Mathematical explanation of “Aumic” Theory :

Mathematically the field Y may be described as an n-dimensional affine space . An “abstract” affine space is a pair of sets . The set of points and the set of vectors. Here the points refer to the “field” of virtual photons and the vectors refer to the eigen energy value of the photon traveling along these points. is the set of real numbers of length n.

The superscript i is not a power but simply an index. The elements of are interpreted as “points” of an “n-dimensional space.” Boldface letters are used to indicate points:

.x = {x¹, x²,………..xⁿ } The numbers xⁱ are called the coordinates of the point x. (Note :A light face letter with an index (e.g., y²) is a generic notation for the coordinate of the corresponding point. )

Sometimes standard coordinates x , y , z maybe used in instead of x ¹, x ² , x ³

Elements of can also be interpreted as vectors. Vectors can be added and multiplied by numbers . There is a distinguished vector “zero” : O = (0,…..0). Vectors are also denoted with an arrow :

Consider the point For an arbitrary vector r : x = O + r are equal to the respective coordinates of the vector r : x = (x¹,……, xⁿ) and r = (x¹,……, xⁿ).

The vector r in such a case is called the radius-vector of the point x. ( Or in greater detail: r is the radius-vector of x w.r.t an origin O.

To summarise : has been considered and its elements interpreted in two ways. Hence we may say that we are dealing with two copies of :

= {points} , = {vectors}

Both points and vectors are represented by the same type of arrays in

The most important properties of the addition of a point and a vector , and of the subtraction of two points , are constant in the formulae:

;

if P + a = Q , then

Thus vectors are treated as directed segments.

A function is differentiable at x if :

f ( x + h ) – f (x ) = A(h) + a (h)|h|

where A(b) = A₁ h¹+……… A_n hⁿ is a linear function of h and a (h) ® 0 when h ®0.

(The function A , depends on x ) The linear function A(h) is called the differential of f at x. Notation: df or df(x) , so df(x)(h) = A(h). The value of df on a vector h is also called the derivative of f along h and denoted ¶ _h f(x) = df(x)(h).

It follows that partial derivatives are just the usual derivatives of a function of a single variable if we allow only one coordinate xⁱ to change and fix all other coordinates.

Consider h = e_i = (0,……..,0,1,0,……..0) (the ith standard basis vector in )The derivative ¶_ei f = df(x) (e_i ) =A_i is called the partial derivative w.r.t. xⁱ . The standard notation:

Suppose x ⁱ are the standard coordinates in so that x = { x¹,……, xⁿ). Then we can understand eⁱ = ¶x /¶xⁱ in a straightforward manner and by differentiation get at each place either 1 or 0 depending on whether we differentiate ¶xⁱ/¶xⁱ for j= i or not : hence e₁ = {1,0,…..,0} , e₂ = {0,1,….0}, …,ev_n ={0,……..0,1}. From the general rule we have recovered the standard basis in . In particular , the elements of the basis eⁱ = dx /dxⁱ can be understood directly as the velocity vectors of the coordinate lines x ⁱ ® x {x¹,……, xⁿ}. All coordinates except x ⁱ are fixed.

An orientation of a point is simply a sign ( plus or minus ) assigned to this point. For example in the straight line segment [PQ] we can assign to P , + 1 and to Q , - 1 , the straight line segment [PQ] is oriented in the direction P ® Q . Similarly consider a domain D in where D = [a,b] is a segment. An orientation is a choice of direction from a to b or from b to a. Similarly where D = {d ^1, d ²………,dⁿ } can be regarded as a set of oriented segments of uniform length .

Similarly:

Let a = {-1, -2,…………….-n} ; Let b = {+1, +2,………..+n}

Let x = ( a,b ) : x = {x¹, x²,………………..,xⁿ }

If we add to x = [a ,b] = x :{ x¹,……, xⁿ } the concept of electric charge such that

a = -q and b = +q , then x = [a ,b] = { x¹,……, xⁿ } translates into :

x = [-aq +,bq] ,

The domain x = [-aq ,+bq] = x : = = now resembles a

series of di-poles , with an orientation from –q to +q separated by a distance . The strength of the di-pole is considered to be the vector directed from the negative charge to the positive charge:

In the case of an infinitesimal di-pole , such as under consideration the distance is considered to be : ¶ =0 while the strength p remains constant .

