Tuesday, December 05, 2006

How induced spinor field and imbedding space coordinates can become non-commutative?

The context of this posting is defined by Jones inclusion N subset M with N defining the measurement resolution. M/N defines the quantum Clifford algebra with N valued non-commutative matrix elements. M/N describes the physics modulo measurement resolution and N takes effective the role of a gauge algebra. The general vision is that the transition to the description modulo measurement resolution means that you unashamedly make everything complex valued that you see around N-valued and thus non-commuting. You do this for unitary matrices (including S-matrix), hermitian operators, spinors, rays of state space, etc..

The question is whether you should make this trick even for some coordinates of the imbedding space so that you would get contact with string models and understand bosonic quantization of strings in terms of finite measurement resolution. The following argument deducing number theoretical braids from finite measurement resolution suggests that this is indeed the case.

The transition M→ M/N interpreted in terms of finite measurement resolution should have space-time counterpart. The simplest guess is that the functions, in particular the generalized eigen values, appearing in the generalized eigen modes of the modified Dirac operator D become non-commutative. This would be due the non-commutativity of some H coordinates, most naturally the complex coordinates associated with the geodesic spheres of CP2 and δ M4+=S2× R+.

Stringy picture would suggest that these complex coordinates become quantum fields: bosonic quantization would code for a finite measurement resolution. Their appearance in the generalized eigenvalues for the modified Dirac operator would reduce the anti-commutativity of the induced spinor fields along 1-dimensional number theoretic string to anti-commutativity at the points of the number theoretic braid only. The difficulties related to general coordinate invariance would be avoided by the fact that quantized H-coordinates transform linearly under SU(2) subgroup of isometries.

The detailed argument runs as follows.

  • Since the anti-commutation relations for the fermionic oscillator operators are not changed, it is the generalized eigen modes of D (with zero modes included) which must become non-commutative and spoil anti-commutativity except in a finite subset of the number theoretic string identifiable as a number theoretic braid. This means that also the generalized eigenvalues become non-commuting numbers and should commute only at the points of the number theoretic braid. This would provide a physical justification for the proposed definition of the Dirac determinant besides mere number theoretic arguments and finiteness and well-definedness conditions.

  • Functions of form pζ-1(z(x))a(x), where a(x) is ordinary matrix acting on H-spinors appear as spinor modes. z is the complex coordinate for the geodesic sphere S2 of either CP2 or δ M4+=S2× R+. z(x) is obtained as a projection of X^3 point x to S2 and an excellent candidate for a non-commutative coordinate. In the reduction process the classical fields z(x) and z*(x) would transform to N-valued quantum fields quantum fields z(x) and z(x).

  • The reduction for the degrees of freedom in M→ M/N transition must correspond to the reduction of number theoretic string to a number theoretic braid belonging to the intersection of the real and p-adic variants of the partonic 3-surface. At the surviving points of the number theoretic string the situation is effectively classical in the sense that quantum states can be chosen to be eigen states of z(xk) in the set {xk} of points defining the number theoretic braid for which the commutativity conditions [z(xi),z(xj)]=0 hold true by definition. The quantum states in question would be coherent states for which the description in terms of classical fields makes sense.

  • The eigenvalues of z(xi) should be algebraic numbers in the algebraic extension of p-adic numbers involved. Since the spectrum for coherent states a priori contains all complex numbers, this condition makes sense. The generalized eigenvalues of Dirac operator at these points would be complex numbers and Dirac determinant would be well defined and an algebraic number of required kind. Number theoretic universality of ζ would fix the eigen value spectrum of z to correspond to ζ(s) at points s=∑k nksk, nk≥ 0, ζ(sk=1/2+ iyk)=0, yk>0. Note however that this condition is not absolutely essential.

  • If a reduction to a finite number of modes defined at the number theoretic string occurs then [z(x),z(y)] can vanish only in a discrete set of points of the number theoretic string. The situation is analogous to that resulting when the bosonic field z(φ) defined at circle has a Fourier expansion z(φ)= ∑m am exp(imφ/n), m= 0,1,...n-1, [am,an] =δm,n. [z(φ1),z2)] is given by ∑mexp(imφ/n), φ=φ12 and in general non-vanishing. The commutators vanish at points φ= kπ, 0< k< n so that physical states can be chosen to be eigen states of the quantized coordinate z(φ) at points φ=kπ/n. One can ask whether n could be identified as the integer characterizing the quantum phase q=exp(iπ/n). The realization of a sequence of approximations for Jones inclusion as sequence of braid inclusions however suggests that all values of n are possible.

  • This picture conforms with the heuristic idea that the low energy limit of TGD should correspond to some kind of quantum field theory for some coordinates of imbedding space and provides a physical interpretation for the quantization of bosonic quantum field theories. M→ M/N reduction is analogous to a construction of quantum field theory with cutoff. The replacement of complex coordinates of the geodesic spheres associated with CP2 with time shifted copies of δ M4+ defining a slicing of M4+ would define the quantization of H coordinates. This notion of quantum field theory fails at the limit of continuum

For a brief summary of quantum TGD see the article TGD: an Overall View.

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