Derivation of the Photon Mass-Energy Threshold

8/14/2019 Derivation of the Photon Mass-Energy Threshold

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Derivation of the Photon Mass-Energy Threshold

Riccardo C. Storti1, Todd J. Desiato

Abstract

An analytical representation of the mass-energy threshold of a Photon is derived utilising

finite reciprocal harmonics. The derived value is < 5.75 x 10-17

(eV) and is within 4.3(%) of the

Eidelman et. al. value endorsed by the Particle Data Group (PDG) < 6 x 10-17

(eV). The PDG

value is an adjustment of theoretical predictions to fit physical observation. The derivation

presented herein is without adjustment and may represent physical evidence of the existence of

Eulers Constant in nature at the quantum level.

[email protected], [email protected]:


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1 ITRODUCTIO

It shall be demonstrated that the Polarizable Vacuum (PV) model of gravitation, [1]

complimenting General Relativity (GR) in the weak field, is capable of predicting the Photon mass-

energy threshold to within 4.3(%) of the Particle Data Group2 (PDG) prediction presented byEidelman et. al. of < 6 x 10

-17(eV). [2]

The derived Photon mass-energy threshold m, based on the physical properties of the

Electron, may be usefully described by a finite reciprocal harmonic series representation as thenumber of harmonic modes approaches infinity, producing the result m < 5.75 x 10

-17(eV).

The proceeding section sets the foundation from which a complete construct may be formed

based on practical modelling methods. The use of physical modelling techniques will be shown to

be highly advantageous in the development of m.

2 MATHEMATICAL MODELLIG32.1 BUCKINGHAM THEORY

Commencing from first principles, we apply Buckingham's Theory (BPT). The

underlying principle of BPT is the preservation of dynamic, kinematic and geometric similarity between a mathematical model and an Experimental Prototype (EP). When applying BPT, one

selects a set of significant parameters avoiding repetition of dimensions as illustrated in table (1) in

accordance with the application of BPT. [3-5]

2.2 FORMULATION OF GROUPINGS

The formulation of groupings begins with the determination of the number of groups to

be formed. The difference between the number of significant parameters (a, B, E, , r and Q) and

the number of dimensions (kg, m, s and C), represents the number of groups required (two).

where,

Significant Parameter Description Units Composition4

a a B E, , r, t,( ) Magnitude of acceleration vector m/s2

kg0

m1

s-2

C0

B B r, t,( ) Magnitude of magnetic field vector T kg1

m0

s-1

C-1

E E r, t,( ) Magnitude of electric field vector V/m kg1

m1

s-2

C-1

r r x y, z, t,( ) Magnitude of position vector m kg0

m1

s0

C0

Q Q r t,( ) Magnitude of electric charge C kg0 m0 s0 C1

Propagation frequency of field Hz kg0 m0 s-1 C0

Table 1, significant parameters.

We may write the general formulation of significant parameters as,

a K0 X( ) Bx 1. Ex 2. x 3. rx 4. Qx 5.

(1)

2A collaboration of leading Nuclear and Theoretical Particle physicists funded by the USDoE,

CERN, INFN (Italy), US NSF, MEXT (Japan), MCYT (Spain), IHEP and RFBR (Russia). [2]3

All mathematical modelling and output was formed using MathCad 8 Professional and appears

in standard product notation.4

The traditional representation of mass (M), length (L) and time (T), in BPT methodology has been

replaced by dimensional representations familiar to most readers (kg, m and s). C denotes

Coulombs, the MKSA units representing charge.


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where, K0(X) represents an experimentally determined dimensionless relationship function and X

denotes all variables, within the experimental environment that influences results and behaviour.

This also includes all parameters that might otherwise be neglected, due to practical calculation

limitations, in theoretical analysis. Hence, the general formulation may be expressed in terms of its

dimensional composition as follows,

kg0

m1

s2

C0

K0 X( ) kg1

m0

s1

C1

x 1. kg

1m

1s

2C

1x 2

. kg0

m0

s1

C0

x 3. kg

0m

1s

0C

0x 4

. kg0

m0

s0

C1

x 5.

