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We show that $3$-dimensional AdS spacetime can be semiclassically unstable due to strongly interacting quantum field effects. In our previous paper, we have pointed out the possibility of such an instability of AdS$_3$ by inspecting linear perturbations of the (covering space of) static BTZ black hole with AdS${}_4$ gravity dual in the context of holographic semiclassical problems. In the present paper, we further study this issue from thermodynamic viewpoint by constructing asymptotically AdS$_3$ semiclassical solutions and computing free energies of the solutions. We find two asymptotically AdS${}_3$ solutions to the semiclassical Einstein equations with non-vanishing source term: the one whose free energy is smaller than that of the BTZ with vanishing source term and the other whose free energy is smaller than that of the global AdS$_3$ with no horizon (thus manifestly zero-temperature background). The instability found in this paper implies the breakdown of the maximal symmetries of AdS$_3$, and its origin is different from the well-known semiclassical linear instability since our holographic semiclassical Einstein equations in $3$-dimensions do not involve higher order derivative terms., Comment: 18 pages, 2 figures

It remains unclear in general how the pole-skipping appears as a physical phenomenon, and we study the issue in the context of the AdS soliton. The pole-skipping has been discussed in black hole backgrounds, but the pole-skipping occurs even in the AdS soliton background. The geometry has a compact $S^1$-direction, and we compute the mass spectrum for the bulk scalar field, the bulk Maxwell field, and the gravitational perturbations with $S^1$ momentum. We show that the pole-skipping leaves its fingerprint in the the normal mode spectrum. The spectrum has some puzzling features because the would-be states are missing at pole-skipping points. The puzzling features disappear once one takes into account these pole-skipping points that we call "missing states.", Comment: 14 pages, 1 figure, ReVTeX4.2; v2: several clarifications, published version, title change to the published version

The pole-skipping has been discussed in black hole backgrounds, but we point out that the pole-skipping exists even in a non-black-hole background, the AdS soliton. For black holes, the pole-skipping points are typically located at imaginary Matsubara frequencies $\omega=-(2\pi T)ni$ with an integer $n$. The AdS soliton is obtained by the double Wick rotation from a black hole. As a result, the pole-skipping points are located at $q_z=-(2\pi n)/l$, where $l$ is the $S^1$ periodicity and $q_z$ is the $S^1$ momentum. The ``chaotic" and the ``hydrodynamic" pole-skipping points lie in the physical region. We also propose a method to identify all pole-skipping points instead of the conventional method., Comment: 15 pages, ReVTeX4.2; v2: comment on alternative formalism added in Sec. 2, physical implication added in Sec. 5, typos corrected, published version

In this paper, we consider how to formulate semiclassical problems in the context of the AdS/CFT correspondence, based on the proposal of Compere and Marolf. Our prescription involves the effective action with self-action term for boundary dynamical fields, which can be viewed as imposing mixed boundary conditions for the gravity dual. We derive the semiclassical Einstein equations sourced by boundary CFT stress-energy tensor. Analyzing perturbations of the holographic semiclassical Einstein equations, we find a universal parameter $\gamma_d$ which controls the contribution from boundary CFTs and specifies dynamics on the AdS boundary. As a simple example, we examine the semiclassical Einstein equations in $3$-dimensions with $4$-dimensional AdS gravity dual, and show that the boundary BTZ black hole with vanishing expectation value of the stress-energy tensor becomes unstable due to the backreaction from quantum stress-energy tensor when the parameter $\gamma_d$ exceeds a certain critical value., Comment: 34 page, 4 figures; one figure and refs added, some clarifications made, version to appear in JHEP

The holographic superconductor is the holographic dual of superconductivity, but there is no Meissner effect in the standard holographic superconductor. This is because the boundary Maxwell field is added as an external source and is not dynamical. We show the Meissner effect analytically by imposing the semiclassical Maxwell equation on the AdS boundary. Unlike in the Ginzburg-Landau (GL) theory, the extreme Type I limit cannot be reached even in the $e\to\infty$ limit where $e$ is the $U(1)$ coupling of the boundary Maxwell field. This is due to the bound current which is present even in the pure bulk Maxwell theory. In the bulk 5-dimensional case, the GL parameter and the dual GL theory are obtained analytically for the order parameter of scaling dimension 2., Comment: 14 pages, ReVTeX4.2; v2: corrected typos and several clarifications

