The Corpusular Struture of Matter, the Interation of Material Partiles, and Quantum Phenomena as a Consequene of Selfvariations. Emmanuil Manousos APM Institute for the Advanement of Physis and Mathematis, 3 Pouliou str., 53 Athens, Greee Abstrat With the term Law of Selfvariations we mean an exatly determined inrease of the rest mass and eletri harge of material partile. In this artile we present the basi theoretial investigation of the law of selfvariations. We arrive at the entral onlusion that the interation of material partiles, the orpusular struture of matter, and the quantum phenomena an be justified by the law of Selfvariations. We predit a unified interation between material partiles with a unified mehanism (Unified Selfvariations Interation, USVI). Every interation is the result of three learly distint terms with learly distint onsequenes in the USVI. We predit a wave equation, whose speial ases are the Maxwell equations, the Shrödinger equation, and the related wave equations. We determine a mathematial expression for the total of the onservable physial quantities, and we alulate the urrent density 4-vetor. The orpusular struture and wave behaviour of matter and their relation emerge learly, and we give a alulation method for the rest masses of material partiles. We prove the «internal symmetry» theorem whih justifies the osmologial data. From the study we present, the method for the further investigation of the Selfvariations and their onsequenes also emerges. Keywords: Partiles and Fields, Quantum Physis, Cosmology.
. Introdution The present study is founded on three axioms: The priniple of the onservation of the four-vetor of momentum, the equation of the Theory of Speial Relativity for the rest mass of the material partiles, and the law of Selfvariations. With the term Law of Selfvariations we mean an exatly determined inrease of the rest mass and eletri harge of material partile. It is onsistent with the priniples of onservation of energy, momentum, angular momentum and eletri harge. It is also invariant under the Lorentz-Einstein transformations. The most immediate onsequene of the law of Selfvariations is that the energy, the momentum, the angular momentum, and the eletri harge of material partiles are distributed in the surrounding spaetime (when the material partile is eletrially harged). In order for the value of the eletri harge to inrease in absolute value, the eletron, in some way, should 'emit' a positive eletri harge in the spae-time environment. Otherwise, the onservation of the eletri harge is violated. Similarly, the inrease of the rest mass of the material partile involves the emission of negative energy as well as momentum in the spae-time surrounding the material partile (spaetime energymomentum, STEM). The law of Selfvariations desribes quantitatively the interation of material partiles with the STEM. Every material partile interats both with the STEM emitted by itself due to the selfvariations, and with the STEM originating from other material partiles. The material partile and the STEM with whih it interats, omprise a dynami system whih we alled generalized partile. We study this ontinuous interation in the present artile. For the formulation of the equations the following notation is used: W the energy of the material partile J the momentum of the material partile m the rest mass of the material partile E the energy of the STEM interating with the material partile P the momentum of the STEM interating with the material partile E the rest energy of the STEM interating with the material partile With the above symbolism, the law of Selfvariations for the rest mass is given by equations
m b Em t. (.) b m Pm in every system of referene (t, x, y, z )., is Plan s onstant, b x y z. onstant, b, b and The the findings resulting from the law of Selfvariations will be referred to as "the Theory of Selfvariations" (TSV). Initially, we present the TSV in inertial frames of referene.. The basi study of the internal struture of the generalized partile We onsider a material partile with rest mass m. That is, we onsider a generalized partile. The rest mass m and the rest energy E given by equations (.) and (.) respetively aording to speial relativity [-4] m W J 4 (.) E E P. (.) We now denote the four-vetors X x it x x x y x3 z (.3) 3
J J iw J J J y J J 3 z J x (.4) P ie P P P x P P y P3 P z (.5) where is the vauum veloity of light snd i is the imaginary unit, i. Using this notation, equations (.), (.) and (.) are written in the form of equations (.6), (.7) and (.8) m x b P m,,,,3 (.6) J J J J m (.7) 3 E P P P P3. (.8) After differentiating equation (.7) with respet to x,,,,3 we obtain J J J J m J J J J m 3 3 x x x x x and with equation (.6) we obtain J J J J b J J J J P m 3 3 x x x x and with equation (.7) we obtain J J J J b J J J J P J J J J 3 3 3 x x x x 4
J b J b J P J J P J x x J J b P J J J b P J 3 3 3,,,,3 x x. (.9) We now symbolize Ji b P J i i,, i,,,3. (.) x With this notation, equation (.9) an be written in the form J J J J3 3,,,,3. (.) We now need the 4 4 matrix T as given by equation 3 3 T. (.) 3 3 3 3 33 With this notation, equation (.) an be written in the form TJ. (.3) We now prove the following theorem: Theorem. For m, and for every i,,,,3 equation (.4) holds Pi P, i,, i,,,3. (.4) x x i Proof. Indeed, by differentiating equation (.6) with respet to xi, i,,,3 we get m b xi x xi and using the identity P m m m xi x x xi we get m b x xi xi P m 5
