The Physical Foundation of a Grand Unified Statistical Theory of Fields and the Invariant Schrödinger Equation.DOC

SOHRAB Robert McCormick School of Engineering and Applied Science Department of Mechanical Engineering Northwestern University, Evanston, Illinois 60208 UNITED STATES OF AMERICA Abstract

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

23

k k

k m νc1.38110 c 1.381 10

   J/K Parallel to de Broglie relation = h/p, the relation = k/pis introduced for

the frequency of matter waves The photon mass, the Avogadro number, and the universal gas constant are

k

m hk / c 1.84278 10

k

N 1/(m c ) 6.0376 10  , and o

k

RN k 1/  8.3379

1

m

Key-Words: Grand unified statistical theory of fields, invariant Schrödinger equation, TOE.

1 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

2 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 fielddescribed

by a distribution function f(u) = f(r, u, t)

drdu, constitutes a "system" that is composed of

an ensemble of "elements", each element is

composed of an ensemble of "atoms" that are

point-mass 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]

p

p

d

(t)

dt   

u

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 5s 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]

GALACTIC CLUSTER LGD (J + 11/2) UNIVERSE

LHD (J + 7/2)

LPD (J + 9/2) GALAXY

LCD (J + 1/2) LMD (J - 1/2)

LSD (J - 5/2)

LED (J + 3/2) LFD (J + 5/2)

LAD (J - 3/2)

LKD (J - 7/2) LTD (J - 9/2)

EDDY

FLUID ELEMENT

HYDRO- SYSTEM PLANET

MOLECULE CLUSTER

ATOM SUB-PARTICLE PHOTON ETD (J -5)

EGD (J +5)

EMD (J -1)

EAD (J -2)

ESD (J -3)

GALACTIC

CLUSTER

EDDY

FLUID

ELEMENT

HYDRO

SYSTEM

PLANET

GALAXY

MOLECULE

CLUSTER

ATOM

SUBPARTICLE

PHOTON

TACHYON

ECD (J)

EKD (J-4 )

EED (J+1)

EFD (J +2)

EHD (J +3)

EPD (J+4 )

Lp

Lh

Lf

Le

Lc

Lm

Ls

Lk

Lt

La

Lg

Lp

Lh

Lf

Le

Lc

Lm

Ls

Lk

Lt

La

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

non-participating 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 Fitzgerald-Lorentz 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)

3 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, andthe system w velocity

at the scale  are [29]

     

    , u= v (2)



    

v (3) The invariant equilibrium and non-equilibrium translational temperature and pressure are

3kT m u ,P   u / 32 ,3kT m V 2,an

dP   V2/ 3, leading to the corresponding

invariant ideal "gas" laws [29]

P R T  and P RT (4)

At the scale of EKD, one obtains the temperature

and pressure of photon gas

kT   m u / 3 m u    u  u  / 3

m (3c ) / 3 m c

P  ρu/3ρcnmcν u / 3 ρu/3ρcnmcν c     n m c νc1.38110 

k k k k k k

n h νc1.38110 n E E

h m c h 6.626 10

Planck constant Following the definition of h by

Planck [54, 55], one introduces the definition of

Boltzmann constant as

23

k k m νc1.38110 c 1.381 10

and the Kelvin absolute temperature scale becomes

equivalent to a length scale, nkTk  nk  k Ek

Also, following de Broglie hypothesis for the

wavelength of matter waves [2]

h / p

where p= mv is the momentum, we introduce the

relation

for the frequency of matter waves Therefore, the

mass of photon is predicted as

k

m (hk / c ) 1.84278 10

that is much larger than the reported value of

51

4 10

 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)

k 0.119935

  m , νc1.38110k  2.49969 10  9Hz

(11)

, the Avogardo number o 2 23

k

N 1/(m c ) 6.0375 10  , the universal gas constant o

k

RN k 1/  8.3379

1

m , and the photon molecular weight

W N m 1.1126 10

4 Derivation of Invariant Schrödinger

Equation from the Invariant Bernoulli

Equation

The invariant forms of the mass and the momentum conservation equations are [29]

ρu/3ρcnmcν (ρu/3ρcnmcν ) 0 t



 



(ρu/3ρcnmcν )

.(ρu/3ρcnmcν ) V t

 





v

v v

For irrotational and incompressible flow, one arrives at the invariant Bernoulli equation

