Black holes and decoherence

Date: Wed, 11 Sep 1996 17:01:57 -0700 (PDT)
From: Lawrence B. Crowell <lcrowell@unm.edu>
To: quantum-d@teleport.com
Subject: Black holes and decoherence


I have written here concerning the problem of decoherence with respect
to black holes and open environments. I argue that the open system
approach is much more realistic than the assumption of quantum black
holes and the like...

---

A central issue for quantum biology is decoherence.  In order for a wave

function to play a real role in biological systems there must be a mechanism

that prevents the phase space volume of the quantum system from being

absorbed into the environment.  At issue is what the mechanism of decoherence

and the process that prevents decoherence, or "recoheres" the system.


The Penrose-Hameroff approach invokes the role of quantum black holes as the

source of wave function collapse.  The result of Stephen Hawking [1] is that

a quantum field will scatter off a black hole with a different value for the

trace of the square of the density matrix.  This is similar to what occurs to

a wave function with a measurement.  This change in entropy involved with

measurements and black holes leads one to question whether there is some

common theme involved with the two different processes.  This is given some

added weight by Wigner's assertion that consciousness is one aspect of nature

that can violate the conservation law of quantum mechanics


d rho^2/dt = 0.


This is not the only manner in which we can get decoherence.  Decoherence

really means that a system is incapable of entering into a recurrence of

behavior.  This can be understood according to open systems.  Below we

briefly discuss these approaches.


The origin of black hole thermodynamics results from an asymmetry in the

wave propagation of a field into verses out of a black hole.  Let a(k) be

an operator for a scalar field that annihilates the vacuum


a(k)|0> = 0.


Now write a field according to


phi = sum_k (a(k)f(k) + a^{dag}f^*(k))


where f(k) is a mode expanded in Kruskal coordinates u,v for inward and

outward geodesics.  Now evaluate the Greens function


G(x, x') = <0|phi(x) phi(x')|0>,


where the vacuum is defined at an asymptotic region removed from the black

hole.  Input the specifics of the Kruskal metric and we find that the Green's

function is


G(x, x') = -(1/2pi) ln(&u&v) & = delta


         = -(1/2pi) ln(cosh(t-t'/4M) - cosh(r - r'/4M)) + other stuff,


         M = mass of black hole.


The curious thing is that this Green's function is cyclic in t --> t + i(8piM).

This thermal Green function has a temperature of T = 1/(8pikM).  The origin

of this thermal behavior is that u is analytic across the horizon, where v

is not.  The fields entering the black hole can carry information in, but those

fields that tunnel out carry no information that concerns the interior

structure.


A closed system is one that can enter into Poincare recurrence.  Such a closed

system has a finite Hilbert space, with modes explicitly defined.  However,

such perfectly closed systems are a fiction taught to first year graduate

students.  In reality the system is coupled to an environment.  A simple

environment to couple to is the QED vacuum {|n(k)>}, for n = 0,1,2,... and k

a continuum.  Our finite system is described by a Hilbert space of finite

dimensions.  An often used example is a two state system with a basis

{|+>, |->}.  An atom with other states can be used with Clebsch-Gordon

coefficients.  The hapless finite state system is incapable of entering

periodic recurrence since it is coupled to this environment with an infinite

number of states.  This is true even if the coupling to the environment is

weak.  The environment then can be described according to reservoir operators

and the dynamics of the density matrix of our system obeys a master equation

for an open system,


id rho/dt = [H, rho] + L rho.


So here we have two models of decoherence.  One that involves black holes and

the thermodynamics of quantum fields in their environment, and an approach

that involves the breakdown of recurrence by coupling a system to an external

environment with an infinite dimensional Hilbert space.  Before passing

judgement on which process is the most relevant to the issue of quantum

biology, assuming experiments bear fruit on this matter, I will first indicate

that the two processes are in some way related.  The black hole has a Hilbert

space associated with the states of quantum gravity, and all other fields.  Yet

this Hilbert space is coupled to the outside world in an asymmetric manner.

Similarly the environment coupled to the finite state system causes

information of the finite system to flow in an asymmetric manner into the

external environment that system is coupled to.


