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I have been reading with pleasure your highly technical Essay and I think that may be useful to highlight some analogies and differences between our respective approaches.

As explained in my Essay, the canonical theory goes beyond less fundamental approaches as the superstring theory. This includes also the matrix formulation of M-theory (based in a Trace dynamics as that summarized in your...

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I have been reading with pleasure your highly technical Essay and I think that may be useful to highlight some analogies and differences between our respective approaches.

As explained in my Essay, the canonical theory goes beyond less fundamental approaches as the superstring theory. This includes also the matrix formulation of M-theory (based in a Trace dynamics as that summarized in your Essay). The general state \hat{rho} of any system in canonical theory is defined on a generalized Liouville space (see pag 6 in my Essay). We can use an extension of Wigner techniques to derive an equivalent Wigner state \hat{rho}_W, but defined on a non-commutative phase space ({x},{p}), with associated N x N matrices like in Trace dynamics. However, the information contained in this non-commutative phase space is redundant and whereas the simplest dynamics (e.g. the S part of the canonical dynamics) can be adequately described by the geometrical star-products, there are difficulties to obtain the more general dynamics.

Indeed, the generator of the time translation in the canonical theory is not given by an operator but by a superoperator (e.g. the coefficients Omega are represented by superoperators in the quantum formalism associated to the post-Keizer form obtained at our Center). Those superoperators are defined in a superoperator space and are associated to (super)matrices of order N^2 x N^2. The evolution is described by an operator (matrix) Hamiltonian only as approximation in the 'pure' limit (see Box 1 in reference 8 in our Essay). From the Hamiltonian, we can obtain Lagrangians and actions using the usual techniques. That is, neither Hamiltonians nor Lagrangians and actions can describe the dynamics of the processes in the general case. Also the usual bosonic and fermionic commutators/anticommutators are valid only in the same 'pure' approximation.

Next, you introduce a statistical mechanics by coarse graining, using a maximum entropy method. There is several difficulties with this approach, which are solved in a natural way in the canonical theory. The first is that the canonical theory belongs to the modern statistical mechanics School of fine grained methods [FG], eliminating the extradynamical coarse graining considerations and their associated paradoxes. The second, is that the assumptions as your "the ensemble does not prefer any one state in the Hilbert space over the other" are derived rather than merely postulated. The third is that these assumptions work for equilibrium but not for far from equilibrium systems; therefore, we can obtain a nonequilibrium statistical mechanics of broad scope, whereas the work of Adler and others (reference 1 in your Essay) is limited to the simple case of (stable) equilibrium. The fourth, is that your work and the work of Adler and others relies on the Liouville theorem, which means that the resulting equations of motion cannot describe the evolution of the non-conserved variables, doing impossible the link with the phenomenological theory of nonlinear nonequilibrium thermodynamics. This is a well-known deficiency of the older approaches to statistical mechanics, which is solved by modern formulations of nonequilibrium statistical mechanics as that by Byung Chan Eu [Eu]. We can derive his nonequilibrium statistical mechanics as an approximation to the more deeper canonical theory.

As you correctly point, the next step in the scheme is to consider fluctuations. Again, at this point the work of Adler is based in further approximations. Effectively, he obtains a stochastic generalization of the Schrödinger equation and this generalization allows us to describe the nonlinear phenomena cannot be described by the usual Schrödinger equation, such as the collapse of wavefunctions. His work and that of others is very fascinating, because instead of the dual structure associated to the Copenhagen formulation of quantum mechanics, we have a single evolution, which gives reduction for measurements and the usual unitary evolution when the system is not measured! However, this kind of work is not fundamental. The canonical theory includes fluctuations (see Box 1 in reference 8 in our Essay) but cover a more broad spectrum. It is only when we approximate the canonical rate theory by its weak-coupling limit and take the Markovian limit that the fluctuations are Brownian. Moreover, Brownian fluctuations are related to what Nico van Kampen named "extrinsic noise", which is due to the fact that the system (the Brownian 'particle') is open. However, there is also another kind of noise, "intrinsic noise", which is related to the fact that the underlying structure of matter is discrete rather than analog. As van Kampen correctly noted, the noise observed in chemical reactions is of the intrinsic kind. It is not surprising that the canonical theory is able to describe both noises.

