Black hole information paradox and Bogoliubov quantum gravity kinetic equations
-
Upload
alec-wilson -
Category
Documents
-
view
44 -
download
1
description
Transcript of Black hole information paradox and Bogoliubov quantum gravity kinetic equations
![Page 1: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/1.jpg)
Black hole information paradox andBogoliubov quantum gravity kinetic equations
Igor V. Volovich
Steklov Mathematical Institute, Moscow
ROUND TABLE ITALY-RUSSIA@DUBNA
Black Holes in Mathematics and Physics Dubna, December 15 -- 18, 2011
![Page 2: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/2.jpg)
• Black Hole Information Paradox
• Time Irreversibility Problem (BH&BB)
• Functional Formulation of Classical Mechanics and Field Theory
• QG Boltzmann and Bogoliubov Equations
![Page 3: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/3.jpg)
Black hole entropy
• Bekenstein-Hawking:
AG
kcS
4
3
Hawking radiation andBogoliubov transformation
![Page 4: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/4.jpg)
Do Black Holes Really Exist?
• Infinite time to form a black hole in Schwarzshild coordinates?
![Page 5: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/5.jpg)
Black hole information paradox
• In 1976 Hawking has argued that the black hole
creation and evaporation could lead to an
evolution of a pure state into a mixed state.
• This is in contradiction with the rules of
quantum mechanics.
• This has become known as the black hole
information loss problem.
![Page 6: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/6.jpg)
Hawking first argument
• If the material entering the black hole
were a pure quantum state,
the transformation of that state into the
mixed state of Hawking radiation would
destroy information about the original
quantum state.
![Page 7: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/7.jpg)
`t Hooft, Susskind, Hawking…
• Quantum gravity, baby universes, AdS/CFT,…
• 1976: Information loss• 2005: No information loss• In 2005 Hawking has suggested that there
is no information loss
in black holes in asymptotically AdS (anti-de Sitter) spacetimes.
![Page 8: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/8.jpg)
Hawking second argument
• The central point in his argument is the
assertion that there is no information loss
if one has a unitary evolution.
• Then, since the AdS quantum gravity is
dual to a unitary conformal
field theory (AdS/CFT duality) there
should be no information loss.
![Page 9: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/9.jpg)
Time Irreversibility problem• An important point in the argument is the
assertion that, if one considers a theory with a unitary dynamics, there is no information loss problem.
• We remind that the problem of how a
unitary reversible dynamics can lead to an
irreversible behavior (i.e. to information loss) is a famous
fundamental problem in statistical physics.
![Page 10: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/10.jpg)
Time Irreversibility Problem
• We consider the black hole information
problem as a particular example of the
fundamental irreversibility
problem in statistical physics.
Black hole vs. Black body.
![Page 11: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/11.jpg)
Time Irreversibility Problem
Non-Newtonian Classical Mechanics
Functional Probabilistic General Relativity
• I. V. Found. Phys. (2011).
• I.V. Theor. Math. Phys (2010)
![Page 12: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/12.jpg)
Time Irreversibility Problem
The time irreversibility problem is the problem of how to explain the irreversible behaviour
of macroscopic systems from the time-symmetric microscopic laws:
Newton, Schrodinger Eqs –- reversible
Navier-Stokes, Boltzmann, diffusion, Entropy increasing --- irreversible
tt
.ut
u
![Page 13: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/13.jpg)
Time Irreversibility ProblemBoltzmann, Maxwell, Poincar´e, Bogolyubov, Kolmogorov, von Neumann, Landau, Prigogine,Feynman, Kozlov,…
Poincar´e, Landau, Prigogine, Ginzburg,Feynman: Problem is open. We will never solve it (Poincare)Quantum measurement? (Landau)
Lebowitz, Goldstein, Bricmont: Problem was solved by Boltzmann
I.Volovich
![Page 14: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/14.jpg)
Boltzmann`s answers to:
Loschmidt: statistical viewpoint
Poincare—Zermelo: extremely long
Poincare recurrence time
Coarse graining
• Not convincing…
![Page 15: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/15.jpg)
Ergodicity
• Boltzmann, Poincare, Hopf, Kolmogorov, Anosov, Arnold, Sinai,…:
• Ergodicity, mixing,… for various important deterministic mechanical and geometrical dynamical systems
![Page 16: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/16.jpg)
Bogolyubov method
1. Newton to Liouville Eq. Bogolyubov (BBGKI) hierarchy2. Thermodynamic limit (infinite number of particles)
3. The condition of weakening of initial correlations between particles in the distant past
4. Functional conjecture5. Expansion in powers of density
Divergences.
