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Information Preservation in Quantum Gravity

  1. Mar 28, 2006 #1
    This is the start of a presentation of some work I have done over the past year and hope to publish. This is how it is that quantum information is preserved in quantum gravity and cosmology. This will involve a number of posts along this thread. The next three posts involve quantum information with black holes
    //
    The conservation of information from black hole radiance has an unsettled history. The initial indications were that information is lost. Yet a formalism of a consistent quantum theory of gravity with information loss is difficult, for such a theory would have to be nonunitary. In a bet recently Hawking conceded to Preskill that information was preserved in black holes. Information would then not be destroyed, but rather scrambled in such as way as to make its retrieval intractably impossible. A tunnelling approach to quantum radiance by Parikh and Wilczek [1] suggests that the process in total has [itex]\Delta S~=~0[/itex], but as recently pointed out in [2] this is the case where the black hole and environment are in thermal equilibrium. However, the negative heat capacity of spacetime means that a black hole slightly removed from equilibrium is unstable and will diverge from equilibrium. This is seen with the evaporation of a black hole, where as its entropy [itex]\Delta S_{bh}~\rightarrow~0[/itex] its temperature becomes large. Thus the Parikh-Wilczek tunnelling theory appears to be a “ measure zero” case.
    //
    A tunnelling process involves an action which is not accessible classically. With the tunnelling of a particle through a square barrier, this involves an imaginary momentum or action. In the case of a tunnelling of a particle from a black hole [itex]M~\rightarrow~M~+~\delta m[/itex], this involves the imaginary part of the action
    [tex]
    ImS~=~Im\int_{r_i}^{r_f}p_rdr.\eqno(1)
    [/tex]
    The Hamilton equation [itex]{\dot r}~=~{{\partial H}\over{\partial p}}[/itex] permits this to be written as
    [tex]
    ImS~=~Im\int_{r_i}^{r_f}\int_{p_i}^{p_i}{{dr}\over {\dot r}}dH.\eqno(2)
    [/tex]
    Along null geodesics the velocity [itex]{\dot r}~=~\pm 1~+~\sqrt{2M/r}[/itex] the action for the classically forbidden path is
    [tex]
    Im S~=~-2\pi\int_0^{\delta m}{{dr~dm}\over{1~+~\sqrt{{2M~-~\delta m}\over r}}}~=~-2\pi\big((M~-~\delta m)^2~-~M^2\big),\eqno(3)
    [/tex]
    which defines [itex]-{1\over 2}(S_f~-~S_i)[/itex]. The imaginary part of the action gives the tunnelling probability or emission rate as [itex]\Gamma~=~exp(-2ImS)~=~exp(\Delta S)[/itex]. For a black hole in equilibrium with the environment the entropy remains on average zero, for with every quanta it emits it will on average absorb another from the environment. [itex]dM~=~dQ[/itex] as the first law of black hole thermodynamics with [itex]dS~=~{{dQ}\over T}[/itex] holds for a reversible process. However, the fluctuations will eventually cause the black hole to diverge from equilibrium, where no matter how small this is it will cause the black hole radiance to diverge, or for the black hole to acquire larger mass arbitrarily. Any change in the state of the black hole, whether by emission or absorption, perturbs the black hole from equilibrium.
    //
    There are a number of physical ways that black hole radiance are presented. A black hole emits a particle since the quanta which make up a black hole have some small but nonzero probability of existing in a region [itex]r~>~2M[/itex]. Another interpretation is that virtual electron-positron pairs near the event horizon may permit one in the pair to fall into the hole while the other escapes to infinity. This view is equivalent to saying that an electron or positron propagates backward through time from the black hole and is then scattered into the forward direction by the gravity field. A related interpretation has that the creation of a positive mass-energy particle is associated with the creation of a negative mass-energy particle absorbed by the black hole. As a result the black hole’ s