Then you go over discussing the role of time and you introduce two concepts of time; at the one hand, the concept of dimensional time, which, as you correctly point out, is associated to a quantum operator of time and, at the other hand, the concept of "affine parameter". This is not very different from the Stuckelberg extension of string and p-brane theory (see reference 15 in my Essay). At this point, your essay looks ambiguous for me and you refer to a reference still in preparation. You say that you introduce operators (q,t) for each particle. However, next you write that this "introduce a non-commutative Minkowski spacetime". I assume that you mean a 4N dimensional non-commutative spacetime (for N particles), which would reduces to a non-commutative Minkowski spacetime only in the one-particle case.

In the Stuckelberg extension of string and p-brane theory, the introduction of a phase space as your (q,t,p,E) yields dynamical redundancies. These are 'solved' (or at least alleviated) by working off-shell. What approach do you take in your reference 11 in preparation?

Another point that I want to comment is when you take a Block Universe view where your "affine parameter" tau is not identified with time. Here your work departs from the Stuckelberg extension of string and p-brane theory (see reference 15 in my Essay). You seem to support what Pavsic names the interpretation "(i)", whereas he prefers the "(ii)". You give none technical reason for your interpretation, whereas others (mainly Horwitz, Fanchi, and others) have given many details on why the interpretation (ii) is the correct. In the references 17 and 18 cited in my Essay I showed that the parameter tau in the Stuckleberg, Horwitz, & Piron theory reduces to Newtonian time for interacting charged particles and for massive particles under gravity respectively, whereas the concept of dimensional time associated to spacetime rigorously fails to reduce in both cases.

You argue that the "transition from the lower analog layer to the upper analog layer also helps understand the

origin of the arrow of time". However, you give no technical detail and merely state that the question of the origin of the arrow of time is related to "why is the initial entropy of the Universe so low?". This is the well-known cosmological argument, but unfortunately it fails when one considers the details. A better approach to understand the origin of the arrow of time is given by the Brussels-Austin School [IRREV]. It seems that the Brussels-Austin School last theory for LPS with Poincaré resonances can be obtained from the canonical theory for systems with bifurcations [IRREV].

[FG] The quest for the ultimate theory of time, an introduction to irreversibility

[Eu] Nonequilibrium Statistical Mechanics, Ensemble Method 1998: Kluwer Academic Publishers, Dordrecht. Eu, Byung Chan.

[IRREV] Trajectory branching in Liouville space as the source of irreversibility

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I read your paper through in detail last night and started writing this. I wrote further today, so this got a bit long. I even looked at Adler huge paper, though being 175 pages in length I of course could not read the whole. The A-M matrices, traceless diagonal elements of i = sqrt{-1} and -i, forms is related to the Kahler matrix. This is a line bundle form of the symplectic matrix. To...

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I read your paper through in detail last night and started writing this. I wrote further today, so this got a bit long. I even looked at Adler huge paper, though being 175 pages in length I of course could not read the whole. The A-M matrices, traceless diagonal elements of i = sqrt{-1} and -i, forms is related to the Kahler matrix. This is a line bundle form of the symplectic matrix. To include commutator structure between position elements [q_i, q_j] = αħ, for α a constant, and similar commutators for momentum, this can be generalized by the Gelfand theorem and Connes' noncommutative geometry.

