Momentum and Energy of a Mass Consisting of. Confined Photons and Resulting Quantum. Mechanical Implications

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1 Adv. Studies Theor. Phys., Vol. 7, 03, no., HIKARI Ltd, Momentum and Energy of a Mass Consisting of Confined Photons and Resulting Quantum Mehanial Impliations Christoph Shultheiss Karlsruhe Institute of Tehnology (KIT), Institute for Pulsed Power and Mirowave Tehnology (IHM), P.O. Box 3640, 760 Karlsruhe, Germany f0695@partner.kit.edu Copyright 03 Christoph Shultheiss. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. Abstrat A one-dimensional model is presented, in whih a photon, onfined in a avity, represents a model of mass. Photon-partile-interation is redued to a free photon-onfined photon interation, for whih hanges in momentum are exlusively oppler-based. The energy onservation itself desribes a storing of the olliding free photon in the avity. It fuses with the onfined photon and deays in spontaneous or delayed fashion by emission of a oppler-shifted free photon. elayed emissions lead to a stop-and-go motion of the avity in aordane with the quantum mehani de Broglie relation between momentum and wavelength. This kind of motion onsumes no energy and overome potentials with minimum energy onsumption. To simulate a massless avity, the model requires the existene of a photon pool similar to the zero-point radiation desribed by Casimir (948). This photon pool and a thermal fration help to stabilize and onserve the avity. Aeleration by means of a swarm of inident free photons at a avity triggers a series of phase mismathes with the onfined photons whih lead to a distortion of the pool distribution. It is expeted that an attrative stop-and-go - displaement of probe masses behind the aelerated avity takes plae an effet onsistent with energy onservation. PACS: t, j, p, w

2 556 Christoph Shultheiss Keywords: Mass, oppler shift, momentum onservation, energy onservation, zero point energy, Compton Effet, Quantum Mehani, inertia. Introdution Currently world wide effort is foused on the questions of what is mass, how mass omes to elementary partiles, and why masses move with veloities below the speed of light. Following a theory of Higgs [] and others, the so alled Higgs field gives elementary partiles suh as Z- und W-Bosons their mass. At the present time the searh for a so-alled Higgs partile oming from the Higgs field is ontinued at the Large Hadron Collider (LHC) in CERN. Of similar importane is the question of why the mass of a proton is larger by a fator of nearly hundred than the mass of the onstituent quarks []. Obviously, the fore that onfines the quarks in the proton is aountable for this mass amplifiation effet. Following the Standard Model [3], the interation of quarks happens via gluons in an exhange proess. An interesting question is of ourse how these gluons generate mass [4]. The model whih is presented here tries to follow up on this question and defines a mass for whih partiles that travel with the speed of light are onfined in something whih will be alled avity in the following. The avity itself should of ourse be massless ; the partiles are proposed to be thermal photons with well known behavior in terms of momentum, energy and oppler Effet. In the proposed model, the thermal photons inside the massless avity are ontinuously refleted bak and forth with 80 -Compton ollisions at mirror walls. Although photons move with the speed of light, if they are loalized, or onfined in a avity, they behave as if they have mass (a fat already pointed out by Einstein [5] when he disussed the mass of heated matter). It an be demonstrated that onfined photons gain momentum and kineti energy in the same manner as an ordinary mass. Therefore, in the following the unit of thermal photons onfined in a avity will be alled photoni mass. This model only works if spae ontributes a photon pool that auses a stabilizing ounter-pressure to the onfining avity. Absorption and emission of free photons by loalized onfined photons allow insight into the mehanism of the onservation of energy and of the resulting quantum mehanial impliations. All onlusions onerning momentum and energy rely on -dimensional relativisti alulations. This means that only perpendiular omponents of the wave propagation with respet to the avity walls are onsidered. The situation of isotropially distributed propagation of the wave needs higher dimensionality in the desription. The same is true with effets suh as spin, polarization et.. It is important to note that results based on this simple model may not desribe reality in detail but may still give useful information. Sine the Compton proess as the standard ollision proess plays a dominant role, the desription starts with the main features of ollision physis.

3 Momentum and energy of a onfined photon 557. Three fundamental equations desribe the elasti Compton ollision uring refletion proesses of photons with harged partiles the Compton shift appears i.e., the wavelength of interating photons is shifted. It is the onlusion of several authors [6, 7, 8, 9] that these shifts are idential to oppler shifts, that is, after the interation a partile moves with the veloity v and the interating photon, oming bak from the partile, is oppler-red-shifted with respet to the relative veloity. This is demonstrated next for the ase of an elasti 80 Compton ollision: With the dimensionless abbreviations h / m h / m k / m, and for an initial veloity 0 of the harged partile with mass m, by means of mutual addition and subtration the well known solutions of both equations: Energy: Momentum: () 0 () are, where prime indiates the situation after the ollision: (4) The oppler equation an be derived if Eq. and are rewritten as: (5) (6) Subtration: (7) Addition: (8) From the definition of γ one easily derives: (9) Eq.7 and 8 result in:. (0) A omparison with Eq.3 gives:, () whih is the oppler equation. This result suggests the following interpretation: (3)

