Information Preservation in Quantum Gravity

[Lawrence B. Crowell]

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 $\Delta S~=~0$, 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 $\Delta S_{bh}~\rightarrow~0$ 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 $M~\rightarrow~M~+~\delta m$, this involves the imaginary part of the action
$$ImS~=~Im\int_{r_i}^{r_f}p_rdr.\eqno(1)$$
The Hamilton equation ${\dot r}~=~{{\partial H}\over{\partial p}}$ permits this to be written as
$$ImS~=~Im\int_{r_i}^{r_f}\int_{p_i}^{p_i}{{dr}\over {\dot r}}dH.\eqno(2)$$
Along null geodesics the velocity ${\dot r}~=~\pm 1~+~\sqrt{2M/r}$ the action for the classically forbidden path is
$$Im S~=~-2\pi\int_0^{\delta m}{{dr~dm}\over{1~+~\sqrt{{2M~-~\delta m}\over r}}}~=~-2\pi\big((M~-~\delta m)^2~-~M^2\big),\eqno(3)$$
which defines $-{1\over 2}(S_f~-~S_i)$. The imaginary part of the action gives the tunnelling probability or emission rate as $\Gamma~=~exp(-2ImS)~=~exp(\Delta S)$. 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. $dM~=~dQ$ as the first law of black hole thermodynamics with $dS~=~{{dQ}\over T}$ 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 $r~>~2M$. 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 $TdS~=~dM$ 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 $dS~>~{{dM}\over T}$, 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 $dS~>~{{dM}\over T}$.
//
The standard “ Alice and Bob” problem is considered. Alice has the set of observables $A$, which she communicates to Bob in a string $x_1x_2\dots x_n$. Bob similarly has the string $y_1y_2\dots y_n$. The von Neumann information each possesses is then $S(X)~=~-Tr(\rho_X~log_2\rho_X)$, for $X$ either $A$ or $B$. The entropy of each is then a compartmentalization on $\rho_{AB}$ for the total quantum information both possesses and $\rho_{A}~=~Tr_B(\rho_{AB})$ and $\rho_B~=~Tr_A(\rho_{AB})$. 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 $\rho_{AB}$ then defines the joint entropy
$$S(AB)~=~-Tr(\rho_{AB}~log_2\rho_{AB}).\eqno(4)$$
If Alice transmits her string $x_1x_2\dots x_n$ to Bob this defines the conditional information or entropy $S(A|B)$ as the information communicated by Alice given that Bob has $y_1\dots y_n$ defined as
$$S(A|B)~=~S(AB)~-~S(B).\eqno(5)$$
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
$$S(A|B)~=~-Tr(\rho_B\rho(A|B)~log_2\rho(A|B)~=~-Tr(\rho_{AB}log_2\rho_{A|B}),\eqno(6)$$
where $\rho_{A|B}~=~lim_{n\rightarrow\infty}\big({\rho_{AB}}^{1/n}({\bf 1}_A\otimes\rho_B)^{-1/n}\big)^n$. Here ${\bf 1}_A$ is a unit matrix over the Hilbert space for Alice’ s quantum information. This means that the entries of $\rho_{A|B}$ 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 $\rho_{AB}$.
//
When the conditional entropy is negative Alice can only communicate information about the complete state by classical communication. The sharing of $-S(A|B)$ 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.

 

[Lawrence B. Crowell]

Information Preservation in Quantum Gravity II

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

 

[sweetser]

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:
$$H=\pi^{\mu}\frac{\partial A_{\mu}}{\partial t}-\mathcal{L}$$
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

 

[Don J]

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:
$$H=\pi^{\mu}\frac{\partial A_{\mu}}{\partial t}-\mathcal{L}$$
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

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

 

[sweetser]

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

 

[Don J]

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.

doug

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.

 

[Lawrence B. Crowell]

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

 

[sweetser]

GR versus EM

Hello Lawrence:

In EM, the Hamiltonian is simple: it is the $T^{00}$ part of the stress tensor:
$$H=\frac{1}{8 \pi}(E^2+B^2)$$
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

 

[ZapperZ]

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 $M~\rightarrow~M~+~\delta m$, this involves the imaginary part of the action
$$ImS~=~Im\int_{r_i}^{r_f}p_rdr.\eqno(1)$$
The Hamilton equation ${\dot r}~=~{{\partial H}\over{\partial p}}$ permits this to be written as
$$ImS~=~Im\int_{r_i}^{r_f}\int_{p_i}^{p_i}{{dr}\over {\dot r}}dH.\eqno(2)$$
Along null geodesics the velocity ${\dot r}~=~\pm 1~+~\sqrt{2M/r}$ the action for the classically forbidden path is
$$Im S~=~-2\pi\int_0^{\delta m}{{dr~dm}\over{1~+~\sqrt{{2M~-~\delta m}\over r}}}~=~-2\pi\big((M~-~\delta m)^2~-~M^2\big),\eqno(3)$$
which defines $-{1\over 2}(S_f~-~S_i)$. The imaginary part of the action gives the tunnelling probability or emission rate as $\Gamma~=~exp(-2ImS)~=~exp(\Delta S)$. 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. $dM~=~dQ$ as the first law of black hole thermodynamics with $dS~=~{{dQ}\over T}$ 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.

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.

 

[Lawrence B. Crowell]

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

 

[ZapperZ]

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

Zz.

 

[Lawrence B. Crowell]

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

 

[Lawrence B. Crowell]

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

 

[ZapperZ]

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

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.

 

[sweetser]

GR could be wrong

Hello Lawrence:

I think that abandoning GR and trying to do everything as electromagnetism is not going to go well with a lot of people.

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.

Beyond that to PPN and beyond GR demonstrates departures from EM theory.

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.

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.

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.

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.

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

 

[Lawrence B. Crowell]

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.

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

 

[ZapperZ]

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.

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

Zz.

 

[Lawrence B. Crowell]

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

 

[Lawrence B. Crowell]

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

Zz.

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

 

[sweetser]

Hello Lawrence:

Where does this bit of group theory come from?

