The non-discovery of supersymmetric particles caused an earthquake in elementary particle physics. String theory, the first serious pure geometric theory compatible with quantum theory, was broken down. This caused a general melancholia to the elite researchers of the field. Many either have stopped working or turned to philosophical “anthropic” ideas. After so many years of hard work, they have accumulated high expertise, which is actually going to be lost. This makes me sad, because 4-d pseudo-conformal field theory (PCFT) may be the road to the unification of the fundamental interactions. It has the same properties with the Polyakov action, being a 2-d PCFT. Simply the surfaces of CP(3) replace the curves of CP(2). Stand up and continue to trust science. String theory has opened a pathway that PCFT extends without supersymmetry.
Einstein failed to extend general relativity two times. The first time was his Kaluza-Klein theory and the second time was the Einstein-Cartan theory. Despite these failures he returned in 1935 with the derivation of the equations of motion (his forth! scientific revolution). The 4-d PCFT is a Cartan extension of general relativity, based on a renormalizable action, which “contains” all the known elementary particles with their interactions.
I created this blog around a (draft) research e-book that anyone can download. To have a first taste of PCFT please read the PREFACE of the e-book.
<<<General relativity and quantum field theory are the successful fundamental theories of the 20th century. The experimental verifications of both theories are so many that no serious question of their validity actually exists. The fundamental mathematical notion of general relativity is the lorentzian metric of the riemannian geometry. It explains very well gravitation but “considers” all the other fields of electromagnetic, weak and strong interactions as external. All the efforts to incorporate them have failed. On the other hand, the fundamental mathematical notion of quantum field theory is the rigged Hilbert space formulated in the general realm of generalized functions (Schwartz distributions). Its last success is the standard model. But the theoretical efforts to generalize it to grand unified and supersymmetric models have not been experimentally verified. The LHC experimental result in connection with dark matter searches strongly indicate that supersymmetric particles do not exist. These experimental results drifted to failure string theory, which was the most serious effort to unify general relativity with quantum field theory. My 4-dimensional lagrangian model consists of a renormalizable generally covariant action based on a special totally real Cauchy-Riemann (CR-) structure. It does not need supersymmetry to incorporate the fermions, which are just distributional solitons.
The euclidean 2-dimensional conformal field theories have been very successful in condensed matter. The Polyakov action is a 2-dimensional conformal field theory based on the 2-dimensional lorentzian metric. In 1986, during my summer stay at CERN, I realized that the “beauty” and the impressive properties of the 2-dimensional conformal field theories come from their metric independence without being topological. That is, the fact that any 2-dimensional metric admits coordinates (the light-cone coordinates (z⁰,z⁰)) such that g_{μν}dx^{μ}dx^{ν}=g₀₀dz⁰dz⁰. Hence, I looked and found a metric independent 4-dimensional action with 4-dimentional metrics, which take the form g_{μν}dx^{μ}dx^{ν}=g_{αβ}dz^{α}dz^{β} with α,β=0,1 in a coordinate system. I new that not all the 4-dimensional metrics can take this form. Only those which admit two geodetic and shear-free null congruences take this form, like the black-hole metrics. I even considered it as an explanation of the fact that only these solutions of the Einstein equations are observed in nature. My second satisfaction came from the realization that the metric independence of my 4-dimensional generally covariant action assures its (formal) renormalizability, simply because it is dimensionless and the geometrical counterterms are not permitted.
My 4-dimensional action has no other physical relation with string theory. It depends on a special totally real Cauchy-Riemann structure (called lorentzian Caucy-Riemann (LCR-) structure) and it contains a compatible SU(N) gauge field connection. It essentially takes the place of the scalar field X^{μ}(τ,σ) of the 2-dimensional conformal action, which string theory interprets as the immersion field of the 2-dimensional surfaces in the 26-dimensional spacetime. This gauge field is finally identified with the gluon field. Graviton and photon naturally emerge from the fundamental lorentzian CR-structure (LCR-structure). Electron and its neutrino are the stable static and stationary solitonic LCR-manifolds, which are directly related to an irreducible and reducible quadric of CP³. They are the massive and massless ruled surfaces with Hopf invariant one, which may be assumed to be the electronic leptonic number. The gyromagnetic ratio of the electron LCR-structure is g=2 even at the classical solitonic level, already computed by Carter. The other generations of fermionic leptons may be those with higher Hopf invariants. The number of the leptonic generations may be restricted to three by the simple fact that the non conformally flat metrics admit no more than four geodetic and shear-free null congruences (Petrov classification). No other leptons exist, up to type III LCR-structure. For every leptonic LCR-manifold, there is a solitonic colored (with SU(N) degrees) configuration, which are interpreted as the corresponding quarks. The “electron” quark is explicitly derived. This derivation-correspondence makes apparent why the quarks are colored copies of leptons. Hence the present action is apparently the extention of general relativity based on a special totally real CR-structure properly defined in the Cartan moving frame (independent tangent vectors) formalism. The most shocking difference with riemannian geometry is that the Hawking-Penrose singularity theorems do not apply and the Penrose censorship hypothesis is not valid, because the elementary particles have gravitational dressings with naked singularities. That is the stars are aggregations of naked singularities, which are well defined in the context of lorentzian CR-structure.
The present theory is called pseudo-conformal field theory (PCFT) following the initial term for the CR-structure, used by E. Cartan, Tanaka, Severi and others, who first worked on this mathematical notion. Besides, this term is compatible with the name of 2-dimensional conformal field theories used in physics. But the 4-dimensional PCFT is invariant under tetrad-Weyl transformations, which is larger than the metric-Weyl symmetry that the quadratic Weyl-tensor action admits. This tetrad-Weyl symmetry is broken (even at the classical level) by the existence of the conserved quantities of charge and energy-momentum. Hence in brief, LCR-structure is the fundamental structure that replaces the lorentzian riemannian structure of general relativity and PCFT is essentially the lagrangian that Einstein was searching to extend his theory of relativity. On the other hand the Polyakov action is the corresponding 2-dimensional PCFT action. Besides the mathematics are algebraically based on surfaces of CP³ in analogy to the well known dependence of string theory on surfaces of CP².
The standard model is derived through the axiomatic quantum field theory (causal perturbative theory) of Stuckelber and Bogoliubov combined with the Epstein-Glaser remark, viewed as an effective field theory. The starting point is the Poincaré covariance of the distributional solitons viewed as elements of the rigged Hilbert-Fock space of the free fields tempered distributions. The renormalizability condition imposes the standard model relations between the coupling constants and the masses. The “internal” U(2) group and its spontaneous symmetry breaking are just effective. They are essentially artifacts of the necessary conditions for the existence of the product of the tempered distributions, which appear in the S-matrix of the causal perturbative field theory as already described by Scharf in his books.
The purpose of the present “research e-book” is to provide the interested researcher with all the details of PCFT.>>>
Any question or new idea is welcome. But the reader has to realize that this is not an ordinary blog of “social media”. This is not for everybody. Only scientists who want to be informed on PCFT are welcome!