4 558 Christoph Shultheiss CONCLUSION. The Compton Effet an be understood as a two step event. The first step is the refletion of the photon, when both mass and photon hange from the laboratory system into the moving, olliding system. Then, in the seond step the retransformation of the photon into the laboratory system takes plae. In this step the oppler red-shift as formulated in Eq. ours. Sine a hange of photon energy is exlusively a result of oppler shifts (see ref. 6-9), the Compton ollision for these kinematis an be derived diretly from momentum onservation and the oppler shift: Using the oppler equation gives: and. () Inserted into the momentum equation 0 the solutions Eq.3 and 4 follow, without making diret use of energy onservation. The elasti Compton ollision proess seems to be reasonably desribed by momentum onservation and oppler shift alone. It is easy to show that oppler- and energy equations yield the same results; all ombinations of two of the three fundamental equations desribe the elasti Compton ollision. Mathematially seen the Compton Effet is overdetermined. The question arises, how these equations relate to the others and why and does it matter? To answer this, I propose a model in whih the partile mass is generated by onfined photons and that allows to simplify the photon-partile interation into a photon-photon interation. As mentioned, in this model of photoni mass the photons are onfined in a avity, reflet bak and forth between two mirrors and reate rest energy. Motions of the avity imply simultaneously oppler red and blue shifts of the onfined photons, and in addition to the rest energy a kineti energy term appears. Therefore a ollision with a free photon eventually leads to oppler shifts of all involved photons no matter whether they are free or onfined. This leads to the onlusion that energy onservation an be regarded as based on oppler shifts, that is, energy onservation is a oppler shift onservation. But this is only part of the truth. Energy onservation desribes a fusion- and deay proess between free photons and partiles, that is, onfined photons, that is, photoni mass (see Set.7). This leads to a quantum mehanial behavior of these partiles (see Set.9) To simulate realisti mass partiles, the assumption of a massless avity that is onstruted from weightless mirrors must be introdued. Mehanial stabilization by rods, springs et., leads to vibrations of the avity, whih annot be observed for real partiles (see Set.3.). To stabilize the geometri arrangement of the avity mirrors (i.e. the mirror distane), a non-losed physial system is taken into aount (see Set.3.). This alls for the ooperation of a surrounding photon radiation field similar to what Casimir [0] desribed as the zero-point radiation of

5 Momentum and energy of a onfined photon 559 spae. The geometrial arrangement of mirrors and photons is the subjet of the next Setion. 3. Confined light in a avity whih is at rest or in uniform motion uring refletion at a mirror, propagating waves build up a standing wave pattern. In a avity with two mirrors with perfet photon-refleting apability, set up in parallel at a distane L, the spetrum of standing waves appears: L n, where n,,... (3) In priniple, eah of the standing waves with wave number n an be populated with an arbitrary number of onfined photons. Fig.: A avity at rest and in motion in a spae-time presentation where the 45 -lines represent the zigzag spread of a wave Figure shows a avity at rest (left) and in motion (right) in a Minkowski spae-time presentation. For the ase n = L/ =, the 45 -lines show the zigzag spread of the wave front (solid line) and the wave end (dotted line). The thik arrows are onnetion lines between wave ends to fronts. In a system at rest they are denoted as wavelength. In a moving system they are slanted and -vetors (sine the model is -dimensional). The half wave moving to the right is blue- and the half wave moving to left is red-shifted sine they reflet at the approahing and reeding sides of the mirror, respetively. On the basis of a ommon zero point, the expressions of the Lorentz transformation are t t x, (4) x x t here prime indiates a different referene system. In Fig., the observer moves to the left. Therefore is negative. The transformed oordinates are:

6 560 Christoph Shultheiss 0,0 0,0 (5) L, 0 L, L (6) 0, L L, L (7) L, L L, L (8) 0, nl n L, n L (9) L, nl L n, Ln (0) The transformed distane between the mirrors whih is related to the wavelength is a -vetor. The orresponding spae and time intervals of the wavelength are: L, 0 0,0 L, L () L, L 0, L L, L () and so on. As an be seen in Fig., right graph, the length ontration of a moving avity is L /, whih is in agreement with standard relativisti models. To show this: The slope of the t line in Fig. is /. Using Eq.6, a line through (L,0) with this slope intersets the line y = 0 at x L L /. 3. Problems with the physis of a real avity In many regards, Fig. does not represent a realisti model for a free partile, onstruted solely from onfined photons loated in a mirror system in a massless avity. To give an example: Assume two photons with a phase differene of in the avity. At any time, the momentum transfer of the left-refleting photons must be ompensated exatly and instantaneously by the momentum transfer of rightrefleting photons to maintain the mirrors at rest. Massless mirror onnetion rods for momentum transfer are physially not available; instantaneous momentum transfer ontradits the finiteness of the speed of light. In addition, any mass-like rods lead to spaing vibrations between the mirrors and inrease the internal energy. In this ase, the energy balane for inelasti ollisions should be: (3) where represents the vibration exitation. In photon-partile (i.e. eletrons or protons) interations the term is zero. Therefore in this model a pool photon senario is hosen where external photons reflet simultaneously with onfined photons to ompensate the mirror momentum. 3. An external photon pool suh as Casmir s zero point radiation generates pressure at the avity walls The proposed mehanism to maintain the distane of the mirrors is an external photon pool, as illustrated in the Minkowski spae-time presentation in Fig. (outside refleting photons with arrows). Massless mirrors are plaed in a standing wave field of pool photons and onfined photons. Refletions take plae simulta-

7 Momentum and energy of a onfined photon 56 neously at the inside and outside of a mirror, so that both mirrors remain at rest or in uniform motion a proess whih is proposed to be alled in-phase. Fig.: Spae-time frame of simultaneous refletions of onfined photons with external photons (thik arrows) for the ases of a avity at rest (left) and in motion (to the right). This proess will be alled in-phase. If a avity is at rest, one would observe that onfined photons k and pool p photons k reflet instantaneously without oppler shift, as shown in Fig.3 (left). If the avity or the observer moves, all photons whih interat with the avity mirrors undergo red and blue shifts, as shown in Fig.3 (right). In this graph the abbreviations and (4) desribe the red and blue shifts of pool- and onfined photons in a moving avity, where the form of the blue shift in Eq.4 an easily be derived using Eq.9. Fig.3: Spae-time presentation of onfined and pool photon refletions at avity mirrors at rest (left) and in motion (right). Sine photon momenta transferred to the mirrors are equal, the mirrors remain at rest or in onstant motion after refletion.

