Is the perturbation theory based on TGD inspired definitions of super fields UV finite?

In the case of infinite-dimensional super-space the definition of the super-fields is not quite straightforward since the super-space integrals of finite polynomials of theta parameters always vanish so that the construction of super-symmetric action as an integral over super-space would give a trivial result. For chiral fields the integrals are formally non-vanishing but in the case that the super-field reduces to a finite polynomial of theta at y^μ=0 the non-vanishing terms in real Lagrangian involve the action of an infinite number of operators D^cα_c (^c denotes overline for D and _c dot for Weyl spinor index) implying the proportionality to an infinite power of momentum which vanishes for massless states. It seems that one should be able to add in a natural manner terms which are obtained as theta derivatives of the product of all theta parameters and that the action should consist of the products of the terms associated with mononomials of theta and monomials of derivatives with respect to theta parameters acting on the infinite product of theta parameters, call it X.

The fact that positive resp. negative energy vacuum is analogous to Dirac sea with negative resp. positive energy states filled suggests a remedy to the situation. This would mean that positive energy chiral field is just like its ordinary counterpart whereas negative energy chiral fields would be obtained by applying a polynomial of derivatives of theta to the product X=∏θ of all theta parameters. The theta integral of X is by definition equal to 1. In integral over theta parameters the monomials of theta associated with positive energy chiral field and negative energy chiral field would combine together and one would obtain desired action. In the following this approach is sketched. Devil lies in the details and detailed checks that everything works are not yet done.

This was what I wrote in the first version of this posting and I was right;-)! Devil indeed lies in the details! The calculations turned out to contain blunder (should I blame flu or market economy for the error or just admit that I have miserable calculational skills?;-)). It became clear that in TGD context the definition of super-covariant derivative reducing to ordinary partial derivative leads to much more elegant theory. In zero energy ontology super-symmetry reduces to analyticity with respect to theta parameters. In standard framework analyticity would not give kinetic terms to the chiral action but now the situation is different.

1. TGD variants of chiral super fields

Consider first the construction of chiral super-fields and of the super-counterpart of Dirac action.

1. Wormhole throats carry a collection of collinearly moving fermions with momentum appearing in the measurement interaction term identified as the total momentum. This suggests that kinetic terms behave positive powers of Dirac operator with one power for each theta parameter.

2. One must be careful with dimensions. The counterpart of Dirac operator is D = σ^k(p_k+Q_k)/M. The mass parameter M must be included for dimensional reasons and changes only the normalization of the theta parameters from that used earlier and changes the anti-commutation relations of the super-algebra in an obvious manner. The value of M is most naturally CP₂ mass defined as m(CP₂) = n× hbar₀/R, where R is the length of CP₂ geodesic and n is a numerical constant.

3. In the case of single wormhole throat one can speak about positive and negative energy chiral fields. Positive energy chiral fields are constructed as polynomials, and more generally, as Taylor series whereas negative energy chiral fields are obtained by mapping positive energy chiral fields to an operator in which each theta parameter θ is replaced with

   ∂_θD=∂_θσ^k(p_k+Q_k)/M .

   This operator acts in the product X of all theta parameters to give the negative energy counterpart of chiral field. The inclusion of sigma-matrices is necessary in order to obtain chiral symmetry at the level of H, in particular the counterpart of Dirac action. In the integral over all theta parameters defining the Lagrangian density the terms corresponding to mononomials M(θ,x) and their conjugates M(∂θ _cD^→,x) are paired and theta integrals can be carried out easily. Here → tells that the spatial derivatives appearing in D are applied to M.

4. There is an asymmetry between positive and negative energy states and the experience with the ordinary Dirac action Ψ^cD^→Ψ-Ψ^cD^←Ψ (^c denotes overline) suggests that one should add a term in which θ parameters are replaced with -Dθ so that space-time derivatives act on the positive energy chiral field and partial derivative ∂_θc appear as such. The most plausible interpretation is that the negative energy chiral field is obtained by replacing θs in the positive energy chiral field with ∂_θs and allowing to act on X. The addition of D would thus give rise to the generalization of the kinetic term.

5. Chiral condition can be posed and one can express positive energy chiral field in as an infinite powers series containing all finite powers of theta parameters whereas negative energy chiral field contains only infinite powers of θ. The interpretation is in terms of different Dirac vacuum. What one means which super-covariant derivatives is not quite clear.
   1. The usual definition of super covariant derivatives would be as

      D_iα=∂_iα+ i(θ_cD)_iα ,

      D^c_iαc=∂_iαc +i(Dθ)_iαc .

   2. A definition giving rise to the same anti-commutators would be as

   D_iα=∂_iα ,

   D^c_iαc=∂_iαc +2i(Dθ)_iαc

   In the recent case D^c does not appear at all in the chiral action since for negative energy chiral field conjugation does not correspond to θ→θ_c but to θ→ ∂_θ and 1→ X. Hence the simplest theory would result using D_α=∂_α.

   3. If one includes into the product of X of theta parameters only θs but not their conjugates, the two definitions are equivalent since the powers of θ_cDθ give nothing in theta integration. This definition of X is be possible using the definition of hermitian conjugation appropriate also for N=∞. This formalism of course works also for a finite value of N.

Consider now the resulting action obtained by performing the theta integrations. The interesting question is what form of the super-covariant derivatives one should use. The following considerations suggests that the two alternatives give almost identical -if not identical- results but that the simpler definitionD_α=∂_α is much more elegant.

