Bit from it. Mathematical Clarification by Michael Alexeevich Popov

SOME DEFINITIONS.

Dear professor Corda,

Because FQXi contest is not pure scientific forum, I'd like to introduce some common definitions on BHIP ( may be, for readers - poets and philosophers if You agree )

Following Hawking, the black hole (BH) information paradox started in 1967 when Werner Israel showed that the Schwarzschild metric was the only static vacuum black hole solution. Later generalizations showed that all information ( i.e. hypothetical quantities about the collapsing body , which we can define as "pseudo-bits of BH information ") ) was lost from the outside region apart from three conserved quantities: the mass, the angular momentum, and the electric charge. This loss of pseudo-bits of information wasn't a problem in the classical theory ( A classical black hole would exist for ever and the information could be thought of as preserved inside it, but just not very accessible ). However, the situation changed when Hawking discovered that quantum effects would cause a black hole to radiate at a steady rate Such sort of the radiation from the black hole would be completely thermal and would carry no pseudo-bits of information. Hence, as is known,

BHI PARADOX : What would happen to all that pseudo-bits of information locked inside a black hole that evaporated away and disappeared completely? It seemed the only way the information could come out would be if the radiation was not exactly thermal but had subtle correlations. No one has found a mechanism to produce correlations but most physicists believe one must exist.

Hawking predicted that if information were lost in black holes, pure quantum states would decay into mixed states and quantum gravity wouldn't be unitary.(1975)

In other words, any information that falls in a black hole ( in anti de Sitter space ) must come out again. But it still wasn't clear how information could get out of a black hole. Later Hawking (and Hartle ) showed that the radiation could be thought of as tunnelling out from inside the black hole." It was therefore not unreasonable to suppose that it could carry information out of the black hole.

As the next step, as is known, Hawking invented new

LAW OF INFORMATION CONSERVATION:"The information remains firmly in our universe. Thus, If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized. It is like burning an encyclopaedia. Information is not lost, if one keeps the smoke and the ashes. But it is difficult to read. In practice, it would be too difficult to re-build a macroscopic object like an encyclopaedia that fell inside a black hole from information in the radiation, but the information preserving result is important for microscopic processes involving virtual black holes. If these had not been unitary, there would have been observable effects, like the decay of baryons" ( 2005 )

Let us take here initial definition of Shannon's foundational principle : "One device with two stable positions can store one bit of information, correspondingly, n such devices can store n bits, since the total number of possible states is 2ⁿ and log 2 2ⁿ = n " (1948). Thus, using Shannon-like association between bit and "one device with two stable positions" ( transistor), we can make global generalizations on entity Information in theoretical physics and philosophy of physics. For example,

we always can translate Hawking law of information conservation in the following form :

Universe could be considered as a set of transistors with two at least stable positions which can store n bit of information. Because it is based on analogy, we can say that the Universe as a set of transistors can store n pseudo-bit of information. Pseudo-bit information remains firmly in our universe. Thus, If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the pseudo-bits of information about what you were like but in a state where it can not be easily recognized. It is like burning an encyclopaedia ( another poetical metaphor ) Information is not lost, if one keeps the smoke and the ashes. But it is difficult to read because there is no such thing as physical measurement of pseudo-bits of BH thermal radiation ( thermal information ). Hence, thus, BHIP could be understood also as pseudo-problem, indeed.

As a consequence, your mathematical solution of BHIP based on non-Weyl solution of Schrodinger equation cannot provide final resolution of this kind of pseudo-problem.

(copy of the comment to Ch Corda)

Gennady,

Doubters may suggest that in comparison with energy conservation law, Hawking's law of information conservation (The information remains firmly in our universe. Thus, If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized( 2005 )) can be violated ? Moreover, there is no such thing as physical measurement of bits of thermal information in physics .

( copy of comment for Gennady Gorelik )

Dear Helmut,

I have some doubts on 'In physics everything is basically bi-polar energy '? As is known, Levi-Strauss, founder of French structuralism ,showed that the whole structure of primitive thought is binary as well. However, tendency to operate with binary counters in all situations in physics could be understood also as taking simplification seriously ?