8 56 Christoph Shultheiss The path of the blue-shifted onfined photon k is obviously longer than the red-shifted one k, an outome that is unexpeted. This issue will be investigated in the next Setion. 3.3 A standing wave in a moving referene system remains a standing wave At first glane, the presene of a standing wave between a pair of moving mirrors (see Fig.) seems to be impossible beause the wavelengths of red- and blue shifted waves are not the same. However, relativity theory ontradits this, as will be demonstrated next: A standing wave in a avity is equivalent to a standing wave in front of a mirror at rest. uring the refletion of a wavetrain, one wave moves toward and the other moves away from the mirror: i t x F x t e / i t x / x t e F Superposition of both waves gives (refletion at a mirror): i t x / i t x / i t y F x t F x t e e e e i t i x / i x / i t e e i e sin x / i x / e i t e (5) (6) i x / e (7) whih results in a standing wave. In the ase of a moving mirror the Lorentz transformation: gives: F F x x t t t x x t F x t i / e t x x t F x t i / e t x The superposition y : y F x t F x t y / t x i / t x e i e e e i i t x t x (8) blue shifted wave red shifted wave i / t x i t y e isin / x t e isin x (9) that is the expression for a standing wave in a moving referene system. A standing wave in a moving referene frame remains a standing wave, even though photons are oppler shifted. Red and blue shifts do not imply a phase mismath of the standing wave in a avity, although the wavelengths differ. Suh onlusions will not appear if the real and imaginary parts of the wave are onsidered in the alulation. This result is also in agreement with the well known invar-

9 Momentum and energy of a onfined photon 563 iane of the wave number as well as with the invariane of the phase in Speial Relativity []. In ontrast to the behavior of free photons, onfined photons in a standing wave in a moving referene frame undergo length ontration (transverse oppler Effet) and time dilation just like moving masses. The overall momentum and the oppler situation in the pool remain unhanged for a moving avity. Refleted, oppler-shifted pool photons suh as k p (see in Fig.3, right) have the same momentum as if they approah from the opposite side and penetrate the avity ( k represents this). Loally, it looks like a transfer from the pool- into onfined photons and then as a transfer bak into pool photons: p p k left k k right p p k left k k right (30) Equations 30 also give a proof that the neutrality of the pool distribution is not disturbed by the presene of a avity, whether it is at rest or in uniform motion. 3.4 A photon pool onsisting of Casmir s zero point radiation and an additional thermal omponent At a first glane, the demands on the quality of spae, as skethed above, seems to be unrealisti. But in the field of quantum eletrodynamis, Casimir and Polder predited properties of zero-point radiation in spae that may fulfill suh requirements []. Spae is filled with eletromagneti zero-point radiation of nearly unlimited energy. Eletrial onduting plates as shown in Fig.4 positioned at a distane L experiene apparent attrative fores. They result from the fat that inside the avity standing waves are established with L, L, L, L... Outside 3 an additional pattern of standing waves with 4L,6L,8L... up to the diameter of spae is present. Therefore, the flux of photons outside the plates is larger than inside the avity. For the pressure P, Casmir s omputation gives: P 4 40 L. (3) It has been possible to measure the so-alled Casimir fore at metalli foils positioned at a distane of a few mirometers [3]. It should be mentioned that the distribution of wavelengths follows a ubi distribution that has an idential shape in any relativisti system [4]. Therefore, relativity is not affeted in this standingwave framework.

10 564 Christoph Shultheiss Fig. 4: Conduting plates at a distane L, where the number of modes inside is smaller than outside with the onsequene that the outside long wavelength modes (i.e., 4 L ) generate an inwardly direted fore on the plates. A avity embedded in zero-point radiation will experiene an overshooting outer pressure oming from the long wavelength spetrum. The pressure on the outside may be balaned by refleting onfined thermal photons on the inside. For instane, the pressure P onfined of n onfined photons in a gap with a width of L, a wavelength L and a nonrelativisti momentum transfer of during refletion, ating on an area of about L is: P onfined n (3) L L A omparison with Eq.3 shows that the pressure desribed in equation Eq.3 is approximately a fator of 40 n / larger; that is, an expansion of the avity would take plae. In addition to the zero point radiation, an outside ating thermal radiation is needed to balane the fore ating on the avity i.e., the photon pressure ating on both sides of eah mirror is the same and the gap distane remains onstant at L: n n L L 40 n (33) L A stabilizing thermal fration (seond term in Eq.33) is required. A spetrum of 5 ultralong wavelength photons with wavelengths up to 0 m (dimension of spae) ould represent the required thermal fration. This isotropi long wavelength spetrum plays a deisive role in a theory of mass attration governed by sattering and sreening of photons as a new model of gravitation [5] (in this model a thermalization via the photoeletri effet seems suppressed sine the osillation time of a photon with 0 5 m is several billion years). If this is true, a unified theory between photoni mass and gravitation may be within reah (see Set.4 Outlook).