We point out little discussed phenomenon in elementary quantum mechanics. In one-dimensional potential scattering problems, the scattering amplitudes are not uniquely determined at special points in parameter space. We examine a few explicit examples. We also discuss the relation with the pole-skipping phenomena recently found in holographic duality. In the holographic pole-skipping, the retarded Green's functions are not uniquely determined at imaginary Matsubara frequencies. It turns out that this universality comes from the fact that the corresponding potential scattering problem has the angular momentum potential., Comment: 14 pages, ReVTeX4.2; v2: several clarifications, published version

We study the pole-skipping phenomenon of the scalar retarded Green's function in the rotating BTZ black hole background. In the static case, the pole-skipping points are typically located at negative imaginary Matsubara frequencies $\omega=-(2\pi T)ni$ with appropriate values of complex wave number $q$. But, in a $(1+1)$-dimensional CFT, one can introduce temperatures for left-moving and right-moving sectors independently. As a result, the pole-skipping points $\omega$ depend both on left and right temperatures in the rotating background. In the extreme limit, the pole-skipping does not occur in general. But in a special case, the pole-skipping does occur even in the extreme limit, and the pole-skipping points are given by right Matsubara frequencies., Comment: 16 pages, ReVTeX4.2; v2: several clarifications, published version

Recently, it is shown that many Green's functions are not unique at special points in complex momentum space using AdS/CFT. This phenomenon is similar to the pole-skipping in holographic chaos, and the special points are typically located at $\omega_n = -(2\pi T)ni$ with appropriate values of complex wave number $q_n$. We study finite-coupling corrections to special points. As examples, we consider four-derivative corrections to gravitational perturbations and four-dimensional Maxwell perturbations. While $\omega_n$ is uncorrected, $q_n$ is corrected at finite coupling. Some special points disappear at particular values of higher-derivative couplings. Special point locations of the Maxwell scalar and vector modes are related to each other by the electromagnetic duality., Comment: 14 pages, ReVTeX4.1; v2: added comments on field redefinitions, published version

We investigate the "pole-skipping" phenomenon in holographic chaos. According to the pole-skipping, the energy-density Green's function is not unique at a special point in complex momentum plane. This arises because the bulk field equation has two regular near-horizon solutions at the special point. We study the regularity of two solutions more carefully using curvature invariants. In the upper-half $\omega$-plane, one solution, which is normally interpreted as the outgoing mode, is in general singular at the future horizon and produces a curvature singularity. However, at the special point, both solutions are indeed regular. Moreover, the incoming mode cannot be uniquely defined at the special point due to these solutions., Comment: 19 pages, PTEPHY; v2: a few clarifications, published version

We investigate a new property of retarded Green's functions using AdS/CFT. The Green's functions are not unique at special points in complex momentum space. This arises because there is no unique incoming mode at the horizon and is similar to the "pole-skipping" phenomenon in holographic chaos. Our examples include the bulk scalar field, the bulk Maxwell vector and scalar modes, and the shear mode of gravitational perturbations. In these examples, the special points are always located at $\omega_\star = -i(2\pi T)$ with appropriate values of complex wave number., Comment: 18 pages, JHEP; v2: added discussion, published version

We construct an analytic solution for a one-parameter family of holographic superconductors in asymptotically Lifshitz spacetimes. We utilize this solution to explore various properties of the systems such as (1) the superfluid phase background and the grand canonical potential, (2) the order parameter response function or the susceptibility, (3) the London equation, (4) the background with a superfluid flow or a magnetic field. From these results, we identify the dual Ginzburg-Landau theory including numerical coefficients. Also, the dynamic critical exponent $z_d$ associated with the critical point is given by $z_d=2$ irrespective of the value of the Lifshitz exponent $z$., Comment: 17 pages, ReVTeX4.1; v2: a few clarifications; dedicated to the memory of Joe Polchinski

The Kibble-Zurek (KZ) mechanism describes the generations of topological defects when a system undergoes a second-order phase transition via quenches. We study the holographic KZ scaling using holographic superconductors. The scaling can be understood analytically from a scaling analysis of the bulk action. The argument is reminiscent of the scaling analysis of the mean-field theory but is more subtle and is not entirely obvious. This is because the scaling is not the one of the original bulk theory but is an emergent one that appears only at the critical point. The analysis is also useful to determine the dynamic critical exponent $z$., Comment: 10 pages, ReVTeX4.1; v2: a few clarifications