and with equation (.6) we have b b Pm i P m x xi m P m P Pi m P m x x x x i i i and with equation (.6) we have b P b P P m m P Pm m i i i x P P m i x xi P x i and sine m, we obtain equation (.4). We now prove the following theorem: Theorem. For every i,,,,,3 the following equation holds i x v b b P P. (.5) vi v i vi x Proof. Indeed, by differentiating equation (.) with respet to xv, v,,,3 we get i J i b x x x x v v v and with identity J i J i xv x x xv we get i J i b x x x x v v v PJ PJ and with equation (.) we have i i 6
b b P J x x x i v i vi i v v b b x x x x P J P J i vi v i i v v i vi i v i v i i v v v P J b J b P b J b P P J P J x x x x x x i vi b Ji b Ji b Pv P Pv P Ji xv x x xv x xv and with equation (.4) we get b J b J P P x x x x i vi i i v v v and with equation (.) we get i b b b b x v vi Pv P Ji i P Pv Ji vi x and we finally have i x v b b vi Pv i P vi x whih is equation (.5). 3. Physial quantities λ i,,i =,,,3 and the onservation priniples of energy and momentum The physial quantities, i,,,,3 are related to the onservation of energy and i momentum of the generalized partile. This investigation we will present in this setion. We prove the following theorem: Theorem 3. If the generalized partile onserves its momentum along the axes xi, i,,,3, that is Ji Pi i onstant. (3.) then the following equation holds 7
b b b (3.) J P J P J J P P i i i i i i i i for every, i,,,3, i. Proof. Combining equations (.4) and (3.) we obtain x J x i J J i i xi J x i and with equation (.) we get b P J b PJ i i i i b J P J P i i i i whih is equation (3.). The rest of equations (3.) are derived taing into aount equation (3.). Equation (3.) holds for i,,i,,,3, sine equation (.4), from whih equation (3.) results is an identity for i and gives no information in this ase. We now prove the following theorem: Theorem 3.. TSV theorem for the symmetry of indies: If the generalized partile onserves its momentum along the axes x i and x with i, the following equivalenes hold J P J P J J P P. (3.3). i i ii i i i i b b b i J Pi JiP i J Ji Pi ip. (3.4). i i, i,,,3, i. Proof. The theorem is an immediate onsequene of equation 3.. We now onsider the four-vetor C, as given by equation 8
C J P. 3 (3.5) When the generalized partile onserves its momentum along every axis, then the four-vetor C is onstant. Also, we denote M the total rest mass of the generalized partile, as given by equation T C C 3 M (3.6) where T C is the transposed of the olumn vetor C. For reasons that will beome apparent later in our study, we give the following definitions: We name the symmetry i i, i,, i,,,3 internal symmetry, and the symmetry i i, i,, i,,,3 external symmetry. We now prove the following theorem: Theorem 3.3. Internal Symmetry Theorem: If the generalized partile onserves its momentum in every axis, the following hold: for every i,,,,3 J, P. i i and C are parallel P J where,. (3.7). For E the following equation holds: m (3.8) 3. For the following equations hold: b K exp x x x x M m E J i M i, i,,,3 3 3 (3.9) (3.) (3.) (3.) 9
i Pi,i,,,3 where K 4. We have i is a dimensionless onstant physial quantity. i i for every,i,,,3 for every,i,,,3. (3.3) (3.4) Proof. Equivalene (3.7) results immediately from equivalene (3.3). For from the last of equivalene (3.7) we obtain Selfvariations of the rest mass. For m P, whih is impossible, sine in this ase the, do not exist, as seen from equation (.6). Therefore, from the last of equivalene (3.7) we obtain P J (.7) and (.8) we obtain and from equations E m 4 whih is equation (3.8). For from the last of equivalene (3.7) we obtain Pi Ji for every i,,,3 and with equation (3.) Ji Pi i we initially obtain equations (3.) and (3.3). Then, ombining equations (.7) and (3.) we get m 3 and with equation (3.6) we obtain equation m M (3.5) and we finally have M m whih is equation (3.). Similarly, ombining equations (.8) and (3.3) we obtain equation (3.). We now prove that funtion is given by equation (3.9). Differentiating equation (3.5) with respet to xv, v,,,3 and onsidering equation (.6) we obtain
b P m v M 3 x and with equation (3.5) we have b P v v M 3 b P v x M v x and with equation (3.3) for i v v we arrive at equation b v, v,,,3. x v By integration of equation (3.6) we obtain (3.6) b K exp x x x x where K 3 3 is the integration onstant, whih is equation (3.9). Combining equations (.), (3.) and (3.3) for,,,3 we obtain J b PJ i i i x i b i i x i b i i x and with equation (3.6) for we obtain i. i b b i i Aording to the previous theorem, internal symmetry is equivalent to the parallelism of the four-vetors J, P. Starting from this onlusion we an determine the physial ontent of the internal symmetry.
In an isotropi spae the spontaneous emission of STEM by the material partile is isotropi. Due to the linearity of the Lorentz-Einstein transformations, this isotropi emission has as a onsequene the parallelism of the four-vetors J, P ([5] par. 5.3). Thus, the theorem of internal symmetry 3.3 holds for the spontaneous emission of STEM by the material partile due to Selfvariations. In the following paragraphs, we will mae lear that the internal symmetry refers to a spontaneous internal inrease of the rest mass and the eletrial harge of the material partiles, independent of any external auses. The onsequenes of this inrease is the osmologial data, as we'll see in Paragraph. Also, the internal symmetry is assoiated with Heisenberg's unertainty priniple. theorem: We start the investigation of the external symmetry with the proof of the following Theorem 3.4. First theorem of the TSV for the external symmetry: If the generalized partile onserves its momentum along every axis, and the symmetry every i, i,,,,3, then: i i holds for. i v iv v i J J J i i v i P P P i i v i (3.7) for every i v, v, i,, i, v,,,3.. i b bv b bv Pv i i Jvi i x v (3.8) for every i,, i,,,,3.. (3.9) 3. 3 3 3 Proof. From equivalene (3.4) we obtain b i i J J i, i,, i,,,3 (3.) Considering equation (3.) we get b iv iv vi i Jv vj vji i Jv v i J Ji.
