A more detailed view of topological qubit in the TGD framework

The Zoom discussion with Tuomas Sorakivi about Microsoft's claimed realization of a topological qubit was very inspiring and led to a generalization of the notion of Majorana qubit characterized by Z₂ group acting as reflection so that one an assign parity to Majora qubit. In TGD Z₂ is replaced by a generalization of the Galois group and this leads to a discrete group bringing in mind anyons with a larger number of internal states. This also involves the notion of Galois confinement discussed earlier. What would be achieved would be a dual interpretation as topological qubit or as number theoretic qubit. This conforms with the notion of geometric Langlands duality realized in the TGD framework as M⁸-H duality (see this and this).

1. Background

The basic idea is that the Majorana fermions of condensed matter are assumed to define a qubit. A Majorana fermion would be a superposition of an electron and a hole. The idea is not pretty because it violates the superselection rule for fermions and the conservation of the fermion number is also questionable. It has also been found that the existence of the Majorana fermion claimed by the Microsoft research group and the superconductivity it requires have not been demonstrated.

A hole must physically correspond to the electron being "somewhere else". In the case of an insulating band, it could be in the conduction band, or in the case of a conduction band, in another conduction band: this description would hold in wave vector space.

In TGD, the electron corresponding to a hole could be in another space-time plane. The equivalent of a Majorana fermion would be a superposition of states where the fermion would be on two space-time sheets. It would be a topological qubit because small deformations of the space-time surfaces do not cause contact between the surfaces. Of course, one can argue that the energies must be the same on different sheets. In the case of condensed matter, this would correspond to the branches of the Fermi surface touching each other.

This idea can be realized concretely: a transfer is an operation that, when repeated, produces the original state, i.e. acts like a unitary operator. The square of the Majorana fermion creation operator is correspondingly a unitary operator. This leads to a concrete model (see this) and the idea that OH-O^-+p qubits could realize topological qubits, at least in biology.

Yesterday's discussion led to a review of holography=holomorphic vision.

2. About Galois groups and their TGD counterparts

How to define a Galois group when we are in dimension 4 and not in the complex plane? Is it possible to define a generalization of the concept of ramified primes: these would give a generalization of p-adic primes that label elementary particles in TGD?

2.1 Space-time surfaces as solutions of the equations (f₁,f₂)=(0,0)

Holography= holomorphy vision leads to the following picture.

1. Space-time surfaces are roots (f₁,f₂)=(0,0) of two complex values functionsf_i defining an analytic map from H=M⁴× CP₂ to C². f_i, i=1,2 is an analytic function of 3 complex coordinates of H=M⁴× CP₂ and one hypercomplex coordinate of M⁴. The Taylor coefficients of f_i are in an extension E of rationals. A very important special case corresponds to a situation in which f_i are polynomials. There are good physical reasons to believe that f₂ is the same for a very large class of space-time surfaces and its roots actually define a slowly varying analog of cosmological constant.

   The roots (f₁,f₂)=(0,0) correspond to space-time sheets, which are algebraic surfaces. The space-time surface need not be connected. The Hamilton-Jacobi coordinates (see this) serve the coordinates of H: there is one hypercomplex coordinate u and its dual v and 3 complex coordinates w for M⁴ and ξ₁ and ξ₂ for CP₂. The coordinate curves for u and v of M⁴ have light-like tangent vectors.

2. Dimensional reduction occur because the hypercomplex coordinates are separated from the dynamics and take role of parameters appearing as coefficients of f_i interpreted as functions of w,ξ₁ξ₂ so that only three complex coordinates ξ₁,ξ₂ and w would effectively remain dynamical. For partonic orbits as the interfaces between Minkowskian regions and CP₂-like regions with Euclidean signature of the induced metric, u= constant would be a natural condition. At these 3-surfaces, the dimensional reduction would be complete: the roots would not depend on u. In the interior of CP₂ like region u would be also constant and Minkowskian contribution to the induced metric would vanish as for CP₂ type extremals.
3. If f_i is polynomial P_i with coefficients in the rational expansion E, analytic flows as analogs of homotopies that take roots as regions of the space-time surface to each other would correspond to a 4-D version of the Galois group. The definition of the Galois group operation would be as a flow rather than an automorphism of an algebraic extension leaving E unaffected as usual. Definition as flow is used in braid representations of groups.