mass is reduced and a particle escapes. In the case of fermions this is in line with Dirac’ s original idea of the anti-particle with a negative mass-energy. In all of these cases there is a superposition principle at work. Quanta within the black hole are correlated with quanta in the exterior region. How these quanta are correlated is the fundamental issue. The imaginary action is a measure of the nonlocal correlation a particle in the black hole has with the outside world. In the case of equilibrium with [itex]TdS~=~dM[/itex] the black hole exchanges entropy with the environment so that the total information of the black hole and environment remains the same. Yet in general the radiance of a black hole will heat up the environment so that [itex]dS~>~{{dM}\over T}[/itex], and the same is the case of the black hole absorbs mass-energy.
    //
    A black hole will absorb and emit observables, where if information is preserved these observables will have a corollation. The corollation will reflect a quantum process which is unitary, or that the emitted observables are nonlocally entangled with the black hole states in such as way as to preserve information. If information is preserved by a black hole, then in principle a black hole is an efficient teleporter of quantum information. Here the black hole is shown to ultimately preserve quantum information even for the case that [itex]dS~>~{{dM}\over T}[/itex].
    //
    The standard “ Alice and Bob” problem is considered. Alice has the set of observables [itex]A[/itex], which she communicates to Bob in a string [itex]x_1x_2\dots x_n[/itex]. Bob similarly has the string [itex]y_1y_2\dots y_n[/itex]. The von Neumann information each possesses is then [itex]S(X)~=~-Tr(\rho_X~log_2\rho_X)[/itex], for [itex]X[/itex] either [itex]A[/itex] or [itex]B[/itex]. The entropy of each is then a compartmentalization on [itex]\rho_{AB}[/itex] for the total quantum information both possess and [itex]\rho_{A}~=~Tr_B(\rho_{AB})[/itex] and [itex]\rho_B~=~Tr_A(\rho_{AB})[/itex]. Here the trace is over the part of the Hilbert space for Alice or Bob to project out the density operator for Bob or Alice. The density operator [itex]\rho_{AB}[/itex] then defines the joint entropy
    [tex]
    S(AB)~=~-Tr(\rho_{AB}~log_2\rho_{AB}).\eqno(4)
    [/tex]
    If Alice transmits her string [itex]x_1x_2\dots x_n[/itex] to Bob this defines the conditional information or entropy [itex]S(A|B)[/itex] as the information communicated by Alice given that Bob has [itex]y_1\dots y_n[/itex] defined as
    [tex]
    S(A|B)~=~S(AB)~-~S(B).\eqno(5)
    [/tex]
    If Alice sends this string into a black hole, this is the entropy measured by Bob as measured by the quantum information the black hole emits. The conditional entropy may be defined by a conditional von Neumann entropy definition
    [tex]
    S(A|B)~=~-Tr(\rho_B\rho(A|B)~log_2\rho(A|B)~=~-Tr(\rho_{AB}log_2\rho_{A|B}),\eqno(6)
    [/tex]
    where [itex]\rho_{A|B}~=~lim_{n\rightarrow\infty}\big({\rho_{AB}}^{1/n}({\bf 1}_A\otimes\rho_B)^{-1/n}\big)^n[/itex]. Here [itex]{\bf 1}_A[/itex] is a unit matrix over the Hilbert space for Alice’ s quantum information. This means that the entries of [itex]\rho_{A|B}[/itex] can be over unity, which also means that the information content of conditional entropy can be negative as well [3]. Thus quantum information can be negative, in contrast to classical information. The conditional entropy determines how much quantum communication is required to gain complete quantum information of the system in the state [itex]\rho_{AB}[/itex].
    //
    When the conditional entropy is negative Alice can only communicate information about the complete state by classical communication. The sharing of [itex]-S(A|B)[/itex] means that Alice and Bob share an entangled state, which may be used to teleport a state at no entropy cost. The negative quantum information is then the degree of “ ignorance” Bob has of the quantum system which cancels out any future information Bob receives. The “ hole” that Alice fills in Bob’ s state ignorance amounts to a merging of her state with Bob’ s.
     