![Page 17: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/17.jpg)
Why Newton`s mechanics can not be true?
• Newton`s equations of motions use real numbers while one can observe only rationals. (s.i.)
• Classical uncertainty relations
• Time irreversibility problem
• Singularities in general relativity
![Page 18: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/18.jpg)
Classical Uncertainty Relations
0,0 pq
0t
![Page 19: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/19.jpg)
Newton Equation
),(
),(
),(
2
2
2
tnRM
txx
xFxdt
dm
Phase space (q,p), Hamilton dynamical flow
![Page 20: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/20.jpg)
Newton`s Classical Mechanics
Motion of a point body is described by the trajectory in the phase space.Solutions of the equations of Newton or
Hamilton.
Idealization: Arbitrary real numbers—non observable.
• Newton`s mechanics deals with non-observable (non-physical)
quantities.
![Page 21: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/21.jpg)
Real Numbers
• A real number is an infinite series,
which is unphysical:
.9,...,1,0,10
1 n
nnn aat
Ftxdt
dm )(
2
2
![Page 22: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/22.jpg)
• Try to solve these problems by developing a new, non-Newtonian mechanics.
• And new, non-Einsteinian general relativity
![Page 23: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/23.jpg)
We attempt the following solution of the irreversibility problem:
a formulation of microscopic dynamics which is irreversible in time: Non-Newtonian Functional Approach.
![Page 24: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/24.jpg)
Functional formulation of classical mechanics
• Here the physical meaning is attributed not to an individual trajectory but only to a bunch of trajectories or to the distribution function on the phase space. The fundamental equation of the microscopic dynamics in the proposed "functional" approach is not the Newton equation but the Liouville or Fokker-Planck-Kolmogorov (Langevin, Smoluchowski) equation for the distribution function of the single particle.
• .
![Page 25: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/25.jpg)
States and Observables inFunctional Classical Mechanics
![Page 26: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/26.jpg)
States and Observables inFunctional Classical Mechanics
Not a generalized function
![Page 27: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/27.jpg)
Fundamental Equation inFunctional Classical Mechanics
Looks like the Liouville equation which is used in statistical physics to describe a gas of particles but here we use it to describe a single particle.(moon,…)
Instead of Newton equation. No trajectories!
![Page 28: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/28.jpg)
Cauchy Problem for Free Particle
Poincare, Langevin, Smolukhowsky , Krylov, Bogoliubov, Blokhintsev, Born,…
![Page 29: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/29.jpg)
Free Motion
![Page 30: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/30.jpg)
Delocalization
![Page 31: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/31.jpg)
Newton`s Equation for Average
![Page 32: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/32.jpg)
Comparison with Quantum Mechanics
![Page 33: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/33.jpg)
Liouville and Newton. Characteristics
![Page 34: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/34.jpg)
Corrections to Newton`s EquationsNon-Newtonian Mechanics
![Page 35: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/35.jpg)
Corrections to Newton`s Equations
![Page 36: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/36.jpg)
Corrections to Newton`s Equations
![Page 37: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/37.jpg)
Corrections
![Page 38: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/38.jpg)
•
• The Newton equation in this approach appears as an approximate equation describing the dynamics of the expected value of the position and momenta for not too large time intervals.
• Corrections to the Newton equation are computed.
• _____________________________• _____________________________
![Page 39: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/39.jpg)
Fokker-Planck-Kolmogorov versus Newton
p
pf
p
ffH
t
f
)(
},{ 2
2
![Page 40: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/40.jpg)
Boltzmann and Bogolyubov Equations
• A method for obtaining kinetic equations from the Newton equations of mechanics was proposed by Bogoliubov. This method has the following basic stages:
• Liouville equation for the distribution function of particles in a finite volume, derive a chain of equations for the distribution functions,
• pass to the infinite-volume, infinite number of particles limit,
• postulate that the initial correlations between the particles were weaker in the remote past,
• introduce the hypothesis that all many-particle distribution functions depend on time only via the one-particle distribution function, and use the formal expansion in power series in the density.