   This is new mathematics for me and perhaps for mathematicians as well. It would be a generalization of the 2-D Galois group.

4. The 4-surfaces corresponding to different roots would have lower-dimensional surfaces interfaces. The hypercomplex sector effectively decouples this gives 2 conditions in 4-D space stating that the complex coordinates, say w, are identical at the boundary so that interfaces are string world sheets. This fixes w(u) at the interface.
   1. The roots as 4-surfaces could correspond to branches of a fold taking place along a string world sheet. This suggests a complexification of a cusp catastrophe. For cusp catastrophe, the catastrophe curve is a V-shaped curve along which two real roots of a polynomial of degree 3 depending on a real coordinate x and real parameters a,b co-incide. Now x is replaced with a complex coordinate w which at the string world sheet depends on the coordinate hypercomplex coordinate u. One can say that the 1-D boundary of V is replaced with string world sheets. What happens in the vertex of V is an interesting question. The boundaries of V having coinciding root pairs as analogs co-incide. Does this mean that two string world sheets fuse. Could this be regarded as a reaction in which strings fuse along their full length?
   2. Could the space-time regions defined by the roots genuinely intersect along a string world sheet? This kind of intersection would be analogous to a self-intersection of a 1-dimensional curve. The basic example is the curve x²-y²=0 splitting to the curves x-y=0 and x+y=0.

      If for instance, f₁=P₁ fails to be irreducible and decomposes to a product P₁=Q₁Q₂ of two polynomials Q_i, the roots Q₁=0 and Q₂=0 intersect at the common root Q₁=Q₂=0. These kinds of intersections are excluded if one allows only irreducible polynomials. The irreducibility can fail for some values of the coefficients of the polynomials.

      The space-time surface would decompose to a union of 2 surfaces represented as roots of Q₁ and Q₂ and do not interact unless they intersect along a string world sheet. The dimensional reduction due to the same Hamilton-Jacobi structure implies that 2 2-surfaces intersect in 6-dimensional space. This does not happen in the generic case. Hence this option does not seem possible.

Analytical flows take the points corresponding to the roots from one sheet to another through string world sheets: here cusp catastrophe helps to visualize. String world sheets correspond to the common values of ξ₁, ξ₂, w. For instance w can serve as coordinate and at the intersection w the value is fixed.

The ends of the strings correspond to complex numbers that depend on the time parameter u: the complex number, say w, would represent the intersection of the space-time sheets as a root. The complex roots depend on u through polynomial coefficients. If one has u=constant at the parton trajectories at which the signature of induced metric changes, the u-dependence disappears at the paths of the string ends at which fermions are attached in the physical picture about the situation. Under very mild assumption about the polynomials P_i(w,ξ₂,ξ₂,u=0), the roots can be algebraic numbers in an extension of E and would characterize the intersections of the roots of the equation (P₁,P₂)=(0,0).

These complex numbers are considered a generalization of complex roots and would be related to quantum criticality, i.e., the fact that the two roots are the same and the system is at the interface between space-time regions. The criticality would correspond to a fold of the cusp catastrophe.

If it is possible to attach a Galois group to the set of string world sheets transforming them to each other, it would transform different string world sheets into each other. Could this group serve as an algebraization for the generalized Galois group represented as a geometric flow?

What about the counterparts of p-adic primes? The product of the differences of the roots defines the discriminant D. Can it be decomposed into the product of powers of algebraic primes of the extension E? If so , this would generalize the concept of a p-adic prime. The intersections of the sheets of the space-time surface, or rather their intersections with partonic 2-surfaces, could be associated with p-adic primes. This has just been a physical picture.

2.2 The analogs of Galois group associated with dynamic symmetries

The descriptions g: C²→ C² define dynamic symmetries f=(f₁,f₂) → g(f) , which produce new space-time surfaces of higher complexity.

1. What happens in the operation (f: H→ C²)→ (g\circ f: H→ C²), f H→ C² and g: C²→ C²? The surface g(f) =0 would correspond to the surface (g₁(f₁,f₂), g₂(f₁,f₂))=(0,0).

