The Physical Foundation of a Grand Unified Statistical Theory of Fields and the Invariant Schrödinger Equation SIAVASH H SOHRAB Robert McCormick School of Engineering and Applied Science Department of Mechanical Engineering Northwestern University, Evanston, Illinois 60208 UNITED STATES OF AMERICA Abstract: A scale-invariant statistical theory of fields from cosmic to tachyonic scales is presented Invariant Schrödinger equation is derived and the invariant forms of Planck constant, de Broglie matter wave hypothesis, and Heisenberg uncertainty principle are presented The definition of Boltzmann constant is introduced as k mνk ck 1.381 10 23 J/K Parallel to de Broglie relation = h/p, the relation = k/pis introduced for the frequency of matter waves The photon mass, the Avogadro number, and the universal gas constant are o 23 o predicted as m k hk / c 1.84278 10 41 kg, N 1/(m k c ) 6.0376 10 , and R N k 1/ k 8.3379 m Key-Words: Grand unified statistical theory of fields, invariant Schrödinger equation, TOE Introduction Turbulent phenomena is a common feature between diverse and seemingly unrelated branches of physical sciences This is in part evidenced by the similarities between the stochastic quantum fields [1-16] on the one hand, and the classical hydrodynamic fields [17-26], on the other Turbulence involves the chaotic motion of galactic clusters [26, 27], fluid eddies [17-26], and photons [28] at cosmic, hydrodynamic, and chromodynamic scales In this paper the scale-invariant model of statistical mechanics [29, 30] is applied to present the physical foundation of a grand unified statistical theory of fields A Scale-Invariant Model of Statistical Mechanics from Cosmology to Tachyon Chromodynamics The invariant model of statistical mechanics [29] for the statistical fields of equilibrium galacto-, planetary-, hydro-system-, fluid-element-, eddy-, cluster-, molecular-, atomic-, subatomic-, kromo-, and tachyon-dynamics corresponding to the scales = g, p, h, f, e, c, m, a, s, k, and t are shown in Fig.1 Also shown are the corresponding non-equilibrium, laminar flow fields Each statistical fielddescribed by a distribution function f(u) = f(r, u, t) drdu, constitutes a "system" that is composed of an ensemble of "elements", each element is composed of an ensemble of "atoms" that are pointmass particles[29] The element (system) at scale (j) becomes the atom (element) of the larger scale (j+1) The evidence for the existence of the statistical field of equilibrium cluster-dynamics ECD (Fig.1) is the phenomena of Brownian motions [23, 31-34] Modern theory of Brownian motion starts with the Langevin equation [23] du p dt u p A(t) (1) where up is the particle velocity The drastic nature of the assumptions inherent in the division of forces in Eq.(1), emphasized by Chandrasekhar [23], results in two physical problems that confront the classical theory First, the viscous effects in the friction term up will lead into dissipation, contrary to the empirical fact that Brownian motion is an equilibrium phenomenon [34] Second, the period of fluctuation of A(t), of the order of 10 21 s [23], is too small as compared with the characteristic time of Brownian motions 10 5 s According to the modified theory [30], the particles are in equilibrium with molecular clusters that themselves possess Brownian motion The fields of ESD and EKD are identified as the well known fields of stochastic electrodynamics SED and stochastic chromodynamics SCD [1-16] Photons are considered to be composed of a large number of much smaller particles called tachyons [35] UNIVERSE GALACTIC CLUSTER EGD (J +5) GALAXY EPD (J+4 ) PLANET EHD (J +3) HYDRO SYSTEM EFD (J +2) FLUID ELEMENT EED (J+1) EDDY LGD (J + 11/2) Lg Lp Lh Lf Le Lc ECD (J) CLUSTER EMD (J 1) MOLECULE EAD (J 2) ATOM ESD (J 3) Lm La Ls SUBPARTICLE Lk EKD (J4 ) PHOTON Lt ETD (J 5) Lp Lh Lf Le Lc Lm GALACTIC CLUSTER LPD (J + 9/2) GALAXY LHD (J + 7/2) PLANET LFD (J + 5/2) HYDRO SYSTEM LED (J + 3/2) FLUID ELEMENT LCD (J + 1/2) EDDY LMD (J 1/2) CLUSTER LAD (J 3/2) La MOLECULE LSD (J 5/2) Ls Lk ATOM LKD (J 7/2) SUBPARTICLE LTD (J 9/2) Lt PHOTON TACHYON Fig.1 A scale-invariant view of statistical mechanics from cosmic to tachyonic scales The physical space is identified as this tachyonic fluid, that is the ether of Dirac [36] “ We can now see that we may very well have an aether, subject to quantum mechanics and conforming to relativity, provided we are willing to consider the perfect vacuum as an idealized state, not attainable in practice From experimental point of view, there does not seem to be any objection to this We must make some profound alterations in our theoretical ideas of the vacuum It is no longer a trivial state, but needs elaborate mathematics for its description.” It is emphasize that space is the tachyonic fluid itself and not merely a container that is occupied by