ABSTRACT This is a model with particularly interesting attributes of a Photon, especially size, shape, navigation & travel in free space, momentum, force developed within the Photon, the instant of Cosmic Creation, subsequent Inflation, and Entropy. This model appears to possess elegance in that given an initial (or starting) energy, the model is well behaved with no singularities, discontinuities, infinities, or point particles/packets except for the Probability function going to Infinity. The model requires NO measured/observed parameters beyond Planck’s Constant and the Speed of Light, both of which can be considered = 1 using an appropriate system of units/measurements. It follows/observes all well-accepted Quantum Physics Principles (known today) including Uncertainty and Wave/Particle Duality although Quantization and Entanglement are briefly mentioned in the analysis. The Model is dependent only on a wavelength, which defines energy, size and shape. A Wave-Function is derived and is readily converted into a Probability Density. This provides the uncertainty required. Size and Shape of a Photon are determined and while they are exact, exhibiting NO uncertainty, the Photon is contained within the Probability Density function rendering the linear path of the Wave Packet to be Uncertain as well as the Photon’s relationship to such path. A review of certain geometric principles starts off the article and these principles must be Ready-To-Hand. Next Photon attributes, expressed by Mathematical equations is presented with derivation omitted. The Geometry and most of the Photon attributes are found in the literature but size and shape have not been found but may very well be in the literature. Next a speculation re a Photon at creation is presented. In an environment containing virtually nothing beyond vacuum energy, creation of a particle from nothing to what we may term large, e.g, 1E100 Joules, is no different from creation of a particle we call small, e.g., 1E-33 Joules. A primal Photon entering a highly curved, small space is analyzed to show a different means of inflation based on a Fourier analysis demonstrating that various frequencies result due to a distortion of the primal Photon and an “immediate” expansion occurs. A rough-and-ready analysis is then performed for the primal particle and then for the sum of the particles “broken” from the primal particle. Finally, Entropy is discussed and seen as invariant over space and time. THE PHOTON Consider a single photon with linear polarization. Unless otherwise specified this will be called a Photon and may have wavelength greater than in the micron region and down to orders of magnitude smaller than the Cosmic ray region. In free Space the Photon travels in a straight line (typically the Z Axis in this treatise) and curvature of space is neglected unless specified otherwise. Photon Energy and Momentum will be as per the Planck Equation of E = hν and |p| = h/ where E = Energy, h = Planck’s constant (6.26E-34 Joule-Sec), ν = Frequency, |p| = Magnitude of Momentum, = Wavelength, c = Speed of Light (2.998 E8 Meters per Second ≈ 3E8 M/s), and ν = c. The Photon has a Wave-Function in Cylindrical Coordinates (, r, z) as per Appendix A1 of 2z − r 2 − 8 −i( − t) 16 () = rz = ( √ ) e e 2 1 ℏc e z ct 2 ( − ) And an Energy Probability Density of P= ℏc 22 − 8 e r 2 − 4 ( ) e z ct 2 ( − ) Where ℏ is the reduced Planck Constant of h/2. 2 problems exist with this/these model(s): 1. The Z Axis is never identified, and in fact, cannot be identified. This is due to Heisenberg Uncertainty of p z = σp σz ≥ ℏ/2 and |p| is precisely known as |p| = h/. However, we will take the Z Axis to be an average of where a Photon might travel and define Probability Density from there. 