( copy of comment for H.A.von Schweizer )

Dear Jan,

my remark :

Shannon and Other Structuralisms

In comparison with Wheeler Structuralism ( 0-1 structure underpins all of our data and all our sciences ) and Category Structuralism ( by Roman Jakobson, Levi-Strauss, Burbaki, Abramsky and Coecke ) where the universals of human culture exist only at the level of structure,'culinary triangles' or other intuitive ordered things ; Shannon bits mathematics is pure mathematics. Shannon mathematical assumption is based on elementary algebra of common logarithm logax, where the logarithm of x to the base 2 is defined by the equation y = log2 x. This definition is of course applicable only when y is RATIONAL NUMBER. Beyond Shannon rational bits there exists new world of real, irrational, imaginary and complex bits, the world of Weierstrass theorem and the world of real lines contained infinitely many imaginary bits.

Thus, it could be difficult to find real differences between Wheeler binarism and Alternative Gospel of structures , indeed. In fact, similarity between Wheeler and Topos Gospel by Doring & Isham philosophies are obvious.

(Copy of comment for Jan Durham ).

Michel,

P e r f e c t s. It's correct. But it is only a beginning of mystery- Plato ( Euler who made reference to this puzzling place in Republic ) predicted an existence of merely three perfect periodic numbers at all, Ramanujan ( note books ) developed more general technoque using similar assumption and recieved somothing fundamentally different ( he had surprisingly quantum "taste").My new definition of "perfectness" in Bul Sci math is based on elementary proof of impossibility of Euler odd perfect numbers as well.

C u b i c ( already sent ) One - way function in cryptology is another definition of P is not NP solution.

R H . An attempt of Latorre-like superposition of all odds ( 2013 )in quantum computations of pi(x) and the Mangoldt's function.

Best

Dear Akinbo,

Shannon was a mathematician and it is logical to follow his context ( not Wheeler brave speculations ) to understand bit.

In arithmetic we start from the positive integers ( 1,2,3,4,5,6,...) and from the ideas of addition, multiplication,substraction and division. It is easy to test that these operations are not always possible ( 4 - 29, 5 - 7, 2 - 8, 4/29,5/7,... etc )unless we admot new kind of integers ( negative numbers, or more generally, rational numbers )If we include root extraction and the solution of equations, we can find some operations are not possible also unless we admit a new kind of numbers. Mathematicians had found that the extraction of the square root - 1 is not possible unless we go further and admit the complex numbers ( as is known, following mathematicians Einstein, Heisenberg and Schrodinger introduced the square root of - 1 in physics ).Thus,it is practical and productive everywhere ( even philosopher Immanuel Kant made an attempt to introduce negative numbers in philosophy ...)

Complex numbers are sometimes called imaginary.Complex number is not number in the same sense as a rational number ( used by Shannon for bits )It is a pair of numbers (x,y), united symbolically in the form z = x + yi . Hence, it is easy to see, that when y = 0 we say that z is real ( special term for 'post-rational numbers' ), correspondingly, when x = 0 then z is pure imaginary.

Next step.

let ax + by +c = 0 be an equation with complex numbers ( coefficients ). If we give x any concrete complex value , we can find the value of y. Set of pairs of real and complex values of x and y which satisfy the equation is called imaginary straight line, the pairs of them usually are called imaginary points and are said to lie on the line. When x and y are real, the point is called a real point; correspondingly, when a, b, c are all real, the line is defined as real line.etc Hence, we can easy deduce answers for questions 2 and 3.

Dear Than,

I suppose that Physics as a science tries to establish truth but not poetics and political sense of argument.R.Feynman also suggested that physicists have a way of avoiding the politics and subjective tastes in science : if you have an apparatus which is capable of telling how many bits of information given thermal energy ( "termal information" )must contain in the terms of physical measurement, then you can say scientifically about entity information, indeed. Because there is no such thing as physical measurement of the bits of thermal information, then Hawking-like law of information conservation (' The information remains firmly in our universe. Thus, If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized '( Hawking ,2005 )) and its consequences might be considered, unfortunately,as popular illusion.

respectfully

Michael

Dear Antony,

Thank you for high rank. Let me imagine a continuation of your n-dimensionality logical game. If rational n-dimensionality ( n = -1,0,1,2,3 and we assume that n is rational number ) is accepted ( i.e. there is a mathematical proof ) we can go further and we may admit a new kind of possible dimensionality, expressed by the square root -1 and complex numbers ( why not ? Einstein and Hawking use the square root - 1 as an imaginary time / complex time variable u in physics ).Hence, new unexpected physical generalizations are deduced.

( copy of my comment for Antony Ryan by 1 Aug 2013 )