this fluid, as in the classical theories of ether [37] Using a glass of water as an example, the physical space is analogous to the water itself, and not to the glass In the ontology of universe according to Newton [38], absolute space was considered to be a nonparticipating container filled with a medium called ether in order to describe the phenomena of light as well as gravitation [38] “Qu 19 Doth not the refraction of light proceeds from different density of this aethereal medium in different places, the light receding always from the denser parts of the medium? And is not the density thereof greater in free space and open spaces void of air and other grosser bodies, than within the pores of water, glass, crystal, gems, and other compact bodies? For when light passes through glass or crystal, or falling very obliquely upon the farther surfaces thereof is totally reflected, the total reflection ought to proceed rather from the density and vigor of the medium without and beyond the glass, than from the rarity and weakness thereof Qu 22 May not planets and comets, and all gross bodies, perform their motions more freely, and with less resistance in this aethereal medium than in any fluid, which fills all space adequately without leaving any pores, and by consequence is much denser than quick-silver or gold ? And may not its resistance be so small, as to be inconsiderable ? For instance: if this aether ( for so I will call it ) should be supposed 700,000 times more elastic than our air, and above 700,000 times more rare, its resistance would be above 600,000,000 times less than of water And so small a resistance would make any sensible alteration in the motions of the planets in ten thousand years If any one would ask how a medium can be so rare, let him tell me how the air, in the upper parts of the atmosphere, can be above hundred thousand thousand times rarer than gold.” The atmosphere of opposition by scientists against the hypothesis of ether was most eloquently described by Maxwell in his treatise [39]: “ There appears to be, in the minds of these eminent men, some prejudice, or a priori objection, against the hypothesis of a medium in which the phenomena of radiation of light and heat and the electric action at a distance take place It is true that at one time those who speculated as to the cause of the physical phenomena were in the habit of accounting for each kind of action at a distance by means of a special aethereal fluid, whose function and property it was to produce these actions They filled all space three and four times over with aethers of different kinds, so that more rational inquirers were willing rather to accept not only Newton's definite law of attraction at a distance, but even the dogma of Cotes, that action at a distance is one of the primary properties of matter, and that no explanation can be more intelligible that this fact Hence undulatory theory of light has met with much opposition, directed not against its failure to explain the phenomena, but against its assumption of the existence of a medium in which light is propagated.” The existence of a medium called ether was found to be indispensable for the proper description of electrodynamics according to Lorentz [40] “ I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with certain degree of substantiality, however different it may be from all ordinary matter,” The participation of ether in the transmission of perturbations as well as the possible granular structure of space were anticipated by Poincaré [41] “ We might imagine for example, that it is the ether which is modified when it is in relative motion in reference to the material medium which it penetrates, that when it is thus modified, it no longer transmits perturbations with the same velocity in every direction.” Also, the notion of ether was considered by Einstein as not only consistent with the General Theory of Relativity, but in his opinion, according to GTR, space without ether is unthinkable [42] “ Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time interval in the physical sense But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time The idea of motion may not be applied to it.” The statement "space without ether" shows that ether was considered as a medium that filled the space, rather than being the space itself Similar idea is also conveyed by "even in empty space" in the following statement by Hawking [43] “ Maxwell's theory predicted that radio or light waves should travel at a certain speed But Newton's theory had got rid of the idea of absolute rest, so if light was supposed to travel at a fixed speed, one would have to say what that fixed speed was to be measured relative to It was therefore suggested that there was a substance called "ether" that was present everywhere, even in "empty" space Light waves should travel