   The intuitive picture is that complexity increases the in these dynamical symmetries. For example, in the case of C, iterations produce fractals. These descriptions would provide a geometric model for the abstraction and can be combined and iterated.

2. If g(0,0)=(0,0) then (f₁,f₂)=(0,0) remains a root and in the "G\"odelian" view of the classical dynamics of the space-time surfaces produces analogies to theorems (see this). Other roots represent more complex space-time surfaces: the non-trivial action of g brings in the meta-level and makes the composition with g provides statements about statements represented by(f₁,f₂)=(0,0). "Simple" spacetime sheets, which do not allow a decomposition to f=g(h) , would represent lowest level statements. The associated magnetic bodies could correspond to the surfaces (g₁(f₁,f₂), g₂(f₁,f₂))=(0,0). Entire hierarchies of meta-levels are possible.

   Magnetic bodies indeed represent a higher level in the number theoretic hierarchies and correspond to larger values of the effective Planck constant as dimension of extension associated with E. In the TGD inspired quantum biology, the magnetic body serves as a controller of the biological body.

Can the concept of Galois group be generalized in this case?

1. The regions of the surface (g₁(f₁,f₂), g₂(f₁,f₂)=(0,0) correspond to roots. 2+2 conditions fix the roots f₁= a and f₂=b are 6-surfaces, and their intersection is a 4-surface.

   If the consideration is restricted to the surface u=constant, assumed to correspond to a partonic orbit, then the roots do not depend on u and can be algebraic numbers and perhaps a generalization of the Galois group could be defined.

   The condition g₂(f₁,f₂)=0 gives f₁ =h(f₂), where h is an algebraic function. The condition g₁(f₁,h(f₁))=0 gives f₁=a and f₂=b, where a and b are algebraic numbers. They correspond to 6-surfaces: the space-time surface is the intersection of two algebraic 6-surfaces. If (a,b) and (c,d) are not identical, then the corresponding surfaces are disjoint.

2. Is it possible to define a Galois group using the algebraic extension of E defined by the roots? The Galois group would permute the surfaces (f₁=a,f₂=b), which would correspond to pairs of complex numbers and would be disjoint.

   Now the element of the Galois group would not correspond to a flow permuting the pairs (a,b). It would seem that it should act as an automorphism of E× E. Is this possible? Can one provide E× E with the structure of a number field? The only possibility that comes to mind is that E× E corresponds to rationals but a and b are complex numbers. This would provide E× E with a product structure. This option does not however seem plausible.

3. Is it possible to generalize the concept of ramified prime? They would define generalized p-adic primes. The discriminant can be defined as the product of the differences of the roots, which would factor into the product of algebraic primes in the extension E. The roots (a,b) would be in E× E so that the structure of the number field would be required. For quaternions the lack of commutativity implies that the product of the root differences depends on their order.

It was already noticed that there are good physical motivations for decomposing WCW to sub-WCWs for which f₂ is fixed. The counterpart of the ordinary Galois group is obtained in the sub-WCWs: g=(g₁,I) reduces to a map g₁: C→ C. The roots of g₁(f₁)=0 are surfaces (g₁(f₁),f₂)=(0,0). g₁ has n surfaces as roots. The transitions between these disjoint surfaces would generate the analog of the ordinary Galois group acting as a number-theoretic dynamical symmetry group. Also ramified primes as primes of algebraic extension of E are obtained.

1. Representations of the Galois group transfer fermions between space-time regions corresponding to different roots of g₁. The Galois group is generally non-Abelian and its elements could appear in topological quantum computation as basic operations for the topological qubits. The analogs of anyons would be irreducible representations of the Galois group.
2. If the degree n is prime, g is a prime polynomial. It cannot be represented as a composite of polynomials, whose degree is a product of smaller integers.

   Remark: If P is irreducible then it cannot be a product, in which case the degree would be the sum of their degrees. Therefore one has two kinds of primeness.

3. The surfaces corresponding to different roots of g₁ are disjoint. If the roots are the same then the surfaces are the same. If g(0,0)=0 then (f₁,f₂)=(0,0) is a root. As two roots approach each other. the two separate surfaces merge into one. What does this mean physically? Should one regard the identical copies of the surface as different surfaces, members of a double, and carrying different many-fermion states? In any case, the order of the Galois group is reduced in this case.

