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An exposition of Planck's Law of Radiation is presented within the context of modern quantum many body theory. In particular the generality of the Planck radiation law is demonstrated to be Valid to all orders of perturbation theory for an interacting system, regardless of the underlying statistics for the particles comprising the matter field and the precise nature of the interaction mechanism. This exposition justifies the use of Planck's law with minor modifications via the refractive index, in the optical properties of semiconductors and insulators. We conclude by clarifying when Planck's law is likely to fail.

The work is based on a singularity-free model of the universe previously proposed by the authors. According to this model, at the beginning of the present expansion phase the universe was filled with prematter for which P = -rho, and rho is about rho(p), the Planck density, and the prematter went over to ordinary matter in the course of time. Two approaches are considered. In the first it is assumed that there are initial perturbations in the prematter. These grow tremendously in the course of time. In the second it is assumed that the prematter has no perturbations, but that perturbations develop as the prematter is converted to ordinary matter. 12 refs.

Gravitational theories can be written in terms of rescaled fields without the Planck mass. The rescaled tetrads acquire the dimension of mass. The actual distribution of energy throughout space-time causes the tetrads to assume vacuum expected values of the order of the Planck mass,mp. Thus the gravitational constant,G=ħc/mP2, may be viewed not as a fundamental constant, but as a mass scale that is dynamically determined by the large-scale structure of the Universe.

Dual string theories of everything, being purely geometrical, contain only two fundamental constants: c, for relativistic invariance, and a length λ, for quantization. Planck's and Newton's constants appear only through Planck's length, a "calculable" fraction of λ. Only the existence of a light sector breaks a "reciprocity" principle and unification at λ, which is also the theory's cut-off.

A theoretical relationship for the electron rest mass is derived in terms of the electron charge, Planck's quantum of action, and the speed of light. The basis for this derivation is an assumption that the electron rest mass is entirely electrostatic in origin, combined with the realization that the maximum action density of the world is simply the speed of light. Planck's quantum of action becomes perspicuously associated with an elliptical microspace.

The effect of quantum fluctuations of gravity on the measurement of proper distances is considered. It is shown that, when the length scales are of the order of Planck length, the concept of a unique distance between points ceases to exist. It is also shown that the quantum expectation value of the proper length is bounded from below by Planck length in any space-time.

Two models are presented of the Planck era in the Big Bang. The first of them starts from the state of an essentially cold and non-expanding universe (assumed to be closed). It is shown that the conditions in the Planck era rendered the total annihilation of the matter with the Planck density absolutely necessary. The annihilation was, further, a necessary condition for the expansion of the post-Planck universe, which, by itself, also was a necessity. The hot stage of the Big Bang in the post-Planck era was, in fact, a necessary consequence resulting from the scheme. The second model is motivated by the weak points of the first one. It is concluded that, owing to the extreme physical conditions, the cosmic medium within the Planck era could only consist of ultrahighly energetic photons. Finally, it is proposed that the second model is to be preferred.

A possible way of building Planck's constant into the structure of space-time is considered. This is done by assuming that the torsional defect that intrinsic spin produces in the geometry is a multiple of the Planck length.

The significance of Planck length in a quantum gravity model is investigated by concentrating on the conformal degree of freedom. It is shown that Planck length is a lower bound to physical proper length in any space-time. It is impossible to construct an apparatus which will measure length scales smaller than Planck length. These effects exist even in flat space-time because of vacuum fluctuations of gravity. It is shown that these fluctuations lead to a high energy cut-off in flat space field theories, thereby removing the divergence problem. The one-loop corrections to a self interacting scalar field is computed and shown to be finite.

The ethereal nature and origin of all the TWs of a particle is emphasized. The notion of superluminal pure momentum carrier radiation is introduced. Planck's constant as fundamental factor of correlation and interconnection of the TWs is presented and discussed.

