### 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 connection: 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.

- In TGD framework the fundamental reason for the 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-time 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. At WCW level spinor fields are formally classical.

Strong form of holography (SH) forced by strong form of General Coordinate 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 =M

^{4}× CP_{2}? Can one deduce it from the q-variants of anticommutation relations for fermionic oscillator operators - perhaps by a suitable transformation of oscillator operators? Is the Fermi/Bose statistics at imbedding space level 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?

- Zero energy ontology (ZEO) means that physical systems are replaced by pairs of positive and negative energy states defined at the opposite boundaries of causal diamond (CD). CDs 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 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).

- Two states are negentropically entangled if their density matrix is proportional to a projection operator and thus proportional to unit matrix. This require also algebraic entanglement coefficients. For bipartite entanglement this is guaranteed if the entanglement coefficients form a matrix proportional a unitary matrix. The so called quantum monogamy theorem (see ) has a highly non-trivial implication for NE. In its mildest form 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. A stronger form of monogamy states that two maximally entangled qubits cannot have entanglement with a third system. It is essential that one has qubits. For 3-valued color one can have maximal entanglement for 3-particle states (baryons). For instance, the negentropic entanglement associated with N identical fermions is maximal for subsystems in the sense that density matrix is proportional to a projection operator.

Quantum monogamy could be highly 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 could the system at this level be 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?

- NE seems to relate also to the statistics. Totally antisymmetric many-particle states with permutations of states in tensor product regarded as different states can be regarded as negentropically entangled for any subsystem since the density matrix is projection operator. Here one could of course argue that the configuration space must be divided by the permutation group of n objects so that permutations do not represent different states. It is difficult to decide which interpretation is correct so that let us considere the first interpretation.

The traced out states for subsystems of many-fermion state are not pure. Could 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 - could one identify them as CDs? Does this demand that the entanglement due to statistics is present only inside CDs/selves?

- At space-time level space-time sheets (or space-like 3-surfaces or partonic 2-surfaces and string world sheets by SH) serve as natural candidates for conscious entities at space-time level. At imbedding space level elementary particles associated with various space-time sheets inside given CD would contain elementary particles having NE forced by statistics. But doesn't this imply that space-time sheets cannot define separate conscious entities?

The notion of finite resolution for quantum measurement, cognition, and consciousness suggests a manner to circumvent this conclusion. One has entanglement hierarchies assignable to the length scale hierarchies defined by p-adic length scales, hierarchy of Planck constants and hierarchy of CDs. Entanglement is defined in given resolution and the key prediction is that two systems unentangled in given resolution can be entangled in an improved resolution. The space-time correlate for this kind of situation are space-time sheets, which are disjoint in given resolution but contain topologically condensed smaller space-time sheets connected by thin flux tubes serving as correlates for entanglement.

The paradoxical looking prediction is that at a given level of hierarchy characterized by size scale for CD or space-time surface two systems can be un-entangled although their subsystems are entangled. This is impossible in standard quantum theory. If the sharing of mental images by NE between subselves of separate selves makes sense, contents of consciousness are not completely private as often assumed in theories about consciousness. For instance, stereo vision could rely on fusion and sharing of visual mental images assignable to left and right brain hemispheres and generalizes to the notion of stereo consciousness making to consider the possibility of shared collective consciousness. An interpretation suggesting itself is that selves correspond to space-time sheets and collective levels of consciousness to CDs.

Encouragingly, dark elementary particles would provide a basic example about sharing of mental images. Dark variants of elementary particles could be negentropically entangled by statistics condition in macroscopic scales and form part of a kind of stereo consciousness, kind of pool of fundamental mental images shared by conscious entities. This could explain why for instance the secondary p-adic time scale for electron equal to T= .1 seconds corresponds to a fundamental biorhythm.

