International Journal of Applied Physics and Mathematics, Vol. 3, No. 6, November 2013
The Relativistic Wave Equation
Ilija Barukčić
Meanwhile, there are several other wave equations (Weyl Abstract—In general, it is well known that the Schrödinger equation, Majorana equation, Breit equation, Proca equation, equation is not compatible with special relativity theory to a Rarita-Schwinger equation, Bargmann-Wigner equations et necessary extent. Thus far, there are already several trials to cetera), each of them with different strength and reliability. formulate versions of the Schrödinger equation to ensure But besides of all, a generally valid and invariant form of a compatibility with special relativity theory, the Klein-Gordon-equation or the Dirac equation are some of these relativity consistent wave equation or a fully self-consistent attempts. relativistic quantum theory is still not in sight. Material and Methods: In this paper, Einstein’s relativistic energy momentum relation is re-analyzed, a normalized relativistic energy momentum relation is derived. The derived II. MATERIAL AND METHODS normalized relativistic energy momentum relation together with the known Schrödinger equation is used as a starting point A. Einstein's Mass-Energy Equivalence Relation to establish a wave equation consistent with special relativity According to Albert Einstein [3], it is theory.
Results: In this publication, based on Einstein’s relativistic energy momentum relation, the historical problem of the v² mm 2 1 (1) “particle-wave-duality” is solved. Furthermore, a special 0 r c² relativity theory consistent wave equation is derived. or equally
Index Terms—Causality, quantum mechanics, special and general relativity, unified field theory. v² m c² m c ² 2 1 (2) 0 r c²
I. INTRODUCTION where m0 denotes the “rest” mass, mr denotes the “relativistic” The famous, long lasting dispute between Leibniz mass, v denotes the relative velocity and c denotes the speed (conservation of vis viva or “kinetic” energy) and Newton of light in vacuum. (conservation of momentum) had its roots in a dispute B. The (Time-Dependent) Schrödinger Equation between Gottfried Wilhelm Leibniz [1] (1646-1716) and The Schrödinger equation [4] for any system, no matter followers of René Descartes (1596-1650) concerning whether relativistic or not, has the form Descartes' law of the quantity of motion, as discussed by
Descartes in his Principia philosophiae of 1644. In fact, the (3) core of this controversy was the dispute about the H i t 0 t conservation of [mr × v × v] (Leibniz) versus the conservation of [mr × v] (Newton [2]) through changes. According to Roger Boscovich (1745) and Jean d’Alembert (1758) both where H is the Hamiltonian and (t) the wave function of the concepts are equally valid. Thus far, even today, it is a point quantum system. The form of the Schrödinger equation itself of dispute, what is preserved through changes. depends on the physical situation. Under certain circumstances, the momentum [mr × v] is C. The Energy Operator part of Schrödinger's equation too. The Schrödinger's In quantum mechanics, the Hamiltonian is a quantum equation is still of importance especially for quantum mechanical operator corresponding to the total energy of a mechanics. But it was Erwin Schrödinger himself who quantum mechanical system and usually denoted by H. recognized that his equation is not relativity consistent and By analogy with classical mechanics, the Hamiltonian is valid only under non-relativistic circumstances. In contrast to the sum of operators corresponding to the potential and this, the principle of relativity, which is the foundation of the kinetic energies (of all the particles associated with a system) theory of special relativity, should be valid in quantum of a quantum mechanical system and can take different forms mechanics too. In other words, can the principles of quantum depending on the situation. The total energy operator is mechanics be reconciled with the principles of special determined as relativity? The Klein-Gordon-equation established by Oskar Klein Hi (4) and shortly afterwards deduced by Walter Gordon is one of t these trials to reconcile relativity and quantum theory, the Dirac equation formulated by Dirac (1902-1984) another one. For our purposes, the (non-relativistic or relativistic) Hamiltonian is corresponding to the total energy of a Manuscript received June 2, 2013; revised December 23, 2013. Ilija Barukčić is with Horandstrasse, Jever, Germany (e-mail: quantum mechanical object. Thus far, using a suitable [email protected]). Hamiltonian operator (i.e. the problem of incorporating spin)