Hello Sweetser
Gravity is SO(3,1) ~ SL(2, C)xZ, where SL(2, C) is SU(2)x*SU(2).

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:

Having talked a lot about what the symmetry structure of general
relativity is *not*, I'd like to conclude my participation in this
thread with a description of what this symmetry structure *is* in my
opinion.

The set X of models of the theory of general relativity consists of all
pairs (M,g), where M is a 4-dimensional manifold and g is a Lorentzian
metric on M. The symmetry structure of the theory cannot adequately be
described by a group. It has to be described by a groupoid.

This groupoid is the groupoid G of all diffeomorphisms between
4-manifolds. G acts from the right on the set X of models, by pullback:
If f:M-->N is an element of G and x=(N,h) is an element of X, then
their "product" x.f is the pair (M,g), where g is the pullback of the
metric h by the diffeomorphism f.

Maybe one should include in the discussion of the symmetry structure of
general relativity also the fact that there is a "submodel" relation on
the set X of all models: (M0,g0) is a submodel of (M1,g1) if and only if
M0 is an open subset of M1 and the metric g0 is the restriction of g1 to
M0. One could therefore argue that each smooth imbedding of one
4-manifold into another should count as a "partial symmetry" of general
relativity (we can pull back metrics via imbeddings).

I think that's more or less all one can say about the symmetry
structure of general relativity.

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

 

[ZapperZ]

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

Can you please give me the exact citations, please?

Zz.

 

[Lawrence B. Crowell]

This says something different, but related. The group here is SO(3, 1) or SL(2, C). What this outlines below is how that group acts as the local action for diffeomorphisms between (M,g).

Why SO(3, 1)? This is because the symmetry must involve four dimensions, which is SO(4) for orthogonal rotations. The signature change changes this to SO(3, 1). The relationship S(3, 1) ~ SL(2,C)xZ_2 is a four space quaternionic version of the double cover relationship between SO(3) and SU(2).

Lawrence B. Crowell

 

[Lawrence B. Crowell]

The Freed & Uhlenbeck book is Springer Verlang 1984. As for Hawking papers, well frankly I think one can do a bit of research

Lawrence B. Crowell

 

[ZapperZ]

The Freed & Uhlenbeck book is Springer Verlang 1984. As for Hawking papers, well frankly I think one can do a bit of research

Lawrence B. Crowell

Yes, but I'd rather have specific citations. You'll understand that I do not have the time to hunt for all Hawking's papers. I'm just surprised you don't have a few handy on the tip of your fingers, considering that you are using it in your formulation here. Don't you have any intention of citing a reference when you actually "publish" this?

Zz.

 

[Lawrence B. Crowell]

Hawking and Hartle used the Euclideanized path integral to compute the quantum transition for cosmology in their seminal paper

S. Hawking & J. Hartle, Wave Function of the Universe, Phys. Rev. D28 (1983)

the term imaginary time was first used by Hawking (as I recall) in

S. Hawking, Quantum Coherence and Closed Timelike Curves, Phys. Rev. D52 (1995)

I presume you are able to further such research. Imaginary time, instantons and spacetime tunnelling has become pretty much standard fair in the past decade or more.

Lawrence B. Crowell

 

[ZapperZ]

Hawking and Hartle used the Euclideanized path integral to compute the quantum transition for cosmology in their seminal paper

S. Hawking & J. Hartle, Wave Function of the Universe, Phys. Rev. D28 (1983)

the term imaginary time was first used by Hawking (as I recall) in

S. Hawking, Quantum Coherence and Closed Timelike Curves, Phys. Rev. D52 (1995)

Thanks. I'll check up on those.

I presume you are able to further such research. Imaginary time, instantons and spacetime tunnelling has become pretty much standard fair in the past decade or more.

Well, I'm an experimentalist. My definition of what is "pretty much standard fair" is obviously quite different than yours, since I require something to be experimentally verified before it becomes a "standard fair".

Zz.

 

[verdigris]

LAWRENCE B. CROWELL said:

"The conservation of information from black hole radiance has an unsettled history. The initial indications were that information is lost"

If a large number of pions decay into photon pairs which travel in
opposite directions, and one member of each pair travels to a fixed
point in space,a black hole would form at the fixed point in space.Since the photon polarizations are coupled, I could get information about the
microstates in the black hole by measuring the polarization angles of
the photons that are outside the black hole.
And by placing a number of polarizing filters in a line, for each
photon traveling outside the black hole, with one photomultiplier per
photon to detect each photon, I could gain information on the
microstates in the black hole at different periods in time.So I would
know more about a black hole than just its total spin,mass and
charge.
It seems to me that the photon polarizations would remain coupled even across the event horizon of a black hole because if they do not remain coupled then we would be saying that quantum mechanics breaks down for a black hole and therefore that Stephen Hawking original calculation of the temperature of a black hole is
faulty.This does not seem likely! I think the only way to resolve the loss of information problem is to assume that as radiation is emitted from a black hole it is coupled to something still in the hole:so
if a gamma ray is emitted with a certain polarization then there is
a corresponding gamma ray with a coupled polarization that still exists in the black hole.

 

[Lawrence B. Crowell]

An EPR pair that enters into a black hole will become entangled with the quantum states of the black hole. This means that the entanglement in the original EPR pair will become lost. The only way it can be preserved is if Alice and Bob correlate their EPR pair with some auxilliary state.

