect to some choice of quantization axes unlike topological quantum numbers. This makes the idea of assigning gauge interactions to topological invariants highly implausible.
These objections do not mean that braids could not be important in fundamental physics and they indeed are in a central role in the fundamental physics predicted by TGD. In TGD one does not give up the notion of continuum but introduces finite measurement resolution as a fundamental notion described in terms of inclusions of hyper-finite factors at quantum level and by discretization at space-time level.

Finite measurement resolution is seen as a property of state rather than a limitation preventing to know everything about it. The solutions of the modified Dirac equation indeed lead to this notion automatically: by conservation of electric charge they are localized to 2-D surfaces (string orbits) of space-time surface defining orbits of space-like braids and partonic 2-surfaces. Their ends at 3-D light-like wormhole throats define light-like braids.

At quantum level inclusion of hyper-finite factor is part of definition of state and leads to "quantum quantum theory" with non-commutative WCW ("world of classical worlds") spinors. Infinite-dimensional space of quantum states is replaced with space of "quantum quantum states" with finite fractional dimension. More concretely:

  1. Finite measurement resolution (and finite resolution of cognition) replaces partonic 2-surfaces with the collections of ends of light-like orbits of wormhole throats. They correspond to external particles of generalized Feynman diagrams and define also their vertices as surfaces at which the wormhole throat orbits are glued to together along their ends like lines of Feynman diagram. Braid strands carry quantum numbers of fermion or antifermion (quark or lepton). String world sheets result when space-like braid strands connecting braid ends at different wormhole throats move.

    The orbits of the ends of space-like braid strands move along 3-D light-like orbit of wormhole throat defining what might be called light-like braid strands. Space-like and light-like braidings become part of fundamental physics. This physics is new physics: there is no attempt to interpret it as standard model physics.

    This new physics becomes interesting only when the number of fermions and antifermions at partonic 2-surface is at least three as also in the original variant of Bilson-Thompson model. Standard model particles in TGD framework contain only single fermion or antifermion per wormhole throat and thus correspond to 1-braids. The experimental braid physics at elementary particle level there would belong to - I cannot say how distant - future in TGD Universe;-).

  2. Partonic 2-surfaces could however have macroscopic size and one can ask whether anyonic systems could correspond to large partonic 2-surfaces containing many-electron states and having non-standard value of Planck constant scaling up the size of the wormhole contact. The electrons would move with parallel momenta and it is not clear whether this can make sense. What is interesting that the fermionic oscillator operator algebra at partonic 2-surface defines SUSY algebra with large value of N and therefore anyonic system would represent SUSY with large N.

To my opinion the basic problematic assumption of "Emergent Braided Matter of Quantum Geometry" is the assumption that discretization is something fundamental. Mathematician would see spin networks without imbedding as something analogous to the attempt to square the circle. The irony is that by introducing braiding the authors are forced to give up the motivating assumption! Why not try to deduce braids from a theory starting directly from space-time?


Ulla said...

This was the guy highlighted by KEA. I noted he has a Higgs double particle (quasiparticle) too.

If you read this Kea, I wish you could continue your work. It isn't important with the Higgs 'discovery'. It highlights the problems with the Higgs mechanism only.

Ulla said...

The microstates must also have a geometry?

Unknown said...

‎"Why not try to deduce braids from a theory starting directly from space-time? "

Because we found out that at small scales at least space (possibly time?) and energy are getting discrete, and continuity is only residing in amplitudes, which means in effect, probabilities calculated from ensembles of elementary events? Isn't this the problem of unification of general relativity with quantum theory? So I don't understand your remark, please explain.

Matti Pitkanen said...

Dear Marcus,

thank you for the objection;-).

I do not see space-time continuity as a problem in unifying general relativity with quantum theory and I listed the many problems emerging when genuine discreteness is assumed. And the irony is that the introduction of braids means giving up the genuine discreteness by assuming imbedding to 3-D space: braiding is something associated with imbedding, to the relation with external 3-D continuum.

I believe that effective discretization is natural counterpart for finite measurement resolution and essential part of theory as it is of course at practical level in QFTs already now. This discretezation is that for imbedding and one has quantum superposition for different discretizations so that no problems with continues symmetries are encountered.

The problems if unification of GRT and quantum theory relate to my view to lack of sound principles and loose conceptualization. The notion of four-momentum is ill-defined in GRT, one has only local density of it. Quantum theory requires it as a global notion.

String model suggests how to proceed. The replacement of space-times with 4-D surfaces meaning that classical fields are obtained by inducing metric and spinor structure of M^4xCP_2 is the basic genuinely new element in TGD approach.

