Sunday, April 12, 2009

Still about the emergence of bosonic propagators and vertices

In TGD Universe only fermions are fundamental particles and bosons can be identified as their bound states. This suggest that in the possibly existing QFT type description bosonic propagators and vertices must emerge from the fermionic propagators and from the fundamental fermion-boson vertex appearing in Dirac action with a minimal coupling to gauge bosons. In the earlier posting I discussed how the emergence can be understood in terms of path integral approach (see also the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix"). The nice feature of the approach is that there are no free parameters in the theory. In particular, the counterparts of gauge couplings are predictions of the theory. In this posting I represent some further comments about the resulting Feynman diagrammatics.

  1. Consider first the exponent of the action exp(iSc) resulting in fermionic path integral. The exponent
    exp[i∫ d4xd4y ξbar(x)GF(x-y)ξ(y)]=
    exp[i∫ d4kξbar(-k)GF(k)ξ(k)]
    is combinatorially equivalent with the sum over n-point functions of a theory representing free fermions constructed using Wick's rules that is by connecting n Grassmann spinors and their conjugates in all possible ways by the fermion propagator GF.

  2. The action of
    exp[i∫ d4x (δ/δ ξbar(x))γ • A(x) (δ/δ ξ(x))]
    = exp[i∫ d4k d4k1 (δ/δ ξbar(k-k_1))γ• A(-k) (δ/δ ξ (k_1))]
    on diagrams consisting of n free fermion lines gives sum over all diagrams obtained by connecting fermion and anti-fermion ends of two fermion lines and inserting to the resulting vertex A(k) such that momentum is conserved. This gives sum over all closed and open fermion lines containing n ≥2 boson insertions. The diagram with single gauge boson insertion gives a term proportional to Aμ(k=0) ∫ d4k kμk-2, which vanishes.

  3. Sc as obtained in the fermionic path integral is the generating functional for connected many-fermion diagrams in an external gauge boson field and represented as sum over diagrams in which one has either closed fermion loop or open fermion line with n ≥2 bosons attached to it. The two parts of Sc have interpretation as the counterparts of YM action for gauge bosons and Dirac action for fermions involving arbitrary high gauge invariant n-boson couplings besides the standard coupling. An expansion in powers of γμDμ is suggestive. Arbitrary number of gauge bosons can appear in the bosonic vertices defined by the closed fermion loops and gauge invariance must pose strong constraints on the bosonic part of the action if expressible in terms of bosonic gauge invariants. The closed fermion loop with n=2 gauge boson insertions defines the bosonic kinetic term and bosonic propagator. The sign of the kinetic terms comes out correctly thanks to the minus sign assigned to the fermion loop.

  4. Feynman diagrammatics is constructed for Sc using standard Feynman rules. In ordinary YM theory ghosts are needed for gauge fixing and this seems to be the case also now.

  5. One can consider also the presence of Higgs bosons. Also the Higgs propagator would be generated radiatively and would be massless for massless fermions as the study of the fermionic self energy diagram shows. Higgs would be necessary CP2 vector in M4×CP2 picture and E4 vector in M8=M4×E4 picture. It is not clear whether one can describe Higgs simply as an M4 scalar. Note that TGD allows in principle Higgs boson but - according to the recent view - it does not play a role in particle massivation.

The diagrammatics differs from the Feynman diagrammatics of standard gauge theories in some respects.

  1. 1-P irreducible self energy insertions involve always at least one gauge boson line since the simplest fermionic loop has become the inverse of the bosonic propagator. Fermionic self energy loops in gauge theories tends to spoil asymptotic freedom in gauge theories. In the recent case the lowest order self-energy corrections to the propagators of non-abelian gauge bosons correspond to bosonic loops since fermionic loops define propagators. Hence asymptotic freedom is suggestive.

  2. The only fundamental vertex is AFFbar vertex. As already found, there seems no point in attaching to the vertex an explicit gauge coupling constant g. If this is however done n-boson vertices defined by loops are proportional to gn. In gauge theories n-boson vertices are proportional to gn-2 so that a formal consistency with the gauge theory picture is achieved for g=1. In each internal boson line the g2 factor coming from the ends of the bosonic propagator line is canceled by the g-2 factor associated with the bosonic propagator. In S-matrix the division of the bosonic propagator from the external boson lines implies gn proportionality of an n-point function involving n gauge bosons. This means asymmetry between fermions and bosons unless one has g=1. Gauge couplings could be identified by transferring the normalization factor of gauge boson propagators to fermion-boson interaction vertices so that bosonic propagators would have standard normalization. The counterparts of gauge coupling constants could be identified from the amplitudes for 2-fermion scattering by comparison with the predictions of standard gauge theories. The small value of effective g obtained in this manner would correspond to a large deviation of the normalization factor of the radiatively generated boson propagator from its standard value.

  3. Furry's theorem holding true for Abelian gauge theories implies that all closed loops with an odd number of Abelian gauge boson insertions vanish. This conforms with the expectation that 3-vertices involving Abelian gauge bosons must vanish by gauge invariance. In the non-abelian case Furry's theorem does not hold true so that non-Abelian 3-boson vertices are obtained.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".

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