2. The model uses a point energy density and does not consider the Volume of the Photon. The probability density P (, r, z) must be integrated over the Photon Volume and that probability convolved over to determine a normalization factor. We will be considering the Photon centered at r = 0 and z = 0 as well as considering variation in P small enough to not affect our analysis. It is known from Photons emitted from distant stars that they do not disperse over distance and/or time, thus, are compactly supported wave packets of energy of definite size and shape. Even though a Photon has properties akin to EM waves as per Maxwell, energy density within the volume is constant. This is because if the Photon is viewed from a Poynting Vector perspective, Energy Area Density as a function of distance (Z Axis here) and/or time, viewed as an intersection of the Photon volume with a plane perpendicular to the Z Axis of Photon travel, will appear as a Cosine function extending from - to + and Zero elsewhere. Technically the function is 1 + Cosine (aZ) or Cosine² (aZ/2) from -/2 to + /2 but it will be called generically “Cosine” here and extended to “Cosine of Revolution” or “COR” herein. See Appendix 3. As per Appendix A2, this Photon model is compactly supported and nicely behaved in that no singularities, discontinuities, infinities, etc. occur. Before continuing, let us consider Geometry as regards The Photon and Curved, Including Severely Curved Space with regard to The Photon: GEOMETRY Consider a ray of length R emanating from an origin and spinning around the origin at a rate . All activity occurs in a plane which never changes. Now consider a line, l, extending from R on the ray to R-l₀ on the ray. As a crutch, it aids to think of R >>>l₀ Since both ends of the line are rotating at an angular rate , each moves through the same angle, in the same time. Thus, the entire line moves through the same angle . Assign R₀ to the outer end of l (at R) and R₁ to the inner end of l (at R – l₀) Using S = R the distance R₀ or R₁ travels in time t as S₀ = R₀ and S₁ = R₁ and since is the same for both, S is different. That is because dS/dt is different with dS₀/dt = R₀ d/dt = R₀ and similarly dS₁/dt = R₁. This can be compared to a human (head to toe) standing on a spinning Earth: Our heads are moving faster than out feet. Now let us change conditions. Assign Ц = dS/dt as Tangential Velocity and require Ц₀ =Ц₁ = dS₀/dt = dS₁/dt For this to hold, cannot be constant as above and must vary. Thus, d₀/dt ≠ d₁/dt and ₀ ≠ ₁ Set up R₀ as a basis so that instead of both R₀ and R₁ rotating, R₀ stays still and R₁ rotates with and angular velocity = = ₁ - ₀ = ₂ As a result, the geometry, thus the length, of l₀ varies as proceeds over time and ₁ varies. l₀ stretches from an initial length of l₀i to a curved length ≈ R₀ at = 180° (or ) and then to length ≈ 2R₀ at = 360° (or 2) and continues to increase as increases and continues to lengthen in a dense spiral. Now change the line, l, to a square with side length l₀. One side of the square may be R₀-R₁ in which case it is slightly “tilted” or it may be a trapezoid that closely approximates the square or other possible variations. Now as , corresponding to , increases, the square is “stretched” at 2 corners making it a parallelogram and if is large enough, the result at = very closely approximates a very long, very skinny parallelogram and eventually returns to the origin with a “parallelogram” shape in curved space. The cycle continues with a “ribbon” forming a tight spiral. Note that the area of the square/parallelogram never changes. Now change the square into a circle and similar action results with a = stretch being very approximately a very long, very skinny ellipse in curved space whose area never changes. Next move into a 3D space and consider a cube and a sphere. The ray