through the ether as sound waves travel through air, and their speed should therefore be relative to the ether.” Space is herein considered to be a compressible medium, in harmony with compressible ether model of Planck [44] Parallel to atmospheric air that becomes compressible when the Mach number Ma = v/a approaches unity, the tachyonic fluid that constitutes the physical space becomes compressible when the Michelson number defined as Mi = v/c approaches unity, with a and c denoting the velocity of sound and light, respectively [45] Thus, Ma >1 (Mi >1) corresponds to supersonic (superchromatic) flow, leading to Mach (Poincaré-Minkowski) cone that separates the zone of sound (light) from the zone of silence (darkness) [45] It can be shown that the compressibility of space, evidenced by the fact that c is finite, will result in the FitzgeraldLorentz contraction [45], thereby providing a causal explanation of relativistic effects [46] in harmony with the perceptions of Poincaré and Lorentz The granular and compressible structure of space is suggested by the cascade of fractal [47] statistical fields from cosmic [48] to tachyonic scales (Fig.1) Invariant Definition of Density, Velocity, Temperature, and Pressure Following the classical methods [49-53], the invariant definitions of the density , and the element v, the atom u, and the system w velocity at the scale are [29] n m m f du v 1m u f du , u= v (2) w= v (3) The invariant equilibrium and non-equilibrium translational temperature and pressure are 3kT m u 2 , P u 2 / , 3kT m V2 ,and P V / , leading to the corresponding invariant ideal "gas" laws [29] P R T P RT and (4) At the scale of EKD, one obtains the temperature and pressure of photon gas Pρ k uk / 3 ρ c k n m kc k ν k (5) k n k hνk k n E k k k E k (6) h k m k k c h 6.626 1034 J-s is the where T Planck constant Following the definition of h by Planck [54, 55], one introduces the definition of Boltzmann constant as k k k mνk ck 1.381 10 23 J/K (7) and the Kelvin absolute temperature scale becomes equivalent to a length scale, nkTk nk k E k Also, following de Broglie hypothesis for the wavelength of matter waves [2] h / p (8) where p= mv is the momentum, we introduce the relation ν k / p (9) for the frequency of matter waves Therefore, the mass of photon is predicted as 41 m k (hk / c ) 1.84278 10 kg (10) that is much larger than the reported value of 10 51 kg [56] This leads to the prediction of the mean-free-path and the frequency of photons in vacuum , i.e EKD field (Fig.1) 1/ k 0.119935 m , ν k 2.49969 109 Hz (11) o 23 , the Avogardo number N 1/(m k c ) 6.0375 10 , o the universal gas constant R N k 1/ k 8.3379 m , and the photon Wk N m k 1.1126 10 o 17 molecular t .(ρ v ) weight kg/mole Derivation of Invariant Schrödinger Equation from the Invariant Bernoulli Equation The invariant forms of the mass and the momentum conservation equations are [29] (12) (ρ v ) .(ρ v v ) V (13) t For irrotational and incompressible flow, one arrives at the invariant Bernoulli equation (ρ ) 2 kTk m u 2 / m u kx u ky u kz /3 m k (3c2 ) / m k c ρ t (ρ ) 2 V Constant = (14) where is the velocity potential, and v The constant in (14) is set to zero since the potential V is only defined to within an arbitrary constant Comparison of Eq.(14) and the Hamilton-Jacobi equation for a particle in classical mechanics [2, 5759] S (S) V 0 t 2m (15) leads to the definition of the invariant modified action [60] S (x, t) (16) the gradient of which is the volumetric momentum density S (x, t)ρ v n p p Next, one introduces the expansions f f o f = , f S , , v , V (17) and defines the wave function of quantum mechanics (x, t) as the perturbed action (x, t) S (x, t) ( x, t) (18) The mean velocity v o is assumed to be constant that v v , such and hence v (x, t) The substitution from the expansion (17) into the Bernoulli equation (14) and the separation of terms with equal powers of results in So t t (So ) 2 Vo So Introducing z x vo t , (19) V the such (20) stationary that coordinate x So zSo and So / t v ozSo , Eq.(19) results in the volumetric total energy density E o of the field E o To Vo = vo2 n m v o2 n E o (21) Following de Broglie [2], the invariant modified de Broglie hypothesis is introduced as o h o / m v o n h o /(ρ vo ) (31) when the volumetric kinetic To , and potential involving ho With the mean velocity expressed as vo = oo, Eq.(22) becomes energy Vo density are Tρ v / 2o2 (nm )(v o o o To v / n m v / n To o Vo v / n Vo (23) By substitutions from Eqs.(24), (29), and (32) in Eq.(26), one obtains the invariant modified timeindependent Schrödinger equation of Eq.