Thus, we get the first of equations (3.7). Similarly, from the other two equalities of equivalene (3.4) we obtain the seond and the third equation of (3.7). Sine i in equivalene (3.4), the physial quantities, i, i,, i,,,,3.,, i i in equations (3.7) are defined for Differentiating equation (3.) with respet to xv, v,,,3 we obtain i b J Ji i xv xv xv and with equation (.) we get i b b b i Pv J v Pv Ji vi xv i b b P v i J J i iv vi x v i x v b b b P J J v i i i v vi and with equation (3.) we obtain i x v b b Pv i i v vi and with the first of equations (3.7) we obtain i v vi v i we get i x v b bv P vi i whih is equation (3.8). The seond equality in equation (3.8) emerges from the substitution P J, v,,,3 v v v aording to equation (3.5). Taing into aount equation (3.) we obtain 3
b 4 3 3 3 J J J3 3J J J 3J J 3 3J J3 J J after the alulations. In the next paragraphs we investigate the external symmetry. 4. The Unified Selfvariations Interation (USVI) Aording to the law of selfvariations every material partile interats both with the STEM emitted by itself due to the selfvariations, and with the STEM originating from other material partiles. In the seond ase, an indiret interation emerges between material partiles through the STEM. STEM emitted by one material partile interat with another material partile. Through this mehanism the TSV predits a unified interation between material partiles. The individual interations only emerge from the different, for eah partiular ase, physial quantity Q whih selfvariates, resulting in the emission of the orresponding STEM.In this paragraph we study the basi harateristis of the USVI. We suppose that for the generalized partile the onservation of energy-momentum holds, hene the equations of the preeding paragraph also hold. For the rate of hange of the four-vetor J m we get J J m J x m m x m x i i i and with equations (.6) and (.) we get J i Ji b b P m P Ji i x m m m and we finally obtain J i i x m m, i,,,,3. (4.) Aording to equation (4.), when i for at least two indies, i,, i,,,3, the ineti state of the material partile is disturbed. Aording to equivalene (3.4) in the 4
internal symmetry it is i for every, i,,,3. Therefore, in the internal symmetry the material partile maintains its ineti state. In an isotropi spae we expet that the spontaneous emission of STEM by the material partile annot disturb its ineti state. Consequently, the internal symmetry onerns the spontaneous emission of STEM by the material partile in an isotropi spae. In ontrast, in the ase of the external symmetry it an be i for some indies, i,, i,,,3. Therefore, the external symmetry must be due to STEM with whih the material partile interats, and whih originate from other material partiles. The distribution of STEM depends on the position in spae of the material partile relative to other material partiles. This leads to the destrution of the isotropy of spae for the material partile. The external symmetry fator will emerge in the study that follows. The initial study of the Selfvariations onerned the rest mass and the eletri harge. The study we have presented up to this point allows us to study the Selfvariations in their most general expression. We onsider a physial quantity Q whih we shall all selfvariating harge Q, or simply harge Q, unaffeted by every hange of referene frame, therefore Lorentz-Einstein invariant, and obeys the law of Selfvariations, that is equation Q x b P Q,,,,3. (4.) In equation (4.) the momentum P,,,,3, i.e. the four-vetor P, depends on the selfvariating harge Q. Two material partiles arrying a selfvariating harge of the same nature interat with eah other when the STEM emitted by the harge Q of one of them interats with the harge Q of the other. In this partiular ase, we denote with Q the harge of the material partile we are studying. The rest mass m is defined as a quantity of mass or energy divided by, whih is invariant aording to the Lorentz-Einstein transformations. The 4-vetor of the momentum J of the material partile is related to the rest mass m through equation (.7). The harge Q ontributes to the energy ontent of the material partile and, therefore, also ontributes to its rest mass. Furthermore, the harge Q modifies the 4-vetor of momentum J of the material partile and, therefore, ontributes to the variation of the rest mass m of the material partile. Consequently, for the hange of the four-vetor J of the material partile 5
due to the harge Q, the four-vetor P of equation (.) enters into equation (4.). The onsequenes of this onlusion beome evident when we alulate the rate of hange of the four-vetor J. Q Theorem 4. Seond theorem of the TSV for the external symmetry:. The rate of hange of the four-vetor given by equation J Q due to the Selfvariations of the harge Q Ji i,, i,,,3 x Q Q. (4.3) is. For i the physial quantities i Q are given by i za i, i,,i,,,3 Q (4.4) where z is the funtion b z exp x x x 3x3. (4.5) 3. For the onstants a i the following equations hold a a a i v iv v i J a J a J a i v iv v i Pa P a P a i v iv v i (4.6) for every i v, v, i, i,,,,,3. i i i i (4.7) 4.,,,,,,3 5. 3 3 3. (4.8) Proof. In order to prove the theorem, we tae J J Q J x Q Q x Q x i i i 6
and with equations (4.) and (.) we get Ji Ji b b P Q P Ji i x Q Q Q J i x Q Q i whih is equation (4.3). Equations (4.) and (.) hold for every,i,,,3. Therefore, equation (4.3) also holds for every, i,,,3. obtain For i,, i,,,3 and v,,,3 equation (3.8) holds and, sine i b bv Q PQ v i Qi x v and with equation (4.) we get i Q x v Q bv i Qi x i i i Q x x Q i bv i xv Q Q v Q b Q and integrating we obtain i b a exp i x x x 3x3 Q Q where a, i,, i,,,3 are the integration onstants, and with (4.5) we get equation i (4.4). Equations (4.6) are derived from the ombination of equations (3.7) and (4.4), taing into aount that zq. Equation (4.7) is derived from the ombination of equation i i, i,, i,,,3 with equation (4.4). Simirarly, equation (4.8) is derived from the ombination of equations (3.9) and (4.4). We will also use equation z b z,,,,3 x whih results immediately from equation (4.5)., we (4.9) 7
For i,, i,,,3 equation (4.4) does not hold. So we define the physial quantities T as given by equation T,,,,3. (4.) zq Taing into aount the notation of equation (4.) the main diagonal of matrix T equation (.) is given from matrix of T T zq T T 33 3 (4.) We now define the three-vetors α and β, as given by equations (4.) and (4.3) respetively x i α y i Q (4.) 3 z i 3 x 3 β y 3 Q. (4.3) 3 z Vetors α and β ontain all of the physial quantities i for i,, i,,,3 sine i. i Combining equations (4.) and (4.3) with equation (4.4), the vetors α and β are written in the form of equations (4.4) and (4.5), respetively x α y iz (4.4) 3 z 3 x 3 β y z 3. (4.5) 3 z 8
