The Gravitational Constant as a quantum mechanical expression

Transcription

1 The Gravitational Constant as a quantum mehanial expression Engel Roza Stripperwei, 555 ST Valkenswaard, The Netherlands engel.roza@onsbrabantnet.nl Abstrat. A quantitatively verifiable expression for the Gravitational Constant is derived in terms of quantum mehanial quantities. This derivation appears to be possible by seleting a suitable physial proess in whih the transformation of the equation of motion into a quantum mehanial wave equation an be obtained by Einstein s geodesi approah. The seleted proess is the pi-meson, modeled as the one-body equivalent of a two-body quantum mehanial osillator in whih the vibrating mass is modeled as the result of the two energy fluxes from the quark and the antiquark. The quantum mehanial formula for the Gravitational Constant appears to show a quantitatively verifiable relationship with the Higgs boson as oneived in the Standard Model. keywords: gravitational onstant; Higgs boson; quark model; pion; geodesi equation.introdution The basi onept in quantum physis is the partile wave duality, whih implies that a partile an dually be desribed by a mehanial equation of motion and by a quantum mehanial wave funtion. This wave funtion is the solution of a wave equation. The wave equation is obtained by a transformation of the partile s equation of motion in a way as oneived by Dira []. Although Einstein s geodesi equation of motion [] is the most generi of all, it is, so far, not adopted as the axiomati base for the partile wave duality desription. This is probably due to the mathematial omplexity of D spae-time. It is also the reason for the failure to unify quantum physis with gravity. Instead, Dira derived his wave equation from the Einsteinean energy relationship of a partile in motion. Therefore, the equation is relativisti, but only speial relativisti and not general relativisti. In this artile I wish to develop the partile wave duality on the basis of Einstein s geodesi equation in D spae-time and to ompare it with Dira s result. Next to that, I wish to motivate that the D spae-time approximation is a justified modeling of a realisti physial proess. This enables to relate gravity and quantum physis to the extent that the Gravitational Constant an be formulated as a quantitatively verifiable expression of quantum physial quantities. The onepts as will be outlined in this artile, are invoked from previous work by the author [3]. Setion is devoted to a omparison of the D quantum mehanial wave equations as derived respetively by the geodesi approah (setion ) and Dira s approah (setion 3). This will result into an expression for the Gravitational Constant (setion 5). In setion 6 a physial proess is seleted that will deliver the quantity values. The result is subjet to a relativisti orretion (setion 7). The final result is disussed in setions 8 and 9. The equations in this artile will be formulated in sientifi notation and the quantities will be expressed in SI units. Spae-time will be desribed on the basis of the Hawking metri (+,+,+,+).

2 . The geodesi approah towards a D wave equation Let us onsider two frames of Cartesian oordinates ( x, x) ( x,it) and (, ) (,i ). The first one is the frame of a stationary ( lab frame ) observer O. The seond one is the ( enter of mass ) frame of an observer o- moving with a partile P. Time t and proper time are normalized on the vauum light veloity suh that t it and i, where i. Under stationary onditions, the D geodesi equation an be written as [], x d g g { x ( ) d g x dt ( ) }, d d xx tt d t d g tt xx g tt x dt ( ). d d () The quantities g xx and g tt are elements of the metri tensor. They detere the way how the frame (, ) of the o-moving observer by onsidering funtions ( x, t) and ( x, t) - is transformed into the frame ( x, t) of the stationary observer. In partiular ( ) x ( ) x g xx and ( ) t ( ) t gtt. () The adopted stationary ondition means that the metri tensor omponents are supposed to be independent of time t. The geodesi equation expresses that the straight spae-time path (, ) of the o-moving observer, is observed as a urved path by the lab frame observer. It is the onsequene of expressing a presupposed field of fores into a metri tensor. The seond part of () an be integrated into dt d k int g tt, (3) where have k i.e., d k is an integration onstant. Considering that dt / d in flat spae-time ( g ), we int int i. In the ase of a onservative field of fores, the loal spae-time interval is invariant, g xx g dt, so that tt tt d ( ) d dt g xx ( ) gtt ( ). () d d From () and (3) we get ( ) d ( g xx g tt ). (5) This result is onsistent with the integration of the first part of () under onsideration of (3). Under the axiomati quantum mehanial hypothesis