Sub-manifold gravity not only solves the energy problem but also brings the rich dynamics related to the shape of space-time as seen from 8-D space. This approach allows the reduction of standard model symmetries to isometries and holonomies of M^4xCP_2 (braiding is topological aspect of same phenomenon for 1-D curves in 3-D space!). The notion of space-time generalizes dramatically and Feynman graphs have space-time counterparts as regions with Euclidian signature of induced metric.

Enormous reduction of local in degrees of freedom takes also place but does not lead to any obvious contradiction since many-sheeted space-time brings in non-local degrees of freedom as a compensation. The super-conformal symmetries of string models generalize: in particular space-time dimension D=4 emerges as a unique dimension.

The essence of TGD is what it means quantum physically to be a sub-manifold. Braiding is one aspect of this.

Matti Pitkanen said...

To Ulla:

I have some sensitive attitude to Higgs particle. TGD counterpart for Higgs gives a dominant contribution only to the masses of weak gauge bosons, not to those of fermions. This solves the hierarchy problem which motivates in turn standard SUSY.

After many turns and twists I dare say that standard SUSY does not emerge from TGD. And it does not seem to emerge from LHC either;-). By the way, I summarize the Higgs Odysseia at and SUSY Odysseia at .

For these reasons it would be somewhat misleading to talk about Higgs although Higgs and others certainly deserve their Nobel. Euclidian M_89 pion does not sound very media sexy. And fermion-phobic Higgs neither;-). Einstein is to blamed for all these head aches related to mass: could it be stupid to call it Alberton so that everyone would knew whom to accuse?;-)

Ulla said...

Alberton is not sexy, rather it is feminine in my ears. ZWION, maybe? Z is a very aggressive letter :)

I struggle hard with his mass...

Topological space as guiding principle, and then the time must be the first one, otherwise time would be emergent and not fundamental? The second is light? Gravity and time are twins?
Dimension 0 for time only,
Another site had a topological dimension to start from -1, because of the math. This is imaginary time? Hawking and Sarfatti talked about that.

In this way the FTL can also be explained.

Sabine Hossenfelder has written about FTL,

Ulla said...

LHC Experiments Shed Light on Primordial Universe

Ulla said...

The theory of superqubits
Kamil Bradler
(Submitted on 14 Aug 2012)
Superqubits are the minimal supersymmetric extension of qubits. In this paper we investigate in detail their unusual properties with emphasis on their potential role in (super)quantum information theory and foundations of quantum mechanics. We propose a partial solution to the problem of negative transition probabilities that appear in the theory and has been previously reported in [arXiv:1206.6934]. The modification does not affect the performance of supersymmetric entangled states in the CHSH game - superqubits provide resources more nonlocal than it is allowed by ordinary quantum mechanics.

I thought it may say something about the Schrödinger cat, but MSSM still??? And non-locality? Seems they begin to expand their theories toward the hierarchy thinking?

Matti Pitkanen said...

Super qubit is an interesting albeit purely formal idea. From hasty reading I understood that unitarity is extended to super-unitarity. The problem is that one ends up with negative probabilities and this is certainly a fatal outcome.

There are many views about supersymmetrization standard SUSY induces super-space extending space-time by adding anticommuting coordinates as a formal tool. Many mathematicians are not enthusiastic about this approach because of the purely formal nature of anticommuting coordinates. Also I regard them as non-sense and there is actualy no need to introduce them as the following little argument shows.

Grassmann parameters (anticommuting theta parameters) are generators of Grassmann algebra and the natural object replacing super-space is this Grassmann algebra with coefficients of Grassmann algebra basis appearing as ordinary real or complex coordinates. This is just an ordinary space: mysterious anticommuting coordinates are not needed. To me this notion is one of the conceptual monsters created by the over-pragmatic thinking of theoreticians speaking native american english;-).

This allows allows to replace field space with super field space, which is completely well-defined object mathematically, and leave space-time untouched. Linear field space is simply replaced with its Grassmann algebra. For non-linear "field space" this replacement does not work. This allows to formulate the notion of linear super-field just in the same manner as it is done usually.

Another manner to realize SUSY in terms of representations the super algebra of conserved super-charges. In TGD framework these super charges are naturally associated with the modified Dirac equation. In TGD framework super-conformal symmetry is defined in this manner and anticommuting coordinates and super-fields do not appear anywhere.

Ulla said...
this is important for the brain function. Remember the CNS circulation and pulse I have talked of earlier.

The team found that glial cells called astrocytes use projections known as “end feet” to form a network of conduits around the outsides of arteries and veins inside the brain.
Those end feet are filled with structures known as water channels or aquaporins, which move CSF through the brain. The team found that CSF is pumped into the brain along the channels that surround arteries, then washes through brain tissue before collecting in channels around veins and draining from the brain.