J x i We write equation (.) in the form b P J,,i,,,3. i i (4.6) The rate of hange of the momentum of the material partile equals the sum of the two terms in the right part of equation (4.6). For, and sine x it, equation (83) gives the rate of hange of the partile momentum with respet to time t, i.e. the physial quantity we all fore. By using the onept of fore, as defined by Newton, we also have to use the onept of veloity. For this reason we symbolize u the veloity of the material partile, as given by equation u u x u u y u. (4.7) u u 3 z Also, we define the 4-vetor of the four-vetor u, as given by equation u u i u u. u u x u u y 3 z (4.8) We now prove the following theorem: Theorem 4.. The rates of hange with respet to time t x it of the four-vetors J and P of the momentum of the generalized partile arrying harge Q are given by equations i dj dq i i J zq u Q uα dx Qdx α uβ (4.9) i dp dq i i J zq u Q uα. (4.) dx Qdx α uβ 9
Proof. The matrix vetors u and β. is given in equation (4.). By uβ we denote the outer produt of We now prove the first of equations (4.9): d J J J J J u u u3 dt Q t Q x Q y Q z Q and using the notation of equation (.3) we get id J J J J J i u u u 3 dx Q x Q x Q x Q x3 Q and with equation (4.3) we get id J i u u u dx Q Q Q Q Q 3 3 d J i 3 u u u 3 dx Q Q Q Q Q d J i 3 u u u 3 dx Q Q Q Q Q dj J dq i 3 u u u 3 Q dx Q dx Q Q Q Q dj dq i J u u u33 dx Qdx and with equations (4.) and (4.) we have dj dq i i i i J zqt Q u u u3 3 dx Qdx whih is the first of equations (4.9) sine i i zqtu zqti zqt. We prove the seond of equations (4.9) and we an similarly prove the third and the fourth: d J x J x J x J x J x u u u 3 dt Q t Q x Q y Q z Q
and using the notation of equations (.3) and (.4) we obtain id J i J J J J u u u3 dx Q x Q x Q x Q x3 Q and with equation (4.3) we get id J i u u u dx Q Q Q Q Q 3 3 d J iu iu iu dx Q Q Q Q Q 3 3 dj J dq iu iu iu Q dx Q dx Q Q Q Q 3 3 dj dq iu iu iu J dx Qdx 3 3 and with equations (4.), (4.) and (4.3), we obtain dj dq i i i J zqt Q Q u3 u3 dx Qdx whih is the seond of equations (4.9). Equation (4.) results from the ombination of equations (4.9) and (3.5). Using the symbol J for the momentum vetor of the material partile J J J x J Jy J J 3 z and taing into aount equations (.3) and (.4) and (4.) the set of equations (4.9) an be written in the form dw dq W zq T Q dt u α Qdt Tu. (4.) dj dq J zq Tu Q dt Qdt α u β Tu 3 3
Equations (4.) are a simpler form of equation (4.9) with whih they are equivalent. The rate of hange of the four-vetor J of the momentum of the material partile is given by the sum of the three terms in the right part of equation (86). The USVI and its onsequenes for the material partile depend on whih of these terms is the strongest and whih is the weaest. The first term expresses a fore parallel to four-vetor J whih is always different than zero due to the Selfvariations. As we will see next, the seond term is related to the urvature of spaetime. The third term on the right of equation (4.9) is nown as the Lorentz fore, in the ase of eletromagneti fields. In many ases a term or some of the terms on the right of equation (4.9) are zero, with the exeption of the first term whih is always different than zero. αβ, From equation (4.9) we onlude that the pair of vetors expresses the intensity of the field of the USVI aording to the paradigm of the lassial definition of the field potential. From equation (.) we derive that the physial quantities, i,,,,3 i have units (dimensions) of g s. Thus, from equation (4.) we derive that if Q is the rest mass, the intensity α N C has unit of. Now we will prove that for field, m s. If Q is the eletri harge, the intensity α αβ the following equations (4.) hold: Theorem 4.3. For the vetor pair αβ, the following equations hold: ibz α 3 3 (a) β (b) has unit of β α () (4.) t 3 3 bz α β 3 3. (d) t 3 3 3 3 Proof. Differentiating equations (4.4) and (4.5) with respet to x,,,,3 and onsidering equation (4.9), we obtain equations
α x b α (4.3) β x b β. (4.4) From equations (4.3) and (4.4) we an easily derive equations (4.). Indiatively, we prove equation (4.b). From equation (4.5) we obtain z z z β 3 3 x x x3 and with equation (4.9) we get bz β 3 3 3 and with the first of equations (4.6) for i, v,,3, we get β. The first of equations (4.6) should be taen into aount for the proof of the rests of equations of (4.). j Considering equations (4.) we define the salar quantity and the vetor quantity, as given by equations ibz α a a a a a 3a3 bz j a a 3a3 a3 a3 a 3 3 3 (4.5) where is a onstant. We now prove that for the physial quantities and j the following ontinuity equation holds: j. (4.6) t Proof. : From the first of equations (4.5) we obtain 3
α α t t α t t and with the seond of equations (4.5) and equation (4.d) we get β t j t whih is equation (4.6). j Aording to equation (4.6), the physial quantity is the density of a onserved physial quantity q αβ, field with urrent density j. The onserved physial quantity q is related to through equations (4.).We will revert to the issue of sustainable physial quantities in the next paragraphs. The density and the urrent density j have a rigidly defined internal struture as derived from equations (4.5). We now onsider the four-vetor of the urrent density the onserved physial quantity q, as given by equation j of j j i j j j j x j j y 3 z (4.7) and the 4 4 matries M M 3 3 3 3 3 3. (4.8) Using matrix M equations (4.5) an be written in the form of equation bz j MC. (4.9) 4
αβ, From equations (4.b,) we onlude that the potential is always defined in the - field of the USVI. That is, the salar potential,,,,,, V V t x y z V x x x x and the vetor potential A 3 A A x t, x, y, z x, x, x, x3 A Ay A A A are defined through the equations A A 3 z β Α Α iα α V V t x. We an introdue in the above equations the gauge funtion the salar potential V the term f. That is, we an add to f t if x and to the vetor potential A the term f for an arbitrary funtion f,,,,,, f f t x y z f x x x x 3 without hanging the intensity αβ, of the field. The proof of the above equations is nown and trivial and we will not repeat it here. For the field potential of the USVI the following theorem holds: 5