(2)

Applying the indicial method [4] yields,

x1 x2 0

x2 x4 1

x1 2 x2. x3 2

x1 x2 x5 0

solve x2, x3, x4, x5, x1 x1 2 x1 1 0

(3)

Substituting the expressions for xn into the general formulation and grouping terms yields,

a

r2.

K0 X( )B . r.

E

x 1

.

(4)

Note that the variable for electric charge has dropped out of the general formulation. This implies

that the acceleration derived is not to be associated with the Lorentz force.

2.3 WAVEFUNCTION PRECIPITATION

Storti et. al.[1] illustrated that the preceding equation may be precipitated into a number ofuseful forms by the application of limits and boundary values. A particular form of importance,

known as the wavefunction precipitation (x1 1, B/E 1/c), may be generated utilising the

preceding equation and defined, such that a aPV and PV as follows5,

a PV K0 PV r, E, B, X,

PV

3r2.

c.

(5)

Where, aPV and PV denote the magnitude of the acceleration vector and harmonic frequency

modes of the PV respectively.

It was illustrated in [1] that the experimental relationship function K0(PV,r,E,B,X),

formed by the method of incorporation may be shown to be,

K0(PV,r,E,B,X) = K0(X) (6)

By application of the Equivalence Principle6, we may determine the value of K0(X) of a

homogenous solid spherical mass by using the weak field approximation to the gravitational

potential [1,6] such that for an observer at infinity,

K0 PV r, E, B, X, e

3G M.

r c2.

.

(7)

5For investigations involving transverse plane wave solutions in a vacuum, by this we mean the PV

backgroundfield, Maxwells equations require E/B = c, when r /2 in the frequency range

0


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where,

Variable Description Units

G Universal Gravitational Constant Nm2kg

-2

M Rest mass of solid spherical object kg

c Velocity of light in a vacuum ms-1

Table 2, variable definitions.

2.4 CONSTANT ACCELERATION

A constant function may be expressed as a summation of trigonometric terms [1,7] over the

longest period in a spectrum of frequencies. It is convenient to model a gravitational field utilizing

modified Complex Fourier Series, according to the harmonic distribution nPV = -n, 2 - n ... n,

where n is a terminating odd number harmonic in the PV model of gravitation. Hence, the

magnitude of the gravitational acceleration vector g may be usefully represented by equation (8)

as7|nPV|,

g r M,( )G M.

r2

n PV

2 i.

n PV.

e n PV

. PV 1 r, M,( ). t. i.

..

(8)

Hence, the amplitude spectrum [7] CPV may be written as,

C PV n PV r, M,G M.

r2

2

n PV.

.

(9)

such that,


PV(1,r,M) Fundamental field harmonic of PV Hz

t Time s


2.5 FREQUENCY SPECTRUM

In accordance with the harmonic representation of g illustrated by equation (8),

K0(PV,r,E,B,X) is a frequency dependent experimental function approximating unity. Hence, an

expression for the frequency spectrum may be derived in terms of harmonic frequency mode. This

may be achieved by applying theEquivalence Principle and assuming that an arbitrary accelerationdescribed by equation (5) aPV is dynamically, kinematically and geometrically similar to the

amplitude of the 1st

harmonic CPV(1,r,M) described by equation (9) as follows,

aPV CPV(1,r,M) (10)

Therefore, utilising equation (5), (7) and (10), it follows that all frequency modes may be

represented by,

PV n PV r, M,n PV

r

3 2 c. G. M.

r.. e

G M.

r c2..

(11)

7 Representing the magnitude of a periodic square wave solution with constant amplitude.


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2.6 ENERGY DENSITY

The gravitational field surrounding a homogenous solid spherical mass may be characterised

by its energy density. Assuming that the magnitude of the field is directly proportional to the mass-

energy density of the object, then the energy density U may be evaluated over the difference

between successive odd frequency modes as follows,

U n PV r, M, U r M,( ) n PV 2

4

n PV

4.

