The number theoretic picture leads to a deeper understanding of a long standing objection against p-adic thermodynamics (see this) as a thermodynamics for the scaling generator L_{0} of Super Virasoro algebra.

If one requires super-Virasoro symmetry and identifies mass squared with a scaling generator L_{0}, one can argue that only massless states are possible since L_{0} must annihilate these states! All states of the theory would be massless, not only those of fundamental particles as in conformally invariant theories to which twistor approach applies! This looks extremely beautiful mathematically but seems to be in conflict with reality already at single particle level!

The resolution of the objection is that * thermodynamics* is indeed in question.

- Thermodynamics replaces the state of entire system with the density matrix for the subsystem and describes approximately the interaction with the environment inducing the entanglement of the particle with it. To be precise, actually a "square root" of p-adic thermodynamics could be in question, with probabilities being replaced with their square roots having also phase factors. The excited states of the entire system indeed are massless (see this).
- The entangling interaction gives rise to a superposition of products of single particle massive states with the states of environment and the entire mass squared would remain vanishing. The massless ground state configuration dominates and the probabilities of the thermal excitations are of order O(1/p) and extremely small. For instance, for the electron one has p= M
_{127}=2^{127}-1≈ 10^{38}. - In the p-adic mass calculations (see this and this), the effective environment for quarks and leptons would in a good approximation consist of a wormhole contact (wormhole contacts for gauge bosons and Higgs and hadrons). The many-quark state many-quark state associated with the wormhole throat (single quark state for quarks and 3-quark-state for leptons (see this).
- In M
^{8}picture (see this and this) tachyonicity is unavoidable since the real part of the mass squared as a root of a polynomial P can be negative. Also tachyonic real but algebraic mass squared values are possible. At the H level, tachyonicity corresponds to the Euclidean signature of the induced metric for a wormhole contact.Tachyonicity is also necessary: otherwise one does not obtain massless states. The super-symplectic states of quarks would entangle with the tachyonic states of the wormhole contacts by Galois confinement.

- The massless ground state for a particle corresponds to a state constructed from a massive single state of a single particle super-symplectic representation (CP
_{2}mass characterizes the mass scale) obtained by adding tachyons to guarantee masslessness. Galois confinement is satisfied. The tachyonic mass squared is assigned with wormhole contacts with the Euclidean signature of the induced metric, whose throats in turn carry the fermions so that the wormhole contact would form the nearby environment.The entangled state is in a good approximation a superposition of pairs of massive single-particle states with the wormhole contact(s). The lowest state remains massless and massive single particle states receive a compensating negative mass squared from the wormhole contact. Thermal mass squared corresponds to a single particle mass squared and does not take into account the contribution of wormhole contacts except for the ground state.

- There is a further delicate number theoretic element involved (see this and this). The choice of M
^{4}⊂ M^{8}for the system is not unique. Since M^{4}momentum is an M^{4}projection of a massless M^{8}momentum, it is massless by a suitable choice of M^{4}⊂ M^{8}. This choice must be made for the environment so that both the state of the environment and the single particle ground state are massless. For the excited states, the choice of M^{4}must remain the same, which forces the massivation of the single particle excitations and p-adic massivation.

One can regard the breaking of conformal invariance induced by the thermodynamical description of entanglement as a TGD counterpart for the breaking of gauge symmetry.

Polynomials P define two kinds of space-time surfaces depending on whether their roots determine either mass or energy shells. For the energy option a space-time region corresponds by M^{8}-H duality (see this and this) to a solution spectrum in which the roots correspond to energies rather than mass squared values and light-cone proper time is replaced with linear Minkoski time. The physical interpretation of this solution spectrum has remained unclear.

The energy option gives rise to a p-adic variant of the ordinary thermodynamics and requires integer quantization of energy. This option is natural for massless states since scalings leave the mass shell invariant in this case. Scaling invariance and conformal invariance are not violated.

One can wonder what the role of these massless virtual quark states in TQC could be. A good guess is that the two options correspond to phases with broken * resp.* unbroken conformal symmetry. In gauge theories to phases with broken and unbroken gauge symmetries. The breaking of gauge symmetry indeed induces breaking of conformal symmetry and its breaking is more fundamental.

- Particle massivation corresponds in gauge theories to symmetry breaking caused by the generation of the Higgs vacuum expectation value. Gauge symmetry breaking induces a breaking of conformal symmetry and particle massivation. In the TGD framework, the generation of entanglement between members of state pairs such that members having opposite values of mass squared determined as roots of polynomial P in the most general case, leads to a breaking of conformal symmetry for each tensor factor and the description in terms of p-adic thermodynamics gives thermal mass squared.
- > What about the situation when energy, instead of mass squared, comes as a root of P. Also now one can construct physical states from massless virtual quarks with energies coming as algebraic integers. Total energies would be ordinary integers. This gives massless entangled states, if the rational integer parts of 4-momenta are parallel. This brings in mind a standard twistor approach with parallel light-like momenta for on-mass shell states. Now however the virtual states can have transversal momentum components which are algebraic numbers (possibly complex) but sum up to zero.
Quantum entangled states would be superpositions over state pairs with parallel massless momenta. Massless extremals (topological light rays, see this) are natural classical space-time correlates for them. This phase would correspond to the phase with unbroken conformal symmetry.

- One can also assign a symmetry breaking to the thermodynamic massivation. For the energy option, the entire Galois group appears as symmetry of the mass shell whereas for the mass squared option only the isotropy group does so. Therefore there is a symmetry breaking of the full Galois symmetry to the symmetry defined by the isotropy group. In a loose sense, the real valued argument of P serves as a counterpart of Higgs field.
If the symmetry breaking in the model of electroweak interaction corresponds to this kind of symmetry breaking, the isotropy group, which involves also a subgroup of quaternionic automorphisms as analog of Galois group, could act as a discrete subgroup of SU(2)

_{L}× U(1). The hierarchy of discrete subgroups associated with the hierarchy of Jones inclusions assigned with measurement resolution suggests itself. It has the isometry groups of Platonic solids as the groups with genuinely 3-D action. In the QCD picture about strong interactions there is no gauge symmetry breaking so that a description based on the energy option is natural. Hadronic picture would correspond to mass squared option and symmetry breaking to the isotropy group of the root.

See the article Two objections against p-adic thermodynamics and their resolution, the chapter TGD as it is towards the end of 2021, or the chapter About TGD counterparts of twistor amplitudes.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD. theory.fi/tgdmaterial.html">Articles and other material related to TGD.

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