Theorem 4.4. αβ,. In the through equations V, A -field of USVI the pair of salar-vetor potentials is always defined β Α Α iα α V ia t x. (4.3). The four-vetor A of the potential A iv A A A x A A y A 3 A z (4.3) is given by equation A i i f z,for i b xi f,for i xi (4.3) where,,,,3, i,,,3 and f is the gauge funtion. 3. For, i,, i,,,3 equation (4.33) holds i 4 z f f,, i,, i,,,3. (4.33) i i i b i Proof. Equations (4.3) are equivalent to equations (4.b, ) as we have already mentioned. The proof of equation (4.3) an be performed through the first of equations (4.6). The mathematial alulations do not ontribute anything useful to our study, thus we omit them. You an verify that the potential of equation (4.3) gives equations (4.4) and (4.5) through equations (4.3) taing also into aount the first of equations (4.6). From equation (4.3) the following four sets of the potentials follow: 6
f A x z f A b x A z f b x z 3 f A3 b x 3 (4.34) z f A b x f A x A z f b x z 3 f A3 b x 3 (4.35) z f A b x z f A b x A f x z 3 f A3 b x 3 (4.36) 7
3 z 3 f3 A b x 3 3 z 3 f3 A b x A 3 z f b x f3 A3 x 3 3 3. (4.37) Indiatively, we alulate the omponents and of the intensity αβ, USVI field from the potentials (4.34). From the seond of equations (4.3) we obtain A i x A x and with equations (4.34) we get of the f z f i x x x b x z i b x and with equation (4.9) we get iz that is we get the intensity A x of the field, as given by equation (4.4). From the first of equations (4.3) we have A x 3 3 and with equations (4.34) we get z3 f z f x b x3 x3 b x 3 z z b x b x and with equation (4.9) we get z 3 3 and onsidering that, we get z 8
z. (4.38) 3 3 From the first of equations (4.6) for i, v,,,3 we obtain a a a 3 3 3 a a a 3 3 3 and substituting into equation (4.38), we see that z 3 that is, we get the intensity of the field, as given by equation (4.5). The gauge funtions f,,,, 3 in equations (4.34)-(4.37) are not independent of eah other. For and i for i,, i,,,3 equation (4.39) holds 4 z f f,, i,, i,,,3. (4.39) i i i b i The proof of equation (4.39) is through the first of equations (4.6). The proof is lengthy and we omit it. Indiatively, we will prove the third of equations (4.34) from the third of equations (4.35) for (4.39) for and i in equation (4.39). For and both equations (4.34) and equations (4.35) hold. From equation and i we get equation 4 z f. (4.4) b f From the third of equations (4.35) and equation (4.4) we get A A z 4 z f b x b z f 4 z b x b x and with equation (4.9) we obtain A z f z b x b z f A b x and sine, we get equation z f A. (4.4) b x 9
From the first of equations (4.6) for i, v,,, we obtain a a a a a a a a a and substituting into equation (4.4) we obtain equation A z f. (4.4) b x Equation (4.4) is the third of equations (4.34). Aording to equation (4.39), if for more than one of the onstants,,,,3, the sets of equations of potential resulting from equation (4.3) have in the end a gauge funtion. In the appliation we presented assuming and for a speifi gauge funtion f in equations (4.34), the gauge funtion f in equations (4.35) is given by equation (4.4). αβ, We onlude the investigation of the potential of the field the following orollary: of USVI by proving Corollary 4.. In the external symmetry, the 4-vetor C generalized partile annot vanish: of the total energy ontent of the C. (4.43) 3 Proof. Indeed, for C we obtain J P from equation (3.5). Therefore, the four-vetors J and P are parallel. Aording to equivalene (3.7) the parallelism of the four-vetors J and P is equivalent to the internal symmetry. Therefore, in the external symmetry it is C. A diret onsequene of these findings is that the potential of the field αβ, of USVI is always defined, as given from equation (4.43). This onlusion is derived from the fat that at least one of the onstants,,,,3 is always different than zero. 3
5. The onserved physial quantities of the generalized partile and the wave equation of the TSV The generalized partile has a set of onserved physial quantities q whih we determine in this paragraph. At first, we generalize the notion of the field, as it is derived from the equations of thetsv. We prove the following theorem: Theorem 5.. ξω,. For the field of the pair of vetors a i a ξ (5.) a 3 a3 ω a3 (5.) a is a funtion satisfying equation where x, x, x, x 3 b J P x,,,3, (, ),,, are funtions of x, x, x, x, the following 3 equations holds (5.3) ω ω ξ t. (5.4). The generalized partile has a set of onserved physial quantities q with density and urrent density j ξ j ω ξ t (5.5) 3
where are onstants, for whih onserved physial quantities the following ontinuity equation holds j. (5.6) t 3. The four-vetors of the urrent density j are given by equation b j M J P. (5.7) Proof. Matrix M in equation (5.7) is given by equation (4.8). We denote J and P the threedimensional momentums as given by equations J J J J 3 (5.8) P P P. (5.9) P 3 For the proof of the theorem we first demonstrate the following auxiliary equations (5.)- (5.5) a3 J a3 (5.) a a3 P a3 (5.) a a a3 J a J a (5.) 3 a 3 a a a3 P a P a (5.3) 3 a 3 a 3
a3 Ja J3a3 J a J a J a (5.4) 3 3 3 a Ja3 Ja 3 a3 P a P3 a3 P a P a Pa. (5.5) 3 3 3 a Pa 3 P a 3 In order to prove equation (5.) we get a3 J a J a J a J a a 3 3 3 3 and with the seond of equations (4.6) for ( i, v, ) (,3, ), we have a J a a 3 3 Similarly, from the third of equations (4.6) we obtain equation (5.). We now get a Ja3 J3a Ja3 J3a J a J a J a J a J a 3 3 3 3 a 3 Ja Ja Ja Ja and with the seond of equations (4.6) we obtain a Ja3 J a J a 3 a 3 Ja whih is equation (5.). Similarly, by onsidering the third of equations (4.6) we derive equation (5.3). Equations (5.4) and (5.5) are derived by taing into aount equations (5.8) and (5.9). Equations (5.4) are proven with the use of equations (5.)-(5.5). We prove the first as an example. From equation (5.) we obtain 33
a 3 ω a3 a and with equation (5.3) we get a3 a3 b b ω J a P a 3 3 a a and with equations (5.) and (5.) we obtain ω. From equations (5.4) and (5.5), the ontinuity equation (5.6) results. The proof is similar to the one for equation (4.6). The proof of equation (5.7) is done with the use of equations (5.)-(5.5), and equation (4.8). Field αβ, presented in the previous paragraph is a speial ase of the field ξω, for. For these values of the parameteres b J P x b J P x and with equation (3.5) we obtain, we obtain from equations (5.3) b x and finally we obtain b z exp x x x x 3 3 and from equations (5.),(5.) and (4.4),(4.5) we obtain ξ = α and ω β. From equation (.) it emerges that the dimensions of the physial quantities,, i,,,3 are i 34