3 p pˆ, with pˆ i x, where p m, (6) d where m is the rest mass of the partile in onsideration, and where is Plank s (modified) onstant, we get from (3) and (5), m i and im f ( x), t x g tt where f ( x) ( ). (7) g xx g tt The two parts of (7) an be joined to a semantially orret quantum mehanial wave equation. This an be done by differentiation of the seond part and addition to the first part after multiplying with / m, resulting in, m df i { ( m f i )}. (8) t m x g tt The equation is temporally of first order and spatially of seond order. It guarantees the positive definiteness of the wave funtion. This means that the spatial integral of squared absolute value of the wave funtion is time-independent. This is required to obey the semantis of the wave funtion, whih states that its squared absolute value expresses the probability that the partile is at a ertain moment at a ertain plae. As this ertain plae must be somewhere, the spatial integral has to be a time-independent. Eq. (8) has a similar format as Shrödinger s equation. The differene is in the right-hand part. In Shrödinger s equation this part is the (spatially dependent) potential energy of the partile in motion. Here, it is replaed by a quantity that expresses the metri urvature of spaetime. 3. Dira s approah towards a D wave equation Dira derived his relativisti quantum mehanial wave equation for a partile moving in free spae from a heuristi elaboration of Einstein s energy relationship E ( m p, (9) ) ( ) where p is the three-vetor momentum ( d s / dt, not to be onfused with p ), squared as E p m p3, () or equivalently, p p, with p p. () m Under onsideration of the required semantis, Dira expanded and transformed this quadrati equation into a set of two linear ones, 3

4 pˆ ˆ p pˆ ˆ p where are the Pauli matries, so that i, p pˆ i and p pˆ i. () ˆ 3 ˆ 3 If the veloity of the partile is small with respet to the light veloity, we have from (), m, so that p3 m, and therefore p ˆ i. (3) p i From (3) and the first part of (), and adopting the us sign to avoid a meaningless result, it follows i ˆ p 3. () After substitution of () into the seond part of (), we get, under onsideration of (6), m. (5) t m x i Mutatis mutandis, and now adopting the plus sign in (3), we get i m and ˆ p 3. (6) t m x i Eqs (5) and (6) an be summarized as m and t m x i m x. (7) Eq. (7) is known as the Pauli-Shrödinger approximation of Dira s equation. The solution is a twoomponent wave funtion, with a doant omponent and a or (spin) omponent that an assume two states. If a partile is subjet to a field of fores, Dira s Equation as formulated in (7) does not hold. Potentially however, its format an be preserved if the operators on the wave funtion are redefined in a suitable way. This is known as the appliation of the Priniple of Covariane. This involves a redefinition of the operators in the wave equation, implying that derivatives of the wave funtion are replaed by ovariant derivatives. In partiular, ga D ( ). (8)

5 where g is a dimensionless generi oupling fator and where, generially, A are the omponents of the four-vetor potential A, A, A, ) that haraterizes the field fores, and where ( 3 A A i / is the salar part of the field. In the ase that the field is haraterized by a salar only, we get for the doant wave omponent from (7) and (8), ih g t m m. (9) x. Comparing the geodesi approah with Dira s approah Comparing (9) with (8), it is onluded that, under non-relativisti onditions, the partile s wave equation in a onservative salar field of fores derived by the geodesi approah and Dira s approah, are equivalent if m df g m ( m f i ), () g tt where f is a funtion of the metri omponents mehanial oupling fator g ). g tt and g xx (not to be onfused with the quantum Let us proeed by onsidering onditions as often assumed in general relativisti gravitational problems. These are, isotropy and weak field ondition, implying g tt ( x), ( x), where x, x. () g xx This enables to rewrite () as g x m. () This expression relates the spae-time urvature with the potential energy of the partile in motion. In the partiular ase that the potential field shows a spatial quadrati dependeny, the wave equation represents a quantum mehanial osillator. This is true if g g ( k k ), (3) 5 x where the normalization parameter is introdued to make the onstant k dimensionless. Elementary algebra shows that () and () are equivalent if k x g, where, provided that k ( ) g. () m k Eq. () imposes an interrelationship between the partile s potential energy and its rest mass. This may seem an over-onstraint. Atually, it is not, beause the rest mass energy urves spae-time and