DOI: 10.7763/IJAPM.2013.V3.242 387 International Journal of Applied Physics and Mathematics, Vol. 3, No. 6, November 2013 is of fundamental importance. m ² v² 0 1 (11) mcr ² ² III. RESULTS A fundamental insight of special relativity is the Rearranging the equation above yields relativistic energy-momentum relation. Even particles which propagate at velocities comparable to the speed of light c will mc² 4 v² m m c² 0 rr 1 (12) obey this natural law. Thus far, extracting some mathematical m² c4 c² m m c² and physical consequences from the relativistic r r r energy-momentum relation is of use while establishing a In general, the “rest energy” is E ² = m ²× c²× c² while the generally valid relativistic wave equation. But a logically 0 0 consistent relativistic description of quantum mechanical “relativistic energy” is Er² = mr²× c²× c² and the “relativistic systems will not be achieved only by inserting the energy momentum” is pr² = v² × mr². Thus far, the equation before operator and momentum operator into the relativistic can be simplified as energy–momentum relation. E ² pc² ² A. The Normalized Relativistic Energy Momentum 0 r 1 (13) EE² ² Relation rr
Thus far, let m denote the “rest mass”, let E = m × c² 0 0 0 The energy of the electromagnetic wave E is known to denote “the rest energy”, let m denote the “relativistic mass”. Wave r be determined as E = p × c. Based on this insight, the In this context, let E = m × c² denote “the relativistic energy”, Wave r r r general form of the normalized relativistic energy momentum let v denote the relative velocity, let c denote the speed of relation follows as light in vacuum. Further, let pr = mr × v denote the
“relativistic momentum”. In the following, let EWave = pr × c EE0² Wave ² (14) denote the energy of the electromagnetic wave. 1 EErr² ² Claim. Q. e. d. In general, it is
A fundamental property of objective reality and thus far EE² ² one of the most important features of quantum theory is that 0 Wave 1 (5) quantum mechanical objects or matter as such can exhibit EErr² ² both, wave and particle properties, simultaneously. Under Proof. different circumstances, either the particle aspects or the Starting with wave aspects of a quantum mechanical object may manifest themselves more strongly. 11 (6) The simultaneous observation of wave and particle properties of quantum mechanical objects may be difficult as it is equally such. The question whether is it possible at all to measure simultaneously (collapse of the wave function) the extent, to what a quantum mechanical object has to be regarded as a 11mm (7) 00 particle or as a wave, is not the topic of this publication. These questions have long been debated and continue to be Based on Einstein's mass-energy equivalence (1) it follows debated. Yet and besides of all, any single quantum that mechanical object is determined by (14) too. Thus far, in order to consider the contradictory nature of v² the particle-wave duality of quantum mechanical objects in mm 2 1 (8) 0 r c² greater detail, equation (14) show us to what extent something must be regarded as a wave and to what extent the Rearranging this equation, we obtain same something must be regarded as being a particle. According to (14), the more a quantum mechanical object is a m v² “particle” the less it is a “wave” and vice versa. It is 0 2 (9) 1 worthwhile to state here that the one (i.e. the extent to which a mcr ² quantum mechanical object is a particle) can be calculated, as
Let us perform a square operation of this equation. The soon as its own other (i.e. the extent to which a quantum result is mechanical object is a wave) is measured. The historic problem of the particle-wave duality, deeply
embedded into the foundations of quantum mechanics, is m0 ² v² (10) 1 solved by (14). mcr ² ² B. The Relativistic Wave Equation or the normalized relativistic energy momentum relation, a probability theory consistent formulation of the particle-wave In general, however, just as proofed before, a “classical” or duality, as relativity based concept of “particle” and “waves” fully
