About Fermi-Dirac and Bose-Einstein statistics, negentropic entanglement, Hawking radiation, and firewall paradox in TGD framework

In quantum field theories (QFTs) defined in 4-D Minkowski space spin statistics theorem forces spin statistics correlation: fermions/bosons with half-odd integer/integer spin correspond to totally antisymmetric/symmetric states. TGD is not a QFT and one is forced to challenge the basic assumptions of 4-D Minkowskian QFT.

1. In TGD framework the fundamental reason for fermionic statistics are anticommutation relations for the gamma matrices of the “World of Classical Worlds” (WCW). This naturally gives rise to geometrization of the anticommutation relations of induced spinor fields at space-tme surfaces. The only fundamental fields are second quantized space-time spinors, which implies that the statistics of bosonic states is induced from the fermionic one since they can be regarded as many-fermion states.

   Strong form of holography (SH) forced by strong form of General Goordinate Invariance (SGCI) implies that induced spinor fields are localized at string world sheets. 2-dimensionality of the basic objects (string world sheets and partonic 2-surfaces inside space-time surfaces) makes possible braid statistics, which is more complex than the ordinary one. The phase corresponding to 2π rotation is not +/-1 but a root of unitary and the phase can be even replaced with non-commuting analog of phase factor. What about the ordinary statistics of QFTs expected to hold true at the level of imbedding space H=M⁴× CP₂. Can one deduce it form the q-variants of anticommutation relations for fermionic oscillator operators – perhaps by a suitable transformation of oscillator operators?

   Does the Fermi/Bose statistics at imbedding space level (one has H=M⁴× CP₂) as an exact notion or does it emerge only at the QFT limit when many-sheeted space-time sheets are lumped together and approximated as a slightly curved region of empty Minkowski space?

2. Zero energy ontology (CD) means that physical systems are replaced by pairs of positive and negative energy states defined at the opposite boundaries of causal diamond (CD). Causal diamonds form a fractal hierarchy. Does this mean that the usual statistics must be restricted to coherence regions defined by CDs rather than assume it in entire H. This assumption looks very reasonable since it would allow to milden the rather paradoxical looking implications of statistics and quantum identify for particles.

Interesting questions relate to the notion of negentropic entanglement (NE).

1. Two states are negentropically entangled if their density matrix is projection operator and thus proportional to unit matrix. This require also algebraic entanglement coefficients. This is guaranteed if the entanglement coefficients form a unitary matrix. So called quantum monogamy theorem has a highly non-trivial implication for NE. It states that if two entangled systems are in a 2-particle state which is pure, the entire system must be de-entangled from the rest of the Universe. As a special case this applies to NE.

   Quantum monogamy could be hihgly relevant for the understanding of living matter. Biology is full of binary structures (DNA double strand, lipid bi-layer of cell membrane, epithelial cell layers, left and right parts of various brain nuclei and hemispheres, right and left body parts, married couples,… ). Could given binary structure correspond at some level to a negentropically entangled pure state and that the system at this level is conscious? Could the loss of consciousness accompany the formation of a system consisting of a larger number of negentropically entangled systems so that 2-particle system ceases to be pure state and is replaced by a larger pure state. Could something like this take place during sleep?

2. NE relates also to the statistics. Totally symmetric and antisymmetric many-particle states are negentropically entangled for any subsystem since the density matrix is projection operator. The traced out states for subsystems are not pure. Could bosonic and fermionic statistics emerge at imbedding space-level from the braid statistics for fundamental fermions and Negentropy Maximization Principle (NMP) favoring the generation of NE? Could CD be identified as a region inside which the statistics has emerged? Are also more general forms of NE possible and assignable to more general representations of permutation group? Could ordinary fermions and bosons be also in states for which entanglement is not negentropic and does not have special symmetry properties? Quantum monogamy plus purity of the state of conscious system demands decomposition into de-entangled sub-systems – perhaps CDs? Does this demand that the entanglement due to statistics is present only inside CDs/selves.

Quantum monogamy relates also to the firewall problem of blackhole physics.

3. There are two entanglements involved. There is entanglement between Alice entering the blackhole and Bob remaining outside it. There is also the entanglement between blackhole and Hawking radiation implied if Hawking radiation is only apparently thermal radiation and blackhole plus radiation defines a pure quantum state. If so, Hawking evaporation does not lead to a loss of information. In this picture blackhole and Hawking radiation are seen as a single pure system.

Since Alice enters blackhole (or its boundary), one can identify Alice as part of the modified blackhole being entangled with the original blackhole and forming a pure state. Thus Alice would form an entangled pure quantum state with both Bob and Hawking blackhole. This in conflict with quantum monogamy. The assumption that Alice and blackhole are un-entangled does not look reasonable. But why Alice, Bob and blackhole could not form pure entangled 3-particle state or belong to a larger entangled state?

In TGD framework the firewall problem is mostly due to the use of poorly defined terms. The first poorly defined notion is blackhole as a singularity of GRT. In TGD framework the analog for the interior of the blackhole is space-time regions with Euclidian signature of metric and accompany all physical systems. Second poorly defined notion is that of information. In TGD framework one can define measure for conscious information using p-adic mathematics and it is non-vanishing for NE. This information characterizes always two-particle system – either as a pure system or part of a larger system. Thermodynamical negentropy characterizes single particle in ensemble so that the two notions are not equivalent. Further, in the case of blackhole one cannot speak of information going down to blackhole with Alice since information is associated with a pair formed by Alice and some other system outside blackhole like objects or perhaps at its surface. Thirdly, the notion is hierarchy of Planck constants, which allows NE in even astrophysical scales. Therefore entangling Bob, Alice, and TGD counterpart of blackhole is not a problem. Hence the firewall paradox seems to dissolve.

This allows to generalize the notion of Hawking gravitation, and one can speak about dark variant of Hawking gravitation and assign it with any material object rather than only blackhole. The generalized Hawking temperature is proportional to the mass of the particle at the gravitational flux tubes of the central object and to the ratio of the Schwartschild radius and radius for the central object. Amazingly, the Hawking temperature for solar Hawking radiation in the case of proton corresponds to physiological temperature. This finding conforms with the vision that bio-photons result from dark photons with h_eff= h_gr. Dark Hawking radiation could be very relevant for living matter in TGD Universe!

Even more, one ends up via strong form of holograpy with the suggestion that the Hawking temperature equals to the Hagedorn temperature assignable to flux tubes regarded as string like objects! This assumption fixes the value of string tension and is highly relevant for living matter since it guarantees that subsystems can become time-reversed with high probability in state function reduction. The frequent occurrence of time reversed mental images makes possible long term memory and planned action and one ends up with thermodynamics of consciousness. This is actually not new: living systems are able to defy second law and the notion of syntropy was introduced long time ago by Fantappie.

It would however seems to me that the TGD analog of dark Hawking radiation along flux tubes is quite essential for communications and even more, for what it is to be Alice and Bob! The flux tube connections of living systems to central star and planets could be an essential part of what it is to be alive as I have already earlier suggested with the inspiration coming from h_eff= h_gr. In this framework biology and astrophysics would meet in highly non-trivial manner.