Bootstrap approach to obtain a unitary S-matrix

In TGD framework S-matrix must be constructed without the help of path integral. The replacement of the loop momenta with light-like momenta does not eliminate UV divergences and the worst situation is encountered for gauge boson vertex corrections. This suggests a bootstrap program in which one starts from very simple basic structures and generates the remaining n-point functions as radiative corrections. The success of twistorial unitary cut method in massless gauge theories suggests that its basic results such as recursive generation of tree diagrams might be given a status of axioms. The idea that loop momenta are light-like cannot be however be taken too seriously. Also massive particles should be treated in practical approach.

The dream

Let us summarize the first variant of the dream about bootstrap approach.

1. In Construction of Quantum Theory: M-Matrix of "Towards M-Matrix" I have discussed how both field theoretic and stringy variants of the fermion propagator could arise via radiative self energy insertions described by a fundamental 2-vertex giving a contribution proportional to p^kγ_k and leading a propagator containing the counterpart as a mass term expressed in terms of CP₂ gamma matrices so that massive particles can have fixed M⁴×CP₂ chirality.

2. In TGD bosons are identified as bound states of fermion and antifermion at opposite wormhole throats so that bosonic n-vertex would correspond to the decay of bosons to fermion pairs in the loop. Purely bosonic gauge boson couplings would be generated radiatively from triangle and box diagrams involving only fermion-boson couplings. Even bosonic propagator would be generated as a self-energy loop: bosons would propagate by decaying to fermion-antifermion pair and then fusing back to the boson. Gauge theory dynamics would be emergent and bosonic couplings would have form factors with IR and UV behaviors allowing finiteness of the loops constructed from them.

As already found this dream about emergence is killed by the general arguments already discussed demonstrating that one encounters UV divergences already in the construction of gauge boson propagator for both light-like and free loop momenta. The physical reason for the emergence of these divergences and also their cure at the level of principle is well-understood in TGD Universe.

1. The description in terms of number theoretic braids based on the notion of finite measurement resolution should resolve these divergences at the expense of locality.

2. Zero energy ontology brings into the picture also the natural breaking of translational and Lorentz symmetries caused by the selection of CD. This breaking is compensated at the level of configuration space since all Poincare transforms of CDs are allowed in the construction of the configuration space geometry.

3. If this approach is accepted then for given CD there are natural IR and UV cutoffs for 3-momentum (perhaps more naturally for these than for mass squared). IR cutoff is quantified by the temporal distance between the tips of CD and UV cutoff by similar temporal distance of smallest CD allowed by length scale resolution. If the hypothesis that the temporal distances come as octaves of fundamental time scale given by CP₂ time scale T₀ and implying p-adic length scale hypothesis, the situation is fixed. A weaker condition is that the distances come as prime multiples pT₀ of T₀.

4. QFT type idealization would make sense in finite measurement resolution and the loop integrals would be both IR and UV finite.

This leads to a modified form of the dream.

1. Concerning propagators there are two options: only fermionic propagators are allowed and bosonic propagators emerge or both fermionic and bosonic propagators appear as fundamental objects. Only boson-fermion coupling characterizing the decay of a wormhole contact to two CP₂ type almost vacuum extremals with single wormhole throat carrying fermion and anti-fermion number would be feeded to the theory as something given and all vertices involving more than one boson would result as radiative corrections. Boson-fermion coupling would be proportional to Kähler coupling strength fixed by quantum criticality and very near or equal to fine structure constant at electron's p-adic length scale for the standard value of Planck constant. If not anything else, this approach would be predictive.

2. This approach could be tried to both free and light-like loop momenta. For free loop momenta the cutoff would be naturally associated with the mass squared of the virtual particle rather than the energy of a massless particle. Despite its Lorentz invariance one could criticize this kind of UV cutoff because it allows arbitrarily small wavelengths not in accordance with the vision about finite measurement resolution.

The following considerations lead to the conclusion that bosonic propagators could emerge from fermionic ones in the quantum field theory type description and that this description is also favored by the basic structure of quantum TGD. This kind of formulation would simplify enormously the definition of the theory.