We present a model of gravity in which the Planck mass is generated spontaneously by a similar mechanism that is responsible for the tension in a recent membrane model of strings. The action describes very floppy fluctuations of the physical space in some large flat embedding space. The coupling constant is dimensionless but the fluctuations are so violet that they produce spontaneously a mass which plays the role of the dimensionally transmuted coupling constant and can be identified with the Planck mass.

A statistically founded derivation of the quanta of energy is presented, which yields the Planck formula for the mean energy of the blackbody radiation without making use of the quantum postulate. The derivation presupposes an ensemble of particles and leads to a statistical interpretation of the Planck constant, which is defined and discussed. By means of the proposed interpretation ofh and as an application of it, the quantum uncertainty relation is derived classically and results as a statistical inequality. On the whole this paper is compatible with the statistical ensemble interpretation of quantum mechanics.

We finally conclude with some speculation concerning when one might expect the extra quantum localization observed in this paper to vanish and for classical and quantum mechanics to correspond in classically chaotic regions. The simple answer is .when flux across most curves approaches h (see, however, the quantum effects observed in ref 5 7 ) . But how does the quantum mechanics approach this limit? The work of ref 57 suggests tunneling which increases with energy might be the route for such a limit. This would be manifested in quantum eigenstates which become more delocalized in phase space as energy is increased, though their peaks might remain in the same location even after a great deal of spreading. Such eigenstates would be consistent with the work of Berry,94 which suggests that Wigner transforms of chaotic eigenstates would have essentially flat distributions. However, we do not necessarily expect this flattening to occur in

According to Planck's first theory (1900) the best expression for the average energy of an harmonic oscillator at high temperature is not kT but kT-1/2h nu . A second theory of Planck (1911) restores the expression kT, and also predicts zero-point energy. A discussion is given in terms of the correspondence principle, which, it is stressed, is active, rather than passive, in nature. Planck's second theory is then discussed in more detail, including the use made of it by Bohr (1918), both in the construction of his atomic theory, and in his development of the correspondence principle itself.

The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin-equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal. The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.

Any attempt at unification of gravity with quantum physics inevitably leads to the Planck length, usually interpreted as defining the distance scale at which quantum corrections to general relativity are expected to become important. Here we arrive at a scalelength of the same magnitude from the cosmological requirement that gravitating vacuum or zero-point energy does not overdominate the dynamics of the Universe. Other cosmological considerations are again seen to imply such a constraining lower scalelength.

The "threshold epoch" of classical cosmology is first discussed from a classical point of view. It is shown that if the density fluctuations at this epoch are proportional to ${N}^{\ensuremath{-}n}$ (where $N$ is the number of particles in a disturbance), then $n$ must exactly, or very closely, equal the value $\frac{2}{3}$ in order that such fluctuations develop into protogalaxies. This result is independent of the value of the threshold epoch, and of the equation of state for the very early universe. Arguments of a speculative nature are then presented which indicate that the threshold epoch occurs at the Planck density. It is also proposed that the initial conditions of galaxy formation are metric fluctuations at the threshold epoch of classical cosmology. The evidence in favor of this theory is (i) the density fluctuation obeys the $n=\frac{2}{3}$ rule, and (ii) the spin fluctuation gives an eventual angular momentum of the order ${N}^{\frac{4}{3}}\ensuremath{\hbar}$, in reasonable agrement with that of the Galaxy.

This article reviews recent work on the possible variability of basic physical constants, such as the charge of the electron e, Planck's constant h and the gravitational constant G. The problem was first raised by Dirac as early as 1937. Its present state is here discussed, together with new suggestions which include Gamow's hypothesis that the fine-structure constant varies directly with time (e2/hc ∞ t), the hypothesis (e2 ∞t, h ∞ t, G ∞ t) and the hypothesis (e4 ∞ t, h2 ∞t, G=constant). The conclusion is that at the present time the invariability of the charge of the electron and Planck's constant is hardly to be questioned, whereas the invariability of the constant of gravitation cannot be considered as finally proved. Some recent geological data point to a possible change in G with time.