- 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 assumed to form 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 seems to be 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 interiors of the blackhole are space-time regions with Euclidian signature of induced metric and accompany all physical systems. Second poorly defined notion is that of information. In TGD framework one can define a 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 albeit closely related. 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. Finally, the notion is hierarchy of Planck constants 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.

- The hierarchy of Planck constants h
_{eff}=n×h connects also with dark quantum gravity via the identification h_{eff}= h_{gr}, where h_{gr}= GMm/v_{0}, v_{0}/c≤ 1, is gravitational Planck Planck constant. v_{0}/c< 1 is velocity parameter characterizing system formed by the central mass M and small mass m, say elementary particle.

This allows to generalize the notion of Hawking radiation (see this, this, and this ), and one can speak about dark variant of Hawking radiation and assign it with any material object rather than only blackhole. The generalized Hawking temperature is proportional to the mass m of the particle at the gravitational flux tubes of the central object and to the ratio R

_{S}/R of the Schwartschild radius R_{S}and radius R 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 (see this

and this) , one ends up via SH to suggest 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 in TGD Universe 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.

- Does one get rid of firewall paradox in TGD Universe? It is difficult answer the question since it is not at all clear that there exists any paradox anymore. For instance, the assumption that blackhole represents pure state looks in TGD framework rather ad hoc and the NE between blackhole and other systems outside it looks rather natural if one accepts the hierarchy of Planck constants.

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 and even for their existence! 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.

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

For a summary of earlier postings see Links to the latest progress in TGD.

## 15 Comments:

See section 4 entitled "Physical Perspectives" of http://arxiv.org/abs/0903.4321 "Eigenvalue Density, Li's Positivity, and the Critical Strip" where the Hamilton-Jacobi (no Bellman here!! :) conditions of classical mechanics form the basis of the quantization conditions. The idea is related to "H=xp" Hamiltonian, with a twist. From the paper also "This tells us that the

Riemann ξ-function (symmetrized Zeta function), up to a factor which does not vanish in the critical strip, is the Mellin transform of a Fermi–Dirac distribution"

Can you please have a look and comment?

--crow

A nice paper written in such a manner that physicist understands.

*The finding that by writing zeta as WKB wave function as a product of modulus and phase with phase given by exponent of action defined by the function S(z) giving the number of zeros at critical line is very interesting since at zeros of zeta S(z) as number of zeros is quantized action!

*The authors propose interpretation of zeros as conformal weights: this is natural since angle is generalized to complex coordinate z and the integrals INT p.dq along closed classical orbit with residue integral over S interpreted as complex 1-form. The interpretation of zeros as conformal weights is also the TGD interpretation.

*The authors mention also the doubling formula of zeta and deduce that Riemann zeta is proportional to Mellin transform of fermionic zeta function. Here I did not understand at all. I see it as Mellin transform of **inverse** of the inverse 1/(1+exp(x)) of fermionic zeta , not fermionic zeta 1+exp(x)!! My first reaction is that they have made a mistake. I cannot hope that I am wrong since in this case I myself had made a horrible blunder!;-).

Let me explain why I believe that I am right. I have proposed different interpretation for the fermionic zeta based on the fact that fermionic zeta zeta_F equals to Z_F=Prod_p (1+p^s) and Riemann zeta to Z_B= Prod 1/(1-p^s) as formal partition functions of a number theoretic many-fermion/many-boson system with energies coming as multiples of log(p) (for fermions only n=0,1 is of course possible, for bosons all positive values of n are possible and this the form of Z_F). Product involves all primes.

In this framework **poles**(not zeros!!) of fermionic zeta zeta_F(s)= zeta(s)/zeta(2s)) (this identity is trrivial to deduce, do it!!) correspond trivial zeros of zeta, the pole of zeta at s=1, and to trivial poles at negative integers.

The interpretation of **poles** (much more natural in physics, and even more so in TGD, where fundamental particles involve only fermions!) of z_F as conformal weights associated with the representations of extended super-conformal symmetry associated with super-symplectic algebra defining symmetries of TGD at the level of "world of classical worlds" is natural.