2

(ρu/3ρcnmcν ) ( ρu/3ρcnmcν )

V





   

where  is the velocity potential, andv    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, 57-59]

2

S ( S)

V 0

leads to the definition of the invariant modified action [60]

the gradient of which is the volumetric momentum density S (x, t) ρu/3ρcnmcν  v np p

Next, one introduces the expansions

o

f     f f   1 , f  S , , , V v  (17)

and defines the wave function of quantum mechanics ( , t) x as the perturbed action

( , t) S ( , t) ( , t)

The mean velocity vo is assumed to be constant

such that  v    v  0, and hence ( , t)

v x  The substitution from the

expansion (17) into the Bernoulli equation (14) and

the separation of terms with equal powers of  results in

2

o





o

S

t







Introducing the stationary coordinate

o

z x v t    , such that x oS z oS

and

o o z o

S / t v  S

, Eq.(19) results in the volumetric total energy density Eo of the field

E   T V=v2o  n m v  2o  n E o (21)

when the volumetric kinetic To, and potential

energy Vo density are

T v / 2 n m v / 2 n T        (22)

2

Also, (E ,T , V )o o o are the atomic total, kinetic,

and potential energy that are related by

Assuming that V  0, the first time derivative

of Eq.(20) and substitution for / t  from Eq

(20) itself, results in the wave equation

2

2 2

o

2 v

t



 

 

Substituting the product solution

( , t) ( ) (t)

 x  x  inEq.leads to

2 2

o

v



  

when is the separation constant The solution

of the temporal part of (26) is

exp( i v t) exp( i t)

suggesting that the quantity voo   2 νc1.38110o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

k k

h m   c  E hνc1.38110 , E nhνc1.38110 P   k (28)

, n is the number of photons per unit volume, one

introduces the invariant expressions

h  m v , Eo h νc1.38110o o

E  n h νc1.38110    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 2 E / ho o

, such that Eq.(27) may be written as

o o

exp( i2 E t / h )

Following de Broglie [2], the invariant modified

de Broglie hypothesis is introduced as

o h / m vo  o n h /(ρu/3ρcnmcν v ) o  o

involving ho With the mean velocity expressed as

vo = oo, Eq.(22) becomes

2

T  ρu/3ρcnmcν v / 2 (n m )(v       ) / 2

= n h νc1.38110 / 2 E / 2 o o  o (32)

By substitutions from Eqs.(24), (29), and (32) in

Eq.(26), one obtains the invariant modified time-independent Schrödinger equation

2 2

2 o

8 m

h







From multiplication of Eq.(33) by  from (30), one

arrives at the time-dependent modified Schrödinger

equation

2









In view of de Broglie hypothesis (8), ho= poo=

h, Eqs.(33)-(34) become the classical Schrödinger

equations [62]

2 2

2

8 m

h





2 2

o











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    h νc1.38110  kT  m u  

k

hνc1.38110

 = 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 vice-versa 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 (νc1.38110 , νc1.38110 ) 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

k

, 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 = m νc1.38110      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 residesT Tk

5 The Invariant Planck Law of Energy

Distribution

To obtain a correspondence between photon gas at

EKD scale and other statistical fields, particles with

the energy   h νc1.38110  hνc1.38110 m v 2are viewed as

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]

3 3

dN 8 hνc1.38110

dνc1.38110

V v exp(hνc1.38110 / kT ) 1





(40)

as well as the invariant Rayleigh-Jeans number

3 2

n   (8 / v )νc1.38110  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]

The identification of the wave function as

ρu/3ρcnmcν 

   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

nature of and the quantity ( )1/ 2 ρu/3ρcnmcν

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 functionmay be separated into translational, rotational, vibrational, and internal parts as =

trvi 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 wave-packets, eddy waves in a fluid element, cluster waves in an eddy, , and so on, as schematically shown in Fig.2 EDDY CLUSTER MOLECULE ATOM ECD (J) EMD (J-1) EAD (J-2) c m a .

Fig.2 Cascade of wave-packets, wave functions

,and particles for equilibrium cluster-dynamic (ECD), molecular-dynamic (EMD), and atomic-dynamic (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   ρu/3ρcnmcν  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) 2 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 valueD =  / (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.”

6 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 atomic-dynamics SAD, subatomic-atomic-dynamics SSD, and kromo-dynamics SKD The predicted mass of

k

m (hk / c ) 1.84278 10 kg

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

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