• Non-Newtonian Functional Mechanics:
• Finite volume. Two particles.
![Page 41: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/41.jpg)
Liouville equation for
two particles
![Page 42: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/42.jpg)
Two particles in finite volume
![Page 43: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/43.jpg)
If satisfies the Liouville equation then
obeys to the following equation
),,( 21 txx
),( 11 txf
Bogolyubov type equation for two particles in finite volume
![Page 44: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/44.jpg)
• Kinetic theory for two particles
• Hydrodynamics for two particles?
![Page 45: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/45.jpg)
• No classical determinism
• Classical randomness
• World is probabilistic
(classical and quantum)
Compare: Bohr, Heisenberg,
von Neumann, Einstein,…
![Page 46: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/46.jpg)
Single particle (moon,…)
})()(
exp{1
|
),,(
2
20
2
20
0 b
pp
a
ab
pq
V
qm
p
t
tpq
t
![Page 47: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/47.jpg)
• Newton`s approach: Empty space (vacuum) and point particles.
• Reductionism: For physics, biology economy, politics (freedom, liberty,…)
• This approach: No empty space. Probability distribution. Collective phenomena. Subjective.
![Page 48: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/48.jpg)
Fixed classical spacetime?• A fixed classical background spacetime
does not exists (Kaluza—Klein, Strings, Branes). No black hole metric.
There is a set of classical universes and
a probability distribution
which satisfies the Liouville equation
(not Wheeler—De Witt).
Stochastic inflation?
),( gM
![Page 49: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/49.jpg)
Functional General Relativity
• Fixed background .
Geodesics in functional mechanics
Probability distributions of spacetimes• No fixed classical background spacetime. • No Penrose—Hawking singularity
theorems• Stochastic geometry? Stochastic BH?
),( gM
),( gM
),( px
![Page 50: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/50.jpg)
Quantum gravity.Superstrings
( , )( , ) [ ]MiS g
M
F g e D Dg The sum over manifolds is not defined.Algorithmically unsolved problem.
![Page 51: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/51.jpg)
Example
singular1
)(,0
02
tx
xtxxx
rnonsingula
},/)1(exp{),( 22
0 qxt
xCtx
![Page 52: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/52.jpg)
Fixed classical spacetime?• A fixed classical background spacetime
does not exists (Kaluza—Klein, Strings, Branes).
There is a set of classical universes and
a probability distribution
which satisfies the Liouville equation
(not Wheeler—De Witt).
Stochastic inflation?
),( gM
![Page 53: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/53.jpg)
Quantum gravity BogoliubovCorrelation Functions
• Use Wheeler – de Witt formulation for QG.
Density operator of the universe on
Correlation functions
))()...((
);,...,(
1
1)(...
11
11
sjiji
ss
jiji
yxgTr
yxf
ss
ss
![Page 54: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/54.jpg)
Factorization
rrrnmji
ss
jiji
yxf
yxf
rrrr
ss
);,(
);,...,( 1)(...11
Th.M. Nieuwenhuizen, I.V. (2005)
![Page 55: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/55.jpg)
QG Bogoliubov-Boltzmann Eqs
),,,,(
),(
gyx
fJf
![Page 56: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/56.jpg)
Conclusions
BH and BB information loss (irreversibility) problem
Functional formulation (non-Newtonian) of classical mechanics: distribution function instead ofindividual trajectories. Fundamental equation: Liouville or FPK for a single particle.
Newton equation—approximate for average values.Corrections to Newton`s trajectories.
Stochastic general relativity. BH information problem.QG Bogoliubov-Boltzmann equations.
![Page 57: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/57.jpg)
Thank You!
![Page 58: Black hole information paradox and Bogoliubov quantum gravity kinetic equations](https://reader036.fdocuments.net/reader036/viewer/2022062408/56813401550346895d9af7d4/html5/thumbnails/58.jpg)
Information Loss in Black Holes
• Hawking paradox.
• Particular case of the Irreversibility problem.
• Bogolyubov method of derivation of kinetic equations -- to quantum gravity.
• Th.M. Nieuwenhuizen, I.V. (2005)