3. On the intersections of 4-surfaces

There are several options to consider.

1. The 2 4-surfaces X⁴ and Y⁴ correspond to different pairs (f₁,f₂). If the Hamilton-Jacobi structures are different so that the hypercomplex coordinates (u,v) are different, the intersection X⁴\cap Y⁴ is a discrete set of points. Field theory suggests itself as a natural description of fermions assigned with the interaction points.

   If the Hamilton-Jacobi structures are the same, the dimensional reduction occurs and one has effective intersection of 2 complex surfaces in 6-D complex space. In the generic case the intersection is empty.

2. One can also consider the analogs of self-intersections as interfaces for 2 4-D roots for the same pair (f₁,f₂). The intersection consists of string world sheets. As found, genuine self-intersection is exclude so that only the analogy of a complexified cusp catastrophe remains.

   String model is a natural description of the interactions of 4-surfaces and the self-interaction of 4-surfaces in the fermionic sector. Fermion propagators can be calculated because the induced spinor field is a restriction of the corresponding H.

The analogy of TGD based physics with formal systems discussed in \cite{btar/Gtgd} led to ask whether the interaction of space-time surfaces involves the fusion of the 3-surfaces with different Hamilton-Jacobi structures to a single connected 3-surface with a common Hamilton-Jacobi structure for the components. Physically the tusion could mean a generation of monopole flux tube contacts between the 3-surfaces.

In the G\"odelian framework, this interaction would have an interpretation as a morphism realized as an action of the composite space-time surfaces on each other. In the connected intermediate state, string model type description might apply in the fermionic degrees of freedom. Even stronger condition would be that fermions reside at the string ends at partonic orbits.

4. Galois group as as group of possible transfer operations for fermions and a generalization of the Majorana qubit

4.1 Roots for the condition (f₁,f₂)=(0,0) as space-time sheets

Generalization of the Galois group. Galois generalizes Z₂ to Majorana fermions. Classical equivalent of the transfer operation between space-time sheets. A particle is transported through a string word sheet corresponding to a common root pair to another sheet.

Topological/number-theoretic qubit. Transfer through a string world sheet. What is the physical interpretation. String 1-D object in 3-space. Could the Riemann surface for z^1/n serve as an analogy. Anyons and braid statistics. Since hypercomplex coordinates are passive, we get effective 2-dimensionality and braid statistics.

4.2 Roots in the special case g=(g₁,Id)

Ordinary roots of a polynomial represented as 4-surfaces. Disjoint or identical. However, the representation of the Galois group of g₁ is non-trivial. These would correspond to abstractions. Fermion transfer between disjoint surfaces Galois group operation represented using oscillator operators.

When does this?

1. This happens only if f₁ allows the decomposition f₁= g₁(h₁). When could this be possible? In the case of polynomials, this means that the degree of f₁ for a given H complex coordinate ξ₁,ξ₂, or w polynomial is the product of the degrees of n₁× n₂× n₃ for the lower degree polynomials n₁,n₂,n₃.
2. If the degrees of the polynomial for different coordinates are primes, then the decomposition is not possible. These would be "prime polynomials". The 3 prime numbers p₁,p₂,p₃ characterize these. If it is a homogeneous polynomial, then one prime number p is enough. These polynomials would be in a special position physically. They would correspond to "elementary particles". The tetrahedra associated with them would be uniform.

4.3 Concrete realization of topological/number-theoretic qubit and generalization of qubits

The TGD based view leads to generalization of bit to n-ary digit or pinary digit, where n or p corresponds to a degree of a polynomial g₁ in g=(f₁,Id) defining a dynamical symmetry and associated Galois group whose elements would correspond to transfers of fermions between different branches of the space-time surface.

1. Roots as regions of an n-sheeted space-time surface correspond to roots (f₁,f₂)=(0,0) and would correspond to different values of an n-ary digit. They are glued together along string world sheets as analogs of folds.

   The functional composition f→ g(f) gives rise to hierarchies of Galois groups. The Galois group, represented as analytic flows, replaces the group Z₂ of the Majorana case. Analytic flows define braiding operations, which define the 4-D Galois group.