still remains in a plane that never changes. The square closely approximates a long, skinny prism at = and the sphere closely approximates a long, skinny ovoid. Remember these strictly geometric principles. THE FORCE The Photon is a confined volume (“Ball”) of energy, with zero mass, and moving at a constant speed of light. Exceptions are not considered here. Force is a vector and defined as F = dp/dt. We consider three different situations re Photon Force. One example of Photon force is Pressure where for P = Pressure and A = Area, F = PA is the phenomena of photons striking a vane(s) on an axle causing the device to rotate. Another example is in the human eye where a Photon strikes a Rhodopsin “tube” causing it to deform, thereby changing an ionic fluid flow (akin to squeezing a hose) which the brain interprets as fundamental light or colors or images which combine to form intelligence/data. In flat space, the Photon travels along a straight line. Upon collision energy is transferred and total Photon momentum, p, decreases from some p₀ to Zero in time T. Since momentum density varies along the Z Axis, increasing from zero to ρ₀ and back down to zero, instantaneous force is not constant. For COR it is of the form d/dt [Cos²(t)] = - Sin(2t). Note the double “bump” of force, one positive and the other negative. Thus, upon collision the Photon force “pushes” for half the time and “pulls” for half the time being the occurrence due to the Photon and separated from anything else. In a slightly curved space, motion and p are vectors tangent to the curvature and dp/dt is a vector, F, perpendicular to curvature (in the direction of the R basis vector), motion, and p. Since the Photon has size and shape, F is also a function of some radius and corresponding center point defining curvature and is non-zero only within the confines of the Photon volume. A higher order effect, readily seen in severely curved space, has been ignored for slightly curved space and adjustment for such can easily be made as below. There are alternate ways to consider dp/dt : #1 - One Vector Tangent to the curve “attempting to go straight” or #2 - the vector sum of a Radial (outward direction) component plus a second component perpendicular to a considered Ray and at an angle d to the Tangent Vector. For Vector #1, |p| is known as E/c = h/. t = s/c = R₀ /c. Chord length from #1 Tangent point to #2 Perpendicular point = 2 R₀ Sin(/2). Ray length from Perpendicular point to Ray #1 Intersection = Sqrt {p² + [2 R₀ Sin(/2)]²} – R₀. Do the two forces one due to curved space and the other due to momentum vector change (direction) oppose each other and sum to become Zero. Just like spinning a ball on a rope. In severely curved space, the Photon volume shape becomes distorted to the point where the Photon forms a ring in the space with elliptical cross-section but not joining with the other “end” of the Photon but spiraling underneath to form a spiral. It is somewhere near the spiraling result that a small, severely curved space cannot sustain the entire Photon which “breaks apart” into Branes. SEVERELY CURVED SPACE Now consider the Photon as moving from a flat space into a curved space and the consideration from GEOMETRY, above taken into account. The space may or may not be uniformly curved, i.e. curvature varies and/or changes to flat, but we will consider for the nonce only an initial portion of the curved space and consider it well approximated (or precisely defined) by a radius and an origin. Moreover, 2D representations will often be used as a convenience for explanation realizing that 3D follows naturally, although with more complexity. Since ALL portions internal to the Photon have equal energy density, each V has an energy of E = ρE V where ρE is Energy density. Further, ALL V (or