i gs,, i,,,3. Thus, from equations (4.), (4.3) and (4.4), (4.5) we obtain the dimensions of the physial quantities Q,, i,,,3. Furthermore, from equation (4.) we obtain the i dimensions of the physial quantities T,,,,3. Thus, we get the following relationships Q i gs i i,,,,,,3, QT gs,,,,3. (5.6) ξω, Using the first of equations (5.6) we an determine the units of measurement of the -field for every selfvariating harge Q. When Q is the eletri harge, we an verify that the field units are V m,t. When Q is the rest mass, the field units are m s, s. The dimensions of the field depend solely on the units of measurement of the selfvariating harge Q. From equation (5.7) and taing into aount that, we an define the dimensions of the physial quantities q through the first of equations (5.6). When Q is the eletri harge, and for, where is the eletri permittivity of the vauum, q is a onserved physial quantity of eletri harge. For, where e the onstant value we e measure in the lab for the eletri harge of the eletron, q is a onserved physial quantity of angular momentum. For, q is a dimensionless onserved physial quantity, that e q. When Q is the rest mass, and for, where G is the gravitational onstant, q 4 G is a onserved physial quantity of mass. Theorem 5. reveals the onserved physial quantities of the generalized partile. One of the most important orollaries of the theorem 5. is the predition that the generalized partile has wave-lie behavior. We prove the following orollary: 35
Corollary 5.. For funtion the following equation holds ji j i x x xi ji j i t x xi (5.7) i,, i,,,3. Proof. To prove the orollary, onsidering that x in the form it, we write equations (5.4) and (5.5) i ξ j ω iω ξ x ω j x iξ. (5.8) We will also use the identity (5.9) whih is valid for every vetor α α α α. (5.9) From the third of equations (5.8) we obtain iω ξ x i ξ ω x and using the identity (5.9) we get i x ξ ξ ω and with the first and fourth of equations (5.8) we get i ξ i j j x x ξ and we finally get 36
ξ i j ξ. (5.) j x x Woring similarly from equation (5.8) we obtain ω ω j. (5.) x Combining equations (5.) and (5.) with equations (5.) and (5.), we get i ji j i x x xi, i,, i,,,3 whih is equation (5.7). Equation (5.7) an be haraterized as the wave equation of the TSV. The basi harateristis of equation (5.7) depend on whether the physial quantity F x t (5.) is zero or not. This onlusion is drawn through the following theorem: Theorem 5.. For the generalized partile the following equivalenes hold t (5.3) if and only if for eah i,, i,,,3 it is ji x j x i (5.4) if and only if ξ t ω t ξ ω. (5.5) 37
Proof. In the external symmetry there exists at least one pair of indies (, i), i,, i,,,3, for whih i. Therefore, when equation (5.4) holds, then equation (5.3) follows from equation (5.7), and vie versa. Thus, equations (5.3) and (5.4) are equivalent. When equation (5.4) holds, then the right hand sides of equations (5.4) and (5.5) vanish, that is, equations (5.5) hold. The onverse also holds, thus equations (5.4) and (5.5) are equivalent. Therefore, equations (5.3), (5.4), and (5.5) are equivalent. In ase that F, that is in ase that equivalenes (5.3), (5.4) and (5.5) hold, we shall refer to the state of the generalized partile as the generalized photon. Aording to equations (5.5), for the generalized photon the ξω, -field is propagating with veloity in the form of a wave. For the generalized photon, the following orollary holds: Corollary 5.: For the generalized photon, the four-vetor onserved physial quantities q, varies aording to the equations j of the urrent density of the j,,,,3. (5.6) t j Proof. We prove equation (5.6) for, and we an similarly prove it for,,3. Considering equation (4.7), we write equation (5.6) in the form j j j j3. (5.7) x x x x 3 Differentiating equation (5.7) with respet to x we get j j j j 3 x x x x x x x3 j j j j 3 x x x x x x3 x and with equation (5.4) we get 38
j j j j x x x x x x3 x3 j x j whih is equation (5.6) for, sine x it. The way in whih equations (5.5) emerge in the TSV is ompletely different from the way in whih the eletromagneti waves emerge in Maxwell s eletromagneti theory [6- ]. Maxwell s equation predit the eqs. (5.5) for j αβ,. The TSV predits waves for j, when eq. (5.4) is valid. Moreover the urrent density j in this ase varies aording to eq. (5.6). We now prove the following orollary of theorem 5.: Corollary 5.3. For the 4-vetor x x x x x3 (5.8) it is M x where j (5.9) M 3 3 3 3 3 3 and j the 4-vetor of the urrent density of the onserved physial quantities of the generalized partile. Proof. From eq. (5.3) and with the notation of eq. (5.8) we have 39
b J P x and multiplying from the left with the matrix M we get b M M J P x and with eq. (5.7) we have M x j whih is eq. (5.9). The eqs. (5.3), (5.7) and (5.9) give the relation of the wave funtion quantities J, P and of the generalized partile. j with the physial One of the most important onlusions of the theorem 5. is that it gives the degrees of freedom of the equations of the TSV. In equation (5.7) the parameters,,(, ) (,) an have arbitrary values or an be arbitrary funtions of x, x, x, x 3. The TSV has two degrees of freedom. Therefore, the investigation of the TSV taes plae through the parameters and of equation (5.7). If we set,, b,, i or i,, b in equation (5.7), we get equations x i J i J. (5.3) For,, b,, i or,, i b we have x i P. i P (5.3) For we have b J P,,,,3 x and with eq. (3.5) we have 4
b,,,,3 x and equivalently we have b x. b C (5.3) Taing into aount that x it and J iw, we reognize in equations (5.3) the Shrödinger operators. Using the marosopi mathematial expressions of the momentum J and energy W of the material partile, we get the Shrödinger equation [-5]. The Shrödinger equation is a speial ase of the wave equation of the TSV. The designation of the degrees of freedom and determines in a large extend the form of eq. (5.7). 6. The Lorentz-Einstein-Selfvariations Symmetry In this paragraph we alulate the Lorentz-Einstein transformations of the physial quantities i, i,,,,3. The part of spaetime oupied by the generalized partile an be flat or urved. TheLorentz-Einstein transformations give us information about this subjet. We onsider an inertial frame of referene O t, x, y,z moving with veloity u with respet to another inertial frame of referene, x, y,z,, and O Ot, with their origins O oiniding at t t. We will alulate the Lorentz-Einsteintransformations for the physial quantities, i,,,,3. We begin with transformations (6.) and (6.) i 4