6 the potential energy is a manifestation of the spae-time urvature (otherwise equating the geodesi approah with Dira s approah would be meaningless). 5. The Gravitational Constant Energeti fields are formats of energy, similarly as massive partiles in rest. All formats of energy are subjet to Einstein s field equation. Aepting the universality of this priniple beyond the realm of gravity, justifies applying Einstein s field equation to nulear energeti fields. One side of this equation is the Einstein tensor ( G ), whih is expressed in quantities that an be derived from the metri tensor. The other side is the energy momentum tensor ( T ), whih ontains quantities that an be derived from the energy present in the spae as haraterized by the metri tensor. There is a proportionality fator G involved, known as the gravity onstant. The full expression is 8G G T with G R Rg. (5) Here, R and R are respetively the so-alled Rii tensor and the Rii salar, whih an be alulated if the metri tensor omponents g are known [,]. Let us start by alulating the energy momentum tensor. This presupposes knowledge of the spatial energy density. e wish to proeed under the assumption that all energy is omprised in the potential energy g of the moving objet and that the energy density w is given as (see Appendix I), w 8 ( ) d. (6) The viability of this assumption will be shown for a partiular physial proess, to be disussed in setion 6. From (6), it follows that T T 8 ( ) d and T T. (7) Note that the index applies to the spatial dimension and to the temporal dimension. It follows straightforwardly from (7) and (3) that, T (k x). (8) 8 ( ) The more diffiult part is the alulation of the Einstein tensor from the Rii tensor R ) and ( R ) the Rii salar R, to be obtained from the metri tensor g( g). Beause of the D isotropy ondition, the alulation results into rather simple expressions [], R R g g R, and (9) g g g R gr. (3) 6

7 Note: a and a is short for differentiation and double differentiation (after x )of the parameter a, and g is the inverse of the matrix g. Appliation of () and (3) on (9) and (3), gives R x so that G, (3) G R Rg R( g) R{ ( )} R x. (3) From (5),(8) and (3), under onsideration of (), the Gravitational ConstantG follows as, G ( ) 3 m ( ) ( k kg s ) kg, with. (33) This results means that it is possible to express the Gravitational onstant into quantum mehanial quantities, provided that we an identify a physial proess that an be modeled as a harmoni D quantum mehanial osillator where the mass of the objet in motion is extrated from the field s potential energy. 6. The physial proess In this setion, it will be laimed that suh a proess exists. It is the basi quark dipole, known as the pi-meson. It is ommonly aepted that the quark masses have their origin from an omni-present salar field of energy, known as the Higgs field. This field is funtionally desribed by a Lagrangian density with two harateristi quantities and, suh that [7, p.363] ( ) U H ( ), with U H ( ). (3) Usually, the soure term is omitted, beause it is simply stated that an unknown soure sustains the field. But let us onsider the onsequenes if we do not wish to aept an inomplete Lagrangian density desription. To do so, let us ompare the Lagrangian density of the Higgs field with the Lagrangian density of the type ( ) U( ), with U ( ). (35) Let us further suppose that the soure is a three-dimensional Dira distribution ( r). Appliation of the Lagrange-Euler equation yields a differential equation for the spatial behavior of the field s potential energy. In this ase, d r dr ( r ) ( r), (36) The solution of (36) is 7