"Conformal confinement" stating that the sum of conformal weights is real is natural assumption in this picture.

The fact that superconformal algebra has a fractal structure implies direct connection with quantum criticality: infinite hierarchy of symmetry breakings to sub-symmetry isomorphic to original one!! Needless to say the conformal structure is infinitely richer than the ordinary one since the algebra in question has infinite number of generators given by all zeros of zeta rather than a handfull of with conformal weights n=-2,...+2). Kind of Mandelbrot fractal realized physically.

*The problem of all attempts to interpret zeros of zeta relate to the fact that zeros are **not** purely imaginary, they have the troublesome real part Res)=1/2. This led me for long time ago to consider coherent states instead of eigenstates of Hamiltonian in my proposal for a strategy to prove Riemann hypothesis.

Also the interpretation as partition function suffers from the same disease: genuine partition function should be real.

In TGD framework the solution of the problem is zero energy ontology (ZEO). Quantum theory is "complex square root" of thermodynamics and means that partition function indeed becomes complex entity having also the phase. Ordinary partition function is modulus squared for it.

yes... if the authors would have done the same analysis with the Hardy Z function (it is totally real valued when its parameter is real) they would have gotten farther or made even deeper insights.

I will think more about the Mellin transform aspect, I am quite comfortable with them by now.

[MP] *The finding that by writing zeta as WKB wave function as a product of modulus and phase with phase given by exponent of action defined by the function S(z) giving the number of zeros at critical line is very interesting since at zeros of zeta S(z) as number of zeros is quantized action!

[SC] Yes, it is quite cool! I am adding more content related to the Hamilton-Jacobi equations to my paper on this topic at http://vixra.org/abs/1510.0475

p.s. see page 5 of the paper at http://www.math.waikato.ac.nz/~kab/papers/zeta2.pdf the flow near the zeros is graphed... I think, if u look on the appendix of my paper at http://vixra.org/abs/1510.0475

I suggest an extension of the Berry-Keating Hamiltonian H(x(t))=((x(t)x'(t)+x'(t)x(t)))/2=-i(x(t)x'(t)+1/2) to something like

𝒥^(2,+)x(t)={(p,X)|x(t+z)⩽x(t)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}

and

𝒥^(2,-)x(t)={(p,X)|x(t+z)⩾u(x)+p⋅z+(X:z⊗z)/2+o(|z|^2) as z→0}

which are sets called the second-order superjets and subjets of x(t) where p∈ℝ^n is the Jacobian of x(t∈Ω)∈C^0(Ω⊆ℝ^n) and X∈𝕊^n is the Hessian of x(t) where 𝕊^n is the set of n×n symmetric matrices. The main idea being that the canonical momentum x˙(t) is replaced with the generalized pointwise derivative which can be written as 𝒥^2x(t) in case both the superjet and subjet exist and actually define the same set.

The "X" in that superjet and subjet definition should really be X_n where n is the n-th Riemann zero, I think that the flow around each zero can be specified in terms of a symmetric operator X... it would algebrically/numerically encode the flow around each zero.

anyway... I'm not personally that interested but it would be something someone else could do.. I really want to work with the Hamilton-Jacobi aspects some more and try applying their analysis to the Z function, since that will get rid of some of the objects you stated. Hardy's Z function is most clearly expressed as

Z(t)=exp(I*(-I*(lnGAMMA(I*t*(1/2)+1/4)-lnGAMMA(1/4-I*t*(1/2)))*(1/2)-(1/2)*ln(Pi)*t))*Zeta(1/2+I*t)

Mellin transform representation is of course completely ok mathematically but the physical interpretation in terms of fermionic c statististics seems utterly wrong. Fermionic partition function is sum n^-s where n is square free integer and I cannot see a connection with representation.

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o?