Ultimately the information is, or should be, preserved. However, this information is far less accessible once the EPR pair is absorbed into the black hole. The Bogoliubov transformations for

A_k = a_k cosh(x) + b^*_{-k}sinh(x)

B_k = b_k cosh(x) + a^*_{-k}sinh(x)

for * = dagger and x the rapidity will obey

[A_k, A^*_k] = [a_k, a^*_k]cosh^2(x) + [b^*_{-k}, b_{-k}] sinh(x)

= cosh^2(x) - sinh^2(x) = 1,

and so the fine grained quantum scale of action is not changed. So ultimately the quantum information is preserved. It just might be highly unavailable. In other words Bob might have to wait around 10^60 years to retrieve the information Alice sent in the EPR pair.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

This is an elementary presentation on the relationship between quantum mechanics and gravitation. We start with the examination of the overlap between a state $|\psi(t)\rangle$ and $|\psi(t)~+~\delta\psi(t)\rangle$. This leads to the expansion

$$\langle\psi|\psi~+~\delta\psi\rangle~=~\langle\psi|\psi\rangle~+~\langle\psi|\frac{\partial\psi}{\partial t}\rangle \delta t~+~\langle\psi|\frac{\partial^2\psi}{\partial t^2}\rangle\delta t^2$$

With the use of the Schrodinger equation $i|\partial\psi/\partial t\rangle~=~H|\psi\rangle$ a modulus square of this expansion is then

$$|\langle\psi|\psi~+~\delta\psi\rangle|^2~=~|\langle\psi|\psi\rangle|^2~-~(\langle H^2\rangle~-~\langle H\rangle^2)\delta t^2.$$

Physically the term $\sqrt{\langle H^2\rangle~-~\langle H\rangle^2}~=~\Delta E$, which is what defines the Heisenberg uncertainty principle. This also defines a phase

$$\phi~=~\int dt\sqrt{\langle H^2\rangle~-~\langle H\rangle^2}~=~\int dt\Delta E$$

which is the geometric or Berry phase. For certain systems the above overlap of states can be a measure of the entanglement of states. This is also the Fubini-Study metric for the projective space $CP^n~\subset~C^{n+1}$. The complex space is projective space $C^{n+1}$ is the $2n~+~2$ dimensional state space for a finite dimensional quantum system, where for $n~=~0$ this defines the Bloch sphere for a spin system. The complex vector space defines a unitary group $U(n~+~1)$, and is an indication of the unitarity of quantum mechanics.

What is of interest is that an elementary example of quantum fields in curved spacetime can be defined. Let the energy eigenvalues of the state space be $E_i~=~\hbar\omega$ which are functions of a one dimensional parameter r, which is a function of time $r~=~r(t)$. Without worry we will let the spectrum become a continuum, and frequencies a continuous function of this parameter. Let this dependency be a Doppler shift, so the frequency spectrum is $\omega'~=~(1~-~nv/c)\omega$, where n is an index of refraction, $v~=~dr/dt$ a velocity and c the speed of light. The index of refraction along the one dimensional space is then assumed to vary according to $n~=~n_0~+~\delta n$. The Doppler equation defines a retarded time $1~-~nv/c~=~\omega/\omega'~=~\nu\tau$ for $\tau~=~t~-~r/v$. The effective frequency $\nu'$ is then

$$\nu'~=~\frac{v}{c}\frac{\partial n}{\partial\tau}~=~\frac{v^2}{c}\frac{\partial\delta n}{\partial r}$$

The frequency $\nu'$ is then related to a non-Doppler shifted frequency $\nu$ by $\nu'~=~(1~-~nv/c)\nu$ for

$$\nu~=~-v\frac{1}{\delta n}\frac{\partial\delta n}{\partial r}~=~-v\frac{\partial ln(\delta n)}{\partial r}$$

We may then write the Berry phase above from the Fubini-Study metric according to a $\Delta E~=~(\omega' d\tau~-~\omega dt)/\hbar$ which according to the frequencies$\nu,~\nu'$ defines the Berry phase

$$\phi~=~\int_{r_i}^{r_f} dr(1~+~\frac{v}{c}\nu)~=~|r_f~-~r_i|~-~\frac{v}{c}ln(n),$$

With the appropriate identification of $vn/c~\rightarrow~GM/rc^2$ the above result reproduces the phase term for a black hole. Further, for the phase to become imaginary the condition is when

$$1~=~\frac{\partial ln(n)}{\partial r}$$

and for the imaginary time identification $t~=~\hbar/kT$ we find the condition for the onset of an imaginary phase angle as

$$kT~=~\frac{\hbar\nu}{2\pi}$$

which with the identification between the velocity in a medium with index of refraction recovers the temperature for Hawking radiation

$$T~=~\frac{\hbar }{2\pi GM kc^2}$$

This simple approach to a derivation of Hawking radiation is based on the notion of a simple bundle fibration over the complex space of quantum states. Here the bundle is just the real line $R$, which parameterizes the index of refraction along the real line. It is then clear that this program can be extended to more realistic groups which include gravitation.

 

[Lawrence B. Crowell]

I have decided to resurrect my little site here I started a year or two ago. I worked up an interesting idea on quantum fields in curved spacetime. This is very simple, only relying upon some basic ideas of geometry in QM and a fibration.

https://www.physicsforums.com/showthread.php?t=115826&page=2

I worked this up in my head as I wrote this, so there might be a boo-boo or two here, but I think the basic idea looks reasonable.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

This is a second part to the post on the Fubini-Study metric and the Hawking effect. The Fubini-Study metric defines the set of projective rays in the complex space $C^{n+1}$, or $CP^n~\subset~C^{n+1}$. The Fubini-Study metric is then for n = 1 the Bloch sphere and further defines quantum entanglements and Berry phases.

A Hermitian function or differential form in $C^{n+1}$ defines a unitary subgroup $U(n+1)~\subset~ GL(n+1,C)$. The Fubini-Study metric is invariant under scaling under group actions of such a $U(n+1)$. Hence the space, a Kahler manifold, is homogeneous. Hence any two Fubini-Study metrics are isometric under a projective automorphism of $CP^n$.