The relationship between the various Planck mean coefficients in the optically thin limit is discussed in terms of isothermal curves of growth. The specification of a general thin limit is shown to require two restrictive statements—one specifies the optical depth to be small and the other states the degree of radiative nonequilibrium. In general the limit is scaled by both the ordinary and modified Planck means. For linearized problems near radiative equilibrium, however, these two means reduce to the linear Planck mean that alone acts as the inverse scaling length.

An exact $c$-number generalized Fokker-Planck equation is derived for six noncommuting variables: the field variables, atomic populations, and atomic polarization of a laser model treated extensively by Lax. Langevin procedures yield the correct Fokker-Planck terms. Adiabatic eliminations are performed on the classical Markoff process given by the Fokker-Planck equation to compare with previous results. It is found that higher-order diffusion coefficients influence the second-order ones slightly during adiabatic elimination.

A dynamical mechanism is discussed to account for the numerical value of the Planck's constant ℏ.

Starting from the hypothesis of a material carrier of the EMF, the Planck formula E = hv is generalized, statistical sense is ascribed to the Planck action quantum, and a general scheme is proposed for removing the divergencies in QFT.

The scattering process of two pointlike particles at CM energies in the order of Planck units or beyond, is very well calculable using known laws of physics, because graviton exchange dominates over all other interaction processes. At energies much higher than the Planck mass black hole production sets in, accompanied by coherent emission of real gravitons.

It is suggested that a third universal constant 1 be added to quantum field theory, in addition to h and c. It is proposed that the h-c-1 theory might explain mass spectra and give coupling constants. Using the h-c-1 form of Wien's law and thermodynamic postulates, an h-c-1 form of Planck's law is derived. (T.F.H.)

Numerical values of the gravitation constant and Planck's constant are derived through a cosmological approach. The gravitation constant is computed though the mass-energy relation in which loss of mass in a region is compensated by a gravitational potential energy change. Similarly, through steady state cosmology, Planck's constant is determined. A possible application to other physical constants is implied. (L.N.N.)

A lower bound for the numerical value of Planck's constant is estimated on theoretical grounds alone, if elementary particles are gravitating entities.

A cosmological model with variable c, h, G is proposed. The characteristic lengths of physics (Compton, Jeans, Schwarzschild) are assumed to vary like R(t). Both light and matter’s worlds are found to obey the same law R≈t2/3. The Planck constant is found to vary like t and the gravitation one like 1/R, while the Planck length vary like R. The particle masses follow m≈R. The Hubble law still applies. The redshifts come from the secular variation of the Planck constant.

If one compares the biographical, historiographical, and philosophical literature about Max Planck with that about Albert Einstein the relative paucity of the former is quite conspicuous. One of the main reasons for this state of affairs is no doubt the historical coincidence that even more brilliant stars soon appeared on the physical scene. Another reason may be found in the personality of Planck himself. Yet there seems to be still another reason which is of a more general nature, being related to sociology rather than to physics. Planck protrudes into our century as a representative of that older generation of physicists for whom materialism in natural science was a matter of course. Hence the various idealistic trends appearing at the turn of the century and affecting the general intellectual climate in increasing measure could not claim Max Planck for their cause. If in recent years the interest in Planck’s scientific work has become more pronounced, one of the main reasons is no doubt the fact that more and more scientists are turning away from positivism which had always been opposed by Planck. In a strictly physical sense, we are today likewise in a much better position to judge the work of Max Planck; most of his problems have meanwhile been solved by the quantum theory of fields in a more satisfactory manner than has been possible by the old [‘classical’] quantum theory or by nonrelativistic quantum mechanics. In view of this the question may be raised whether Planck’s aversion to quantum mechanics, shown up to the end of his life, is solely due to his maintaining the metaphysical ideal of deterministic causality or perhaps to other, and maybe progressive, reasons.