Quantitative realization of UV finiteness in terms of p-adic length scale hypothesis and finite measurement resolution

p-Adic fractality suggests an elegant realization of the notion of finite measurement resolution implying the finiteness of the ordinary Feynman integrals automatically but predicting divergences for light-like loop momenta.

1. For the four-momenta above cutoff-momentum scale defined by the measurement resolution characterized by p-adic mass scale one cannot detect any details of the wave function of the particle inside sub-...-sub-CDs in question. Only the position of sub-...-sub-CD inside CD can be measured with a resolution defined by the cutoff scale. Therefore the number of detectable momentum eigen states does not anymore increase as the momentum scale is doubled but remains unchanged.

2. Unitarity realized in terms of the Cutkosky rules and in consistency with the finite measurement resolution requires that the density of states factor d³k/2E receives a reduction factor 2^-2 as the momentum scale is doubled above the resolution scale in the Feynman integral. This gives an effective reduction factor μ^-2L to the Feynman integral.

3. The cutoffs will be posed on both mass squared and hyperbolic angle. This conforms with the p-adic length scale hypothesis emerging from p-adic mass calculations and with the geometry of CDs. p-Adic length scales come as L_p propto p^1/2, p≈ 2^k rather than L_p propto p as the proportionality T(p)= pT(CP₂) of the temporal distance between tips of the CD combined with Uncertainty Principle would suggest. The reason is that light-like randomness of partonic 3-surfaces means Brownian motion so that L_p propto T(p)^1/2 and M_p propto T(p)^-1/2 follows. To avoid confusions note that for the conventions that I have used T(p) corresponds to the secondary p-adic length scale T_p,2= p^1/2T_p. For electron T(p) corresponds to .1 seconds.

Definition of loop integration

Consider now definition of the integration measure for loop momenta.

1. It is far from obvious whether the usual definition based on Wick rotation of the Euclidian variant of the integral makes sense in the recent case. The definition based on Wick rotation would eliminate the divergence in the hyperbolic angle leave only a cutoff in k² > 0 and give quadratic resp. logarithmic divergences for n=1 resp. n=2. This prescription is not favored by the picture suggested by the geometry CDs.

2. The most natural integration measure is just the standard M⁴ volume element d⁴k, which can be written as

   d⁴k=k³dk× sinh²(η)dη dΩ , k=(k^μk_μ)^1/2

   for time-like momenta and

   ⁴k=k³dk×× cosh²(η)dη dΩ, k=(-k^μk_μ)^1/2

   for space-like momenta. The original calculations contained a silly error due to the naive generalization of the Euclidian integration measure by replacing sin³(θ) with sinh³(η).

3. The geometry of CDs requires IR and UV cutoffs in both mass squared and hyperbolic angle giving
   p_max^-1/2 ≤ (m/m(CP₂)≤ p_min^-1/2 ,
   sinh(η)≤ sinh(η_max).
4. The primes p_max and p_min correspond to IR and UV cutoffs and p_min≥ 2 holds true naturally in QFT limit since stringy excitations having mass scale given by CP₂ mass are not included. This means that all loop integrals are finite if also hyperbolic cutoff is present.
5. The justification for the cutoff in |sinh(η)| comes either from the requirement that the Lorentz transformed sub-CDs to which the fermion loop can be associated remain inside CD within the measurement resolution for temporal distance in the scale corresponding to T(k) or from the condition that the decomposition of the gauge boson to a pair of fermion and antifermion at opposite wormhole throats restricts the range of the virtual momenta to momenta almost at rest in the rest system of boson. The condition that coupling constant evolution is realistic fixes the form of hyperbolic cutoff for time-like momenta in high precision and it remains to be seen whether quantum criticality can be used to predict the hyperbolic cutoff from first principles.
6. Hyperbolic cutoffs can and must be different for time-like and space-like momenta and the cancellation of the mass term from the bosonic propagator fixes the relationship between these cutoffs uniquely. Hyperbolic cutoffs can and must depend on p-adic length scale so that the integral over loop momenta decomposes to integral over momenta corresponding to p-adic half octaves with a fixed hyperbolic cutoff in each half octave.