8 exp( r). (37) r If and Q /( ), where Q is the eletri harge of a pointlike soure and is the free spae eletri permeability, this expression represents a Coulomb field. Generially it represents a field with a format that orresponds with the potential as proposed by Yukawa [5] to explain the short range of a nulear fore. It an also be viewed as a sreened Coulomb field, whih originates if the free Coulomb flux is suppressed by a surrounding spae harge, suh as first desribed by Debije [6]. The Debije shielding is observed in a wide variety of physial proesses, partiularly in plasma physis and in astrophysis. In these proesses the Debije shielding length / may range from values as large as 5 m to values as small as -9 m. Unfortunately, the high non-linearity of the potential energy term U H () in the Lagrangian density of the Higgs field, as shown, in (3) prevents the analytial derivation of a spatial desription of the Higgs field from a pointlike soure ( r). But if we annot solve the spatial field equation analytially, why not solving it numerially? The way how to do has been doumented in previous work [3a], in whih it has been shown that the potential (r) that satisfies the wave equation derived from (3), is losely approximated by, exp[ r] exp[ r] ( r) ( ) r r with.6 and 3.3. (38) This field shows far field harateristis orresponding to the Yukawa ones, next to near field harateristis with opposite diretion that represents a short-range fore. e identify the far field fore as the fore that is ommonly known as the weak fore (short for fore responsible for weak interation) and the salar near field fore as the strong fore (short for fore responsible for strong interation). The nulear fore is the ombination of the two, suh that the field, as shown by (38), is given by s w with s 8 exp[ r] and ( r) exp[ r] w. (39) r If we identify the quark as the soure of this nulear field, a nie piture is obtained. Any quark ouples to the field of any other quark with the oupling fator g, suh that it feels a nulear fore F desribed as F g, () r where the quantum mehanial oupling is supposed to be equal to the square root of the eletromagneti fine struture onstant ( g /37, see Appendix I). This means that any quark is repelled by any other quark under influene of the far field, but attrated by the near field. As a onsequene, strutures are possible to exist that are omposed by two quarks or three quarks, whih are holding eah other in a stable equilibrium. Similarly as Lorentz did in the past for eletrons, we may suppose that the masses of these strutures will primarily, if not all, be detered by the fields that result from the potential fluxes from the omposing quarks. In this model, two quarks, positioned at a spaing d apart, ompose a struture, the enter of mass of whih will vibrate around the position just half-way the two quarks. Therefore, this quark dipole, to be identified as meson, an be modeled effetively as an equivalent single-body quantum mehanial osillator with a ertain effetive mass that is built by the energeti fluxes from the quarks.

9 The far field omponent (weak fore) an be oneived as the salar part of the four-vetor potential A, A, A, ), of a field with the harateristis of Proa s generalization of a Maxwell field, ( 3 A desribed by a Lagrangian density of the type, L F F A A J A with F A A, () x x where the vetor omponents represent the soures of the field, possibly exlusively onsisting of the pointlike well ( r). In the past, suh a field has been onsidered as a andidate to explain the origin of nulear fores indeed. It has been abandoned beause the term spoils the gauge onstraint that is required to obtain the ovariant format of Dira s equation. Stuekelberg, however, has shown that the gauge onstraint remains valid in the ase that the vetorial Proa field is supplemented by an additional salar field. In Stuekelberg s time, there was no rationale for this artifiial esape. ithin the ontext of this artile it makes sense, beause the salar near field in (39) may serve the purpose. The justifiation for it is shown in Appendix II. A side effet of this solution is the removal of a possible renormalization problem assoiated with the inverse square potential format of the salar near field. The meson is subjet to a quantum mehanial wave equation, whih will be developed within the enter of mass frame. As noted before, the two onstituting quarks will hold eah other in a stable equilibrium. This enables us to develop a one-body equivalent of a two-body osillator. Although suh osillator resembles a lassial one, there is a fundamental differene. In the lassial ase, we have two masses and a (potential) field in between. In the lassial ase, the energy aptured in the two masses is muh larger than the energy represented by the field. In the model to be developed here, the other extreme is adopted: the bare mass of the bodies is supposed to be negligible as ompared with the field energy. It makes the osillator relativisti, beause the mass in the wave equation is no longer the mass of the two bodies, but it is an equivalent mass that aptures the energy of the field. In spite of the relativisti nature of the model, the enter of mass view allows applying the Pauli-Shrödinger approximation of Dira s wave equation, desribing a linear motion of an effetive mass between the quark enters. Therefore, we write, d { U( d x) U( d x)} E. () m m Here, m the non-relativisti effetive mass of the enter, V( x) U( d x) U( d x) its potential m J energy and E the generi energy onstant, whih will be subjet to quantization. From (3) we have, exp( x) exp( x) U( x) g ( ). (3) ( x) x The potential energy V (x) an be expanded as V( x) U( d x) U( d x) g ( k k x...), where () k exp( d) exp( d) ( ), and ( d) d 9