More talk here http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/physics2.htm

Corrected version of the comment which contained something which should not belong to it.

########

I was thinking further about the idea that physically the poles of fermionic zeta are the appropriate notion. The partition function for poles would diverge as intuition suggests. I want to explain more precisely this intutiion.

Since the temperature interpreted as 1/s is not infinite this means that one has analog of Hagedorn temperature in which degeneracy of states increases exponentially to compensate exponential decreases of Boltzman weight so that partition function is sum of infinite number of numbers approaching to unity. Hagedorn temperature relates by strong form of holography to magnetic flux tubes behaving as strings with infinite number of degrees of freedom. One would have quantum critical system. Supersymplectic invariance, etc..

The real part of temperature is real part of 1/s and given by T_R= 1/2/(1/4+y^2) and approaches zero for large y as it should. Also imaginary part T_Im approach zero. One has infinite number of critical temperatures.

An interesting question is whether zeros of zeta correspond to critical values of Kahler coupling strength having interpretation as inverse of critical temperature?!

But what about negative values of poles of z_F at n=-1,-2,...? They would correspond to negative temperatures -1/n. No problem! In p-adic thermodynamics p-adic temperatures has just these values if one defines p-adic Boltzmann weight as exp(-E/T)---> p^(-E/T), with E=n conformal weight!! The condition that weight approaches zero requires T negative! Trivial poles would correspond to p-adic thermodynamics and non-trivial poles to ordinary real thermodynamics!

This would be marvellous connection between TGD and number theory. Riemann zeta would code basic TGD and of course, all of fundamental physics! ;-)

It would be interesting to look the behavior of inverse of complex temperature defined by zeros of zeta and its pole: this for for fermionic partition function.

I have assumed the Kahler coupling strength is real but one might consider in spirit of electric magnetic duality also complex values. This would allow to consider the identification as poles of fermionic zeta as values of 1/alpha_K. Just for fun of course!

Trivial zeros would correspond to p-adic temperatures T=-1/n in the convention defined by strict formal correspondence of p-adic Boltzmann weight with real Boltzmann weight. I have earlier defined p-adic temperature as T=1/n.

The inverse of real critical temperature correspond to 1/alpha_K=T=1 (pole of Riemann zeta), p-adic temperatures T=1/n, and inverses of complex temperatures to inverses of non-trivial zeros of zeta which approach to zero at the limit y--> infty. This could have interpretation as asymptotic freedom since alpha_K would go to zero. At infrared 1/alpha_K approach to the lowest zero 1/2+iy, y=14.1.. so that it does not diverge anywhere.

The very naive guess would be that the real or imaginary part of some nontrivial zero corresponds to fine structure constant: this guess might be wrong of course The first estimate shows that this cannot be the case very accurately although zeros . The smallest value of alpha_K would correspond to 1/14: color coupling strength?

One obtains also alpha_K = about 127 for one of the zeros - fine structure constant at electroweak scale. Could the values of 1/alpha_K be identified imaginary parts of zeros of zeta and assigned with p-adic length scales?

Magnetic coupling would correspond to real part and be equal to -1/2,-1, and n=1,2,3,.... Kahler electric coupling would have values vanishing for real zeros and pole and imaginary part of zero at critical line. Does this make any sense? Difficult to say!

If this crazy conjecture makes sense then both super-symplectic conformal weights and complex inverse of Kahler coupling strength would have poles of zeta_F as their value spectrum. The different values for zeros of zeta could in turn naturally correspond to number theoretical coupling constant evolution with values of coupling strength associated with different algebraic extensions of rationals.

http://www.nature.com/ncomms/2015/151117/ncomms9818/full/ncomms9818.html The new way to look at symmetry and deformation. Compare to

http://scitation.aip.org/content/aip/magazine/physicstoday/article/68/11/10.1063/PT.3.2980 and the older

http://scitation.aip.org/content/aip/magazine/physicstoday/article/62/1/10.1063/1.3074260

and http://motls.blogspot.fi/2015/11/fq-hall-effect-has-vafa-superseded.html

Moreover, the big strength of Chern-Simons and topological field theories is that you may put them on spaces of diverse topologies so even the weaker topological claim about the AdS space can't be considered an absolute constraint for the theory.