We investigate a holographic model of superfluid flows with an external repulsive potential. When the strength of the potential is sufficiently weak, we analytically construct two steady superfluid flow solutions. As the strength of the potential is increased, the two solutions merge into a single critical solution at a critical strength, and then disappear above the critical value, as predicted by a saddle-node bifurcation theory. We also analyze the spectral function of fluctuations around the solutions under a certain decoupling approximation., Comment: 10 pages, 6 figures

It is known that time-dependent perturbations can enhance superconductivity and increase the critical temperature. If this phenomenon happens to high-T_c superconductors, one could obtain room-temperature superconductors, but this is still an open issue experimentally. Meanwhile, we would like to understand this phenomenon from gravity dual and see if the enhancement is possible for holographic superconductors. Previous work (arXiv:1104.4098 [hep-th]) has studied this issue by adding a "time-dependent chemical potential," but their analysis is questionable as a true dynamic equilibrium. In particular, the AdS boundary does not supply energy to the bulk spacetime in their setup. A more appropriate way to discuss the enhancement is to add a time-dependent vector potential, i.e., a time-dependent electric field. However, the enhancement does not occur for holographic superconductors., Comment: 26 pages, 3 figures, JHEP; v2: added comment in Sec. 4.3 to the published version (v1)

We study the proposal by Bredberg et al. (1006.1902), where the fluid is defined by the Brown-York tensor on a timelike surface at r=r_c in black hole backgrounds. We consider both Rindler space and the Schwarzschild-AdS (SAdS) black hole. The former describes an incompressible fluid, whereas the latter describes the vanishing bulk viscosity at arbitrary r_c, but these two results do not contradict with each other. We also find an interesting "coincidence" with the black hole membrane paradigm which gives a negative bulk viscosity. In order to show these results, we rewrite the hydrodynamic stress tensor via metric perturbations using the conservation equation. The resulting expressions are suitable to compare with the Brown-York tensor., Comment: 18 pages, PTEPHY; v2: added discussion

We investigate static inhomogeneous charged planar black hole solutions of the Einstein-Maxwell system in an asymptotically anti-de Sitter spacetime. Within the framework of linear perturbations, the solutions are numerically and analytically constructed from the Reissner-Nordstr\"{o}m-AdS black hole solution. The perturbation analysis predicts that the Cauchy horizon always disappears for any wavelength perturbation, supporting the strong cosmic censorship conjecture. For extremal black holes, we analytically show that an observer freely falling into the black hole feels infinite tidal force at the horizon for any long wavelength perturbation, even though the Kretschmann scalar curvature invariant remains small., Comment: 11 pages, 8 figures, introduction has been clarified and typos have been modified. Some references are added. To be published in PRD

In dynamic critical phenomena, singular behaviors appear not only in the order parameter but also in the other transport coefficients (due to the mode-mode coupling). However, this effect has not been observed in the AdS/CFT duality. We point out that this mode-mode coupling is suppressed by 1/N^2 in the large-N gauge theories which correspond to model H in the classification of Hohenberg and Halperin. Thus, the effect cannot be seen in the classical supergravity approximation. We illustrate this point using the example of the N=4 super-Yang-Mills theory at a finite chemical potential. We also discuss the implications of this result to heavy-ion collisions., Comment: 10 pages, ReVTeX4.1; v2: published version

We investigate energy dissipation associated with the motion of the scalar condensate in a holographic superconductor model constructed from the charged scalar field coupled to the Maxwell field. Upon application of constant magnetic and electric fields, we analytically construct the vortex flow solution, and find the vortex flow resistance near the second-order phase transition where the scalar condensate begins. The characteristic feature of the non-equilibrium state agrees with the one predicted by the time-dependent Ginzburg-Landau(TDGL) theory. We evaluate the kinetic coefficient in the TDGL equation along the line of the second-order phase transition. At zero magnetic field, the other coefficients in the TDGL equation are also evaluated just below the critical temperature., Comment: 10 pages, 2 figures