Now for the issues of quantum biomolecules it seems best to take the open

systems approach.  There are a number of reasons for this.  The first is simply

one of scale.  A model that involves decoherence according to the existence

of virtual black holes requires the tie in between physics at the scale of

10^-33cm to that of macromolecules of a cell 10^-7 cm.  Physics usually

requires that things at a high ultraviolet frequency are cutoff from things

at an infrared domain.  This involves some things about renormalization

group theory, and quantum gravity has not been conclusively shown to be

renormalizable.  Yet it is at least still reasonable to assume that things on

the Planck scale have little to do with biochemistry, molecular biology, and

the processes involved communication between cells.


The issue is which approach is the most fruitful for problem that involve

nonrelativistic wave propagation in a molecular wave guide.  To be honest

I think the open systems approach is the most reasonable.  Further, I am

sufficiently familiar with people in the molecular biology field to know

that there is little interest in some marriage of quantum black holes to

molecular biology.  Just getting basic quantum mechanics, Schrodinger

equation, density matrices, etc, married into molecular biology is going

to be tough enough, without straining things to include black holes,

strings, or unification of gauge fields.


I think that methods of quantum trajectories, master equations, and the

rest are much more likely to bear real fruit.  There is nobody in quantum

optics that is invoking quantum gravity to understand the problems of

decoherence in cavity QED, chaos and measurement.  Similarly I think that

q-black holes are a domain of theoretical physics that is renomalized, or

scaled out, so that this high frequency regime is sealed off from problems

of quantum molecular physics.


The alpha and beta tubulins linked together in microtubules and centrosomes

are subjected to an environment with thermal noise and the QED-vacuum.  It

is easy enough to sum over these influences as delta function correlated

processes in a Langevin equation and its associated Fokker-Planck equation.

The processes involved with the quantum gravity vacuum are at such a high

frequency end of the scale as to be negligibly apparent.  If quantum

gravity is involved with this scale of physics then why not SU(5) GUT

physics?  After all the electron in the hybrid bonds do have some nonzero

probability of interacting with a quark in a nucleus.  However, that

perturbation is vanishingly small, not to mention that SU(5) is not likely

to be the right GUT, but the idea is still there.  If not GUT physics then

why not QCD? After all a self bound field knot of electric and magnetic

QCD field lines do have some probability of tunneling out of the nucleus,

for only a brief period of time due to Heisenberg, and coupling to the

electrons in molecular bonds.  Yet these things are ignored as being too

small.

A microtubule is composed of alpha and beta tubulin that can carry phonons

and exhibit polarizations.  Microtubule associated proteins (MAPs) bind to

microtubules (MT's) to mediate the interaction of MT's with the rest of the

cellular structures.  As such the MAPs act to pump energy into the MTs.

Now let us assert the hypothesis that this energy that is pumped into the

MT drives the polarization of the tubulin molecules.  These polarizations can

be expanded into cylindrical Bessel functions with a mode expansion.



P(x,t) = sum_n p_n(x,t) B_n(y,z)e^{iw_n}


where the microtubule is oriented with its axis of azimuthal symmetry along

the x axis.  Now Maxwell's equation gives that the electric field satisfies

the equation


{grad^2 - (1/c^2)&^2/&t^2}E(x,t) = (4pi/c^2)&^2P(x,t)/&t^2 & = partial.


The electric field is expanded in a similar set of modes as the polarization.

Now the electric displacement vector D = epsE.  Now the medium is a solution

ions, and the index of refraction is often for such media intensity dependent


n = n_0 + n_2|E|^2.


Some analysis leads one to the equation of motion for the envelope of fields

in the MT as


i&E/&t = -(1/w)((1/2)&^2E/&t^2 - n_2|E|^2 E,


which is the cubic Schrodinger equation.


Physically this means that the fields within the tube are self focusing for

n_2 > 0.  For a few number of photons the interactions between the photons

are nonlinear.  The "gas" of photons forms a quantum wave packet through this

self-focusing process.  The solution to the cubic Schrodinger equation is


E = |E_0|sech(x/d)e^{iw'_n},


for w'_n = (k_0/2)n_2|E|^2 and d = (1/k_0)|E|sqrt(n_0/n_2).  The important

fact is that the solution is a soliton.  This soliton is stable under

perturbations.  This makes MT's a possible quantum transmission line that

utilizes the infinite number of symmetries of a soliton as the fundamental

bit, or q-bit, carrier.  This part of the problem appears to be in hand,

except for the issue of decoherence.  This is an issue that I am currently

working on.  The cubic Schrodinger equation must now have two terms added on.