(12)Where,

U r M,( )h

2 c3. PV 1 r, M,( )

4.

(13)


U(nPV,r,M) Energy density per change in odd harmonic mode Pa

h Plancks Constant Js


2.7 CUT-OFF MODE AND FREQUENCY

Utilizing the approximate rest mass-energy density of a homogenous solid spherical object

Um, the terminating harmonic mode of the PV, n may be derived as follows,

U m r M,( )3 M. c

2.

4 . r3. (14)

Storti et. al. conjectured in [1] that experimental investigations into the characteristics of thePV may be conducted at a specific frequency mode provided dynamic, kinematic and geometric

similarity is preserved between a mathematical model and an EP.Therefore, assuming that the total rest mass-energy density may be characterised by one

change in harmonic mode |Um(r,M)| = |U(nPV,r,M)|, equation (12) may be solved for the

maximum value of |nPV|. This is termed the harmonic cut-off mode n and may be applied toequation (8) as the terminating mode in the finite reciprocal harmonic Fourier Series as follows,

n r M,( ) r M,( )

12

4

r M,( )1

(15)

Where, (r,M) is termed the harmonic cut-off function,

r M,( )

3

108U m r M,( )

U r M,( ). 12 768 81

U m r M,( )

U r M,( )

2

..

(16)

Consequently, the upper boundary of the PV frequency spectrum , termed the

harmonic cut-off frequency, may be calculated as follows, r M,( ) n r M,( ) PV 1 r, M,( )

.(17)

The derivation of equations (15-17) is based on the compression of energy density to one change in

odd harmonic mode whilst preserving dynamic, kinematic and geometric similarity in accordancewith BPT. The subsequent application of these results to equation (8) acts to decompress the energy

density over the Fourier domain yielding a highly precise reciprocal harmonic representation of g.


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3 PHYSICAL MODELLIG

3.1 CONJUGATE PHOTON PAIR POPULATIONS

The PV spectrum is conjectured to be composed of mathematical wavefunctions, over the

symmetrical frequency domain -


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m g N g2

N g

St g n re m e,.

(21)

where,


Ng Photon pair population None

re Classical Electron radius m

me Rest mass of an Electron kg


Evaluating equation (21) assuming that the population of conjugate Photon pairs is mode

normalised to unity (Ng = 1) yields,

mg 1.2 x 10-15

(eV) (22)

3.2 PHOTON MASS-ENERGY THRESHOLD

To predict the mass-energy threshold of a Photon m, we shall utilise the conjugate Photon

pair population principles defined above. Firstly, we shall establish some useful mathematical

relationships that facilitate the concise representation of m.

It has been illustrated that the summation of the odd harmonic modes are representative ofthe magnitude of the acceleration vector g. [1] Therefore, summing the spectrum over the odd

modes across both sides of the spectrum leads to the following representation with vanishing error,

[8] proportional to the sum of all modes on the positive side of the spectrum as |nPV| n and

n >> 1,

n PV

1

n PV

ln 2( )

1

n r M,( )

n PV

1

n PV=

ln 2 n r M,( ).

(23)

where,

i. The LHS8 of the preceding equation denotes the summation of all odd modes across the entirespectrum, symmetrical about the 0

thmode, following the sequence nPV = -n, 2 - n ... n.

ii. The middle expression of the preceding equation represents the summation of all odd and evenmodes on the RHS side of the spectrum following the sequence nPV = 1, 2 n.

iii. On the RHS of the preceding equation denotes Eulers Constant.There are half as many odd modes as there are odd + even modes when |nPV| n.

Hence, we may deduce m by the following ratio,

m g

m

1

2ln 2 n re m e,

. .>

(24)

Performing the appropriate substitutions and recognising that the preceding equation may be further

reduced by usefully approximating the exponential term in equation (11) to unity, yields the Photon

mass-energy threshold to be,

m

512 h. G. m e.

c re. 2.

n re m e,

ln 2 n re m e,.

.

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Derivation of the Photon Mass-Energy Threshold