  2. jcsd
  3. Apr 4, 2006 #2
    Information Preservation in Quantum Gravity II

    This is the full presentation on information preservation in quantum gravity.
     

    Attached Files:

  4. Apr 21, 2006 #3
    The Hamiltonian for gravity

    Hello Lawrence:

    Let me ask a basic question. I thought there was a problem writing out the Hamiltonian for general relativity. If it was straightforward, there would be no need for all the struggles with quantum gravity in the first place.

    For all other areas of physics where things are working well, the Hamiltonian can be found by direct calculation if the Lagrange density is known:
    [tex]H=\pi^{\mu}\frac{\partial A_{\mu}}{\partial t}-\mathcal{L}[/tex]
    Hilbert figured out the Lagrangian back in 1915. The problem as I understand it, is that for any point in spacetime, one could choose a coordinate system so that energy was zero there but not everywhere, known as the Riemann normal coordinates. This is the energy localization problem for general relativity. This is accepted as an interesting insight into gravity, not a deadly technical flaw of general relativity.

    Personally, I do not think the underlying math of gravity should somehow be different from the rest of physics. It is a BIG RED FLAG if calculating the Hamiltonian is somehow different on a technical level for gravity than every other working theory in physics.

    doug
     
  5. Jun 26, 2006 #4
    Hello Doug
    What do you think about Arnowitt, Deser and Misner (ADM) 3+1 Form for solving the problem
    http://www.tat.physik.uni-tuebingen.de/~koellein/bericht-WEB/node19.html
     
  6. Jun 28, 2006 #5
    The ADM-split of the Einstein field equations into 3 dimension of space, and one for time, is not relevant to the issue I raised. ADM is used by people who do numerical relativity, stuff like the path followed by a nuetron star spiralling into a black hole. I think it shines no light on finding the Hamiltonian.

    The gauge symmetry in my GEM proposal is different from that found in GR and GEM, so you would have to go that thread and see if you understand my viewpoint (it is not appropriate to go into details in Lawerence's thread. I hope he is OK, as I have not seen him around these pages for a while).

    doug
     
  7. Jun 28, 2006 #6
    Thanks for the precision...I admit I am a little rusty after some months out of the discussion.
    I need to refresh some notions about your theory.
     
  8. Aug 7, 2006 #7
    responses

    I have not been here for a while. I thought I would return to see what has been up.

    The lagrangian for GR is the Hilbert-Palatini action which is of the form

    L = sqrt{g} k^{-1}R,

    where k = 8piG/c^4. If one is involved with issues of inflation this constant is replaced by k^{-1} ---> k^{-1} + phi^2/6, where phi also obeys a Higgian field equation.

    In the space plus time approach the action is

    S = sqrt{g}pi_{ab}dg^{ab} - NH - N_iH^i

    where H = 1/2pi^{ab}pi_{ab} - R^{(3)} and H_i = nabla_jpi^{ij}, which are the Hamiltonian and momentum constraints and N and N_i are the Lagrange multipliers for the theory. This leads to the dynamical equation H = 0. This is extended to the Wheeler-Dewitt equation by canonical quantization.

    This issues with quantum gravity are rather subtle issues of what it means for HPsi = 0, where this is a wave functional over some closed or compact set of spatial surfaces. The issue is how is that one can make this define a set of diffeomorphisms over spatial surfaces.

    Lawrence B. Crowell
     
  9. Aug 8, 2006 #8
    GR versus EM

    Hello Lawrence:

    In EM, the Hamiltonian is simple: it is the [itex]T^{00}[/itex] part of the stress tensor:
    [tex]H=\frac{1}{8 \pi}(E^2+B^2)[/tex]
    Square the two fields involved, at it up, and one is done. When gravity is described by general relativity, it is much harder to understand. This means one of two possible things: it is worth the effort to work at appreciating the subtleties, or GR is wrong. One of the strengths of GR is the simplicity of the Hilbert action, but it does not continue through the Hamiltonian. EM is giving us a clear message: the Hamiltonian should be simple so the theory can be qunatized. With GR it is not, the issues are subtle. This is why I look to EM for guidance, and have the courage to clearly abandon GR.

    doug
     
  10. Aug 8, 2006 #9

    ZapperZ

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    Why would a square barrier be valid in this case? Since when is the gravitational potential can be accurately represented by a square barrier?