In the AdS/CFT duality, it is often said that a local symmetry in a bulk theory corresponds to a global symmetry in the corresponding boundary theory, but the global symmetry can become local when one couples with an external source. As a result, the GKP-Witten relation gives a response function instead of a Green function. We explore this point in details using the example of holographic superconductors. We point out that these points play a crucial role to interpret the holographic London equation properly., Comment: 11 pages, ReVTeX4.1; v2: added discussion

We investigate the vortex lattice solution in a (2+1)-dimensional holographic model of superconductors constructed from a charged scalar condensate. The solution is obtained perturbatively near the second-order phase transition and is a holographic realization of the Abrikosov lattice. Below a critical value of magnetic field, the solution has a lower free energy than the normal state. Both the free energy density and the superconducting current are expressed by nonlocal functions, but they reduce to the expressions in the Ginzburg-Landau (GL) theory at long wavelength. As a result, a triangular lattice becomes the most favorable solution thermodynamically as in the GL theory of type II superconductors., Comment: v2: minor changes, references added; 11 pages, 2 figures: version to appear in PRD

We study "holographic superconductors" in various spacetime dimensions. We compute most of the static critical exponents in the linear perturbations and show that they take the standard mean-field values. We also consider the dynamic universality class for these models and show that they belong to model A with dynamic critical exponent z=2., Comment: 13 pages, 5 figures, ReVTeX4

In critical phenomena, singular behaviors arise not only for thermodynamic quantities but also for transport coefficients. We study this dynamic critical phenomenon in the AdS/CFT duality. We consider black holes with a single R-charge in various dimensions and compute the R-charge diffusion in the linear perturbations. In this case, the black holes belong to model B according to the classification of Hohenberg and Halperin., Comment: 17 pages, ReVTeX4; v2: added references and discussion

We investigate in more detail the holographic model of a superconductor recently found by Hartnoll, Herzog, and Horowitz [Phys. Rev. Lett. 101, 031601], which is constructed from a condensate of a charged scalar field in AdS_4-Schwarzschild background. By analytically studying the perturbation of the gravitational system near the critical temperature T_c, we obtain the superconducting coherence length proportional to 1/\sqrt{1-T/T_c} via AdS/CFT correspondence. By adding a small external homogeneous magnetic field to the system, we find that a stationary diamagnetic current proportional to the square of the order parameter is induced by the magnetic field. These results agree with Ginzburg-Landau theory and strongly support the idea that a superconductor can be described by a charged scalar field on a black hole via AdS/CFT duality., Comment: 9 pages, no figure; v2: typos corrected; v3: version to appear in PRD, an early discussion based on convensional superconductor with dynamical photon removed and an argument about the type of the holographic superconductor added

We obtain new transport coefficients of causal hydrodynamics for the M2 and the M5-brane using a Kubo-like formula proposed by Baier, Romatschke, Son, Starinets, and Stephanov (arXiv:0712.2451 [hep-th]). The relaxation times agree with the ones obtained from the "sound mode" in our paper (arXiv:0712.2916 [hep-th])., Comment: 6 pages, ReVTeX4

In this paper we investigate radiation by an accelerated quark in a strongly coupled gauge theory via AdS/CFT correspondence. According to AdS/CFT dictionary, we can read off energy density or energy flux of the radiation from asymptotic gravitational field in AdS bulk sourced by an accelerated string trailing behind the quark. In the case of an oscillating quark with frequency $\Omega$, we show that the time averaged energy density is asymptotically isotropic and it falls off as $(g_{\text{YM}}^2 N)^{1/2} \Omega^4/R^{2}$ with distance $R$ from the source. In a toy model of a scattered quark by an external field, we simply estimate Poynting vector by the bremsstrahlung radiation and show that the energy flux is anisotropic outgoing radiation. Based on these investigations, we discuss the properties of strongly coupled gauge theory radiation in comparison with electromagnetic radiation., Comment: 16 pages, no figures, accepted for publication in Phys. Rev. D

This is a comment to hep-th/0703243. The paper determined the relaxation time tau_pi of the shear viscous stress for the N=4 SYM from AdS/CFT correspondence. The purpose of this comment is to point out that the value of tau_pi is 3 times larger than their result if one takes into account an additional term in the hydrodynamic equation., Comment: 2 pages, ReVTeX4