388 International Journal of Applied Physics and Mathematics, Vol. 3, No. 6, November 2013 describes the behavior of quantum-scale objects too. determined as To put it slightly differently, nearly all physicists and ^ Let E denote the quantum mechanical operator of the philosophers will agree that the total energy of a physical p system S cannot be different in quantum theory and in special relativistic potential energy. For our purposes it is important relativity theory. to note that the relativistic potential energy is the energy of a (high speed) particle (or of a system) due to the position of 1) Definitions this (high speed) particle (or of this system). Thus far, in what follows, let us consider the following. Let The definition of the relativistic potential energy Ep is m0 denote the “rest mass”, let E0 denote “the rest energy”, let backgrounded by Einstein's publication in 1907. According mr denote the “relativistic” mass, let EWave denote the energy to Einstein [6], it is: of the electromagnetic wave, let Er denote “the relativistic “Jeglicher Energie E kommt also im Gravitationsfelde eine energy”, let v denote the relative velocity, let c denote the Energie der Lage zu, die ebenso groß ist, wie die Energie der speed of light in vacuum, let Ek denote the relativistic kinetic Lage einer 'ponderablen' Masse von der Größe E/c².” energy, let mp denote the “relativistic potential” mass, let Ep Translated into English: denote the relativistic potential energy, let H denote the “Thus, to each energy E in the gravitational field there Hamiltonian, let (t) denote the wave function, let i denote corresponds an energy of position that equals the potential the imaginary unit and let h denote the reduced Planck energy of a "ponderable" mass of magnitude E/c².” constant. Let us define the relativistic kinetic energy Ek as 2) The relativistic wave equation
E ² Claim. E Wave k The total relativistic energy of a particle, Er, is the sum of Er relativistic kinetic energy, Ek, and relativistic potential energy, Ep. Based on this assumption, the relativity theory consistent 2 mvr relativistic wave equation can be derived as
1 ^^ 2 ( mv 2 ) Epk t E t H t i t (17) 2 r t
(15) m Proof. 0 v2 Starting with v² 2 1- c² 11 (18)
v² it is according to (10) at the end m0 v² 2 1- m ² v² c² 0 1 (19) mcr ² ²
E0 ² Ep Er Rearranging (19) yields
2 m0 ² c² c² mvr ² mcp 1 (20) mrr² c² c² m c²
m00c² m c ² The relativistic kinetic energy Ek was defined just before as mcr ² (16) Ek = mr × v²(15). Substituting this in (20), we obtain
m00 c² m c ² EE² 0 k 1 (21) m0 c² EErr² v² 2 1 c² The relativistic potential energy Ep was defined above (16) as Ep = E0² /Er In other words, it is E0² = Ep × Er (16). Based v² on (16) and (21) we obtain mc 2 1 2 0 c² EE E prk 1 (22) ^ EEEr r r Let Ek denote the quantum mechanical operator of the relativistic kinetic energy. The quantum mechanical operator Inside (22) Er cancel out. It follows that of relativistic kinetic energy is related to the historical concept of vis viva [5] and originally proposed by Gottfried E E Wilhelm Leibniz (1646-1716). p k 1 (23) EE Further, let the relativistic potential energy Ep be rr
389 International Journal of Applied Physics and Mathematics, Vol. 3, No. 6, November 2013
Multiplying (23) by E yields ^ r relativistic kinetic energy E and the quantum mechanical (24) k EEEp k r ^ operator of the relativistic potential energy E have to be p The “total” relativistic energy Er is the sum of the identified. relativistic potential Ep and the relativistic kinetic energy Ek. The quantum mechanical operator of the relativistic kinetic Multiplying (23) by the Hamiltonian H and the wave function ^ energy E k can be constructed in the following way. Under (t) yields some circumstances, the kinetic energy operator of quantum mechanics is known to be defined as E E p H t k H t H t (25) ^ ^ ^ EErr 2 2 2 ^ p p p T x y z (29) In special relativity theory, the total energy (of a quantum 2 mr mechanical system) is corresponding to the “relativistic energy” Er. Thus far, Klein-Gordon's [7], [8] and Dirac's [9] Recall, it is approach in mind, we equate 2 2 2 2 (30) 2 2 2 EHr (26) x y z
Rearranging (25), the (time-dependent) relativistic wave Under these circumstances, the kinetic energy operator of equation [for a single relativistic particle] (H and Er cancel quantum mechanics can be written more concisely as out) follows as ^ 2 T 2 (31) (27) 2mr Epk t E t H t i t t In general, the quantum mechanical operator of the The relativistic wave equation as derived above can be relativistic kinetic energy is related to the kinetic energy expressed in terms of quantum mechanical operators too. In operator of quantum mechanics in the following way. The terms of quantum mechanical operators, the special relativity formula of the kinetic energy in Newtonian (EKN) sense is consistent wave equation follows as known to be
^^ (28) 1 Epk t E t H t i t E m v² (32) t KN 2 r
Q. e. d. while the relativistic kinetic energy (15) is determined as In plain language, the (time-dependent) relativistic wave equation means the total relativistic energy E equals r 1 (33) relativistic kinetic energy E plus relativistic potential energy Ek 2 ( mr v mr v k 2 Ep. The following figure illustrates this relationship (24).