    If you try to do this via the WKB approximation, then you will have to deal with the tunneling matrix element, which is the most accurate way of getting the tunneling probablity. I do not see where you have considered any of this.

    Zz.
     
  11. Aug 11, 2006 #10
    One can model black hole tunnelling in this fashion. However, this is more of an approximate or heuristic sort of model. My purpose is to illustrate something about information in black hole radiation. So in order to do this I use the action in this way.

    Let there be an initial quantum state

    |psi> = 1/sqt(2)(|1>_u|2>_v + |2>_u|1>_v)

    where the EPR type states 1 and 2 are in a superposition according to whether they are on the Kruskal coordinates u or v. The v coordinates are not analytic across the horizon where the u coordinates are. This means that a particle pair from the polarized vacuum are in an entanglement, but as we know the u coordinates enter the black hole. Hence states on u end up being absorbed into the black hole interior within a "time" ~ pi M/2. If the black hole state is given by |M> we then have that

    |psi> ---> 1/sqt(2)(|2>_v|M_1> + |1>_v|M_2>)

    where |M>_1 = |1>_u|M> and similarly for the "2" state. However, the v coordinates are not analytic across the horizon so in general the outside observer does not have access to |M_n>. This is similar to the whole issue of wave function collapse. This loss of information defines the conditional entropy S(A|B) in the Alice-Bob teleporation problem.

    Yet we know that S(A|B) = S(AB) - S(B), so in principle given an appropriate accounting Alice may actually teleport his states through the black hole to Bob. A negative S(A|B) corresponds to an entanglement between the Alice and Bob states. This is the gist of my short paper.

    This means that black holes may absorb quantum states along the v coordinate and the entanglement appears to be effectively lost with an outgoing state along the u coordinate. However, all this means is that the entanglment measure has shifted to the states of the black hole, and the information arriving along the u coordinate appears random because information has been encrypted. The black hole quantum gravity states must encrypt quantum information in a way which preserves them from the wild quantum fluctuations near the singularity.

    The nature of this encyption is the core of what I am working on. The structure of quantum gravity in order to preserve quantum information must involve symmetries which preserve quantum information, and at the same time be the field symmetry of gravity. This necessitates that quantum gravity be a Goppa and/or Golay code system. The first has the advantage of being over algebraic varieties which define event horizons as projective varieties. The geometric content is given by Golay codes which have E_8 and higher lattice descriptions. The E_8 error correction system effectively exists in the heterotic string.

    So all of this is a prelude to further work. I hope this helps for now.

    Cheers,

    Lawrence B. Crowell
     
  12. Aug 11, 2006 #11

    ZapperZ

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    All you did was convince me that it isn't "tunneling".

    Zz.
     
  13. Aug 11, 2006 #12
    I think that abandoning GR and trying to do everything as electromagnetism is not going to go well with a lot of people. The two theories have a similar structure up to the post-Newtonian (PN) term. Beyond that to PPN and beyond GR demonstrates departures from EM theory.

    Electromagnetism is the "easy" field theory because the photon carries no charge. This is why it is abelian U(1), with a nice linear structure. The problem is that other field theories are not so simple. In the case of gravitation such an abelian structure can't work in general. Think of Newtonian gravity, where the potential energy is V = -GMm/r. for M and m small enough the mass equivalence of this m_grav = Vc^2 is very small. However, for very large masses the m_grav becomes large enough, though negative, to contribute to the gravity field. In other words the gravity field gravitates. This has been called the "lightness of gravity," for the field for large masses tends to actually reduce the field. From analogues with electromagnetism it would be as if the photon has a charge and can radiate photons. In QCD and the weak interactions this sort of gauge field exists.

