The construction of many-particle states as zero energy states defining scattering amplitudes and S-matrix is one of the basic challenges of TGD. TGD suggests two approaches implied by physics as geometry and physics as number theory views to TGD. Geometric vision suggests Yangians of the symmetry algebras of the "world of classical worlds" (WCW) at the level of H = M4 × CP2. Number theoretic vision suggests Galois confinement at the level of complexified M8. Could these approaches be M8-H duals of each other?
The states would be constructed from fermions and antifermions as modes of WCW spinor field. An idea taking the notion of symmetry to extreme is that this could be done purely algebraically using generators of symmetries.
- For a given vacuum state assignable to a partonic 2-surface and identifiable as a ground state of Kac-Moody type representation, the states would be generated by Kac-Moody algebra. Also super-Kac-Moody algebra could be used to construct states with nonvanishing fermion and antifermion numbers. In the case of super symplectic algebra, the generators would correspond to super Noether charges from the isometries of WCW and would have both fermionic and might also have bosonic parts.
- The spaces of states assignable to partonic 2-surfaces or to a connected 3-surface are however still rather restricted since it assumes in the spirit of reductionism that the symmetries are local single particle symmetries. The first guess for many-particle states in this approach is as free states and one must introduce interactions in an ad hoc manner and the problems of quantum field theories are well-known.
- In the TGD framework, there is a classical description of interactions in terms of Bohr-orbit like preferred extremals and one should generalize this to the quantum context using zero energy ontology (ZEO). Classical interactions have as space-time correlates flux tubes and "massless extremals" connecting 3-surfaces as particles and topological vertices for the partonic 2-surfaces.
- The construction recipe of many-particle states should code automatically for the interactions and they should follow from the symmetries as a polylocal extension of single particle symmetries. They should be coded by the modification of the usual tensor product giving only free many-particle states. One would like to have interacting many-particle states assignable to disjoint connected 3-surfaces or many-parton states assignable to single connected space-time surfaces inside causal diamond (CD).
Yangian algebras are especially interesting in this respect.
- Yangian algebras have a co-algebra structure allowing to construct multi fermion representations for the generators using comultiplication operation, which is analogous to the time reversal of a Lie-algebra commutator (super algebra anticommutator) regarded as an interaction vertex with two incoming and one outgoing particle. The co-product is analogous to tensor product and assignable to a decay of a particle to two outgoing particles.
- What is new is that the generators of Yangian are poly-local. The infinitesimal symmetry acts on several points simultaneously. For instance, they could allow a more advanced mathematical formulation for n-local interaction energy lacking from quantum field theories, in particular potential energy. The interacting state could be created by a bi-local generator of Yangian. The generators of Yangian can be generated by applying coproducts and starting from the basic algebra. There is a general formula expressing the relations of the Yangian.
- Yangian algebras have a grading by a non-negative integer, which could count the number of 3-surfaces (say all connected 3-surfaces appearing at the ends of the space-time surface at the boundaries of causal diamond (CD)), or the number of partonic 2-surfaces for a given 3-surface. There would also be gradings with respect to fermion and antifermion numbers.
There are indications that Yangians could be important in TGD.
- In TGD, the notion of Yangian generalizes since point-like particles correspond to disjoint 3-surfaces, for a given 3-surface to partonic 2-surfaces, and for a partonic 2-surface to point-like fermions and antifermions. In the TGD inspired biology, the notion of dark genes involves communications by n-resonance. Two dark genes with N identical codons can exchange cyclotron 3N-photon in 3N-resonance. Could genes as dark N-codons allow a description in terms of Yangian algebra with N-local vertex? Could one speak of 3N-propagators for 3N cyclotron-photons emitted by dark codons.
- In quantum theory, Planck constant plays a central role in the representations of the Lie algebras of symmetries. Its generalization assignable to n-local Lie algebra generators could make sense for Yangians. The key physical idea is that Nature is theoretician friendly. When the coupling strength proportional to a product of total charges or masses becomes so large that perturbation series fails to converge, a phase transition increasing the value of heff takes place. Could this transition mean a formation of bound states describable in terms of poly-local generators of Yangian and corresponding poly-Planck constant?
- For instance, the gravitational Planck constant ℓgr, which is bilocal and proportional to two masses to which a monopole flux tube is associated, could allow an interpretation in terms of Yangian symmetries and be assignable to a bi-local gravitational contribution to energy momentum. Also, other interaction momenta could have similar Yangian contributions and be characterized by corresponding Planck constants.
- It is not clear whether ℓgr and its generalization can be seen as a special case of the proposal heff = nh0 generalizing the ordinary single particle Planck constant or whether it is something different. If so, the hierarchy of Planck constant would correspond to a hierarchy of polylocal generators of Yangian.
The above discussion was at the level of H=M^4× CP2 and "world of classical worlds" (WCW). M8-H duality predicts that this description has a counterpart at the level of M8. The number theoretic vision predicting the hierarchy of Planck constants strongly suggests Galois confinement as a universal mechanism for the formation of bound states of particles as Galois singlets.
- The simplest formulation of Galois confinement states that the four-momenta of particles have components which are algebraic integers in the extension of rationals characterizing a polynomial defining a 4-surface in complexified M8, which in turn is mapped to a space-time surface in H=M^4× CP2, when the momentum unit is determined by the size of causal diamond (CD).
The total momentum for the bound state would be Galois singlet so that its components would be ordinary integers: this would be analogous to the particle in box quantization. Each momentum component "lives" in n-dimensional discrete extension of rationals with coefficient group, which consists of integers.
In principle one has a wave function in this discrete space for all momentum components as a superposition of Galois singlet states. The condition that total momentum is Galois singlet forces an entanglement between these states so that one does not have a mere product state.
- Galois confinement poses strong conditions on many-particle states and forces entanglement. Could Galois confinement be M8-H dual of the Yangian approach?
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.