11 Momentum and energy of a onfined photon 565 In priniple, the number n of onfined thermal photons in the avity an be un- k j limited. This is also true with higher modes, with j,,3,... and so on. L However, on the basis of a given density of thermal pool photons (Eq.33, seond term), the avity will expand in a spaing between the mirrors proportional to L 4 (see Eq.3) to redue the expansion pressure. 3.5 Mass of onfined photons in a avity The number n of onfined thermal photons existing in a massless avity at rest represents a mass m * with n / photons propagating to the left and n / photons propagating to the right: m n k n kleft right n k, (34) In the simplest ase, k * k is the wave vetor with the magnitude. In this L ase, the photons builds a standing wave pattern with synhronous right- and leftpropagating half waves. As mentioned above, the number of onfined photons n an have values n, and so on. The mass of the avity at rest inreases proportionally to the number n of onfined photons. Of ourse, a proportional inrease of mass will also take plae with higher modes of a onfined photon k j L with j,,3, Behavior of a photoni mass versus a lassial mass We now disuss the momentum of onfined photons in a moving avity: In Fig.3 (right) one an see that onfined photons in a moving avity are opplershifted. The avity walls remain in uniform motion and the mirror distane remains onstant but is ontrated by the γ-fator. A onfined photon in a moving k avity with mass m has the momentum when propagating parallel and k when propagating antiparallel to the avity s diretion of motion. Sine in the derivation of Eq.5-9, the behavior of a standing wave is onserved when the avity is in motion, the averaged magnitude of the momenta of the onfined photons (in the ground state i.e., k L ) is inreased and an be written using Eq.4 as: n k n n k k * n k m, (35) The averaged magnitude of the momenta of a photoni mass is of ourse related to the kineti energy. Equation 35 shows that the mass of a photoni mass inreases in the same manner as an ordinary mass. The physial meaning of the

12 566 Christoph Shultheiss momentum m * is, if all onfined thermal photons k n esape in one diretion, the resulting reoil momentum of the esaped photons is m *. In analogy to Eq.35, the resulting momentum of onfined thermal photons gives the resulting momentum of the photoni mass: n k n n k k n k m As an be seen in Eqs.35 and 36, the terms on the right, suh as kineti energy and momentum, relate to an ordinary mass and the terms on the left relate to orresponding oppler shifts of ounter-propagating, onfined thermal photons in a photoni mass: (37) (38) Equations 37 and 38 have the form of ordinary mass photoni mass equivalene equations. (36) 5. Prinipal onsiderations for a veloity hange of a photoni mass Let a free photon with wave vetor k ollide with a avity or with a photoni mass at rest. This is a situation that an be desribed by a 80 Compton ollision, but details where the free photon ollides is the subjet of Set.7. After the ollision, the avity reoils. If one lets further free photons ollide with a moving avity, during a subsequent time interval then a model of mass aeleration is reated. Of ourse, the aeleration is not ontinuous but stepwise. In QE fields and fores are produed by a harge via exhange of virtual photons with other harges. The presented model onsiders veloity hanges of partiles as a result of photon ollisions. As an be seen from Fig.5, a transition of a avity from rest to motion represents a hange in the spaing between the mirrors from L to L L /. A simple onnetion framework is shown in Fig.5. Fig.5: The free photon ollision leads to a disontinuous transition of a avity at rest into one in motion

13 Momentum and energy of a onfined photon 567 This framework illustrates that after the front mirror starts to move, the opposite rear side of the avity starts moving with a delay. It resembles slightly the deformation of a kiked ball. The path of the olliding free photon has not been inluded, sine the following investigations show that neither the kik at the front nor the one at the rear side is the loation at whih the refletion takes plae. Nevertheless, the kik model is the simplest desription of the envelope of a partile. By means of elementary geometry, the transition time t is found to be: L (39) However a ritial point is synhronization between the external ollision of the Compton photon and the internal ollisions of the onfined photons without destroying the standing wave senario. We assume propagating and ounterpropagating photons with a phase orrelation π as shown in Fig.6 and a sudden inlination as shown in Fig.5. Generally the introdution of the onfined photons into the inlined tube hanges the phase relation, exept for a speial onfiguration as shown in Fig.6 (left) that onserves the phase relation π. In this ase the front mirror starts to move with the delay t x after the last refletion of the onfined photon at this mirror. Fig.6: Three examples how a onfined photon (standing wave situation) transits from a avity at rest into a avity in motion. Only the example on the left gives the phase onservation orretly.

14 568 Christoph Shultheiss The line through L L L, with the slope rosses the vertial line x 0 at L the position t x : t x (40) The following onlusions an be drawn from this framework: Only by hane will a free photon ollide with the avity in the right phase with respet to the phase of the onfined photon. Most transitions do not allow a ollision. A way out is to assume that at a ollision onfined photons are exhanged with pool photons from Casmir s zero-point radiation with orret momentum and orret phase relation. Before pursuing this further in Set.0, a more detailed understanding of the Compton Effet in the ontext of a photoni mass i.e. a mass that results from a thermal photon onfined in a massless avity must be sought. 6. Compton ollision of a free photon with a photoni mass The onservation of energy and momentum in a Compton ollision (Eq. and ) an be rewritten for a so-alled photoni mass. The onfined thermal photons are divided into right- and left-propagating omponents. Energy and momentum have the form: n n n n n n n n, (4) A omparison of Eq. with Eq. with Eq.4 shows that n the number of onfined thermal photons must have the value. Only one photon is onfined at the initial state L and reates a standing wave with ounter-propagating half waves in a phase relation π. This unit interats with a thermal free photon. etails of the interation are subjet of Chap Energy onservation is equivalent to momentum transfer between free-photon and onfined-photon half waves and the intrinsi oppler shift To see what happens in a ollision, the left side of the averaged magnitude of momenta i.e., the energy onservation equation (Eq.4, n = ) is treated as if the involved photons denoted in urly brakets are preparing the ollision; that is, fititious momentum portions denoted in square brakets are already separated to hange the photon momentum values.