u t t x u x x t y y z z (6.) y z W W uj u J x J x W J J J J y z x y z E E up u P x Px E P P y P P z x (6.) u where. We then use the notation (.3), (.4), (.5) and obtain the transformations (6.3) and (6.4) u i x x x u i x x x x x 3 x x J J J J 3 3 3 u J J i J u J J i J 3 3 (6.3) u P P i P u P P i P. (6.4) P P P P 4
for We now derive the transformation of the physial quantity i we get for the inertial referene frame Ot, x, y,z. From equation (.) J b PJ x and with transformations (6.3) and (6.4) we obtain u u b u u i J i J P i P J i J x x J u J u J u J b u b u b u b i i P J i P J i PJ PJ x x x x and replaing physial quantities J J J J,,, x x x x from equation (.) we get b u b u u b u u b ( P J i P J i i PJ i PJ u b u b u b u b P J i P J i PJ PJ and we finally obtain equation u u u i i. Following the same proedure for,i,,,3 we obtain the following 6 equations ) (7) for the Lorentz-Einstein transformations of the physial quantities u u u i i u u u i i u i u 3 3 i 3 i : 43
u u u i i u u u i i u i u 3 3 i 3 (6.5) u i u i 3 3. The first two of equations (6.5) is self-onsistent when equation (6.6) Then by the seond of equations (6.5) we obtain. Aording to equivalene (3.4) these transformations relate to the external symmetry, in whih it holds that i for i, i,,,,3. Thus, we obtain the following i transformations for the physial quantities, i,,,,3 i 33 33 u i u 3 3 i 3. (6.7) 3 3 u 3 3 i 3 u i 44
Taing into aount equations (4.4), (4.) and that the physial quantity zq is invariant under the Lorentz-Einstein transformations, we obtain the following transformations for the onstants i, i,, i,,,3 and the physial quantities T,,,,3 T T T T T T T T 3 3 u i u 3 3 i 3 3 3 u 3 3 i 3 u i. (6.8) Equation (6.6) orrelates the physial quantities and in the same inertial frame of referene. Taing into aount equation (4.) we obtain T T. Thus, when transformations (6.8) hold, T T with respet to the referene frame t, x, y, z also holds. The referene frame t, x, y, z moves with onstant veloity along the x. -axis. If we assume that the motion is along the y - or z -axis, the generalization of equation T T follows; the Lorentz-Einstein transformations lead to the following equation T T T T3. Thus, we derive the following two orollaries. Corollary 6.. When the portion of spaetime oupied by the generalized partile is flat, it is T T T T3. (6.9) Corollary 6.. When T (6.) for at least one,,,3 the portion of spaetime oupied by the generalized partile is urver and not flat. 45
Notie that from the way of proof of orollary 6. it follows that the onverse is not true. For external symmetries whih have T T T T3, spaetime may be either flat or urved. In paragraph 9 we have shown how to he if spaetime is flat or urved for external symmetries with T T T T3. In the external symmetry it is i Thus, in external symmetry it is i for at least on pair of indies i,,,,3. only for some pairs of indies i,,,,3. The Lorentz-Einstein transformations reveal that in flat spaetime this annot be arbitrary. Let s assume that it is for every inertial frame of referene. Then, we obtain and with transformations (6.8) we obtain u i and sine it is we obtain that it also holds. Woring similarly with all of the transformations (6.8) we end up with the following four sets of equations of external symmetry in the flat spaetime: 3 3 3 (6.) 46
3 3 3 3 (6.) 3 3 3 3 (6.3) 3 3 3 3 3 3. (6.4) The symmetry that equations (6.)-(6.4) express will be referred to as the symmetry of the Lorentz-Einstein-Selfvarlations. These symmetries hold only in ase that the part of spaetime oupied by the generalized partile is flat. 7. The Fundamental Study for The Corpusular Struture of Matter in external symmetry. The Π-Plane. The material partiles are in a onstant interation between them (via the USVI) beause of STEM. This interation has onsequenes in the internal struture of the generalized partile, inluding the distribution of its total energy and momentum between the material partile and the surrounding spaetime. The internal struture of the generalized partile is determined by the relations among the elements of the matrix T. The same holds for the rest mass m of the material partile, the 47
rest energy E of STEM, with whih the material partile interats, and the total rest mass M of the generalized partile. In this paragraph, we study this relation among the elements of the matrix T. We now prove the following theorem: Theorem 7.. For the elements of the T matrix it holds that: T TT T T T T T T T TT TT T T. (7.) 3 3 3 3 3 3 3 Proof. We develop equation (.3), obtaining the set of equations J J J J 3 3 J J J J 3 3 J J J J 3 3 J J J J 3 3 3 3 33 and from equations (4.4) and (4.) we have J zqt J zq J zq J zq 3 3 J zq J zqt J zq J zq 3 3 J zqa J zqa J zq J zqa 3 3 J zq J zq J zq J zqt 3 3 3 3 3 and sine it holds that zq, we tae the set of equations J T J J J 3 3 J JT J J33. (7.) J J J T J 3 3 J J J J T 3 3 3 3 3 The set of equations given in (7.) omprise a 4 4 homogeneous linear system of equations with unnowns the momenta J, J, J, J 3. In order for the material partile to exist, the system of equations (7.) must obtain non-vanishing solutions. Therefore, its determinant must vanish. Thus, we obtain equation T TT T T T T T T T TT TT T T 3 3 3 3 3 3 3 ( 3 3 3 ) and with equation (4.8) we arrive at equation (7.). We onsider the 4 4 N matrix, given as: 48
N 3 3 3 3 3 3. (7.3) Using the matrix N, we now write equation (4.6) in the form of NC NJ NP. (7.4) We now prove Lemma 7.: Lemma 7.. The four-vetors C, J, P satisfy the set of equations NC N J N P. (7.5) Proof. We multiply the set of equations (7.4) from the left with the matrix N, and equations (7.5) follow. Using lemma 7. we prove theorem 7. : Theorem 7.. For M it holds that:.. MN NM. (7.6) M N I (7.7). (7.8) 3 3 3 Here, I is the 4 4 identity matrix. 3. For the matrix M has two eigenvalues and, with orresponding eigenvetors and, given by: i 3 i 3 3 3 3 3 3 3 (7.9) 49
i 3 i. (7.) 3 3 3 3 3 3 3 4. For orresponding eigenvetors the matrix N has the same eigenvalues with the matrix M, and two n and n, given by: i n 3 3 i 3 3 3 3 3 3 3 3 (7.) i n 3 3 i. (7.) 3 3 3 3 3 3 3 3 5. 6. (7.3) 3 3 3 M C M J M P. (7.4) Proof. The matries M and N are given by equations (4.8) and (7.3). The proof of equations (7.6), (7.7), (7.9), (7.), (7.) and (7.) an be performed by the appropriate mathematial alulations and the use of equation (4.8). obtain We multiply equation (7.7) from the right with the olumn matries C, J, P, and 5