What is an orientation of the path g ? An orientation of path g : x = x(t) is given by the direction of the velocity vector x.

If we take the path g as originating at origin O ,

x = [a,b] : = then represents a path consisting of a series of polarised dipole segments oriented in the direction of the path g : X = x(t) with its ends resting on infinity. The eigen energy value of the real photon would then be represented as The eigen energy value would be considered as traveling along the fixed points of :

such that ev_n ={0,……..0,1}.

(1) Since the velocity of is constant at c , and given g : x = x(t) to determine at any time the exact position of the eigen energy value .

(2) ) By observation we perceive that the intensity of the photon decreases directly in proportion to the square of the distance from the origin O. Thus at twice the distance from the source the intensity would be reduced by four times and at three times the distance from the source the intensity would be reduced by nine times and so on. This phenomenon is found to apply to all electromagnetic radiation regardless of frequency , wave-length or polarization. All types of electromagnetic radiation from radio-waves , to gamma rays and lasers experience a loss in intensity inversely proportional to the square of the distance from the origin: i = e/r^2. A mathematical description of the inverse square law can be made using the following formula:

To start with the solution to the problem we slice the surface of the cone with planes parallel to the yz – plane and review a derivation of the integral formula for surface area.

The Reimann sum procedure is to partition [a,b] into n sub-intervals [x_{i – 1 ,}x_i] , with the length Dx_i , for i = 1 to n , and then select a sample point - the mid-point will be convenient . The graph of f can be approximated by n line segments L_i connecting (x_{i – 1 ,} f(x_{i – 1})) to (x_{i ,} f( x_i )). Revolving each segment L_i gives a truncated cone C_i , which approximates to a slice S_i of the surface of revolution . Each C_i has surface area :

p(f(x_i-1 ) + f(x_i )) where abbreviates f( x_i ) - f(x_{i – 1}) . This well known formula for the area of a truncated cone can be derived without calculus.) The average :

is the distance from to the midpoint of L_i which can be approximately fixed by . Then the approximate area of C_i and the slice S_i , is:

The slice S_i can itself be subdivided into 2N pieces by n planes through the x – axis. When n and N are large , each of these pieces can be treated as a point approximating the inverse square law.

It should be noted that the above statement applies only to single unshielded photons and is by no means immutable , the amount of shielding ( i.e., photons surrounded by other real photons ) plays a large role in the amount of energy shared out to “virtual” photons. Hence the difference in propagation distances for masers and lasers etc., and the subsequent modification to the inverse square law resulting in an increase or decrease of the q angle at the source.

(a) This is a precise description of the manner in which a single photon may be thought of as propagating according to “Aumic” Theory. The Virtual photons comprising the field , in the presence of a real photon undergo polarization. Each virtual photon may therefore be thought of , with regard to the photon model , as a di-pole . The di-poles comprising the field are polarized , under the influence of the force exerted by a real photon , forming a chain whose ends rest on infinity , the energy of the real photon travels along this line of oriented virtual photons. Thus no energy is lost as the photon travels along this path. The orientation of the virtual photons comprising the “Virtual photon field” maybe thought of as being similar to the manner in which atoms link into di-pole chains under the influence of an electric field .

We are now faced with a paradox when dealing with the propagation of a single photon. Observation shows that all electromagnetic radiation regardless of its energy or frequency is subject to the inverse square law. Even high energy gamma rays found in cosmic rays are subject to this law. At the same time a study of the model of the photon shows that it is not possible for a photon to share energy laterally unless it is connected to a system through which energy is flowing or which has a surplus of energy , such as is present in an electromagnetic field surrounding a conductor through which a current is flowing. The paradox is that because of its structure the photon is compelled to laterally link up with photons of the virtual photon field , while at the same time the photon structure precludes promotion of those virtual photons into real photons since it lacks surplus energy. In this situation it can only be concluded that the energy of the photon is gradually leached away to the virtual photon field until it transforms into a virtual photon and the propagating structure breaks down.

An inevitable conclusion of this theory is that individual photons cannot propagate for great distances . This lends credence to the existence of a “virtual photon field “ since all matter is continually emitting “virtual photons” which of necessity must be isolated or single photons. It also explains how the theory could be applied to gravitation , since the force could be considered to be a series of discontinuous events taking place in a continuous manner and incremental in tiny amounts with regard to alignment of the photons of the “virtual photon field.” Thereby resulting in a tiny but uncompensated force between objects.

