M Theory Lesson 185
A recent talk by Yong-Shi Wu points out that multiple qubit quantum circuits are closely related to the Jones invariant at a fourth root of unity. The factor of four comes from the 4 Bell states, or rather the $2^{2}$ MUB case.
For any prime $p$, M theoretic quantum information likes to specialise the knot invariants to associated roots of unity. For example, the trefoil knot at a cubed root of unity is always 1, and this normalises torus knots. This follows from the categorical $\hbar$ hierarchy, which insists that $q$ take on a fixed value determined by the categorical dimension. If this dimension were given by the number of knot crossings, as it is in Khovanov homology, it suggests a study of the numerical Jones polynomial for $q$ fixed at a primitive root of unity corresponding to the number of crossings. This is not usually done. One often encounters studies of fixed values of $q$ for all knots, but not a grading by crossing number.
A grading by strand number, however, is common in the connection between MZV algebras, knots, Feynman diagrams and chord diagrams, originally due to Kreimer but now studied by many mathematicians. The strand number is also 3 for a trefoil, or 2 for a basic braid generator associated to a qubit, so this grading is important in the analysis of quantum circuits.
For any prime $p$, M theoretic quantum information likes to specialise the knot invariants to associated roots of unity. For example, the trefoil knot at a cubed root of unity is always 1, and this normalises torus knots. This follows from the categorical $\hbar$ hierarchy, which insists that $q$ take on a fixed value determined by the categorical dimension. If this dimension were given by the number of knot crossings, as it is in Khovanov homology, it suggests a study of the numerical Jones polynomial for $q$ fixed at a primitive root of unity corresponding to the number of crossings. This is not usually done. One often encounters studies of fixed values of $q$ for all knots, but not a grading by crossing number.
A grading by strand number, however, is common in the connection between MZV algebras, knots, Feynman diagrams and chord diagrams, originally due to Kreimer but now studied by many mathematicians. The strand number is also 3 for a trefoil, or 2 for a basic braid generator associated to a qubit, so this grading is important in the analysis of quantum circuits.
posted by Kea | 5:37 PM
5 Comments:
Another good post! Re: Your earlier comment.
"I would appreciate the criticism more if it came from people who actually had some clue what I was talking about, but strangely enough, such people tend to be encouraging."
My experience talking with the senior people in physics (Turner, Blandford, Perlmutter, John Huchra, even Alan Guth) is that they are smart enough to nod and listen. Attacks would make them look silly if someday Riofrio were proven to be right.
At the risk of drawing ire of those who make negative attacks, they uniformly come from very small peoplr who have little future in science. You are already a fine ambassador of M-theory.
Thanks again, Louise. Of course, as usual, I am not so certain that I have a future in science, but they will have to shoot me (literally) before I give up trying.
Dear Kea,
this is off topic but I cannot remain silent;-). While writing lectures to Hungary I got new results relating to the speculative "must-be-trues" of quantum TGD.
I took as a challenge to systematically compare handful of first principle approaches to quantum TGD.
*Geometrization of quantum physics using classical spinor fields in the "world of classical worlds" consisting of light-like 2-surfaces in M^4xCP_2
*TGD as almost TQFT for light-like 3-surfaces.
* Three number theoretical visions (unification of real and p-adic physics; the purely number origin of theoretic standard model symmetries and M^4xCP_2; the construction of infinite primes as repeated second quantization of arithmetic QFT)
*Quantum TGD from hyper-octonionic conformal field theory for HFF factor of type II_1.
*M-matrix in zero energy ontology as almost unique "complex square root" of density matrix from the finite resolution of quantum measurement using Connes tensor product.
It seems that hyper-octonionic conformal field theory produces almost all speculative pieces of quantum TGD.
a) Associativity for arguments of n-points functions of hyper-octonionic conformal fields having values in hyperfinite factor of type II_1 (Clifford algebra valued functions in the world of classical worlds creating physical states) restricts the arguments to hyperquaternionic plane so that 4-D Minkowski space emerges from associativity.
b) Commutativity restrict the arguments to some hyper-complex plane (2-D Minkowski space) of M^4. If 2-D partonic ends of space-time surfaces belong to boundary of M^4_+/- in hyper-octonionic space, their points associate. Same for light-like 3-surfaces (tiny wormhole throats representing elementary particles) if they belong to M^4_+/-. Different points in the interior space-time surface cannot associate in general.
c) The commuting points at partonic 2-surfaces belong to the intersection of light-like rays from the tip of M^4_/- with partonic 2-surface at delta M^4_+/-. Unique discrete set identifiable in terms of number theoretical braids results for each choice: one must integrate over them. Under mild assumptions the points at the ends of number theoretic braids are rational/algebraic as points of M^8.
At light-like 3-surfaces 1-D curves, strands of number theoretical braid, correspond commutative sub-manifolds.
Recall that discretization required is required by TGD as almost TQFT approach, by p-adicization and by identification of discretization as space-time correlate for finite measurement resolution. Associativity and commutativity gives all this!
d) Quite generally, associativity combined with the existence of global commutative plane M^2 in the tangent space of hyperquaternionic tangent plane of hyper-quaternionic space-time surface implies that hyper-quaternionic surfaces of M^8 can be mapped to surfaces in M^4xCP_2.
Thus global choice of plane of non-physical polarizations at classical level made also in string models and gauge theories has purely number theoretic interpretation and implies the duality between descriptions based on M^8 and M^4xCP_2 (number theoretical compactification).
This also assigns to light-like 3-surfaces inM^4xCP_2 unique 4-D plane and this fixes the initial conditions for the extremal of Kahler actions so that one space-time surface as analog of Bohr orbit is fixed uniquely.
Essentially all speculative "must-be-trues" follow automatically from associativity and commutativity.
For details see
my blog.
Associativity and commutativity gives all this!
Hi Matti, your comments are most welcome. It is good to hear you have made yet more progress tying different ends of TGD together. The importance of assoc/commmut is something I can relate to from the categorical point of view.
Yes, the power of these conditions is amazing. Already in ordinary conformal theory associativity plays key role. About the role of associativity and commutativity in category theoretical approach I have only dim ideas. In hyper-octonionic situation the implications are purely geometric.
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