The conclusion is that the definition of loop integrals as Euclidian integrals would lead to a catastrophe via the generation of gauge boson mass proportional to the cutoff mass whereas the Minkowskian definition with the notion of cutoff motivated by p-adic length scale hypothesis and hierarchy of causal diamonds keeps gauge bosons massless provided the cutoffs for hyperbolic angle for time-like and space-like loop momenta are related in a unique manner and the only contribution to the boson mass comes from mass terms in the fermionic propagators.

Could bosonic propagators emerge?

My views about whether bosonic propagators can emerge or not have been fluctuating wildly during last weeks. The following argument however suggests that emergent bosonic propagation is a mathematically consistent notion and conforms with the special features of quantum TGD.

1. In basic quantum TGD modified Dirac equation containing induced spinor connection as induced gauge boson field defines the theory and the exponent of Kähler action emerges as Dirac determinant. The natural guess is that this structure is preserved in the sense that Feynman diagrammatics is defined by Dirac action coupled to gauge potentials but containing no kinetic term for gauge potentials with kinetic terms emerging from the fermionic loops and the values of gauge couplings following as predictions of the formalism.

2. One can try to formulate this idea in terms of path integral formalism. Couple gauge bosonic field A resp. Grassmann valued fermion fields Ψ to external currents j resp. Grassmann valued external currents ξ and perform Legendre transform giving the exponent of the effective action as a functional integral over Ψ and A. The functional derivatives of the effective action with respect to the ξ and j allow to deduce N-point functions.

3. The exponent Z=exp(G_c(j,ξ,ξbar)) defined as the functional Fourier transform of the action exponential is the key quantity since its functional derivatives at origin define the connected Green's functions. Z can be calculated in two steps. At the first step one functionally integrates over Ψ and its conjugate. This can be done perturbatively and gives the generating functional for connected fermionic Green's functions for ξ and its conjugate as a functional of gauge fields A. This functional is also analogous to action and contains bosonic kinetic term which is of correct form by the preceding observations. Also interaction terms for A are included and since the original system is gauge invariant also the effective action must be gauge invariant and should reduce to Yang-Mills action in the lowest orders. Perturbation theory is therefore possible and one can calculate effective action by performing the functional integral over A using the induced propagators and vertices. At this step fields ξ are in the role of non-dynamical external fields just as A was at the first step and all propagators are bosonic. From the resulting exponential one can generate connected Green's functions as functional derivatives with respect to the sources.

4. It seems that the proposed description avoids the most obvious divergences. In particular, the tadpole term from A_μΨbar(x)γ^μΨ(x) proportional to the fermion propagator D_F(x,x) proportional to an integral of form ∫d⁴k k^μ/k² and thus vanishing.

5. The bosonic kinetic term would be proportional to the over all gauge coupling g² if one expresses gauge potential in the form gA. This decomposition is however not natural in TGD since the induced spinor connection corresponds to gA with no explicit value of g being specified. In the case of simplest tree diagram describing 2→ 2 fermion scattering that the g² coming from the ends of the boson line is canceled by the 1/g² coming from the bosonic propagator so that the predictions of the theory do not depend on the value of g in the lowest order. This looks strange but would conform with the absence of bosonic kinetic term in the primary action making it impossible to identify the value of g in standard manner. One can however say that the numerical coefficient given by the fermionic loop integrals defining the bosonic propagator predicts the values of gauge couplings g through the comparison of their values with the prediction of standard gauge theory for say 2→ 2 scattering. The sign of the kinetic terms comes out correctly thanks to the minus sign assigned to the fermion loop. This picture would conform with the vision that TGD predicts all gauge couplings. Maybe the emergence of gauge boson propagators and vertices could be seen as one aspect of quantum criticality.

These arguments suggest that the notion of emergent gauge boson propagation makes sense mathematically and is favored also by the general structure of quantum TGD. On the other hand, the preceding arguments allow the presence of the bosonic propagators as fundamental objects and do not force to take seriously the idea about emergent gauge boson propagation. This motivates the attempt to debunk the notion once and for all. Consistency with p-adic mass calculations might provide the needed killer argument.