E) are moving at the speed of light, c, along the Z Axis or along a curve distinctive to the curved space. The COR now becomes distorted into roughly an ovoid with 2 ultimately tiny, conical peaks opposite each other. During this transition/distortion, the spectral content of energy changes from purely sinusoidal at one frequency to a sum of frequencies (each generating a unique particle based on E = ℏ = hν) and related to a radius of curvature, R >>>…>>> for consideration here even though such is not generally required. SOME ATTRIBUTES OF THE PHOTON dp/dt => local force within only the Photon volume but akin to an EM or gravity in that it varies (decreases with distance) but only over its volume and not to infinity or a singularity at zero as gravity or an EMF. Consider a very tiny volume on the Planck order of size, so small that it is reasonable (at least for some finite time) that QM would not have particles/stuff created/annihilated in that volume. Then the primal Photon enters that severely curved space creating dp/dt and a distorted Photon. Start with a flat space, then proceed to a slightly curved space, finally to a severely curved space.. Is this how gravity is created if there are many Photon-like particle with energy <<<…<<< the primal energy? Is the force of gravity quantized but so close (sub-Planck) that we cannot measure it? Entropy of the Photon remains constant since stats over position and time never change. Per Shannon, H = - pi Log (pi) or - ∫ pi Log (pi) and it is known that Photon dispersion = 0. Consider the dual/double Photon of + and - in the model. Each has the same probability density allowing for a separation along the Z Axis. However, they are NOT required to travel along the same line (in opposite directions), but, rather, along 2 parallel lines within P and travel along the same line is possible but very improbable. The significance of this is that in a curved space, the two Photons of + and - will not collide at 180° from their starting point but will distort until they simultaneously break free from the curved space. Momentum in a flat space is always zero for the sum, thus, energy was created but momentum was not. In a curved space, using 2D Cartesian coordinates, since the Photons will travel in the same plane or parallel p[lans slightly displaced, px = 0 whereas py varies periodically as the Photons travel around the circle of curved space (assuming curvature does not change), however, py over one period = 0. The assumed values below are determined based on “desired” end results here and examples use “cherry-picked” values, thus, are truly speculative and not a true scientific process. However, the result is intriguing and invites a rigorous study. Consider a Photon with 6.6E69 Joules energy. Using E = ℏ = hν = hc/ we get = hc/E h = 6.6E-34 Joule Seconds c = 3E8 Meters/Second Thus, = 3E-95 Meters The Photon is in a curved space of R₀ Note: This Photon might be considered as having been created in flat space and entering the curved space some finite time after its creation or it might have existed “forever.” It has only one frequency. The instant of entry into curved space is defined as t = 0. The Photon also might have been formed in the curved space and that instant is defined as t = 0. More critical/fundamental Fourier frequencies exists. A full Fourier analysis will not be performed but estimates made instead. The above Photon stretching occurs at = 2 and it forms “almost” a circle with ribbon thin circumference. If R₀ ≈ 1.6E-34 Meters in a curved space, then a Photon of 6.6E69 Joules ( = 3E-95 Meters) will enter that space and in 3.3E-42 seconds stretch to 2 = 18.85E-95 Meters. To stretch one complete revolution in R₀ curvature (S = 2R₀) n revolutions are required where n = R₀/ = 1.6E-34/3E-95 = 5.3E60 with a