The definition of a projective space is the set of elements $z_i~=~Z_i/Z_0$, where the magnitude of $Z_0$ is artitrary, or more formally

$$CP^n~=~\big{\{z_i,~i=1\dots n}:~z_i~=~Z_i/(Z_0~\ne~0)\big}$$

A point or in $CP^n$ is a line or ray in $C^{n+1}$. An arbitrary vector in the projective space may then be represented in the "bra-ket" notation of quantum mechanics as

$$|\psi\rangle~=~\sum_{i=1}^\infty z_i|e_i\rangle~=~[z_1:~z_2:~\dots~z_n]$$

where the square braket notation is commonly used for projective spaces. The basis vectors $|e_i\rangle$ are the basis vectors for the Hilbert space ${\cal H}~=~C^{n+1}$. There there are two such vectors $|\psi\rangle,~|\phi\rangle$, then the quanatum mechanical overlap defines the distance between the two points in $CP^n$, which in the Hilbert space is a plane. The modulus square of the quantum overlap appropriately normalized

$$\frac{\langle\phi|\psi\rangle\langle\psi|\phi\rangle}{\langle\psi|\psi\rangle\langle\phi|\phi\rangle}~=~cos(\theta(\psi,~\phi))$$

defines the distance between the two as $\theta(\psi,~\phi)$. For n = 1 this is the angle on the Bloch sphere, and defines a general quantum angle.

The overlap between a vector $|\psi\rangle$and its infinitesimal displacement $|\psi~+~\delta\psi\rangle$defines the Fubini-Study metric on $C^n$ with the proper normalization

$$ds^2~=~\frac{\langle\delta\psi|\delta\psi\rangle}{\langle\psi|\psi\rangle}~=~\frac{\langle\delta\psi|\psi\rangle\langle\psi|\delta\psi\rangle}{|\langle\psi|\psi\rangle|^2}$$

With the coordinate notation for projective spaces the Fubini Sudy metric is

$$ds^2~=~\frac{1}{2}\frac{[z_i,~dz_j][{\bar z}^i,~d{\bar z}^j]}{(z_k{\bar z}^k)}$$

where $[z_i,~w_j]$ is a commutator. We might think of the basis elements of the Hilbert space as a Fock basis and the components $z_i$ as due to the application of raising and lowering operators or a field amplitude of the form $A_i~=~a^\dagger_ie^{i\theta}~+~ia_ie^{-i\theta}$. The projective space is a form of algebraic variety, and these commutators are defined as Grassmannian varieties. The form of this metric indicates that the space is a Kahler manifold where the Ricci curvature is given by a potential $\Phi$

$$R_{ij}~=~\frac{\partial^2\Phi}{\partial z_i\partial{\bar z}_j}$$

which is proportional to the metric.

For any two dimensional subspace of $CP^1~\subset~CP^n$, the space is the block sphere of two real dimensional where the Fubini-Study metric reduces to

$ds^2~=~\frac{dzd{\bar z}}{(1~+~z^2)(1~+~{\bar z}^2)}~=~d\theta^2~+~sin^2\theta d\phi^2$

with the state $|\psi\rangle~=~cos(\theta)|0\rangle~+~e^{i\phi}\sin(\theta)|1\rangle$, which is the single qubit state. This may then be generalized to higher dimensional kahler manifolds for hexacode and higher quantum codes and algebras.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

The last equation I quoted in this lastest thread should read

$$T~=~\frac{\hbar c^3}{8\pi GM k}.$$

That is what you get when you do math on an editor instead of on paper.

L. Crowell

 

[Lawrence B. Crowell]

A slice through spacetime will contain some set of black holes and likely have some cosmological event horizon. Each of these event horizons will then satisfy a quadratic polynomial for the coordinates specified in that region. Since the event horizon is an invariant it does not matter what coordinates are used, so long as the spacetime has some consistent set specified by some coordinate condition or the spacetime analogue of a gauge. Hence there now exists a subspace of the universe ${\cal P}$, a set of all projective rays that join at ${\cal I}^\pm$. Event horizons are those which satisfy a certain polynomial condition $P(x_i)~=~0$ which identifies them as event horizons. This is then a projective variety defined by a set of homogenous polynomials $\cal S$, such that some points of $\cal P$ satisfy

$$Z({\cal S})~=~\{x~\in~{\cal P}|P(x)~=~0,~\forall~P~\in~{\cal S}\}$$

This subset of the projective space is a projective algebraic set,where the irreducible set defines a projective algebraic variety. This set has the Zariski topology by declaring all algebraic sets to be closed.

A Zariski topology is a nonHausdoff topology for algebraic varieties, which describes their algebraic content and with weak geometric content. Zariski topology does occur in general relativity. Gravitation has the group structure $SO(3,~1)~=~Z_2\times SL(2,~C)$. The group $SL(2,~C)$ may in turn be written as

$$SL(2,~C)~=~SU(1,~1)\times SU(2).$$

The algebra for this is then $su(2)$ given by the standard Pauli matrices $\sigma_{\pm},~\sigma_3$ and $su(1,~1)$ has the elements $\sigma_{\pm},~\tau_3~=~i\sigma_3$. The latter change gives the pseudoEuclidean nature to spacetime.

Now consider a connection one-form

$$A~=~A^+\sigma_+~+~A^3\sigma_3,$$

and a gauge transformation determined by the group action of $g~\in~{\cal G}$, $g~=~e^{i\lambda\sigma_3}$. The gauge transformed connection is then

$$A^\prime~=~g^{-1}Ag~+~g^{-1}dg~=~e^{-2\lambda}A^+\sigma_+~+~A^3\sigma_3$$

where $d\lambda~=~A^3$. Thus $\lambda$ is a parameterization of the gauge orbit for this connection $A^'~=~A(\lambda)$. This leads to the observation

$$\lim_{\lambda\rightarrow\infty}A(\lambda)~\rightarrow~A^3\sigma_3,$$

where $A^+\sigma_+~+~A^3\sigma_3$ and $A^3\sigma_3$ have distinct holonomy groups and thus represent distinct points in the moduli space $\cal M$. However by the last rquation we must have

$$F_\mu(A^+\sigma_+~+~A^3\sigma_3)~=~F_\mu(A^3\sigma_3),$$

and similarly for any gauge invariant function. Hence there exist two distinct point in the moduli space that define the same set of gauge invariant functions. Hence there does not exist a measure over these two points that separates them, and $\cal M$ is then nonHausdorff and has Zariski topology.