1. The resulting bosonic mass squared would be in the lowest order sum over products of masses of fermion pairs coupling to the boson. It is far from clear whether this prediction is quantitatively consistent with the predictions of the p-adic mass calculations. This possibility is not of course excluded: boson mass squared is quadratic in fermion masses coupling to the boson and the p-adic primes associated with the fermions are naturally those associated with the boson rather than free fermions so that at least the mass scale comes out correctly. This picture conforms also qualitatively with the fact that mass squared is identified as conformal weight and the eigenvalue of modified Dirac operator related closely to the ground state contribution to the mass can be regarded as complex squares root of conformal weight.

2. Note that even photon is predicted to be massive unless the fermion and antifermion associated with photon and other massless particles are massless or in so low p-adic temperature that the thermal mass is negligible. Also the p-adic prime associated with massless bosons could be so large that the mass is small.

3. Boson masses are of course emergent in the sense that they are determined by the masses of the fermion and anti-fermion, which they consist of. The question is whether the emergence of masses takes place via loops rather than p-adic mass calculations in the proposed sense and whether these pictures are equivalent. That loops could provide the fundamental description for boson masses is suggested by the asymmetry between bosons and fermions in the recent form of p-adic mass calculations. The p-adic temperature for bosons must be T_p ≤ 1/2 whereas T_p=1 holds true for fermions, and for fermions the analog of Higgs contribution is negligible whereas for gauge bosons it dominates.

4. It could be also possible to code p-adic thermodynamics into the Feynman diagrammatics in a more refined manner so that loops would give only corrections to the masses obtained from p-adic mass calculations. Instead of simply feeding in the results of p-adic mass calculations as mass parameters of the fermionic propagators, one could replace S-matrix with M-matrix involving the square root of density matrix describing the real counterpart of the partition function characterizing p-adic thermodynamics. Zero energy state would represent a square root of thermodynamical ensemble involving massless ground states and their conformal excitations rather than only ground states with thermal masses.

The emergence of fermionic Feynman propagator

The emergence of the fermionic propagators from the fundamental propagator 1/D defined by the modified Dirac equation is an attractive starting point for the improved variant of the dream.

1. The fundamental two-vertex would basically reflect the non-determinism of Kähler action implying the breaking of the effective 3-dimensionality (holography) of the dynamics, and would generate the fermion propagator from the propagator 1/D associated with the modified Dirac action behaving as Minkowski scalar and expressible in terms of CP₂ gamma matrices. The vertex would be characterized as p^kγ_k. This would give

   G_F= i/[p^kγ_k-D] .

   This expression is consistent with cut unitarity.

2. The propagator G_- is usually identifiable in terms of classical propagators as G_-=G_ret-G_adv and it seems that one an assume that this propagator is just i×(γ^kp_k-D)δ(p²)sign(p₀). It is perhaps needless to restate that light-like loop momenta do not lead to a finite theory under the assumptions motivated by p-adic length scale hypothesis.

From this Feynman propagator and its bosonic counterpart one can build all diagrams and get finite results for a finite momentum cutoff forced by the finite measurement resolution. One could of course worry whether the introduction of the p-adic length scale hierarchy might lead to problems with analyticity and unitarity. It is now clear that the idea about massless loop momenta fails. The idea did not however live for vain since it led to the first concrete quantitatively precise conjecture about how gauge theory could emerge as an approximation of quantum TGD from the basic physical picture behind TGD. I am of course the first admit that the proposed scenario looks horribly ugly against the extreme elegance of gauge theories like N=4 SYM. The tough challenge is to find an elegant mathematical realization of the proposed physical picture and twistor approach might be of considerable help here.

Note added: I have updated and shortened this posting several times as the vision about bosonic emergence has evolved. The recent picture seems rather stable. I have left out detailed formulas and encourage the reader to consult the summary of the recent situation concerning bosonic emergence in quantum TGD framework given in the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".