resulting time duration of 3.3E-42 x 5.3E60 = 17.5E18 Seconds. This is a very long time wrt 1 Revolution so Fourier components related to this time/frequency will be ignored here. A Fourier Transform will show a frequency spectrum containing 3E42 Hz and harmonics. That fundamental is an energy of E ≈ 2E8 Joules. Compare to UHECR Cosmic Rays ≈ 50 Joules. The original Photon has been transformed/converted into a sum of many smaller energy Photons traveling in many directions and colliding. At issue here is that the primal Photon will never get to a “length” S = 2R₀ = 1E-33 Meters since Fourier components will start forming well before that might occur. Thus, the primal Photon “breaks apart” as it lengthens with longer wavelength constituent parts leaving the small curved to enter a larger space, flat or curved. Thus, “Inflation” occurs in a very short time, with a linear size increase of 3E-95 to 1E-33 Meters in 3.3E-42 Seconds. See http://www.physicsoftheuniverse.com/topics_bigbang_inflation.html Note radius change in the region of 1E-35 to 1E-33 Seconds. Note that Physics as we believe it to be today is used and not violated. R₀ is a Planck length and 6.6E69 Joules is a possible estimate of “positive” energy in the observable universe. SPECULATION The assumed values below are determined based on “desired” end results here and examples use “cherry-picked” values, thus, are truly speculative and not a true scientific process. However, the result is intriguing and invites a rigorous study. Consider a Photon with 6.6E69 Joules energy. Using E = ℏ = hν = hc/ we get = hc/E h = 6.6E-34 Joule Seconds c = 3E8 Meters/Second Thus, = 3E-95 Meters The Photon is in a curved space of R₀ Note: This Photon might be considered as having been created in flat space and entering the curved space some finite time after its creation or it might have existed “forever.” It has only one frequency. The instant of entry into curved space is defined as t = 0. The Photon also might have been formed in the curved space and that instant is defined as t = 0. More critical/fundamental Fourier frequencies exists. A full Fourier analysis will not be performed but estimates made instead. The above Photon stretching occurs at = 2 and it forms “almost” a circle with ribbon thin circumference. Speculation: If original creation of energy (matter) were a single particle, that particle could have started in a curved space with R₀ = 1.6E-34 M or started in another space and eventually entered a space with R₀ = 1.6E-34 M the result being what is called “The Big Bang” following a short time (order of 1E-42 Seconds) after creation which here is considered entry/start in curved space. OPEN ISSUES Where did the original energy/momentum come from? Why should space be curved? Flat? What is the meaning of the Photon dp/dt forces seeing as they are confined to a specific volume? Can these forces be considered quantized? Is “Primal Breaking” related to Waveguide-Beyond-Cutoff phenomenon? Does the dual/double Photon of + and - apply to Entanglement? Can the “broken” primal Photon particles form Higgs Bosons? See http://physics.info/standard/ Does the “Primal Breaking” exhibit attributes of life? Appendix A1 DERIVATION OF PHOTON WAVE-FUNCTION Reference: Using the methods in THE PHOTON WAVE FUNCTION Iwo Bialynicki-Birula and The Photon Wave Function Joseph Cugnon we obtain: 𝑝⃗ = - iℏ 𝜕 0 0 ⃗⃗⃗⃗⃗ 𝑆𝑥 = 0 0 0 𝑖 and 𝜕𝑥 0 −𝑖 0 Si ‘s are the spin matrices for a spin-1 particle 𝑝⃗ • 𝑆⃗ = pxSx + pySy + pzSz 𝑝⃗ • ⃗⃗⃗⃗⃗ 𝑆𝑥 = - iℏ = -iℏ [ 𝜕 [ 𝝏 𝝏𝒙 ⃗⃗ + 𝒙 𝜕 –𝑖 𝜕𝑦 𝜕𝑧 𝝏 𝝏𝒚 ⃗⃗ + 𝒚 ] =ℏ[ 𝜕 ⃗⃗ 𝒛 ] 00 𝜕 ] 𝝏 𝝏𝒛 – 𝜕𝑧 𝜕𝑦 0 0 0 𝑖 0 −𝑖 0 [ ⃗⃗⃗⃗⃗ 𝑝⃗ • 𝑆 𝑦 = - iℏ [ 𝑝⃗ • ⃗⃗⃗⃗ 𝑆𝑧 = - iℏ 𝝏 𝝏𝒙 𝝏 𝝏𝒙 ⃗⃗ + 𝒙 ⃗⃗ + 𝒙 𝝏 𝝏𝒚 𝝏 𝝏𝒚 𝑝⃗ = (px, py, pz) = - iℏ ⃗⃗ + 𝒚 ⃗⃗ + 𝒚 𝝏 𝝏𝒛 𝝏 𝝏𝒛 ⃗⃗ 𝒛 ] 0 0 0 𝑖 0 = -iℏ 0 [ −𝑖 ] 0 −𝑖 𝑖 0 0 0 0 0 = -iℏ 0 [𝑖 𝜕 