(20) and substitution for / t from Eq (20) itself, results in the wave equation vo22 (25) Substituting the product ( x, t) (x ) (t) inEq.leads to v 2o solution 2 (26) when is the separation constant The solution of the temporal part of (26) is (27) suggesting that the quantity v o o 2ν o is a circular frequency Following Planck [54, 55], parallel to the expressions for the quantum of action, the energy per photon, and the volumetric energy density (pressure Pk) of radiation field h mk kc 2 E hν , E nhν Pk (28) , n is the number of photons per unit volume, one introduces the invariant expressions h o m o v o , E o hνo o E o n hνo P (29) The fact that amongst (c, e, h), the Planck constant h may be found to be related to other fundamental constants of physics was anticipated by Dirac [61] Using the result in Eq.(29), one obtains o vo 2E o / h o o o , such that Eq.(27) may be written as exp( i2E o t / h o ) (30) 8 m (E o Vo ) h o2 (32) (33) From multiplication of Eq.(33) by from (30), one arrives at the time-dependent modified Schrödinger equation iho t h2o 2m Vo (34) In view of de Broglie hypothesis (8), ho= poo= h, Eqs.(33)-(34) become the classical Schrödinger equations [62] exp( i vo t) exp( io t) E o/2 = n hνo o/2 Assuming that V 0 , the first time derivative t o)/ o2 (22) Also, (E o , To , Vo ) are the atomic total, kinetic, and potential energy that are related by E o To Vo (24) 2 o ih t 8 m h2 h2 2m (E o Vo ) (35) Vo (36) A physical reason for the validity of de Broglie hypothesis may be based on the criteria of thermodynamic equilibrium between radiation and matter fields, i.e the equality of the temperature of photons and matter particles, written as kT m u 2x hν kT k m k u kx hν k = constant (37) If the above criteria is not satisfied, then condensation of matter into light i.e photons, or evaporation of light into matter will take place until their respective temperatures become identical The metamorphosis of matter into radiation and viceversa was first recognized by Newton [38] “Qu 30 Are not gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition? The changing of bodies into light, and light into bodies, is very comfortable to the course of Nature, which seems delighted with transmutations And among such various and strange transmutations, why may not Nature change bodies into light and light into bodies?” At a given T, the time scale (ν , ν k ) uniquely determines the length scale ( , k ) Hence, if the characteristic time of photons is chosen as the unit of time t = tk, such that ν = ν k (38) , then Eq.(37) reduces to de Broglie hypothesis h = h Equation (37) also suggests that, parallel to the classical result of Heisenberg [63], one may express an invariant Heisenberg uncertainty principle as h = mν p h (39) that applies to all statistical fields in Fig.1 In view of Eqs.(37) and (38), the physical interpretation of (39) is that the temperature of matter field must be always greater than that of the radiation field within which it resides T Tk The Invariant Planck Law of Energy Distribution To obtain a correspondence between photon gas at EKD scale and other statistical fields, particles with hν m v 2 are viewed as the energy hν virtual oscillators [64], that act as composite bosons [65] and follow the Bose-Einstein statistics From the classical expression for translational degeneracy of monatomic gas [66], it can be shown that when the harmonic translational, rotational, and pulsational motions of oscillators in six dimensions (x , x ) , ( , ) , and (r , r ) are taken into account, one arrives at invariant Planck law of energy distribution [55] dN V 8hν 3 v exp(hν / kT ) 1 dν The identification of the wave function as ρ resolves some of the fundamental difficulties concerning the dual and incompatible characteristics associated with the objective versus the subjective nature of [1-3] This is because the density accounts for the physical or objective 1/ nature of and the quantity ( )ρ is related to the probability of localization The velocity potential that is complex accounts for the non-physical or subjective nature of , allowing it to be normalizable, thereby accounting for the success of Born's [69] probabilistic interpretation of The subjective part also accounts for the nonlocal nature of thus resolving the classical paradoxes of quantum mechanics such as the double-slit experiment, or the EPR [2] Following the classical practice [66], the wave functionmay be separated into translational, rotational, vibrational, and internal parts as = trvi Next, i of scale is identified as -1 of the next smaller scale to give the invariant expression t r v i = t r v 1 (41) At the scale of EGD, the wave function g will correspond to that of a stationary and homogeneous universe discussed by Hartle and Hawking [70] The wave-particle duality of galaxies is evidenced by their observed quantized red-shifts [71] The solution of Schrödinger equation for given in Eq.(41) results in a cascade of wavepackets, eddy waves in a fluid element, cluster waves in an eddy, , and so on, as schematically shown in Fig.