We study causal hydrodynamics (Israel-Stewart theory) of gauge theory plasmas from the AdS/CFT duality. Causal hydrodynamics requires new transport coefficients (relaxation times) and we compute them for a number of supersymmetric gauge theories including the N=4 SYM. However, the relaxation times obtained from the "shear mode" do not agree with the ones from the "sound mode," which implies that the Israel-Stewart theory is not a sufficient framework to describe the gauge theory plasmas., Comment: 20 pages, ReVTeX4; v3: the sound mode results for SAdS_{4,7} added, an early interpretation based only on the shear mode changed; v6: corrected solution in Appendix B

We study the screening length of a heavy quark-antiquark pair in strongly coupled gauge theory plasmas flowing at velocity v following a proposal by Liu, Rajagopal, and Wiedemann. We analyze the screening length as the direction of the plasma winds vary. To leading order in v, this angle-dependence can be studied analytically for many theories by extending our previous formalism. We show that the screening length is locally a minimum (maximum) when the pair is perpendicular (parallel) to the plasma winds, which has been observed for the N=4 plasma. Also, we compare AdS/CFT results with weak coupling ones, and we discuss the subleading dependence on v for the Dp-brane., Comment: 20 pages, 4 figures, JHEP3; v2: discussion added and modified

We study the screening length L_s of a heavy quark-antiquark pair in strongly coupled gauge theory plasmas flowing at velocity v. Using the AdS/CFT correspondence we investigate, analytically, the screening length in the ultra-relativistic limit. We develop a procedure that allows us to find the scaling exponent for a large class of backgrounds. We find that for conformal theories the screening length is (boosted energy density)^{-1/d}. As examples of conformal backgrounds we study R-charged black holes and Schwarzschild-anti-deSitter black holes in (d+1)-dimensions. For non-conformal theories, we find that the exponent deviates from -1/d and as examples we study the non-extremal Klebanov-Tseytlin and Dp-brane geometries. We find an interesting relation between the deviation of the scaling exponent from the conformal value and the speed of sound., Comment: 21 pages, 1 figure, JHEP3; v3&v4: minor improvements & corrections

In AdS/CFT duality, it is often argued that information behind the event horizon is encoded even in boundary correlators. However, its implication is not fully understood. We study a simple model which can be analyzed explicitly. The model is a two-dimensional scalar field propagating on the s-wave sector of the BTZ black hole formed by the gravitational collapse of a null dust. Inside the event horizon, we placed an artificial timelike singularity where one-parameter family of boundary conditions is permitted. We compute two-point correlators with two operators inserted on the boundary to see if the parameter is reflected in the correlators. In a typical case, we give an explicit form of the boundary correlators of an initial vacuum state and show that the parameter can be read off from them. This does not immediately imply that the asymptotic observer can extract the information of the singularity since one cannot control the initial state in general. Thus, we also study whether the parameter can be read off from the correlators for a class of initial states., Comment: 17pages, 5 figures, accepted in Physical Review D

We compute the strong coupling limit of the shear viscosity for the N=4 super-Yang-Mill theory with a chemical potential. We use the five-dimensional Reissner-Nordstrom-anti-deSitter black hole, so the chemical potential is the one for the R-charges U(1)_R^3. We compute the quasinormal frequencies of the gravitational and electromagnetic vector perturbations in the background numerically. This enables one to explicitly locate the diffusion pole for the shear viscosity. The ratio of the shear viscosity eta to the entropy density s is eta/s=1/(4pi) within numerical errors, which is the same result as the one without chemical potential., Comment: 11 pages, 5 figures, ReVTeX4; v2: minor improvements; v3: explanations added and improved; v4: version to appear in PRD

The nonextreme Dp-brane solutions in type II supergravity(in the near-horizon limit) are expected to be dual to (p+1)-dimensional noncompact supersymmetric Yang-Mills theories at finite temperature. We study the translationally invariant perturbations along the branes in those backgrounds and calculate quasinormal frequencies numerically. These frequencies should determine the thermalization time scales in the dual Yang-Mills theories., Comment: 13 pages, 3 figures, revtex4; v2 & 3: references added, minor changes

We discuss the string creation in the near-extremal NS1 black string solution. The string creation is described by an effective field equation derived from a fundamental string action coupled to the dilaton field in a conformally invariant manner. In the non-critical string model the dilaton field causes a timelike mirror surface outside the horizon when the size of the black string is comparable to the Planck scale. Since the fundamental strings are reflected by the mirror surface, the negative energy flux does not propagate across the surface. This means that the evaporation stops just before the naked singularity of the extremal black string appears even though the surface gravity is non-zero in the extremal limit., Comment: 15 pages