𝜕𝑧 𝜕 𝜕𝑦 +0+𝑖 -𝑖 𝜕 𝜕𝑥 𝜕 𝜕𝑥 ]= ℏ [ ]= ℏ [ 𝜕 𝜕𝑧 𝜕 𝜕𝑥 𝜕 𝜕 – 𝜕𝑧 – 𝜕𝑥 ] ] and 𝑆⃗ = (Sx, Sy, Sz) 0 0 0 0 0 0 −𝑖 𝑝⃗ • 𝑆⃗ = (px, py, pz) 0 0 −𝑖 0 𝑖 ⃗⃗ 𝒛 0 0 −𝑖 0 𝑖 0 −𝑖 0 0 𝑖 0 0 0 0 0 0 0 0 = [(0 ipz -ipy ) (-ipz 0 ipx) (ipy -ipx 0)] = ℏ[(0 +∂z -∂y)(-∂z 0 +∂x)(-∂y +∂x 0)] These ( ∂ ) operators are scalars to be used later with appropriate vector bases. iℏ iℏ 𝜕 𝜕𝑡 𝜕 𝜕𝑡 iℏ ( 𝑝⃗ • 𝑆⃗ 0 = Hf and Hf = c ( ) 0 𝑝⃗ • 𝑆⃗ 𝑝⃗ • 𝑆⃗ 0 𝑝⃗ • 𝑆⃗ 0 ⃗⃗ =c( ) =c( ) (𝐸 + 0 𝑝⃗ • 𝑆⃗ 0 𝑝⃗ • 𝑆⃗ 𝐸⃗⃗ − ⃗⃗⃗⃗⃗ (𝑝 • 𝜕+ /∂t ) = c ( − 𝜕 /∂t ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ 𝑆) 𝐸⃗⃗ −(𝑝 • ⃗⃗⃗⃗⃗⃗ 𝑆) 𝐸⃗⃗ ⃗⃗ 𝑖𝐻 ) ⃗ ⃗ 𝑖𝐻 ⃗⃗⃗⃗⃗ • ⃗⃗⃗⃗⃗⃗ ⃗⃗ 𝑖(𝑝 𝑆) 𝐻 𝑑 + ) = iℏ ( ) ( − ) 𝑑𝑡 ⃗⃗⃗⃗⃗ • ⃗⃗⃗⃗⃗⃗ ⃗⃗ 𝑖(𝑝 𝑆) 𝐻 where = (x + y) z and + = E + iH and − = E – iH 𝑑 iℏ ( ) 𝑑𝑡 𝑥+ 𝑦+ 𝑧+ −𝑥 −𝑦 −𝑧 = c 0 +𝜕𝑧 −𝜕𝑦 −𝜕𝑥 0 +𝜕𝑥 0 0 0 +𝜕𝑦 −𝜕𝑥 0 0 0 0 0 0 0 0 −𝜕𝑧 +𝜕𝑦 0 0 0 𝑑 𝑑𝑡 𝑑 𝑑𝑡 𝑑 +𝑥 +𝑦 +𝑧 𝑥− 𝑦− 𝑧− ] i ( ) +𝑧 = (c/ℏ) [ -∂𝑥+ /∂y + ∂+𝑦 /∂x 𝑑𝑡 0 0 0 −𝜕𝑦 +𝜕𝑥 0 ] i ( ) +𝑥 = (c/ℏ) [ -∂+𝑦 /∂z + ∂𝑧+/∂y i ( ) +𝑦 = (c/ℏ) [ +∂+𝑥 /∂z - ∂+𝑧 /∂y 0 0 0 +𝜕𝑧 0 −𝜕𝑥 ] Solving and converting to Cylindrical Coordinates produces: 1 = rz = ( ℏc √ )e 2z − r 2 − − t) ( ) 16 8 −i( 2 e e z ct 2 ( − ) Starting with above a Probability Density, P = • * can be determined as: P= ℏc 22 − 8 e r 2 − 4 ( ) e z ct 2 ( − ) ∞ 𝑁𝑜𝑡𝑒: ∫− 𝑑 = 2 This is probability density at a point but the Photon is a size with a shape – a compactly supported wavepacket. The significance here is that considering a Gaussian integral, the exponents tell the volume under the curve so that using the standard form for a Normal function 1 2 1 𝑒 −(2)𝑥 √2 Thus, 𝑒 1 −(2)𝑥 2 1 ⟺𝑒 −(2)( 2 ) 𝑟 2 4 When 𝑟 2 = When 𝑟 2 = 2 When 𝑟 = Simiarly, 𝑒 42 +1 , x = 1 and ∫−1 𝑑𝑟 = .6826 , x = 2 and ∫−2 𝑑𝑟 = .9544 82 122 1 2 +2 +3 , x = 3 and ∫−3 𝑑𝑟 = .9974 −( )𝑥 2 1 ⟺𝑒 −(2)( 2 ) 𝑧 2 4 +1 When 𝑧 2 = 42 /, x = 1 and ∫−1 𝑑𝑧 = .6826 +2 When 𝑧 2 =82 /, , x = 2 and ∫−2 𝑑𝑟 = .9544 +3 When 𝑧 2 = 122 /, x = 3 and ∫−3 𝑑𝑟 = .9974 Now all 3 probabilities (, r, and z) are multiplied. can be anywhere between 0 and 2, thus P = 1. Pr = Pz = .9109 or 91.1% for 2σ and for 3σ => .9945 = 99.5% IF the Z Axis can be determined . Thus, it is very probable that the Photon will be found within a radius of 2.26 around the Z axis and within ±1.59 of a zero point along the Z axis. This corresponds to a 4.5 diameter and a 3.2 length. If 3σ certainty is desired then r = 2.76 and z = ± 1.95 or a 5.5 diameter by 4 length. Appendix A2 THE PHOTON Wave-function for a single, linearly polarized Photon in free space using cylindrical coordinates, r, , z is: 2z − r 2 − 8 −i( − t) 16 () = rz = ( √ ) e e 2 1 ℏc e z ct 2 ( − ) Note: Very high energies, many orders of magnitude greater than Cosmic Rays, will be considered below, however, these wave packets, for simplicity, will be termed “Photons.” The Wave-function above, unfortunately considers the Photon as a point source and we wish to consider the Photon as a Size and Shape within the Gaussian uncertainty as above. Even though uncertainty is associated with the Photon, it can be described in Cylindrical coordinates as a Cosine of Revolution (COR) about the Z Axis and extending in length along the Z Axis a distance of ± 2 or from - to + corresponding to ± /2 or - /2 to + /2. The maximum diameter of this COR, i.e. at z = 0, is . Note the format is that of a Traveling Wave. The Photon travels in a straight line for consideration here in that any space curvature is small enough for the space to be considered flat. Below, a severely curved space will be considered but that will be explicitly stated. We know that the energy in a Photon does not disperse, but stay confined to a volume as measured in Photons emitted by distant stars. The energy density within this volume is constant and even though it is similar in nature to EM waves it DOES NOT vary with r nor extend to