10 k exp( d) exp( d) (6 d 8d) ( d ). (5) 5 ( d) ( d) d The two quarks in the meson settle in a state of imum energy, at a spaing detered from the ondition d d, exp( d) d, (6) so that d d.856; k -/ and k.36 (at d d ). The quantum mehanial osillator as represented by () is subjet to exitation. This means that the energy onstant E is subjet to exitation. Under the approximation of the potential energy by a quadrati polynomial as (), the onstants assoiated with the vibration energy ( n / ), are equally spaed. The frequeny is given by the generi basi relationship [9], E n m m g k. (7) A partile-antipartile onjuntion, as in the ase of a meson, has partiular harateristis. All mass is in the binding energy between the two partiles. This allows the onlusion that the spaing d between the quark and the antiquark is detered by half the wavelength of a single harmoni energeti standing wave (boson) with phase veloity, so that ( ) d, (8) ( ) where is a dimensionless onstant with order of magnitude, introdued for orretions beause of the rude modeling and where is the energy of a bosoni mass partile responsible for hange of energeti states of the meson. This boson is known as the weak-interation boson. Its energeti value is known in the lab frame from experimental evidene of nulear deay proesses, but will be subjet to relativisti orretion in the enter of mass frame. From (7) () and (8) it follows that k g. (9) (Note: the value of the effetive mass mm is irrelevant within the sope of this paper. Eventually it omes manifest in the lab frame as the rest mass of the meson). Under onsideration of (8) and (9), the gravitational onstant, as expressed by (33), an thus be written as, G g k ( ) ( ) k ( ) d 3 [m ] ). (5) [kg][s ] (

11 This analysis has been made in the enter of mass frame, i.e., the frame of the o-moving observer. The energy is known in the lab frame as m 8. GeV. The o-moving observer experienes the lab frame veloity as the lab frame speed of the pi-meson. Owing to this partile-antipartile struture, the pi-meson flies at near light veloity v. Therefore, (5) is subjet to a major relativisti orretion, so that, effetively, G g k ( ) m k ( ) d 3 [m ] ), with [kg][s ] ( ( v / ). (5) The relativisti orretion will be onsidered in the next setion. 7. The relativisti orretion The omposite field of three quarks in a baryon, relatively far from the internal struture, as built-up by three ontributions of the type as defined by (39), shows the same behavior as that of a single quark. It has both the harateristis of the attrating near field and those of the repulsive far field. If the inter-baryon interation is represented by a massive bosoni partile with a ertain spatial range and if this range is one or more orders of magnitude larger than the range of the intra-baryon interation bosons, the inter-baryon behavior is just a saled behavior of the intra-baryon behavior. In that ase a similar potential funtion an be defined for the remnant field of the baryon similarly as (39), with the only differene that the -value has to be saled. Therefore, supposing that, the inter-baryon field an be defined as exp( r) exp( r) b ( r) ( ). (5) r r Other baryons may ouple to this field, with some dimensionless oupling fator g. Identifying the interating bosons as pions with rest mass energy m (= 35./39.6 MeV), we have m d. (53) ( ) Aording to this model, in non-exited state, the distane d between bound baryons therefore is, typially, d d (.6 fm). (5) The pions travel at near light speed. As shown by atkins [], this speed v an be detered from the half life value t, in proper time denoted as. It is based upon the relationship between the temporal deay rate and the spatial deay parameter. The deay behavior of N( ) partiles in proper time is given by N ( ) N exp( ). (55) From the half life definition, given by