    Due to this and the fact that the gravity field has to be Lorentz covariant the structure of classical gravitation is what Einstein laid down in general relativity.

    Lawrence B. Crowell
     
  14. Aug 11, 2006 #13
    It's tunnelling. For the elementary case with a potential V > E we have that

    E = p^2/2m + V

    and so

    p = sqrt(2m)*sqrt(E - V)

    which is imaginary valued. So for a wave function given by

    psi ~ exp(ipx/hbar),

    where px is the action, this action is imaginary valued. A tunnelling state or instanton is given by an imaginary action. In my paper the problem involves an imaginary action.

    Lawrence B. Crowell
     
  15. Aug 11, 2006 #14

    ZapperZ

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    Just because you have an "imaginary action" does not mean you are dealing with "tunneling phenomena".

    For example, there's nothing here that calculates the probability of transmission. If you have an initial state, and a final state that you want to tunnel to, where is the transition probability? You seem to have neglected the tunneling matrix element that couples the two states together (look in Merzbacher if you don't believe me).

    This, again, is still neglecting the validity of a "square potential" in the first place as a reasonable substitute for an actual potential. Why isn't any realistic gravitational potential used here? Just so you can simplify the mathematics?

    Zz.
     
  16. Aug 11, 2006 #15
    GR could be wrong

    Hello Lawrence:

    Agreed. A rank 1 field theory is not in the literature. Take something like a scalar-tensor theory. There are lots of papers on the topic. Many good books on GR have a section of a chapter on the topic. Clifford Will's living review article talks about scalar-tensor theories, and does not bring up the possibility of the simplest rank 1 field theory (I asked him directly). Seeing a blind spot takes work that I do not believe a lot of people are willing to take. People earn their income by looking at GR. Tenure is suppose to give people the freedom to explore alternative ideas that is not exercised enough.

    If you are referring to my efforts, the agreement is to first-order PPN accuracy. At this time we have zero data to confirm or reject GR at second-order PPN accuracy, where my approach predicts 12% more bending around the Sun. This is a very rare and good thing: a solid test for two theories of gravity consistent at the current level of measurement.

    Agreed. What can electromagnetism do to 2 unlabeled particles? Well, they could be attracted to each other with a classical 1/R^2 force, or they could repel from one another due to a 1/R^2 force, or they might do nothing. This is simple, only three possibilities exist.

    What can gravity do to 2 unlabeled particles? Well, they can only be attracted to each other with a classical 1/R^2 force. Our explanation of gravity must be simpler than EM.

    There certainly are more complicated gauge field theories out there, such as those for the weak and the strong forces that you cite. Gravity should be simpler. It is also true the GR is not a simpler field theory than EM. There is no direct experimental data to show that gravity fields gravitate. The nonlinearity is far too subtle - I suspect way beyond second-order PPN accuracy, but I do not know the actual numbers. It is my belief that gravity fields do not gravitate. If in Nature, gravity fields do not gravitate, then GR must be abandoned, it is a logical consequence, nothing more. I am saying that all sorts of forms of energy contribute to the 4-momentum charge as a source of gravity - kinetic energy, binding energy, etc. - but that the energy of the gravity field itself does not add to the charge. Gravity, at least classically simpler than EM, is thus like EM in this way, and not like the weak or the strong force which involve more force particles and charges.

    I believe this is referencing work by Weinberg and others where if one wants to make Newton's classical gravitational force and make it consistent with special relativity, then you end up at Einstein's field equations. That path does exist and is valid. A different approach is, well, different. There is no way a scalar field theory can explain light bending around the Sun since one term in the metric get bigger than one, the other less than one. It may be an error to try and make something broken more consistent. Better to start with something that can be valid no matter what.