For gravity this is clearly an important aspect of quantum gravity. Of course we have a lot of funny ideas about this. In your paper you have the “digital” as a world with quantum gravity. This appears evident by just looking at the Schwarzschild metric element 1 – r_0/r, for r_0 = 2GM/c^2, the Killing vector K_t = (1/sqrt{1 – r_0/r})∂_t. This forms a natural operator for a Schrodinger type of equation Hψ = -iħK_tψ. However, the momentum operator p_r = -iħ∂_r does not commute with K_t, and there is a noncommutative geometry. Another way of looking at this is the element r_0 exhibits fluctuations so that r_0 = + δr_0, where δr_0 = f*sqrt{Għ/c^3} = f*L_p for f \in [0, 1]. So a fluctuation of mass-energy in a region of space there is then a fluctuation in the proper time ds ~ (1 - δr_0/r)dt^2, which is a noncommutative situation in energy and time. So there are curiously two different ways of looking at this.

We have two snags with our ideas of quantum gravity. One in string theory, where the action is formulated as

L = ∫d^nx sqrt{-1}(R + α’R_{abcd}R^{abcd} + O(α’^2))

Requires there to be a classical background, or R_{ab} = λg_{ab}. This background dependence is a major criticism which has been lodged at string theory. However, the LQG folks who raise this complaint have problems of their own. In the assignment of a degree of freedom with each strut in a discrete spacetime there is a vacuum E = 3kT/2 element, which when summed up results in a huge entropy to spacetime. For this reason LQG is not able to reconstruct classical spacetime. String theory on the other hand employs holographic principle which vastly reduces the number of degrees of freedom to horizons and boundaries and these problems are avoided.

The string perturbation series is also problematic. It appears almost incomputable. We may then be able to work with some finite series, as in an effective theory. The near horizon for a black hole in an AdS_n is where the spacetime becomes AdS_2xS^{n-2}. The AdS_2 has an equivalency to the CFT_1 with isometries of the SL(2,C) group. This is the elementary group which constructs the quantum SLOCC quantum bit structures equivalent to BPS black holes. The CFT_1 is the Diff(S^1) which is the Virasoro algebra, and in this case with two copies bounded on a conformal map of S^1 to a strip. This defines the Hartle-Hawking quantum states.

The Hartle-Hawking state is constructed by a map from Calabi-Yau three-fold. This constructs the states according to a type of modular form which is related to the partition of integers. This modular form in a Dirchlet L-series has the Riemann zeta function, where its zeros determine the eigenvalues.. The 3-fold in the conjugacy classes of a maximal tori on the F_4 gives the cycle [0, e^{2π/3}, e^{4π/3}], which defines the Eisenstein series E(z), E(2z) E(3z) and the partition function for the quantum states of the AdS_2 ~ CFT_1 spacetime. This is quantum gravity to one loop. This is also equivalently determined by the G_2 group, which in the E_8 is the centralizer of the F_4 group. This extends the work which I present in the paper I wrote for FQXi Building up to AdS_7 will take us up to 6 loop calculations, and extended to 11-dimensional SUGRA to 7 loops.

What comes after that? Frankly, nothing for in effect we run out of algebra. However, there is something which is going on. The hyperbolic dynamics on H_2 ~ AdS_2 is S-dual to a quartic theory of fermions. This is not my work, but was demonstrated by Zamolodchikov (among the other amazing things that guys did), and physically it means the underlying physics of strings in the AdS_2, or equivalently on the boundary as CFT_1, is that of a fermion condensate.

The high temperature domain for the string is the Hagedorn temperature. The density of states for a string with respect to modes n is

η(n) ~ exp(4πn sqrt{α’})

that defines a partition function Z =~ ∫ η(n)exp(-n/T)dn. The Temperature is computed by 1/T = ∂Z/∂n and the path integral diverges for a temperature greater than

T_H = 4π sqrt{α’}

which is the Hagedorn temperature. This is proportional to the reciprocal of the string length. The entropy of the system is the logarithm of the density of states the S ~ 1/nT_H, which in the large n limit is zero. The modes number is given by n = 1/(sqrt{d}M_s), for d the number of degrees of freedom and M_s the string mass. String theory on the AdS_2 transitions into a theory of fermions at this high energy. Strings are then similar to the topological states, such as Skyrmion states.

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