In order to overcome ambiguity problem on identification of mathematical objects in noncommutative theory with physical observables, quantum mechanical system coupled to the NC U(1) gauge field in the noncommutative space is reformulated by making use of the unitarized Seiberg-Witten map, and applied to the Aharonov-Bohm and Hall effects of the NC U(1) gauge field. Retaining terms only up to linear order in the NC parameter \theta, we find that the AB topological phase and the Hall conductivity have both the same formulas as those of the ordinary commutative space with no \theta-dependence., Comment: 7 pages, no figures, uses revtex4; 8 pages, conclusion changed, Appendix added

In order to investigate whether space coordinates are intrinsically noncommutative, we make use of the Hall effect on the two-dimensional plane. We calculate the Hall conductivity in such a way that the noncommutative U(1) gauge invariance is manifest. We find that the noncommutativity parameter theta does not appear in the Hall conductivity itself, but the particle number density of electron depends on theta. We point out that the peak of particle number density differs from that of the charge density., Comment: 6 pages, no figure; v2: References added. Typos corrected

When phase space coordinates are noncommutative, especially including arbitrarily noncommutative momenta, the Hall effect is reinvestigated. A minimally gauge-invariant coupling of electromagnetic field is introduced by making use of Faddeev-Jackiw formulation for unconstrained and constrained systems. We find that the parameter of noncommutative momenta makes an important contribution to the Hall conductivity., Comment: 11 pages, no figures, uses ptptex.cls

When coordinates are noncommutative, the Hall effect is reinvestigated. The Hall conductivity is expressed with noncommutative parameters, so that in the commutative limit it tends to the conventional result., Comment: 7 pages, no figure, uses ptptex.cls

When we have noncommutativity among coordinates (or conjugate momenta), we consider Wigner's formulation of quantum mechanics, including a new derivation of path integral formula. We also propose the Moyal star product based on the Dirac bracket in constrained systems., Comment: 10 pages, No figures, Latex

In a gedankenexperiment about the generalized second law (GSL) of black hole thermodynamics, the buoyant force by black hole atmosphere (the acceleration radiation) plays an important role, and then it is significant to understand the nature of the buoyant force. Recently, Bekenstein criticizes that the fluid approximation of the acceleration radiation which is often used in the estimation of the buoyant force is invalid for the case that the size of the target is much less than a typical wavelength of the acceleration radiation, due to the diffractive effect of wave scattering. In this letter, we argue that even if it is correct that we should calculate the buoyant force as a wave scattering process, its implication in the GSL strongly depends on whether there exists any massless scalar field, that is, S-wave scattering. By reconsidering the diffractive effect by S-wave scattering, we show that if some massless scalar field exists, then the GSL can hold without invoking a new physics, such as an entropy bound for matter., Comment: 9 pages, latex, no figures

We propose that Strominger's method to derive the BTZ black hole entropy is in fact applicable to other asymptotically AdS_3 black holes and gives the correct functional form of entropies. We discuss various solutions in the Einstein-Maxwell theory, dilaton gravity, Einstein-scalar theories, and Einstein-Maxwell-dilaton theory. In some cases, solutions approach AdS_3 asymptotically, but their entropies do not have the form of Cardy's formula. However, it turns out that they are actually not "asymptotically $AdS_3$" solutions. On the other hand, for truly asymptotically AdS_3 solutions, their entropies have the form of Cardy's formula. In this sense, all known solutions are consistent with our proposal., Comment: 21 pages, LaTeX; v2: added discussion for section 3.3

Strominger has derived the Bekenstein-Hawking entropy of the BTZ black hole using asymptotic Virasoro algebra. We apply Strominger's method to a black hole solution found by Martinez and Zanelli (MZ). This is a solution of three-dimensional gravity with a conformal scalar field. The solution is not $AdS_3$, but it is asymptotically $AdS_3$; therefore, it has the asymptotic Virasoro algebra. We compute the central charge for the theory and compares Cardy's formula with the Bekenstein-Hawking entropy. It turns out that the functional form does agree, but the overall numerical coefficient does not. This is because this approach gives the "maximum possible entropy" for the numerical coefficient., Comment: 26 pages, LaTeX; v2: minor corrections