12 N( ) N( ) / (56) It follows from (55) and (56) that ln(). (57) The relationship between spatial deay and temporal deay is frame independent and given by v (58) so that from Eqs. (55)- (58) after relativisti orretion for, ln() t, where v ( v / ), (59) so that, for v, under onsideration of (53), ( v / ) ( )ln(). (6) d m t 8. Result Summarizing, we have (5) and (6), G g k ( ) m k ( ) d 3 [m ] ), where [kg][s ] ( ( )ln(). d m t The result of this analysis is summarized in table I. The left-hand olumn shows the values of the physially known quantities. The middle olumn shows the known dimensionless onstants as they are established in this theory. The right-hand olumn shows that the known value of the gravitational onstant is obtained for the parameter value.69. This value is in agreement with the presupposed order of magnitude. physis this theory alulated 93 MeV fm.69 m 39.6 MeV k.36.59x -6 t.63x -8 s k -/ G 6.67x - m 3 kg - s - m 8. GeV d.856 g /37 Table I. Numerial result for the gravitational onstant, alulated from quantum mehanial quantities.

13 In view of the order of magnitude of the quantities involved, this attempt to unify gravity with quantum mehanis yields a surprising fit of the theoretially alulated value of the gravitational onstant with the experimentally established onstant of nature. The redibility of this result an be underlined with an interesting observation. It has to do with the 6,5 GeV partile, disovered in by CERN and identified as the Higgs boson of the Standard Model. The energeti value of its mass is known to be a funtion of the quantity in the funtional definition of the Higgs field as given by (38). In partiular [7, p.36] m ( ). (6) H The Standard Model, however, is unable to establish theoretially based values for the two Higgs quantities and. That prevents a alulation of (6). Interestingly, though, the theory as outlined in this artile, may do. How? Similarly as the boson of the eletromagneti field, neither the boson of vetorial far field, nor the boson of the salar near field are observables. They an only show up by signatures. There is no reason why the view on the Higgs field as developed in this artile would exlude the possibility that photons interat with the Higgs field as spread by quarks. Suh interations would produe new partiles of the same kind as predited in the state-of-art theory. Let us try to alulate the mass equivalent of suh new partiles under appliation of the relationships as developed in this artile. As shown by (38), the equivalents of the quantities and in the ommon funtional desription of the Higgs field are the spatial field quantities (for is strength) and (for its spatial range). It has been shown in Eq. (9) that the strength quantity an be detered from the weak interation boson m (= 8. GeV), as m, (6) g k and that the spatial range quantity follows from the ratio /, alulated from (8) and (9) as, ( ) g k d m. (63) (Note that, unlike the individual quantities and, the numerial value of this ratio is frameindependent). It has to be emphasized that the results (6) and (63) have to be redited to the general relativisti view on the physial quark dipole. Applying (38) on (6), we have. ( ). (6) m H From (6-6), and table I, it follows that, m H d m 7 GeV. (65) This theoretially established energy value mh of the Higgs mass is lose to the measured value by CERN. ithin the ontext of this artile, it may be regarded as a onfirmation of the viability of the relationship between gravity and quantum physis as expressed by (5), (6) and the numerial result given in Table I. 3