2. Also the dynamical symmetries g give rise to an analog of a Galois group. The non-vanishing roots of g are disjoint. It seems that the Galois group can be defined only if one has g= (g₁,I).

   For OH-O^-+p qubits (see this and this) they could correspond to different pairs because h_eff would be of different magnitude.

4.4 Generalization of a bit to n-ary digit and pinary-digit

The replacement of bit with n-ary digit would take place when the degree d of the polynomial P₁ (or g₁ in g=(g₁,Id)) is d=n and bit → pinary digit when the d is a prime: d=p. Polynomials for which the degrees with respect to complex coordinates of H are primes are primes with respect to the functional composition and could physically correspond to fundamental objects appearing at the bottom of the hierarchy obtained by a functional composition with maps g.

These primes should not be confused with ramified primes. One can of course ask whether the p-adic primes appearing in p-adic mass calculations could actually correspond to these primes.

This allows us to consider a possible definition for a topological/number-theoretic qubit. For g(0)=0, the original surface is included in the set of g\circ f=0 surfaces. In the case of OH-O^-+p qubits, the magnet monopole flux tubes could correspond to the non-vanishing root f\neq 0 of g. In this case the Galois group of g would be Z₂ and correspond to the parity of Majorana fermions. In the general case more complex Galois groups are possible.

4.5 A more precise connection to the Majorana qubit of condensed matter

The definition of a Majorana qubit involves the observation that when two branches of the Fermi surface that correspond to an insulator and to a conduction band touch each other, the gap energy disappears. In superconductivity, this gap energy is very small but non-vanishing. If this energy vanishes, Majorana type excitation becomes possible and is interpreted as a quantum superposition of an electron and a hole.

What could this situation correspond to or how could it generalize in TGD?

1. M⁸-H duality (see this) strongly suggests that Fermi surfaces determined as an energy constant surface in momentum space have space-time counterparts.
2. The group Z₂ defining the parity of Majorana qubit would be generalized to Galois group and and one can consider two options corresponding 1) to the 4-D Galois group realized as analytic flows assignable to a connected 4-surface (f₁,f₂) and 2) to the Galois group assignable to g= (g₁,Id) acting as a dynamical symmetry.

Consider option 1) first.

1. The Galois group would relate string world sheets to each other. The branches of the Fermi surface could at the space-time level correspond to 2-D string world sheets at which the roots associated with the different space-time surface sheets (f₁,f₂)=(0,0) coincide . One could move from one branch of the space-time sheet to another through the string world sheets. Each string world sheet would correspond to a discrete complex point (ξ₁,ξ₂,w).
2. The E³ projection of the string world sheet would be a string, which would have apparent ends at the "boundary" of the 3-surface. The 2-D "boundaries" of the 3-surfaces are surfaces, where the 3-surface has a fold, i.e. the normal M⁴ coordinate has a maximum value. One can say that the string effectively ends at these surfaces although it actually has a fold.

   String would sheet would also have an end at the partonic orbit, where the signature of the space-time metric changes. Since the coordinate u would be constant inside the CP₂ type extremals, the 2-D string world sheet reduces to a 1-D light-like curve inside it.

   In the case of topological qubits, the superconducting wire could correspond to the string identifiable as the superconducting wire whose ends correspond to the points of the Fermi surface at which the branches of the Fermi surface touch. The ends of the wire, assumed to carry Majorana fermions, would correspond to the real ends of the string at partonic orbits to which fermions are assigned or to an apparent end at the fold.

3. The situation would correspond to quantum criticality, since even a small perturbation will move the particle to one of the branches.

For option 2), the space-time surfaces related by the Galois group for g=(g₁,Id) would be disjoint. This does not conform with the assumption that Fermi surfaces touch at a point. This picture could however work for OH-H^- topological qubits for which the two surfaces related by Z₂ Galois group for g=(g₁,Id) would have different "internal" Galois groups represented as flows leaving the space-time surface invariant.

See the article The realization of topological qubits in many-sheeted space-time or the chapter Quartz crystals as a life form and ordinary computers as an interface between quartz life and ordinary life?.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.