The explanation for the manner in which large numbers of photons propagate can be put more definitely. An important aspect of this theory is that the manner in which all electromagnetic waves propagate is unified into a single theory. In the chapter on electrical conduction it has been observed that the lines of force around a conductor through which an electric current is flowing consist of lines of linked or oriented virtual photons. The intensity of the current flowing through the conductor depend upon the number of lines of force flowing through the conductor. In the propagation of light, intensity of the light is due to the number of photons present in each line of force. The process works as follows , an excited atom emits a photon causing the virtual photons in line with the direction of propagation of the emitted photon to align themselves into a line whose ends rest on infinity. The atom once it has emitted a photon may continue to emit other photons most of which would follow the easiest path and propagate via the line of force which is already aligned to the electron. The number of photons which an electron emits per second represents the frequency of the photon , the intensity of the electromagnetic radiation is dependent on the number of photons which follow in the path of the first emitted photons (i.e., the same line of force.) This represents not only the frequency of the emitted electromagnetic wave but also its intensity , the intensity depending upon the number of photons present in a particular line of force. Thus both the intensity and frequency of the electromagnetic radiation are independent of the eigen value of the photon although the frequency is closely related to the eigen value in the sense that higher energy photons are emitted by electrons in more rapid succession than are lower energy photons . This may be seen as a natural consequence of the energy to which the electron is being subjected to , if the electron is being subjected to a higher level of energy it emits that energy mare rapidly. The lines of force (linked virtual photons ) through which the energy of the photons propagate are open ended , with their ends resting on infinity.

Thus close to the source , the energy may be thought of as an almost solid entity , lines of force being formed in close proximity and heavily populated with photons. As the photons move along the lines of force and the wave begins to spread out the gaps formed due to the geometrical spreading of the photon front are filled by photons in the line of force. Thus virtual photons present in gaps in the photon front are promoted to real photons and the resulting line of force are also promoted to real lines of force. The energy for promoting virtual photons into real photons is provided by photons present in any particular line of force. As a virtual photon is formed at the front of the wave a real photon is transformed into a virtual photon at the back of the wave.

If mathematical calculations are made it is found that this process is in exact conformation with the inverse square law.

This model of the propagation of light , offers a comprehensive explanation for all observed phenomenon connected with the propagation of light , it shows that the intensity of light varies inversely according to the square of the distance from the source , while at the same time preserving the eigen value of the photon , most importantly it differentiates between , frequency , intensity and eigen values of the photon. The theory also provides insight into how the “virtual photon field” is constantly replenished.

Credence is given to the existence of a “virtual photon field” since all matter is continually emitting “virtual photons” which of necessity must be isolated or single photons. It also explains how the theory could be applied to gravitation , since the force could be considered to be a series of discontinuous events taking place in a continuous manner and incremental in tiny amounts with regard to alignment of the photons of the “virtual photon field.” Thereby resulting in a tiny but uncompensated force between objects.

Another qualification that “Aumic” Theory makes and which should be observed here is that the reduction in intensity as applicable to electromagnetic radiation and stated in the Inverse Square Law has nothing to do with the photon itself but rather has to do with the manner of its interaction with the “virtual photon field”. If the photon could be shielded in some way from sharing energy with “virtual photons “ parallel to itself , the photon could propagate for much larger distances , as happens with lasers. Other factors also influence the propagation of photons such as the intensity per area of radiation from the source , the overall area over which radiation takes place , the time over which the radiation takes place , the composition of the radiation i.e the number of frequencies being propagated and their intensities and so on .

Yet according to quantum mechanics both (1) and (2 ) above cannot be considered. In QM it is taken as axiomatic that with regard to (1) it is axiomatic that the position of a photon cannot be determined until it is detected or absorbed. The phoon according to QM can follow an infinite number of possible paths between the occurrence of these two events. This might be explained as follows :

First rule :

We assume that there are several physically indistinguishable ways (paths ) in which a photon can move from s- state to f- state . In this case the resulting transition amplitude is the sum of the amplitudes corresponding to the different modes of transition:

( the index i denotes the i-th mode of transition.)