14 9. Disussion It will be lear that the validation of a formula that expresses the Gravitational Constant in quantum mehanial quantities is an important ontribution to the on-going hallenge in present theoretial physis to relate gravity with quantum physis. It will also be lear that a laim that this an be done without oneiving a new theoretial framework beyond the established ones, will be regarded as ontroversial. Nevertheless, in this artile an attempt to do so has resulted in a formula that an be easily numerially verified in an outome that niely fits with the well-known established experimental evidene. This numerial result annot be denied. The theoretial steps might be subjet to ritiism, but the ritiism should then answer the question why suh a result is possible if something is wrong with the basi approah. So, let me summarize the approah, thereby stipulating different angles of view as ompared to present theory. Among the various steps taken, there are three major ones. One of these is the view that Einstein s field equation is universal, therefore also valid beyond the realm of gravity and, therefore, appliable to nulear energeti fields as well. The seond one is the view that Dira s approah for deriving a quantum mehanial wave equation from an equation of motion an be made more fundamental by taking Einstein s geodesi equation as a starting point rather than restriting it to Einstein s expression for relativisti energy. The third one has to do with the Higgs field. Similarly as in present-state quantum theory, the Higgs field is onsidered as a field from whih energy an be subtrated for the purpose to give mass to partiles. The view taken, however, is that the lak of a soure term in the formulation of its Lagrangian density is unaeptable. Aepting a pointlike soure as in onventional field theory, enables the derivation of a spatial field desription, albeit that a numerial approah is required to do so. The logial step taken is, to identify the quark as the pointlike soure of energy. This is not all. The most essential element in this third step is the split of the Higgs field into two different omponents: a bosoni vetorial far field in terms of Proa s generalization of the Maxwell field and an additional bosoni salar near field with a narrow spatial reah. Owing to these harateristis, there is no need to explain the origin of massive nulear bosons as a onsequene of the interation of mass less partiles with a salar field. Instead, the nulear fore bosons ome forward by definition, similarly as photons show up in a Maxwell field. Finally, I want to emphasize the differene between the term, whih speifies the spatial range of the nulear fore, and a physial mass term m / h, suh as in Proa s original formulation. Owing to the invariane of the ratio /, the energy of a H-type Proa boson remains, similar as a photon, the same in any inertial frame. If somebody tries to bring the Proa boson to rest, the term hanges in oherene with the hange of. It is for that reason that the quantum of the Higgs field annot be identified as an observable massive boson. Experimental data on observables (like many fermions) are hard, but experimental data on non-observables, like bosons flying at (near) light speed, are soft. They show up as signatures, whih are interpreted with a theory in d. A signature that supports the theory as developed in this artile is the theoretially derived value for the mass of the 6,5 GeV Higgs partile, whih in present-state theory needs to be empirially established. This is a seond major result of the analysis presented. Appendix I In the physial proess as desribed in setion 6, the nulear field is oneived as an energeti field similar to that of an eletromagneti field. The spread of an energeti flux by pointlike wells auses reates a spatial field, whih, in Maxwell s theory, is haraterized by its energy density, i.e. the amount of field energy per unit of volume. The nulear equivalent of the Maxwellian field density an

15 be found on the basis of the unifiation hypothesis. ithin the ontext of our physial proess this hypothesis is, in SI units, formulated as: e e g and e g, (A) where e and are the salar parts of respetively the eletromagneti potential and the nulear potential, e is the elementary eletri harge and the free spae eletri permeability. The hypothesis states that the square of the nulear oupling fator g is equal to the eletromagneti fine struture onstant ( g /37). The justifiation of the unifiation hypothesis has to be provided by experimental evidene. For its validation, see [3]. The energy density of the nulear field an now expressed in similar terms as the eletri energy density of the eletromagneti field, i.e., as w e. (A) 8 Appendix II In 938, Ernst Stuekelberg [,] showed that, under partiular irumstanes, the elegany of the Priniple of Covariane on the basis of imum substitution, an be maintained for Proa type fields. This will be the ase if, next to a Proa field, an auxiliary salar bosoni field B, will be present, suh that Proa s Lagrangian is modified into, [], L = F F ( A ) ( A B). (B) 5 B This modified Lagrangian density remains unaffeted under the gauges, i A A s ( x, x, x3, x ) g i B B s ( x, x, x3, x ). (B) g Therefore, it is allowed to use the ovariant Dira equation in a Proa field of fores if an auxiliary salar field is present as well. Let us try to identify the near field (strong fore) as defined in (39) as Stuekelberg s auxiliary phantom salar field. First of all, we have to ope with the r -term in the denoator of the near field expression, whih is not ompatible with a pointlike soure. This would suggest that the soure of the salar field is a dipole rather than a monopole. Unfortunately, although the dipole shows an inverse square behavior of the energeti flux indeed, it shows an angular dependeny as well. The esape omes from a reonsideration of the axiomati priniple as adopted by the author in his sequene of papers. It has to do with the numerial fit of the spatial expression (38) with the funtional expression of the Higgs field. Curiously, another expression gives a fit with a similar auray. The fit is obtained by exp( pr) exp( qr) ( r) = { a b } r r with a 5.8; b.737; p.95 and q.88. (B3)