    I could make more banal arguments, citing cases where theories we thought were true turned out to be in error, but I like the specific technical nature of this critique.

    I hope to get the Hamiltonian calculation done next week for my work. The calculation is direct, but I want to get all the LaTeX in place.

    doug
     
    Last edited: Aug 11, 2006
  17. Aug 11, 2006 #16
    You wrote:

    Just because you have an "imaginary action" does not mean you are dealing with "tunneling phenomena".

    In the case of an imaginary action one has an instanton, this is pretty standard. In this case time is imaginary t ---> it or in the more familiar parlance k^2 < 0.

    As for a calculation of a transmission rate look below equation 3.

    As for intial and final states, these are in the entropies. In particular the conditional entropy gives the entanglement of states between Bob and Alice which determines how much quantum information may be teleported.
    between them.

    As for the gravitational potential, the action is given by a pdr which in turn is expressed according to the Hamiltonian for the system. There is no need for an ad hoc square potential substitution.

    Lawrence B. Crowell
     
  18. Aug 11, 2006 #17

    ZapperZ

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    Could you cite me where such a thing has been used as a valid description for a "tunneling" phenomena?

    Zz.
     
  19. Aug 11, 2006 #18
    Hello Sweetser

    As for gravity attracting, this is due to the group structure of the theory. Electromagnetism is a U(1) theory, which is just the circle on the complex plane. The two roots of the theory are +1 and -1 on the circle which are the two charges.

    Gravity is SO(3,1) ~ SL(2, C)xZ, where SL(2, C) is SU(2)x*SU(2). SU(2) is the standard rotation group with the sigma_z matrix containing the eigenvalues +-1. The *SU(2) is the same but where sigma_z ---> i*sigma_z, and so the group is not an elliptic group of rotations, but rather hyperbolic transformations corresponding to the Lorentz boosts. The roots of the *SU(2) are i, -i, which correspond to positive and negative mass. The difference with SU(2) is that there is no continuous rotation between i and -i. In other words in a universe containing positive mass-energy that "sticks," and there are no processes which can generate negative mass-energy. There is a bit I can go on with the Hawking-Penrose energy conditions, but I will leave that for later.

    On this basis gravity is purely attractive and anything with mass-energy will be associated with an attractive field.

    I am aware of your theory of gravity. You want to include symmetric terms. However, this proposal runs afoul with some basic issues of differential geometry or even the theory of vector spaces. I other words it runs afoul with basic "div-grad-curl" mathematics. We have been over this before, so I don't feel like bringing this up in detail again. The curvatures in general relativity have antismmetric structure for the same basic reason that B = -curl A. The differences are really formalistic, but at the core it involves the structure of spaces in differential geometry.

    It is my sense that as a classical theory GR is the theory. This is similar to classical EM, which is the working picture for EM.

    Lawrence B. Crowell
     
  20. Aug 11, 2006 #19
    Look in "Instantons on Four Manifolds" by Freed & Uhlenbech. Hawking uses this trick in half the papers he publishes. This is pretty standard fair these days.

    Lawrence B. Crowell
     
  21. Aug 11, 2006 #20
    Hello Lawrence:

    Where does this bit of group theory come from?
    Discussions of the group theory structure of GR did not make it into MTW as far as I can tell. I did find a quote from Prof. John Baez that the symmetry of GR was the group of Diffeomorphisms on a manifold, Diff(M), the group of all continuous transformations of the manifold. I had a sense that Diff(M) is far larger than SO(3,1), however you choose to represent it. Here is a thread discussing the issue that sounds intellegent to me:

    http://groups.google.com/group/sci....lativity+group+theory&rnum=8#87e436bb685a91a9

    and here is a relevant quote:
    I think I see how the Diff(M) symmetry can work in my theory. I don't see how comments from John Baez I have read but not cited, and this quote, work with your claim (which I bet has a bunch of clarifying conditions that were omitted).

    doug
     
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