This moduli space, a space of gauge equivalent connections, indicates the underlying topology for curvatures. In a b-completeness sense the group action could be $g~=~e^{i\lambda\sigma_3~+~\alpha}$, for $\alpha$ a constant. The gauge connections for $\lambda~\rightarrow~\infty$ converge to $e^{-2\alpha}A^+\sigma_+~+~A^3\sigma_3$, which for a given $\alpha$ can describe a set or congruence of null geodesics. A Cauchy-like sequence of $\lambda_n$ can describe a set of geodesic which converge to the null set. In this manner the underlying topological structure of gravitation has this nonHausdorff topology.

This structure then indicates that quantum gravity if more fundamentally described by algebraic varieties, with minimal geometric content. An abstract algebraic variety is a generalization of a scheme, which geometrically is a correspondence with some class of rings. In this case the polynomials define some ring. A scheme is a space with a local ring structure such that every point has a neighborhood, a locally ringed space, isomorphic to a ring spectrum. An algebraic variety is a scheme whose structure sheaf is a sheaf of K-algebras with the property that the rings $R$ that occur above domains which are all finitely generated K-algebras, i.e., quotients of polynomial algebras by prime ideals.

The system works over any field $F$, and the topology permits one to piece together varieties on open sets and their definition permits them to be on a projective space. This leads to the topic of sheaf cohomology in algebraic geometry. Another mathematical issue of particular interest here is that a special case of an algebraic variety is an algebraic curve. This leads to the Goppa code. For $X$ non-singular, or there exist points $x_1,~x_2,~\dots,~x_n$ fixed among the points of $X$ defined over $F$, and that $G$ is a divisor on $X$, also defined over $F$. There is a finite-dimensional subspace $L(G)$ of the function field of $X$, consisting of the rational functions $f$ on $X$ with zeroes and poles subject to $G$. This means that $G$, which is a formal sum of points of $X$ over the algebraic closure of $F$, bounds the divisor made up of the zeroes and poles of $f$, counted with appropriate multiplicity.

An example of an algebraic variety is an algebraic curve of unity dimension. Simple examples are ellipses, conic sections and the more enigmatic elliptical curves. A curve has at most a finite number of singular points, where if that is zero the curve is nonsingular. This requires that the curve exists in a projective space and be over an algebraically closed field. The theory of nonsingular curves over complex numbers is equivalent to the theory of Riemannian surfaces, where the genus of the curve is that of the two-manifold. This genus is then a result of the Riemann-Roch theorem.

This defines structures which are codes, which preserve the number of quantum bits. To preserve this information it must process this information according to an error correction code, similar to those used in computer data transmission.

What has been developed here is commesurate with a Goppa code, which is a linear code constructed from algebraic curves over a finite field $F$. Let the curve $C$ be nonsingular with a set of fixed points $p_i$, $i~=~1.~\dots~,n$. There then exists a divisor $F$ on the curve $C$ over the field $F$. Given rational functions $f$ over the curve with poles and zeros on $G$, this defines a space $L(G)$ of finite dimension defined as the function field of $C$. Thus the divisor $G$ is a formal sum of point on $C$ that bounds a divisor made from the poles and zeros of $f$, and is algebraically closed in $F$. Thus $L(G)$ has some basis of functions $f_1,~\dots,~f_k$ over the field, where the Goppa codes is found by the evaluation of the fixed points $p_i$, eg. $f_i(p_1),~f_i(p_2),~\dots,~f_i(p_n)$, for $i~=~1,~\dots,~k$. This defines a vector space $f^n$ spanned over $F$. This is then given by the map

$$\mu:L(G)~\rightarrow~F^n,$$

by $f~\mapsto~f(p_1),~\dots,~f(p_n)$. For a divisor defined as $D~=~\sum_{i=0}^np_i$ the Goppa codes is then represented as $c(D,~G)~=~L(G)/ker(\mu)$.

For a function $f~\in~ker(\mu)$, for $f(p_i)~=~0$ the function as a divisor is less than $D$ and so $ker(\mu)~=~L(G~-~D)$. For $f~\in~L(G~-~D)$ since $p_i~<~G$ then $div(f)~>~D$ it is clear that $f(p_i)~=~0$. A distance between code words is the Hamming weight of $\mu(f)$ called $d$. Thus $f(p_i)~=~0$ for $n~-~d_i$ $p_i$s. The points are designated $p_{i_1},~\dots,~p_{i_{n-d}}$. Thus

$$f~\in~L(G~-~p_{i_1}~-~\dots~-~p_{i_{n-d}}),$$

and that $div(f)~<~ G~-~p_{i_1}~-~\dots~-~p_{i_{n-d}}$. By taking the degree of these on both sides the degree of $G$ then satisfies $d~\ge~n~-~deg(G)$ The dimensions of the Goppa code is then $k~=~L(G)~-~L(G~-~D)$. This result is the Riemann-Roch theorem.

The Riemann-Roch theorem indicates how many moduli exist. A moduli space is a space of solutions that defines gauge equivalent gauge vectors. The moduli space is the set of all self-dual connections (instantons) with degrees of freedom removed by gauge conditions. This space is $\omega_{SD}/{\cal G}~=~{\cal M}_{mod}$. The number of metric moduli and conformal Killing vectors of the moduli space is obtained with the Riemann-Roch theorem. Given an operator ${\cal O}$ that is covariantly constant on a geodesic,

$$\partial g_{ij}~=~ (2\delta\omega~-~ \nabla_k\delta \sigma))g_{ij}~-~ 2({\cal O}\delta \sigma)_{ij},$$

where $\omega$ is the conformal factor, $\sigma$ is a spatial parameter. Here the metric is defined on the complexified coordinates $z,~{\bar z}$. ${\cal O}$ may be derived by a variation of the metric $g_{ij}$

$$\delta g_{ij}~=~(2\delta\omega~-~\nabla_i\delta\sigma_j)g_{ij}~-~\nabla_j\delta\sigma_i$$
$$=~(2\delta\omega~-~\nabla_k\delta\sigma_k)g_{ij}~-~({\cal O}\delta\sigma)_{ij},$$