16 This an be rewritten as, exp( pr ) exp( r ) ( r) = { a }, (B) r r where q ; bq ; p p / q and a a / b. This allows to inorporate the soures of the vetorial far field and the salar near field into a omplete Lagrangian desription. Moreover, we may invoke the generalization proposed by t Hooft and Veltman, by inluding an additional real parameter, [,3 p.76], suh that, L = F F ( A B ) ( A B) J A B. (B5) Applying the Euler-Lagrange Equation yields two wave equations. Desribing the pointlike soures for the far field and near field, respetively, as ( r) and a ( r), and defining B s, we get for the salar part of the time independent (vetor type) far field, d r dr ( r ) ( r), (B6) and for the time independent (salar type) near field we get s d r dr ( r ) s s a ( r), (B7) By omparing (B7) and (B), obviously p. It may seem that both equations have the same format as the Klein Gordon equation, whih has erroneously been derived for fermions as a forerunner of Dira s Equation. The presene of the soure term, however, make these two bosoni equations different. It will be lear now that, under the modifiation (B3), the Stuekelberg mehanism allows to give a spatial desription of the funtionally defined Higgs field. The field an be assigned to a single omposite soure that produes a vetorial Proa type (weak fore) far field and a salar type (strong fore) near field. For reasons of simpliity, I wish to stik to the two-parameter formulation as expressed by (38) rather than by the four-parameter equivalent (B). Referenes [] A. Einstein, Relativity: The Speial and General Theory, H. Holt and Company, New York (96, translation 9) [] P. A. M Dira, Dira, Proeedings of the Royal Soiety A: Mathematial, Physial and Engineering Sienes 7 (778): 6. (98) [3] E. Roza, (a) Phys.Essays,, 7 (); (b) Phys.Essays,, 395 (); () Phys. Essays 5, 9, (); (d) Phys. Essays, 5, 39 (); (e) Phys. Essays, 6, 38 (3); (f) Phys. Essays, 7, 3 () [] T.A. Moore, A General Relativity orkbook, University Siene Books, (3) 6

17 [5]H. Yukawa, Pro. Phys. Math. So. Jpn 7, 8 (935); H. Yukawa and S. Sakata, Pro. Phys. Math. So. Jpn 9, 8 (937) [6] P. Debye and E. Hukel, Physikalishe Zeitshrift, vol., no.9, p.85-6 (93) [7] D.J. Griffiths, Introdution to Elementary Partiles, iley-vch, einheim Germany () [8]P.. Higgs, Phys. Rev. Lett. 3, 58 (96) [9] Quantum Harmoni Osillator [] T. atkins, Yukawa and the Pi Mesons, [] E.C.G. Stuekelberg, Helv. Phys. Ata,, 5- (938) [] H. Ruegg and M. Ruiz-Altaba, arxiv:hep-th/35v (3) [3] J.C. Taylor, Gauge theories of eak Interations, Cambridge University Press, Cambridge (976) 7