What is your opinion of AdS space other than y don't understand it :) How then interpret results gained with it?

Thank you. A lot of reading. I hope that I had time. The notion of distortion is new to me. It looks interesting. I did not understand what antisymmetry in this context could mean.

AdS is a mathematical representation of holography but with too small conformal symmetry. One cannot actually forget that space-time is 10-D in it and this makes condensed matter applications questionable. For instance, in applications one must assume that the additional dimensions are large. This makes no sense. One cannot take just some features of the system and forget things like actual space-time dimension. This is typical left-brainy thinking plaguing the theoretical physics today.

One could see AdS an alternative description for 2-D conformal theory with 2-surfaces imbedded in 10-D spacetime with boundary. This is purely mathematical approach and personally I see the attempts to describe condensed matter as 10-D blackholes as a horrible waste of time. AdS has made no breakthroughs where it has been applied. There is diplomatically incorrect bread and butter view about this and I cannot prevent my alter ego from stating it;-). string theory was a failure but string theorists have a lot of methods and they want to apply them and receive also funding for this activity: why not condensed matter!

To get something better one must generalise the notion of conformal symmetry from 2-D to 4-D. Light-bone boundary allows hugely extended superconformal symmetry and supersymplectic symmetry. Now 10-D space is replaced with 4-D space-time surface in M^4xCP_2 and also twistors essential for conformal invariance enter the game. Holography becomes strong form of holography.

In holography 3-D surface could could for 4-D physics. Now 2-D partonic 2-surfaces and string world sheets do it. Strings are of course essential also now and in TGD inspired quantum biology and thermodynamics of consciousness the Hagedorn temperature seems to be a key player. I must also mention the magnetic flux tubes which are everywhere.

If I were a dictator or Witten, it would took five minutes and people would be busily making TGDt;-).

https://books.google.fi/books?id=hyx6BjEX4U8C&pg=PR9&hl=sv#v=onepage&q&f=false

A Quantum Approach to Condensed Matter Physics

AvPhilip L. Taylor,Olle Heinonen, 2002

AdS is a mathematical representation of holography but with too small conformal symmetry. One cannot actually forget that space-time is 10-D in it and this makes condensed matter applications questionable.

This I don't get, condensed matter is not in 10D?

This is a book, http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/pdfs/1.pdf

gauge fields in condensed matter

The most promising support of this theory is the AdS/CFT correspondence [25], which explicitly connects via a one-to-one correspondence the framework of a 5D string theory in anti-deSitter space with a conformal quantum field theory on the 4D boundary.

Maldacena, J., Adv. Theor. Math. Phys.2 (1998) 231;

Maldacena, J., Int. J. Theor. Phys.38 (1998) 1113;

Petersen, J. L., Int. J. Mod. Phys. A 14(1999) 3597;

+ additional introductions to and overviews of the AdS/CFT

+ http://arxiv.org/abs/hep-th/0309246

Condensed matter is certainly not 10-D: this I am saying. String theorists however want some use for their methods;-)! So that it is not so big deal to decide that it is 10-D! A lot of papers and long curriculum vitae: this is after all more important than physics;-).

Condensed matter gives a lot of applications for the high levelled mathematical physics: I have been developing them for more than decade.

I have explained again and again that the problem of string theory approach is that the realisation of holography is doomed to fail because conformal invariance is too restricted. The extended conformal invariance of TGD requires 4-D space-time and 4-D Minkowski space. This is physics.

Maldacena has done nice mathematical work but unfortunately it has very little to do with physics of any kind.

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