with $({\cal O}\delta\sigma)_{ij}~=~\nabla_i\delta\sigma_j~+~ \nabla_j\delta\sigma_i~-~\nabla_k\delta\sigma_k)g_{ij}$. ${\cal O}$ defines the conformal Killing equation $({\cal O}\delta \sigma)_{ij}u^j~=~0$. Thus ${\cal O}$ defines the number of conformal Killing vectors. The transpose of ${\cal O}$, ${1\over 2}({\cal O}^tu)_i~=~-\nabla^ju_{ab}$ defines the metric moduli. The Riemann-Roch theorem states that the difference between the numbers of the metric moduli and the number of conformal Killing vectors is $-3\chi$, where $\chi$ is the Euler characteristic for the manifold. This difference is then

$${1\over 2}\big(dim~ ker({\cal O})~-~dim~ ker({\cal O}^t)\big)~=~-3\chi~=~6g(g~-~1)$$
${\cal O}$ determines the number of instanton in the theory. The size of the Goppa code is then a topological invariant given by an Euler characteristic of Floer cohomology.

This metric applies to a spatial surface with $\delta g_{ij}~=~(\partial g_{ij}/\partial t)\delta t$, for $t$ a parameter labelled as "time." The Ricci flow equation

$${{dg_{ij}}\over{dt}}~=~-2(R_{ik}~+~\nabla_i\nabla_k\omega)g_{kj},$$

is an eigenvalued equation for the Ricci flow for $k~=~i$. The time derivative of the functional $F[g,~\omega]$ is the low energy effective action in string theory and $\omega$ is a dilaton field, similar to that over a string world sheet. The metric may then be rewritten according to a conformal gauge $g_{ij}^\prime~=~exp(2\omega)g_{ij}$ and the Ricci curvature changed by
${g^{\prime}}^{1/2}R^\prime~=~g^{1/2}(R~-~2\nabla^2\omega)$. By an appropriate diffeomorphism, or gauge choice, the governing equation for the metric can recover the Hamilton equation for
Ricci flow.

 

[Lawrence B. Crowell]

The physics of the projective Hilbert space and its bearing upon spacetime physics is illuminated here by the role this plays with quantum phase transitions. The quantum critical point and the change of phase associated with it exhibits scale invariance, such as with high temperature superconductivity, and theoretically this physics is poorly understood. Current theory associates the quantum critical point with the "breakdown" of the quasi-electron and a divergence in its mass. In this critical state however the system maintains a self-similar or scale invariance.

A quantum critical state is one where fluctuations drive a phase transition at or near absolute zero temperature. To treat this sort of physics the statistical organizing principles of bosonic and fermionic states must be considered. Bosonic quantum states are by far the easiest to consider, and the correspondence between the quantum physics and the classical physics, and the emergence of different phases, are far closer and understood. A similar correspondence between fermionic quantum states and any classical physics does not exist, and the emergence of new states of matter are far less well understood. Yet in recent times the electronic states of certain transition-actinide metal crystaline systems have been measured to exist in a new scale-invariant phases, which occur at the "breakdown" of a Fermi or Landau electron liquid state and where the mass of the associated quasi-particle state diverges. This appears to be a new domain of quantum physics.

The Fubini-Study metric determines the uncertainty principle by the Berry phase as

$$\phi~=~\int dt\sqrt{\langle H^2\rangle~-~\langle H\rangle^2}~=~\int dt\Delta E$$

This geometric phase has the content of $\Delta Et/\hbar$, and for this phase bounded by [itex]\sim~2\pi[/tex] the time is then a measure of the fluctuation strength. We may of course encounter a situation, as I demonstrated in the start of this presentation, a situation where this phase may become imaginary valued. This obtains when $\langle H^2\rangle~-~\langle H\rangle^2~<~0$. An imaginary phase may be seen as a case where the time becomes an imaginary time with $t~=~\hbar/kT$. For a high temperature system this imaginary time is very small, but as temperature $T~\rightarrow~0$ there is then more of this imaginary time available for the system to exist in an instanton or tunnelling states where this quantum behavior is detectable. We may then consider the imaginary time as analgous to the classical temperature, where for low real temperature the imaginary time assumes is analogous to a classical temperature that heats up material in the space plus imaginary time "spacetime." Thus time in general is a complex valued vector ${\vec t}~\in~C^1$, where ordinary time lies on the real axis of the Argand plane and temperature on the imaginary axis. For a low enough of a standard temperature the imaginary time as an internal temperature will act to heat up the quantum states of the system and they will then exhibit a phase transition or "melting," which results in a fluid behavior that is scale invariant.

We then consider the case similar to the one dimensional space considered last week, indeed on 3/27/2008. Instead of a simple one dimensional space plus time model we consider the boost space of $SU(1,~1)$ with the connection coefficient

$$A~=~A^1\sigma_1~+~A^2\sigma_2~+~iA^3\sigma_3,$$

The field strength tensor is $F_{ab}~=~\partial_{[b}A_{a]}~+~[A_b,~A_a]$, where if we consider a flat connection the comonents of the curvature tensor are

$$F_{21}~=~[A_1,~A_2],~F_{32}~=~[A_2,~A_3],~F_{13}~=~[A_3,~A_1]$$

where the physical fields are found with the application of $\epsilon^{abc}$ and so the time-like field is $F^3~=~\epsilon F_{12}$. In the case of a moving field in more than one dimension the velocity determines an action of the form

$$S~=~\int_{r_1}^{r_2}p_rdr$$

The Hamilton equation ${\dot r}~=~\partial H/\partial p$ permits this to be written as