1. For D_iα=∂_iα the propagators are just inverses of D^d where d is the number of theta parameters in the monomial defining the super-field component in question so that the Feynman rules for calculating bosonic propagators and vertices are very simple. Only the spinor and vector terms corresponding to degree d=1 and d=2 in theta parameters behave in the expected manner. This conforms with the collinearity. In particular, for spin 2 states the propagator would behave like p^-4 for large momenta. This conforms with the prediction that graviton cannot correspond to singlet wormhole throat but to a string like object consisting of a superposition of pairs of wormhole contacts and of wormhole throats. If this expansion makes sense, higher spin propagators would behave as increasingly higher inverse powers of momentum and would not contribute much to the high energy physics. At energies much smaller than mass scale they would give rise to contact terms proportional to a negative power of mass dictated by the number of thetas.

2. For D_iα=∂_iα+i(θ_cD) _iα the formulas become considerably more complex due to the infinite exponentials exp(i&theta_cDθ), and for N= ∞ one obtains infinite factors given essentially by N multiplying the propagators and vertices. These factors however cancel in the chiral loops defining bosonic vertices and propagators. Also a factor depending on momentum appears but cancel in these loops. The deviations from the first option are small but it seems that this option is so ugly that it can be safely forgotten.

2. TGD variant of vector super field

Chiral super-fields are certainly not all that is needed. Also interactions must be included, and this raises the question about the TGD counterpart of the vector super-field.

1. The counterpart of the chiral action would be a generalization of the Dirac action coupled to a gauge potential obtained by adding the super counterpart of the vector potential to the proposed super counterpart of Dirac action. The generalization of the vector potential would be the TGD counterpart of the vector super field. Vector particle include M⁴ scalars since Higgs behaves as CP₂ vector and H-scalars are excluded by chiral invariance.

2. Since bosons are bound states of positive and negative energy fermions at opposite wormhole throats it seems that vector super field must correspond to an operator slashed between positive and negative energy super-fields rather than ordinary vector super-field. The first guess is that vector super-field is an operator expressible as a Taylor series in which positive energy fermions correspond to the powers of θ and negative energy fermions correspond to the powers of derivatives ∂_θ. Naively, D in ∂_θD is replaced by D+V. Vector super-field must be hermitian (V=V⁺) with hermitian conjugation defined so that it maps theta parameters to the partial derivatives ∂_θ and performs complex conjugation. A better guess is that D appearing in the definition of the kinetic term is replaced with D+V where V is a hermitian super-field. This definition would be direct generalization of the minimal substitution rule.

3. It is difficult to imagine how a kinetic term for the vector super-field could be defined. This supports the idea that bosonic propagators and vertices emerge as one performs functional integral over components of the chiral fields.

4. There is also the question about gauge invariance. The super-field generalization of the non-Abelian gauge transformation formula looks more like the generalization of Dirac action to its super-counterpart: D→ D+V everywhere. Positive energy chiral field would transform as Φ₊→ exp(Λ)&Phi₊;, where Λ is a chiral field. The negative energy chiral field would transform as Φ_-→ exp(Λ⁺)&Phi_-; with hermitian conjugation (denotes by +) involving also the map of thetas to their derivatives. Each theta parameter would represent a fermion transforming under gauge symmetries in a manner dictated by its electro-weak quantum numbers (the inclusion of color quantum numbers is not quite trivial: probably they must be included as a label for quark modes). As in the case of Dirac action, the transformation formula for vector super-field would be dictated by the requirement that the derivatives of Λ coming from exp(Λ) are canceled by the derivative terms in the transformation formula for the vector super-field.

3. Is the perturbation theory UV finite?

Also for the proposed TGD inspired identifications of chiral super-fields and vector super-fields, the cancelation of UV divergences should be essentially algebraic and due to the cancelation of chiral contributions from the loops contributing to the vector super-field propagators and vertices. Also for the emerging bosonic effective action same mechanism should be at work.

The renormalization theorems state that the only renormalizations in SUSYs are wave function renormalizations. In the case of bosonic propagators loops therefore mean only the renormalization of the propagator. In the recent case only the chiral loops are included so that the situation is analogous to Abelian YM theory or N=4 super YM theory, where the beta functions for gauge couplings vanish. Hence one might hope that also now wave function renormalization is the only effect so that the radiatively generated contribution should be proportional to the standard form of the vector propagator. The worst that can occur is logarithmically diverging renormalization of the propagator which occur in many SUSYs. The challenge is to show that logarithmic divergences possibly coming from the θ^d, d=1,2, parts of the chiral super-field cancel. The condition for this cancelation is purely algebraic since the coupling to k=2 part is gradient coupling so that the leading divergences have same form. It could happen that the lowest contributions cancel but the contributions from field components with d>2 give a non-vanishing and certainly finite contribution.

It could happen that the d<1 contributions cancel exactly as they do in SUSYs but the contributions from field components with d> 2, give a non-vanishing and certainly finite contribution. If this were the case then the exotic chiral field components with propagators behaving like 1/p^d, d> 2, would make possible the propagation for the components of the vector super-fíeld.

For the proposed SUSY limit of TGD see the new chapter Does the QFT Limit of TGD Have Space-time Super-Symmetry? of the book "Towards M-Matrix".