M C N C C M J N J J M P N P P and from equations (7.5) we obtain M C C M J J. (7.5) M P P eigenvalue Aording to the set of equations (7.5), and for, the matrix that the four-vetors with orresponding eigenvetor C, J, P M has as. From equations (7.5) it is evident are parallel to the four-vetor, hene they are also parallel to eah other. This is imposssible in the ase of the external symmetry, aording to Theorem 3.3. Therefore, eigenvetor. If the ase it is, so that the matrix M M does not have the four-vetor as an from equations (7.5) we get C J P and beause is J we again have. Thus, we arrive at equation (7.3). Then, from equations (7.5) we arrive at equations (7.4), sine it holds that. The matrix M, for theorem 7., it holds that tr M orollary 7.. Corollary 7.. For the matrix T M, is a 4 4 the physial quantities i, i,, i,,,3 is nonzero. Proof. Let us suppose that the matrix symmetri matrix. Furthermore, aording to. An immediate onsequene of theorem 7. is of external symmerty, it is not possible that exatly one of T has exatly one nonzero element physial quantity i, i,, i,,,3. From equation (4.8) we see that M, and from equation (7.8) we obtain i. This annot hold, aording to equation (7.3). From theorem 7. it follows: 5
Corollary 7.. For the four-vetor j of the onserved physial quantities q it holds that: Mj (7.6) Nj. (7.7) Proof. We multiply equation (5.7) by matrix M from the left and obtain b Mj M J M P and with the seond and the third of equations (7.4) we have Mj. We multiply the terms of equation (5.7) from the left with the matrix N, and obtain b Nj NM J P and with equation (7.6) we tae Nj. In the equations of the TSV there appear sums of squares that vanish, lie the ones appearing in equations (3.6) and (7.3). Writing these equations in a suitable manner, we an introdue into the equations of the TSV omplex numbers. From equation (3.6), and for M, we obtain 3. M M M M Therefore, the physial quantities 3,,, M M M M belong in general to the set of omplex numbers. This transformation of the equations of the TSV is not neessary. It suffies to remember that within the equations of the TSV there are sums of squares that vanish. We now prove theorem 7.3, whih also interorrelates the elements of the matrix T : 5
Theorem 7.3. In the external symmetry and for the elements of the matrix T it holds that: Ta i i,, i, i,,,,,3. (7.8) Proof. We differentiate the seond equation of the set of equations (4.6) J J J i i i i,, i, i,,,,,3 with respet to xj, j,,,3. Considering equations (.) and (4.4), we have b b b Pj Ji zq ji i Pj J zq j i Pj J zq j b P J J J zq j i i i ji i j i j and with the seond equation of the set of equations (4.6), and taing into aount that zq, we obtain ji i j i j. (7.9) i,, i, i,,, j,,,3 Inserting into equation (7.9) suessively i,,,,,,,3,,,3,,,3 j,,,3, we arrive at the set of equations and T 3 T 3 T T T 3 T 3 T T 3 T 3 T 3 T 3 T 3. (7.) The set of equations (7.) is equivalent to equation (7.8). 53
Theorem 7.3 is one of the most powerful tools for investigating the external symmetry. This results from orollary 7.3 : Corollary 7.3. For the elements of the matrix T of the external symmetry the following hold:. For every i,, i,, i,,,,3 it holds that i i. (7.) i,. T 3 3 T T 3 3 3 3 T 3. (7.) Proof. Corollary 7.3 is an immediate onsequene of theorem 7.3. From theorem 7.3 the following orollary follows, regarding the elements of the main diagonal of the matries of the external symmetry: Corollary 7.4. At least one of the elements of the main diagonal of the matrix T is equal to zero. Proof. If T for every,,,3, from equations (7.) we obtain for every set of indies i,, i,,,3, and from equation (7.) we have TTT T3. This annot hold, sine we assumed that T for every,,,3. Therefore, at least one element of the main diagonal of the matrix T is equal to zero. We present a seond way for proving this result. In the ase of T for every,,,3, we obtain from equations (7.) that, for every i,, i,,,3. Thus, the matrix T taes the form i i 54
T T T zq T T3. From equation (.3) we tae T J T J T J T3 J3 Sine we assumed that TTT T3 we obtain J J J J3. Thus, the material partile does not exist. We onsider now the three-dimensional vetors 3 τ (7.3) 3 3 n n n. (7.4) n 3 3 In the ase of the T matries with τ and n, we define the vetor μ from equation 33 μ. (7.5) 3 3 3 3 3 Combining equations (5.), (5.) with equations (7.3) and (7.4) we obtain ξin (7.6) ωτ. (7.7) 55
The field ξ is parallel to the vetor n and the field only variable quantity of the field For every vetor ξω, ω is parallel to the vetor τ. Moreover the. is the funtion x, x, x, x 3 α 3 whih is determined by the physial quantities of the TSV, we define the physial quantity T 3 α α α. (7.8) Here, the matrix T α is the transposed matrix of the olumn matrix α. From equations (7.3) and (7.4) we obtain τn. 3 3 3 Also, from equation (4.8) we have τn. (7.9) Therefore, the vetors τ and n are perpendiular to eah other. Considering also equation (7.5), we see that the triple of the vetors { μ, n, τ } forms a right-handed vetor basis. From equation (7.3) we have 3 3 3 and with equations (7.3), (7.4), and using the notation of equation (7.8), we obtain n τ and finally we obtain n i τ. (7.3) From equation (7.5) we have μ nτ 56
and sine the vetors τ and n that are perpendiular to eah other, we obtain from equation (7.9) μ n τ and using the notation of equation (7.8) we have μ n τ μ n τ and from equation (7.3) we tae μ i n τ. (7.3) In the ase of the T matries, where n, and from equation (7.3), it follows that τ, μ. In these ases we an define the set of unit vetors { ε, ε, ε 3 }, given by ε ε ε 3 μ μ n n τ τ n. (7.3) The triple of vetors { ε, ε, ε 3 } forms a right-handed orthonormal vetor basis. to the vetor In the ases of the T matries with τ, we define with the plane perpendiular τ. In the ases where moreover n, we obtain from equation (7.5) that μ.in these ases the vetors n and μ are perpendiular to the vetor τ, as implied by equations (7.5) and (7.9). Therefore, the vetors n and μ belong to the plane, and they also form an orthogonal basis of this plane. We note that the vetors of the TSV, whih may belong to the plane, are given as a linear ombination of the vetors n the ondition for and μ. Therefore, τ is not suffiient, in order for the plane to aquire a physial meaning. Also, we note that beause of equation (7.3), the plane, when it is defined, is not a vetor subspae of 3. We now prove theorem 7.4: 57