$$S~=~\int_{r_1}^{r_2}\int_{p_1}^P_2}\frac{dr}{\dot r}dH$$

which permits this action to be expressed according to $S~=~\int tdH~\simeq~t\Delta E$, for $\Delta E$ the Fubini-Study metric distance. We then split the action into a real and imaginary part according to a complex velocity field $u^a~=~(U^\mu,~iu^5)$, where the fifth entry corresponds to the imaginary portion of time. The field theory is then extended to five dimensions with indicial entries $a~\rightarrow~(\mu,~5)$. For the sake of Lorentz invariance, where in general this would impose a preferred frame from which to observe the field, the complex velocity is restricted to the complex time in $C^1$. The field is extended similarly to a fifth dimension as well. The Hamiltonian for the above action then determines the action

$${\cal L}~=~-\frac{1}{4}F_{ab}F^{ab}~-~\frac{\mu^2}{2\hbar^2}u^au^bg^{cd}F_{ac}F_{bd},$$

where $\mu$ has units of energy (mass). What has been done is to generalize the non-compact group into a composition of the two according to

$$u:A_3~|\rightarrow~(A_3^\prime,~iA_3^{\prime\prime}),~A_3^\prime~\in~SU(1,1),~A_3^{\prime\prime}~\in~SU(2)$$

or so that the complex elements are generalizations of the boosts in $SU(1,1)\oplus SU(2)$. This means that the time-like potentials and fields have a probability of being Lorentzian or Euclidean.

The above Lagrangian gives the dynamical equations

$$\partial_aF^{ab}~=~\frac{\mu^2}{\hbar^2}(u_cu^b\partial_aF^{ca}~-~u^cu^a\partial_aF^{cb})$$

which when broken into its component parts is

$$\partial_\mu F^{\mu 5}~=~0,~\partial_\mu F^{\mu\nu}~=~-(1~+~\beta^2)\partial_5 F^{5\nu},$$

for $\beta~=~\mu u/\hbar$. The term $\partial_5 F^{5\nu}$ is a source for the field and the term $\beta$ is then a term which amplifies the source. In the case of gravitation, which this is a "B-F" gauge like theory of, this source is the mass. The connection terms can be written according to basis elements of the manifold $A^\mu~=~\partial_\nu g^{\mu\nu}$, which for a bimetric theory in a reduced metric form or the traceless transverse (tt) terms for weak gravity fields gives the gravity equation

$$\square{\bar h}_{\mu\nu}~=~\beta^2\partial^2_5{\bar h}_{\mu\nu}$$

where the righthand side is the source $G\rho$ and the left hand side is the Laplacian of the potential. This equation is then reduced to the Poisson equation form of Newton's law of gravity.

It is interesting to compute the dispersion relationship in the gauge $A_5~=~0$ the spatial connection with the Fourier expansion $A^\mu~=~A_0^\mu e^{ik_\mu x^\nu~+~ik_5 x^5}$ determines the dispersion relationships with the above dynamical equations as

$$k_5k_\mu A_0^\mu~=~0,~\big(k_\nu k^\nu~+~(1~+~\beta^2)k_5^2\big)A^\mu~-~k^\mu k_\nu A^\nu~=~0$$

One of the motivating physical ideas here is with the quantum critical point for electrons in solids. The Dirac equation for the above complex time is

$${\cal L}_d~=~i{\bar\psi}\gamma^a\partial_a\psi~-~m{\bar\psi}\psi~-~i\frac{\mu^2}{\hbar}u^au^b{\bar\psi}\gamma_a\partial_b{\psi},$$

which gives the dynamical wave equation

$$i\gamma^a\partial_a\psi~-~m\psi~-~i\frac{\mu^2}{\hbar}u^au^b\gamma_a\partial_b\psi~=~0$$

Again if the field is expanded in the Fourier mode $\psi~\sim~e^{ik_\mu x^\nu~+~ik_5 x^5}$ this gives the dispersion relationship

$$-k^ak_a~-~2\frac{\mu^2}{\hbar}(u^ak_a)^2~-~4\frac{\mu^2}{\hbar}u^au_a(u^bk_b)^2~=~m^2$$

which when reduced to four dimensions is

$$-k^\mu k_\mu~=~m^2~+~(1~+~\beta^2)k_5^2.$$

The last term for a complex time can become large and will renormalize the mass of the fermion, where as $\beta~\propto~1/T$ will become large as the temperature approaches zero. Thus the complex time has the effect of "heating up" the internal states of the system and the mass of the fermion, in the case of the quantum critical points for the breakdown of the Landau electron liquid making the quasi-fermions "heavy." This then links aspects of a fermionic system and its quantum critical point with the instanton states of gravitation. This behavior should have a univerality to it, where it applies for all quantum fields from scalars, spinors, vectors, Rarita-Schwinger, and spin = 2 gravitons. It is also a property which determines a $\beta$ renormalization term that is invariant with respect to scale.

 

[Lawrence B. Crowell]

A wormhole in spacetime is theoretically understood to result in some strange physics, such as time machines. Wormholes for a number of reasons probably don't exist as classical spacetime structures, but their role in quantum gravity is probably equal to that of quantum black holes. The results of Ford and Roman as well as the apparent logical contradictions implied by closed timelike curves likely means they don't exist as classical structures. Other authors such as Thorne and Visser disagree with this. However, this is a look at wormholes as an aspect of quantum gravity and the issue of their classical existence is not directly addressed.

The Skyrme model of QCD defines a knot topology to gluon fluxes, which in the infrared domain or a confined vacuum are scale invariant. These flux tubes wrap around each other like snakes in a mating ball. If we were to take the quarks and pull them apart with "tweezers" the flux tubes would define a braid group. I could write this in a more complete mathematical form, but I will spare myself that effort for now. So what does this have to do with quantum gravity? Conformal gravity is in the Euclidean form $SU(4)$ and we can embed the $SU(3)$ of QCD in this group. Then conformal gravity with the coupling constant $/alpha^2~\sim~G~\rightarrow$ unity at unification appears remarkably QCD-ish with an extra group rank or dimension. We then consider the analogue between a quark-gluon plasma and a multiply connected spin-net. Each of the spinor fields, or spin-flags, on the manifold are analogous to quarks. In a quark-gluon plasam each quark is connected to each other through a tangle of gluon tubes which are braided and in a nontrivial topology. In the conformal gravity analogue the system each spin is connected to each other by noncommutative terms. We may think of the manifold as being inherently noncommutative, such as obeying an $SU(3)$ group, and that there is an auxilliary $U(1)$ gauge group which "carries" the connection terms. These is a matter of additional roots, but we will ignore that matter for the moment.