The first is a -Gam E for the dissipation of the field envelope, and the

second is a term F = sum_i f_i for a driving force that restores the field

envelope.


MT's are the communication channels within a cell. They organize the cellular

environment by acting as motors and information conveyances.  If they are

communication channels of quantum information they might provide the

"wholeness" that maintains cellular integrity.  There is still the issue of

how these q-bits are process by the macromolecular machinery at the send and

receive end of the MT's.  It is at the ends of the MT's that the problem

becomes actually quite difficult.  It is here that J. Sarfatti's back-action

should rear it head, if this hypothesis is correct.  Again, this process is

likely to involve the quantization of a nonlinear system.  Back-action is a

process that should initially be examined according to quantum chaos, since

back-action implies some degree of emergent complexity or emergent behavior

that occurs with chaotic or nearly chaotic systems.  Back-action would then

be quantum chaotic dynamics with nonlocality that gives rise to emergent

behavior and structure that are quantum correlated.


In much of my interest in this matter I take the Bohm approach since I

think it is highly workable for problems of quantum chaos.  Classical

chaos is formulated around the  separation of particles in a dynamical

system, i.e. Lyapunov exponents. Bohr and Bohm are complementary view of

quantum mechanics, ala deBroglie's wave particle duality.  Bohm's approach

garnered bad press since he formulated this approach to find locality in

QM. Bell's theorem came about and blew that hope into the wind.  Yet Bohm's

approach is a largely unexplored method for doing QM.  Unexplored areas

are to me much more interested than areas that are highly populated.


The advantage that Bohm's approach has is that the reduction of the phase

space volume of a quantum system can be examined as a dynamical process.  The

QHF can be modeled to bleed into an environment.  If the QHF is completely

absorbed by the environment, or measuring apparatus, then the particle is

left in a single point in space.  This is the end result of all measurement

processes.   A spot shows up on a photoplate.  The Bohr approach involves the

use of projector operators and other idempotent operators that stand outside

the proper set of quantum operators.  This approach is a sort of ad-hoc way

of by hand forcing a measurement collapse.


My approach has been to look at the quantization of Hamiltonian quantum

systems, then to later examine the quantization of strange attractor

physics.  The notion I have of Jack's back-action is as a property a

system has for negative feedback.  By quantizing strange attractor physics

I am looking for observables that describe the fractal growth of the

quantum hydrodynamic fluid (QHF)and the Lyapunov exponent for the

particle. To be honest I treat both of these entities as mathematical

objects.  I am not as concerned with the ontological existence of these

things.  Anyway I hope to find criteria where the chaotic motion of the

particle and the fractal evolution of the QHF exert a negative feedback on

each other.  From here Jack's case for back-action should have some

physical justification.


To conclude it should be more reasonable to treat the issue of wave function

collapse as a process where a measurement apparatus or environment couples

into a quantum system with a finite number of states.  The result is that the

recurrence of the finite system is prevented and the system is reduced to

a single state.  This is true if we are talking about coupling a system to a

vacuum with an infinite number of unoccupied modes.  The result is spontaneous

emission.  The other environmental impact is that of thermal noise.  This

more "accessible physics" approach is far more likely to succeed than positing

quantum black holes and spacetime foam as the source of collapse.  I have been

quite involved with the issue of string theories and quantum gravity for some

time.  Yet the problem with these topics is that they concern a domain of

reality that is almost hopelessly removed from the experimental front.  This

lack of falsifiability of theories of quantum gravity and strings is what

gave me the cause to leave these subjects, at least as a future means of

support and employment.  I see in the future a continual constriction of

physics involved with high energy physics, particles, gauge fields, and

gravity.  I see physics, and science in general, becoming more involved with

providing a basis for technological competitiveness for nations in the global

economic matrix.  Molecular biology and genetics is the largest fundamental

science that fits into this emerging political reality.  Unfortunately I do

not quantum gravity fitting into this, but other aspects of physics might fit

in quite well.


Lawrence B. Crowell