Second Rule .We assume that there are several final states {f_{1 ,} fs_{,……………….}fi } and that we are considering the probability of transition to any f these states , no matter which state it is . In this case the resulting transition probability | < f | s>|² is the sum of the transition probabilities.

Third Rule : Let us assume that the transition s ® f takes place through some intermediate state (v-state). In this case we include the idea of the amplitude of the successive transitions s ® v and v ® f ( corresponding to the amplitudes < v | s > and

< f | v > ) ; the resulting amplitude is the product of these amplitudes :

< f | s > = < f | v > < v | s >

In other words if the transition is broken up into successive steps , the transition amplitude is expressed by the amplitudes of the separate steps. (i.e., a photon can be absorbed and emitted many times during the course of its journey , yet still retain its original eigen value. )

The path of an electron in a cathode tube is often quoted in support of these statements , namely that it is impossible to state exactly where the electron in the cathode beam will land on the detecting screen except approximately. The “Aumic” Theory explanation for this anomaly is accounted for by the existence of the “virtual photon field “ which must exert some influence on the path of the electron.

Thus according to QM if we have a photon with its origin at O and moving towards the point of detection P the photon retains the eigen value it possessed at O because the eigen values of all the possible paths that the photon could have followed are

realized at the point of detection P. This is not in keeping with observations.

Observations show that if this were indeed the case , there would be large gaps in the radiation of electromagnetic radiation as it spreads out where no radiation is detected. This is not the case the identical levels of radiation (in a free space ) are detected at every point on the sphere of propagation when dealing with an isotropic radiator.

Again with regard to (2) according to QM it is axiomatic that a photon is either absorbed or will survive with its original eigen value intact regardless of the distance covered.

To elaborate on this point:

“ If an object is moving in a straight line it will continue moving in a straight line forever unless it is acted upon by something else (“a force”) At that time its direction and speed will be altered , depending upon the direction and magnitude of the force which it encounters . Furthermore , every action is accompanied by an equal and opposite reaction.”

The above statement relates to Newton’s Laws of motion which have been repeatedly verified over the course of many centuries . Experiments conducted in space have proved the validity of the Laws of Motion and has shown that they apply even at the atomic and sub-atomic level through the study of cosmic rays , the majority of which consist of accelerated particles such as protons.

However , a real problem arises when an attempt is made to apply the Laws of motion to photons. Although the photon is considered to be both a particle and a wave , and so should in theory because of its particle like properties be able in the absence of any opposing forces to travel in a straight line forever , in reality this does not happen. Quantum mechanics states that a photon once it leaves the electron can travel forever or until it meets another electron and is absorbed. This statement is not in keeping with the observed behaviour of electromagnetic radiation.

A genuine particle with an observed and measurable mass such as the proton follows the Laws of motion , for instance if a proton is accelerated to a certain velocity and then released in space , it will travel in a straight line forever , subject to the forces it encounters. Similarly if a stream of bunched up protons is accelerated and released in a similar manner they will all reach their destination in the same formation , provided of course that no external forces are encountered , in other words there occurs no diminution either in the number of protons or in their energies. With photons a completely different manner of propagation takes place , it appears that (a) the number of photons gets attenuated (b) that this occurs in a manner following the inverse square law i.e the reduction in the intensity is proportional to the square of the distance from the source. This means that the intensity of the original beam of photons is reduced although the energies of individual photons remains intact. This is completely different to the manner in which particles propagate. Take for example a shower of meteors , if they are released at the same time and with the same velocity , it can be reasonable to expect that all of them will arrive , always supposing that they are not subject to outside forces , at the same place and at the same time.

It seems that photons propagate more in the manner of waves than in the manner of solid matter , waves also tend to spread out and dissipate , what sets the photon apart is the ability of individual photons to reach their destinations with their original energies intact. “Aumic” Theory accounts for all aspects of this behaviour.

“Aumic” Theory states that the energy of a photon propagates through a line of linked “virtual Photons “ which orient themselves in the direction of propagation of the photon . The wave nature of electromagnetic radiation might be thought of as the frequency at which the photons are emitted from a source thus creating periods of dispersion and concentration in the surrounding “virtual “ field. Thus according to “aumic” theory , the energy of a photon and its frequency are separate.

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