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EDDY c ECD (J) CLUSTER (40) as well as the invariant Rayleigh-Jeans number n (8 / v 3 )ν 2 Hence, the distinction between matter, radiation, and space diminish, and to borrow a phrase from Boltzmann [67], they are all made of the same cloth At the EGD scale, the model is harmonious with the observed equilibrium cosmic background radiation at about T = 2.75 K [68] m EMD (J1) MOLECULE a EAD (J2) ATOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig.2 Cascade of wave-packets, wave functions , and particles for equilibrium cluster-dynamic (ECD), molecular-dynamic (EMD), and atomicdynamic (EAD) scales As one moves to smaller scales, one always finds a continuum because each element is composed of an entire statistical field (see Fig.1), such that one can again define a velocity potential , and thereby define a new wave function ρ The cascade of particles as singularities embedded in guidance waves, as shown in Fig.2, is in exact agreement with the perceptions of de Broglie concerning interactions between the particle and the "hidden thermostat" [3] “ Here is another important point I have shown in my previous publications that, in order to justify the well-established fact that the expression (x,y,z,t) d gives, at least with Schrödinger's equation, the probability for the presence of the particle in the element of volume d at the instant t, it is necessary that the particle jump continually from one guidance trajectory to another, as a result of continual perturbation coming from subquantal milieu The guidance trajectories would really be followed only if the particle were not undergoing continual perturbations due to its random heat exchanges with the hidden thermostat In other words, a Brownian motion is superposed on the guidance movement A simple comparison will make this clearer A granule placed on the surface of a liquid is caught by the general movement of the latter If the granule is heavy enough not to feel the action of individual shocks received from the invisible molecules of the fluid, it will follow one of the hydrodynamic streamlines If the granule is a particle, the assembly of the molecules of the fluid is comparable with the hidden thermostat of our theory, and the streamline described by the particle is its guiding trajectory But if the granule is sufficiently light, its movement will be continually perturbed by the individual random impacts of the molecules of the fluid Thus, the granule will have, besides its regular movement along one of the streamlines of the global flow of the fluid, a Brownian movement which will make it pass from one streamline to another One can represent Brownian movement approximately by diffusion equation of the form t = D , and it is interesting to seek, as various authors have done recently, the value of the coefficient D in the case of the Schrödinger equation corresponding to the Brownian movement I have recently studied (14) the same question starting from the idea that, even during the period of random perturbations, the internal phase of the particle remains equal to that of the wave I have found the value D = h/ (3m) , which differs only by a numerical coefficient from the one found by other authors This concludes the account of my present ideas on the reinterpretation of wave mechanics with the help of images which guided me in my early work My collaborators and I are working actively to develop these ideas in various directions Today, I am convinced that the conceptions developed in the present article, when suitably developed and corrected at certain points, may in the future provide a real physical interpretation of present quantum mechanics.” Concluding Remarks The hydrodynamic origin of the invariant Schrödinger equation suggests that the phenomena of superfluidity [72], superconductivity [73], and superluminosity (laser action) may be identified as transitions from turbulent (dissipative) to laminar (non-dissipative) flows of atoms, subatomic particles, and photons in the stochastic atomicdynamics SAD, subatomic-dynamics SSD, and kromo-dynamics SKD The predicted mass of 1/ 41 photon m k (hk / c ) 1.84278 10 kg will have an impact on the classical problem of dark matter in cosmology The new theory appears to conform not only to the large body of empirical observations, through the invariant Schrödinger equation, but also provides a more unified and harmonious description of many branches of natural sciences across diverse spatio-temporal scales According to Fig.1, not only the Almighty plays dice, but appears to be playing with an infinite cascade of embedded dices Acknowledgments: This research has in part been supported by the NSF grant No 8820077 and NASA grant No NAG3-1863 References: [1] [2] [3] [4] [5] Broglie, L de, C R Acad Sci Paris 183, 447 (1926); 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