15 Momentum and energy of a onfined photon 569 (4) Of ourse, the fititious momentum portions denoted in square brakets add up to zero. They are arranged on the left as losses and on the right as a gain: (43) By fatoring out the fator (-), the equation gains the form of a fusion between the free photon α and the half wave of the onfined photon: " " Fusion (44) Consequently the ounter-propagating seond half wave (seond in Eq.4 left) appears as a deay after multiplying α in Eq.44 by the oppler fator : " " eay (45) CONCLUSION. In ontrast to the oppler framework (see onlusion ), in the energy framework the free photon enters the system it ollided with and undergoes a fusion proess with the first half-wave of the onfined photon. This leads to a blue shift. The seond half wave deays to the oppler-shifted free photon and suffers a orresponding red-shift. The deay of the seond half wave an be regarded as an intrinsi oppler shift whih takes plae within the photoni mass and not in the spae between the olliding - and the laboratory system. By addition and subtration of equations 44 and 45 and substitution of the expressions by the mass photoni mass - equivalene equations 37 and 38, equations and re-emerge: " " ) " ( " Momentum Conservation Energy Momentum of Magnitude (46) Of ourse the fusion and deay equations (Eq.44 and 45) are also solutions of the Compton formula, sine they depend solely on (see Eq.3 and 0): (47)

16 570 Christoph Shultheiss 8. Geometri requirements during fusion and deay of a Compton free photon for an in-phase situation In Fig.7, the geometrial situation of a Compton ollision for the in-phase situation as shown in Fig.6 (left graph) is skethed. Together with equations 44 and 45 and the substitution k / m k / k, where k and k * are the wave vetors of free- and onfined photons, the following geometrial proess ours: The fusion formula (Eq.44) and the energy onservation behind it suggest that the free photon k must penetrate into the avity although, for onfined photons the avity walls are impenetrable. It fuses with the front half-wave k and indues a blue shift to k (see dotted line in Fig.7). At the same time, when the free photon enters the avity, the seond half-wave k reflets from the rear side of the avity unshifted beause the wall is still at rest. Thus, this senario differs from the one shown in Fig.3, in whih the rear side moves with β* and the halfwave undergoes a oppler red shift ( ). Therefore, to fit into a slanted avity k struture, the seond half wave must deay as desribed in Eq.45. After the ollision, the avity moves with and with respet to the observer at rest all photons have undergone oppler shifts. A phase synhronization between the olliding and the onfined photon in a Compton ollision is unavoidable. The ase with phase mismath is the subjet of Set.0. Before this, the fusion equation (Eq.44) as understood in this Setion leads to a number of interesting onlusions about quantum mehanial properties.

17 Momentum and energy of a onfined photon 57 Fig.7: Sketh of a Compton ollision of a free photon k in phase with the onfined photon of a avity. The onfined photon k is split into two half-waves in the framework of Eq.4. The free photon fuses with the front half-wave k (dotted line) and inreases its momentum. The seond half-wave (dotted line) is transported from a avity at rest into a moving avity. Therefore it deays and emits the reoil photon k. etails are desribed in the text. 9. Motion by stop-and-go displaement - a key for understanding quantum mehanial probability density? Let us assume that a free photon is absorbed by the avity and not refleted immediately. It stays fused with the first half-wave of the onfined photon. In this phase, the avity moves with a veloity orresponding to (see eq.4). If for any reason the seond half-wave does not deay spontaneously with the emission of k (the deay is shown in Fig.7), its refletion stops the motion. Then the first half-wave reflets at the interior side of the front mirror and the period starts again. The stop-and-go motion ontinues as long the free photon remains absorbed. Figure 8 shows a omplex Minkowski spae-time presentation of this ase for a number of refletions.

18 57 Christoph Shultheiss Fig.8: Sketh of `stop-and-go motion of a avity after fusion of a free photon with the first half-wave of the onfined photon and ontinuous refletions This onept seems to lead diretly to a well-known quantum mehanial desription. The reoil veloity an be determined using Eq.4 and the - equation. We have: if (48) After the fusion of the free photon, the avity moves with the veloity v. The flight time of the first half-wave to the opposite mirror is t L t / and bak to the front mirror is. t L / Thus, with Eq.48 the time-averaged veloity ˆ of the avity is ˆ / / / / 0 / 0 ˆ if L L L L t t t t (49) or re-written in physial terms:

19 Momentum and energy of a onfined photon 573 mvˆ k (50) whih is equivalent to the well-known expression of the de Broglie relation between momentum and wavelength. The onlusion is that the average momentum of the photoni mass is equal to the momentum of the absorbed free photon. However the veloity of a stop-and-go avity has a fititious meaning sine it is either zero or represents a Compton reoil. The same is true for its averaged magnitude of momentum i.e., for the kineti energy of the photoni mass. It is also fititious. It is interesting to note that the disontinuous stop-and-go motion of the avity hanges into a forward-bakward vibration if the observer itself moves with the veloity ˆ. The advantage of this onsideration is that the avity has at eah in- stant of time a definite kineti energy, orresponding to ˆ. It turns out, that under the above restrition the time-averaged kineti energy derived from Eq.49 is a fator of 4 smaller than that of the Compton ollision mode (, see Eq.48). Now the avity is plaed into a box with infinitely deep retangular potential walls with side length L. The initial state is an eletromagneti wave with the harateristi wavelength L. Using ˆ (see Eq.49), the time-averaged kineti energy E o m ˆ m is: E 0 h 8m L k m (5) This is exatly the well-known value of the zero-point energy per axis [6]: k E0 j, (5) m where j = orresponds to the ground state of a partile with the mass m, whih is onfined in an area with the harateristi length L. After the fusion, the mass moves in the box in a stop-and-go-manner with the average veloity ˆ, but with a limited lifetime beause of ollisions with the box wall. In a simple approximation, the statistial lifetime is proportional to the remaining flight path between partile after flight reversal and wall of the box. After the deay, the free photon moves with the speed of light, reflets at the box wall and returns to the mass. In the ourse of this proess, the mass moves bak and forth with the fititious veloity ˆ and tends to be present with a maximum probability around the enter of the box as an be seen in Fig.9. Higher modes of the wave ( j, 3..) lead, beause of j (see Set.3.5), to a quadrati inrease of energy in Eq.48 as well as in Eq.5. We an say:

20 Probability of Presene 574 Christoph Shultheiss CONCLUSION 3: Energy onservation suggests that photoni masses absorb free photons (see Set.7). This is true for the time span of refletion and even for the time span of many internal refletions. In the latter ase the de Broglie relation is valid. The partiles move in a stop-and-go-mode with a time-averaged momentum equal to that of the absorbed free photon.,5,0,5,0 0,5 0,0 0,0 0, 0,4 0,6 0,8,0 Position of Partile in the Box Fig.9: Numerial simulation of the probability of presene of a photoni mass partile in its initial state in a box with side length L= (solid line) versus the quantum mehanial probability sin ( x / L) (dashed line) The stop-and-go motion also yields what ould be alled a tramp model, beause the momentum of an absorbed photon is used to move a photoni mass for a arbitrarily long time period with v ˆ h / m. If the photon is emitted in the forward diretion, the photoni mass is at rest again. In priniple, this effet makes it possible to over distanes in spae without energy onsumption. This is in aordane with energy onservation, beause the potential differene between start and end points of motion is zero. In ases with potential differenes 0, the aptured free- as well as the onfined photons hange momentum in aordane with the energy onservation. This is subjet of Setion. 0. Frame of fusion and deay during a phase mismath and phase adaptation. isontinuous motion The problem formulated in Set.5 (Fig.6, left graph) deals with the possibility that a free photon ollide a onfined photon in a avity with just the proper phase that is in-phase as shown in Fig.7. All other phase situations lead to mismathes of

21 Momentum and energy of a onfined photon 575 the standing wave situation in the avity. As mentioned above, a way to handle this problem is to assume that in these ases a spontaneously generated pair of pool photons having the same momentum and orret phase appears in the avity and fuses with the inoming free photon (see Fig.0). p In the ourse of this fusion, the pool photon k (thik solid line, left hand with arrow) is onverted into a onfined one with two half-waves * * ( k P k k ). The seond pool photon moves in the opposite diretion (thik solid line to the right) and ompensates the momentum of the esaping onfined photon on the left k + k P k (dotted line), whih esapes the avity together with its seond half-wave. Fig.0: Effet of a signifiant phase mismath in terms of φ during the ollision of a free photon with a avity. It requires the exhange of the onfined photon (whih esapes together with its seond half-wave to the left) with a pair of pool photons, whih appear with the orret phase (thik solid lines). The half-wave of the leftward propagating pool photon k p fuses with the intruding free photon to the blue shifted half wave / k. The other half-wave reflets at the rear avity wall and deays afterwards into a red-shifted free photon and a red-shifted onfined photon / k. The seond photon k p moves to the right and joins with the onfined photon (whih esapes t / later to the left) to form a pair that hanges the pool distribution (see text).

22 576 Christoph Shultheiss However, neutrality in the pool is not onserved perfetly, sine there is a phase delay in terms of φ between the absorption of the pool photon (time point: free photon intrusion) and the emission of the phase-delayed onfined photon into P the pool ( k P k ). This means that while the seond pool photon k (thik arrow) moves with the orret phase in the diretion of aeleration, the onfined photon esapes with a delay of t = φ/ in the ounter-aeleration diretion into the pool. A probe partile (i.e., a proton, not shown in Fig.0) behind the aelerated photoni mass ollides with a regular pool photon moving to the right but misses the ounter propagating pool photon moving to the left sine this is absorbed by the photoni mass. Thus, the probe moves in the diretion of aeleration until it is stopped by the onfined photon after the delay time t x /. This motion will be alled disontinuous motion as a variation of the stop-and-go-mode desribed in Set.9. epending on the magnitude of phase mismath the magnitude of the 5 displaement ranges between 0 and r m (radius of the proton) and is, on average, ½ r. This proess propagates leftward with the speed of light and is unlimited in time sine the phase distortion is onserved. It ould be mistaken for a fore of attration but is not really suh a fore. As desribed in Set.9, the disontinuous motion of a probe mass reflets the prinipal property of spae, whih allows motion without energy onsumption. Eah inoming free photon triggers a disontinuous motion. It is lear that refletions at single partiles are governed by the Thomson ross-setion. All Compton photons (free photons) with a wavelength omparable to the dimension of the photoni mass ( L r0 ) ollide with a probability of about and free photons with lower energy ollide with a redued probability proportional to r 0 /, unless a mirror-like onglomeration of photoni masses is larger than the wavelength. In this ase, the probability of refletion is also. The power P in arried by a free photon with energy h / that arrives within a time interval t at a photoni mass is: P in h (53) t The probability of refletion at the photoni mass as desribed by Thomson redues P in to P R. Exeptions are the upper mentioned mirror-like onglomerations of photoni masses. If they are larger than the wavelengths of free photons then the fator in M R P remains equal to ( M stands for mirror): 0 r M PR Pin P R P in (54) Even for a low-energy free photon, eah refletion desribed by Thomson sattering at a single harge or at mirror-like onglomerations triggers the delayed emission of a high-energy onfined photon (having the momentum h / r0 ) with the power of:

23 Momentum and energy of a onfined photon h r h P r P 0 r M (55) 0 r0 t r0 t Using Eq.53 to replae the fator t in Eq.55, we have: r0 M Pr P 0 in Pr P 0 in (56) r 0 Finally the photon flux j as well as j M with the dimension: number of photons with momentum h / r0 - per area F and per seond is: r0 r0 j Pin j M j (57) F h r A probe partile (proton) with a ross-setion of r is exposed to this photon flux (r 0 is the lassial eletron radius). It ollides with a high-energy pool photon and a time φ/ later with a high-energy one moving in the opposite diretion. As already mentioned, a displaement dˆ with average value of r 0 will take plae during eah ollision with a probe partile. Therefore the expressions for the flux in Eq.57 will be alled the displaement flux. The produt j gives the number of ollisions. It is a measure for the average displaement per seond: 9 o 0 4 ˆ Pin r0 r0 P r d r0 j dˆ M M in 0 r0 j (58) 8 F h 8 F h The average displaements per seond are tiny even for strong fluxes (see Eq.58, right-hand formula). For instane, for visible light with a power density P in of kw / mm the disontinuous motion per seond is far below r 0. However, with high power laser beams in the MW/GW-area, measurable effets an be expeted, provided the mehanism an really be set up. It is possible that suh effets an already be explored and desribed. A likely andidate is the enigmati photophorese first desribed by F. Ehrenhaft [7], although it was later interpreted as a radiometer effet [8]. Another phenomenon desribed by referenes [9, 0] gives information about motion effets in the viinity of aelerated masses omparable to first order effet in terms of β. A displaement flux resulting from aelerated masses ould be dedued as follows: eviating from the onept desribed in Equations 53 to 58, we start with the senario that mass points m in solid matter of mass M are positioned at fixed distanes from eah other and the oupling of an external fore is done by hypothetial refleting lattie photons. To first order in β, the veloity hange Δv during the time interval / depends on the aeleration A and the momentum transfer of the photon refletion: 3

24 578 Christoph Shultheiss v A t h m v m A t m A h m A The wavelength is virtual, sine for weak aelerations the wavelength far exeeds real lattie distanes. epending on the phase situation (see Fig.0), eah refletion an lead to the emission of a high-energy onfined photon. The resulting displaement flux j A is proportional to / F, to the number of mass points (M/m) in the aelerated mass and to the reiproal refletion time / : M j A F m / (59) (60) To demonstrate the magnitude of the displaement flux, the equivalent of M kgneutrons (with the density of water) is aelerated with A m / s. The wavelength of the fore-oupling photons is 0m and the displaement flux in a square of 36 0m x0m is j 0 displaements per seond. A probe near the A square with a ross-setion of about r 0 is displaed with eah ollision (in the diretion of aeleration) by r 0. Let d be the sum of displaements in one seond: d r 0 0 A r j (6) It is true that the flux is extremely high but the Thomson ross-setion redues the effet onsiderably. Under ertain irumstanes suh small effets an be 30 measured i.e., the very low proton ross-setion of approximately 0 m an be inreased by alignment effets of single rystals.. Inertia of a photoni mass If one applies Newton s third axiom of inertia to a photoni mass then the following happens: uring the refletion the free photon transfers momentum to the photoni mass. Simultaneously the photoni mass transfers momentum to the free photon of equal value and of opposite diretion to the free photon. Let us examine this in a gedankenexperiment where photons are onfined in a box. Via a spring a onstant fore shall at on the box. An observer in the system of the aelerated box states that photons whih reflet at the side of the spring gain a blue shift and those refleting at the opposite side gain a red shift (see Fig.3). The resulting momentum during a refletion period auses a ounterfore against the spring. The momentum obeys Eq.36 and is equal to a photoni mass in motion but here it belongs to an aelerated system. If the aeleration ends the blue- and red shift proesses have also ended. The wavelengths of the photons remain unhanged in omparison to the beginning of the experiment. However an