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Inspired by Jacobson's thermodynamic approach[gr-qc/9504004], Cai et al [hep-th/0501055,hep-th/0609128] have shown the emergence of Friedmann equations from the first law of thermodynamics. We extend Akbar--Cai derivation [hep-th/0609128] of Friedmann equations to accommodate a general entropy-area law. Studying the resulted Friedmann equations using a specific entropy-area law, which is motivated by the generalized uncertainty principle (GUP), reveals the existence of a maximum energy density closed to Planck density. Allowing for a general continuous pressure $p(\rho,a)$ leads to bounded curvature invariants and a general nonsingular evolution. In this case, the maximum energy density is reached in a finite time and there is no cosmological evolution beyond this point which leaves the big bang singularity inaccessible from a spacetime prospective. The existence of maximum energy density and a general nonsingular evolution is independent of the equation of state and the spacial curvature $k$. As an example we study the evolution of the equation of state $p=\omega \rho$ through its phase-space diagram to show the existence of a maximum energy which is reachable in a finite time., Comment: 15 pages, 1 figure, minor revisions, To appear in JHEP

Recently, a new model for flat-metric expansion of the universe that eliminates several problems arising in the expansion theory of standard cosmology was presented. In the original version of this dark source flux (DSF) model, Big Bang is manifested by a mathematical singularity with infinite initial matter density and infinite initial expansion rate. However, an integration constant (a time shift) with extremely small value was for simplicity set equal to zero in this theory. A closer analysis reveals that if this constant is assigned a very small nonzero value, Big Bang will not have started from a mathematical singularity, but from a primordial state of extremely high, yet limited density and expansion rate. This primordial state constitutes a non-singular Planck epoch of extremely short duration. It turns out that all primordial state parameters assume values of the order of Planck units. Thus, the Planck unit system may have found an attractive physical significance: it relates to the primordial Planck state of the universe where the three fundamental theories of physics merge. There is a well known, puzzling ratio (of the order 10120) between ground state (‘vacuum’) density values based on quantum field theory on one hand, and experimental values derived from ΛCDM (or DSF) on the other. This discrepancy has been called the worst theoretical prediction in the history of science. The present article offers a possible explanation to this issue.

In this paper, charging and discharging processes of different capacitors in electrical RC circuit are considered theoretically and experimentally. The non-local behaviors in these processes, arising from the time fractality, are investigated via fractional calculus. In this context, the time fractional differential equation related to electrical RC circuit is proposed by making use of Caputo fractional derivative. The resulting solution exhibits a feature in between power law and exponential law forms, and is obtained in terms of Mittag-Leffler function which describes physical systems with memory. The order of fractional derivative characterizes the fractality of time and being considered in the interval 0 < ? ? 1. The traditional conclusions are recovered for ? = 1, where time becomes homogenous and system has Markovian nature. By using time fractional approach, the discrepancies between the experimentally measured data and the theoretical calculations have been removed.

A revised theory of relativity was proposed by J. Palacios, which avoids the clock paradox of special relativity by regarding a reference system A to be in a state of absolute rest. It was stated that the Planck constant h would have to undergo a transformation. It is now shown that the noninvariance of Planck's constant is no longer required, even if Palacios' transformations are correct.

A random survey of twelve well-known chemistry texts revealed that only one took the trouble to add a quantity unit to the value quoted for Planck's Constant, and other shocking findings.

By adding the trivial term $\ensuremath{-}d{t}^{2}$ one can convert the four-dimensional (positive-definite) Newman-Unti-Tamburino line element into a static solution of the five-dimensional vacuum Einstein equations. Interpreted $\stackrel{`}{a}\mathrm{la}$ Kaluza and Klein, the solution describes a monopole with a charge-to-mass ratio of $\sqrt{2}$ in rationalized Planck units. Although it is a perfectly regular five-geometry it appears singular from a four-dimensional perspective.