The group is then of course made Lorentzian $SU(4)~\sim~spin(6)~\rightarrow~spin(5,1)$. The analogue of the gluon flux tubes are then multiply connected topologies between spin-flags, which are for no better term worm holes. Then as the universe expands and the coupling constant of gravity becomes weak these are removed. The manifold becomes simply connected and the connection between the spin-flags is simple. The "dual field," and in this picture QCD has a duality with gravity, the coupling constant remains large at low energy (infrared confinement) and the field theory is also Euclidean. So the quark-gluon plasma or confinement "bag" is a vacuum region with a knot topology. The braided structure of gluons are then a dual structure which are a quantum foam.

This of course is not the end of the story, for these need to be embedded in an $SO(8) \rightarrow_{Z_2}~spin(7,1)$ or $spin(8)$ which are a Clifford valued algebra $Cl(8)$. The two parts are extended into $F_4$ and $G_2$ and this gives the 248 elements of the 256 N = 8 SUSY, and is the heterotic group $E_8$. Another approach is to use the face that $E_8$ containes $spin(16)~\sim~ SO(32)$, which is a representation for the closed string. For the spin group decomposition of these heterotic groups consistently contain the DeSitter group. The $spin(16)$ group has 128 generators. The additional 112 roots (from the total 248 in $E_8$) define a $D_8$ group (in terms of root system not lattice), which is an acceptable gauge theoretic model $SO(8)$, which also contains the deSitter $SO(3,2)$ under suitable change of signature. Similarly $E_6$ and $E_7$ sit inside $E_8$. $E_6 \times SU(3)/(Z/3Z)$ and $E_7\times SU(2)/(Z/2Z)$ are maximal subgroups of $E_8$. The simple heterotic group $E_6$ may be decomposed into $SU(6)\times SU(2)$, where $SU(6)$ contains the conformal group for gravity $SU(4)$ by

$$SU(4)\times SU(2)\times U(1)~\subset~SU(6),$$

which is remarkably similar to the decomposition of $SU(5)$ in the minimal GUT decomposition into the standard model.

With the group theory preliminaries out of the way we return to physical concepts. Consider a wormhole with openings in different regions of spacetime with different gravitational potentials. The vacua at the two openings are related by a Bogoliubov transformation between their respective raising and lowering field operators. Yet for the observer which traverses the openings the vacuum structure either abruptly changes. However since the observer in a local inertial frame carries their vacuum with then there is then a mismatch between the vacuum the observer carries in their local frame and the vacuum for a coincident local observer in that region. The multiply connected topology of this spacetime has an effective branch cut for the complex valued amplitudes of a quantum field which cycle through the wormhole. This suggests that the openings of a wormhole, or a quantum wormhole define a source for fields. Further, as the openings are multiply connected this is similar to a topologically distinct flux tube between sources. In a strong conformal gravity domain these flux tubes in $SU(2)$ are similar to gluon flux tubes in $SU(3)$ QCD, which connect points in the conformal spacetime to each other through "extra-dimensions. The multiply connected "tube" between the two openings then takes place in extra dimensions, such as the string/brane bulk, or for a LQG spin-net embedded in a larger space.

This then suggests that there are two principles at work here. The source, or vacua ambiguity, means that in a region of spacetime on a small enough scale will exhibit this as an uncertainty principle intrinsic to the geometry of the spacetime. This then suggests that the quantum geometry is noncommutative according to gauge potentials in a region of space or spacetime. Further, this noncommutivity is determined by a gauge potential which defines the multiply connected topology of the spacetime, similar to gluon flux tubes. In what follows these two theoretical observations are brought together to define a new approach to quantum gravity.

We set up the construction as follows. For the group $SU(4)\times SU(2)$ we have the gauge connection $A^\mu$, which is an $SU(2)$ connection, and the basis vectors $e_a,~a~=~1,~2,~,3$ for the space of $SU(2)$ and $\mu~=~1,~2,~\dots,~6$ for $SU(4)$. We may then impose a gauge condition on the system as

$$D_\mu e_a~=~(\partial_\mu~+~{\vec A}_\mu\times)e_a.$$

The gauge vector the carries an added index $a$, and acts as an internal vector space on the $SU(4)$. The vector potential contracted with the basis vectors $A^\mu~=~e^a A^\mu_a$ is a chromo-electric potential for a color charged $SU(2)$ theory. The $SU(2)$ basis vectors defines a bundle section and the quantum vacuum state for the according to the pure gauge element ${\bf e}\wedge d{\bf e}$ by

$$\Omega^k_\mu~=~{\epsilon_{ij}}^ke_i\partial_\mu e_j~=~B^k_\mu e_k$$

The vacuum potential term $B^k_\mu$ then determines the vacuum fields $\underline\underline\Omega~=~d\inderline\Omega$ according to

$$\Omega_{\mu\nu}~=~\partial_{\nu}\Omega_{\mu}~=~\partial_{\mu}\Omega_{\nu}~+~g\Omega_\nu\wedge\Omega_\mu~=~\big(\partial_\nu B^k_\mu~-~\partial_\mu B^k_\nu~+~g{\epsilon_{ij}}^kB^i_\nu B^j_\mu\big)e_k$$

With the definition of the gauge potential $A^\mu~=~e^a A^\mu_a$ the gauge potential

$${A^\prime}^k_\mu~=~A^k_\mu\cdot e_a~+~{\epsilon_{ij}}^ke_i\partial_\mu e_j~=~(A^k_\mu~+~B^k_\mu)e_k$$

for a gauge condition with zero ${A^\prime}^k_\mu$ we clearly have the $SU(4)$ gauge potential defined as $A_\mu~=~-B^k_\mu e_k$.