25 Momentum and energy of a onfined photon 579 observer in the laboratory system measures a resulting photon momentum as desribed in Eq.36. This behavior is also visible in Fig.7. In analogy with the aelerated box, the fusion proess (eq.44) leads to a blue shift and the deay proess leads to the red shift (eq.45) of the onfined photon. The point is that all these proesses are affeted by the free photon itself. To translate these fats into the marosopi: If one throws away (role of free photon) a stone onsisting of photoni masses, then ombined blue and red shifts generate the inertial ounter fore whih is experiened by the hand as a ounter pressure. The inertia of photoni masses shows no differene with respet to ordinary masses. However, it ould be demonstrated that olliding free photons are diretly responsible for the inertia of a photoni mass and not reation proesses in the mass itself. This might be a progress in the understanding of inertia.. A photoni mass in the gravitational field It is well known that under the ation of the gravitational field eletromagneti waves suffer wavelength shifts []. The momentum loss k k of a photon is proportional to the strength g of the gravitational field and to the path s : g s k k (6) The same is true for the onfined photon in the avity. Although it reflets bak and forth, if it passes against the gravitation field it loses energy and momentum as desribed in Eq.6. Sine the dimension of the avity is the avity onsequently also expands. If the avity moves into the gravitational field the situation is reversed []. To relate to Se.9 and to the remarks on the tramp model an absorbed free photon of ourse undergoes the same wavelength shifts as the onfined photon. Assume a photoni mass with an absorbed free photon moving with its fititious veloity against a gravitational field. The question arises what is the ondition for a standstill i.e. the stop-and-go motion against the field is exatly ompensated by the falling distane s. The fall s of the photoni mass in the time interval t L / is s g t where L is the dimension of the avity. We have standstill if: L vˆ g t g (63) For realisti parameters suh as L 0 5 m and g 0 m / s the fititious veloity of the order of 0 0 m/ s. Using the de Broglie relation the wavelength of the absorbed free photon for this ase is 0 m. All wavelengths smaller than the stand- 3 still ondition lead to a propagation of the photoni mass with de Broglie veloity against the gravitational field. The loss mehanism for the absorbed free- and on

26 580 Christoph Shultheiss fined photons with effetive mass m * is the derease of their momenta desribed by the energy loss m g s in Eq.6. Thus, if nature allows it, aligned stop-and-go motions of partiles with fititious de Broglie momentum make it possible to overome gravitational potentials with minimum energy onsumption. This has to be distinguished from the so alled photon roket [3] whih was first ited by H. Sänger [4]. The repulsion in a photon roket is provided by the emission of photons. This is an extremely energeti proess, sine after eah emission of a photon and momentum transfer its energy ounts as a loss. If no potentials are present the ratio energy/momentum is while in the ase of stop-and-go motion this ratio is zero. If gravitation potentials are present the ratio g s / is still low. 3. Conlusion Both, lassial masses and masses onsisting of onfined thermal photons in a avity have the same behavior with respet to momentum and kineti energy. This is the essene of the mass onfined photons equivalene equations (eq.37 and 38). The avity model depends on the existene of a photon pool, whih ompensates the onfined thermal photon refletion momenta at the avity walls by inphase refletions. For example, the postulated photon pool an onsist of the Casimir zero-point radiation together with a yet to be speified thermal ontribution (see Set.3.4 and 4) Set. demonstrates that momentum onservation and oppler shift desribe the Compton Effet of photoni masses exatly. Viewed from this perspetive, the energy onservation seems to be unneessary for 80 -ollision proesses with photoni masses. However it turns out that: Energy onservation requires that for ollision proesses with photoni masses the sum of all oppler shifts of the involved photons has to be zero (eq.43). This is interpreted as the fusion between the free- and the first half wave of the onfined photon (blue shift, see eq.44). Afterwards the unshifted seond half wave of the onfined photon deays to a red shifted free- and a red shifted seond half wave of the onfined photon as it is desribed in eq.45 In priniple the fusion-deay proess of the reoil allows a separation of absorption and emission proess. uring the period of separation the motion of the photoni mass is onsistent with the de Broglie relation between momentum and wavelength. However statements about motions with a fititious veloity together with fititious kineti energy were required.

27 Momentum and energy of a onfined photon 58 Collisions of free photons with photoni masses will generally suffer a phase mismath between free and onfined photons. This will lead to a distortion of the pool photon distribution with the effet that a probe mass behind the aelerated photoni mass experiene a disontinuous motion with fititious veloity (see Set.0). So far, differenes between the statements of Ordinary Relativity Theory [5] and the model presented here are evident. Ordinary Relativity predits in third order of β a weak indued motion of a probe masses in the diretion of aeleration in the environment of aelerated masses, while a photoni mass indues an aeleration dependent disontinuous motion of a probe with a fititious veloity. The inertia of photoni masses results from the fat that the free photon that intrudes into the avity fuses with the onfined photon and auses a blue and red shift as desribed in Eq.36 whih indiates the inertial ounterfore and refletion. So seen the free photon arries the ounterfore into the photoni mass Sine photons experiene oppler shifts in gravitational fields, onfined photons as well as absorbed free photons hange momentum as well. The onsequene is a field dependent spaing of the avity mirrors. It is demonstrated that in priniple the stop-and-go motion of partiles with fititious de Broglie veloity an be used to overome gravitational potentials with minimum energy onsumption. 4. Outlook The presented model lives from an outer pressure to ompensate the interior pressure of the avity (see Set.3.4). The zero point radiation is by far not suffiient to provide this (a fator of 76 too small). The outer pressure an be provided by a ~ 5 dense, isotropi flux j of thermal ultralong wavelength photons ( 0 m ) in the spae as reported in ref.5. This flux auses attration fores between harges similar to gravitation by sattering and mutual sreening. A set of parameters adapted to gravitation suggests that eah harge is penetrated by about photons/s and the diameter of the ross setion for sattering is as small as the Plank length. If one ompares the expansion fore of a onfined thermal photon (see Eq.3) with the ompression fore generated by the flux j of ultralong wavelength photons given in ref.6 (p.48, point ), we have: h h = j L ~ (64) ~ j = 0 63 photons/s (65) 4 L This indiates that both models might have a ommon basis. This gives hope that a theory ould be found where mass, gravitation and quantum mehani an be harmonized.