infinity. The Photon has no reference as does an electron in a Hydrogen atom which is referenced or “tied to” a proton, thus, we will define a Photon reference as the Z Axiz being the straight line path the Photon follows. This ZAxis is parallel to the Z Axis in the model and has uncertainty based upon Pr = r • r*. This Photon model is compactly supported and nicely behaved in that no singularities, discontinuities, infinities, etc. occur. Even though uncertainty exists as ||² = • * the energy of the Photon can be seen as uniformly distributed throughout this volume when viewed from a perspective of a Poynting Vector traveling along the Z Axis with energy density integrated over areas larger than the COR diameter (along its existence wrt Z) is constant and does not diminish unless the integration is over a circular area with radius < /2. This can be performed despite uncertainty in the Photon position/location since in flat space it travels in a straight line defined as Z Axis – it is merely a frame. The Photon is now a traveling wave of sorts in that energy is confined but uncertainty exists. Appendix A3 THE PHOTON SPECTRAL CONTENT A COR as a function of time is considered and a Fourier Transform obtained. The Fourier Transform is defined as: +∞ F{ g(t)} = ∫−∞ 𝑔(𝑡)𝑒 𝑖2𝑓𝑡 +∞ = ∫−∞ 𝑔(𝑡)𝑒 𝑖𝑡 Where g(t) = ½[1 + cos (2ct/)] for -/2c ≤ 𝑡 ≤ +/2𝑐 and g(t) = 0 for t < − 2𝑐 , 𝑡 > /2𝑐 /2𝑐 F{ g(t)} = ½∫−+/2𝑐 [1 + cos (2ct/)]𝑒 𝑖𝑡 dt +/2𝑐 = ½∫−/2𝑐 [1 + ½ 𝑒 +𝑖2𝑐𝑡/ + ½ 𝑒 −𝑖2𝑐𝑡/ )]𝑒 𝑖𝑡 dt F{ g(t)₁} = (/2c) Sinc (/2c) = (1/2ν) Sinc (/ν) for the “1” portion where the ν is from E = h ν. F{ g(t)₂,₃} = (/4c) [/2𝜈1 + ] Sin (/2ν + ) + (/4c) [/2𝜈1 − ] Sin (/2ν - ) = (1/4ν) [ Sinc (/ν + ) + Sinc (/ν - )] Giving F{ g(t)} = (₀/4c) [2 Sinc (₀/2c) + Sinc (₀/2c + ) + Sinc (₀/2c - )] = (1/4ν₀) [2 Sinc (/ν₀) + Sinc (/ν₀ + ) + Sinc (/ν₀ - )] Note that all ’s are in units of ν. Note the Sinc Spectrum of ½ magnitude each at = ± 2ν₀ due to the Cosine and the third Sinc Spectrum at = 0 for the rectangular pulse of width ₀ based on E = hν₀ = h (c/₀). Thus, analyzing F{ g(t)} we see that it is comprised of the original Cosine consisting of 2 components at ±ν₀, but due to multiplication by a unit pulse, a spread of spectrum related to unit pulse width to form a Sinc function. It can be viewed quite simply as a modulation. This is combined with the spectrum of a unit pulse. If instead of a COR we consider Size & Shape to be Spherical, a “modulation” of the sphere radius can be determined to produce a Traveling Wave ala Poynting Vector, however, the radius cannot be permitted to go to Zero which would require an infinite energy density, thus, a singularity. Further, should such a radius modulation occur, energy density must vary and no rationale for that exists. Other shapes, etc. appear to possess the same difficulties. Appendix A4 PHOTON VOLUME AND ENERGY DENSITY Repeating from above: Technically the function is 1 + Cosine (aZ) or Cosine² (aZ/2) from -/2 to + /2 but it will be called generically “Cosine” here and extended to “Cosine of Revolution” or “COR” herein. Consider a COR, solid of revolution, of length (± /2 about z = 0) and maximum diameter at z = 0 of or ± /2. A disc at the origin (z = 0) has area A = (/2)² and along the z avis from -/2 to +/2 has area A(z) = [(/2) Cos²(z/)]² = (/4)² Cos4(z/) +/2 +/2 And Volume V(z) = ∫−/2 𝐴(𝑧)𝑑𝑧 = ∫−/2 z (4) 2 Cos⁴ ( ) dz V(z) = (3/32)³ Using Energy, E = h ν = hc/ And Energy Density ρE = E/V = (hc/)/(3/32)³ ρE = (32 hc)/(3 ⁴) For visible light = 400 – 700 nM thus, EViolet ≈ 4.95 E-19 Joules, VViolet ≈ 1.88 E-20 M³ and ERed ≈ 2.8 E-19 Joules, VRed ≈ 1.01 E-19 M³ and ρEViolet ρERed ≈ 26.26 Joule /M³ ≈ 2.8 Joule /M³ For X-Rays, Gamma-Rays, and Cosmic Rays ( = 6E-10, 1E-12, 3E-15) ρEX ≈ 5.19 E10 Joule /M³ ρE ≈ 6.72 E23 Joule /M³ ρEC ≈ 8.3 E33 Joule /M³