Combining (31) and (32) together yields the quantum The relativistic kinetic energy Ek mechanical operator of the relativistic kinetic energy as
^ ^ ^ 2 2 2 ^^p p p 2 x y z 2 (34) E ETk 2 r mm rr
while the relative velocity squared can follow as
The relativistic potential energy Ep ^ ^ ^ Fig. 1. Illustration of the relativistic energy Er. p2 p 2 p 2 2 v² x y z 2 (35) mr m r mr m r IV. DISCUSSION In quantum mechanics, conditions, circumstances or the The finite ways of physics to manage the infinite are still a independently of human mind and consciousness existing source of many difficulties in physics and in science as such. objective reality is described by waves (the wave function) For our purposes it is important to note that with only a little rather than with discrete particles. The establishment of further effort, the quantum mechanical operator of the quantum mechanical operators associated with the ^ relativistic potential energy E p can be obtained in a similar parameters needed to describe a quantum mechanical system way. in a relativity consistent way is of fundamental importance in this respect. Still, the quantum mechanical operator of the In our understanding, mr follows form the energy operator
390 International Journal of Applied Physics and Mathematics, Vol. 3, No. 6, November 2013 as interpretation dominated quantum mechanics like Heisenberg's uncertainty principle [10], Bell's theorem [11] and the CHSH inequality [12] are already refuted, but a fully, i relativity consistent quantum theory is still not in sight. m t (36) r c² This publication has solved the historical problem of the wave-particle duality. Based on Einstein's mass-energy For our purposes, according to (16) and (27), the quantum equivalence and the normalized relativistic energy ^ momentum relation, a relativity consistent wave equation is mechanical operator of the relativistic potential energy E p derived. The results of this publication can serve as the follows as foundation of a relativity consistent quantum theory.
REFERENCES ^ cc² ² (37) [1] G. W. Leibniz, “Brevis demonstratio erroris memorabilis Cartesii et Ep m00 m aliorum circa legem naturalem, secundum quam volunt a Deo eandem i t semper quantitatem motus conservari, qua et in re mechanica abuntur,” Acta Eruditorum, vol. 3, pp. 161-163, 1686. [2] I. Newton, Philosophiae Naturalis Principia Mathematica, London, The relativistic wave equation in terms of quantum 1687. mechanical operators follows i.e. as [3] A. Einstein, Annalen der Physik, vol. 18, pp. 639-641, 1905. [4] E. Schrödinger, Physical Review, vol. 28, pp. 1049-1070, 1926. [5] G. W. Leibniz, “Specimen dynamicum pro admirandis Naturae legibus ^ ^ ^ circa corporum vires et mutuas actiones detegendis, et ad suas causas 2 2 2 revocandis,” Acta Eruditorum, vol. 4, pp. 145-157, 1695. m m c² c² px p y p z (38) 00 t t i t [6] Albert Einstein, “Über das relativitätsprinzip und die aus demselben mt i r gezogenen folgerungen,” Jahrbuch der Radioaktivität und Elektronik, t vol. 4, pp. 411-462, 1907. [7] O. Klein, Zeitschrift für Physik, vol. 37, no. 12, pp. 895-906, 1926. [8] W. Gordon, Zeitschrift für Physik, vol. 40, pp. 117-133, 1926-1927. The square of the wave function | |² corresponds more or [9] P. A. M. Dirac, “Series A, Containing papers of a mathematical and less to the necessity or to the probability of a particle to be at a physical character,” in Proc. the Royal Society of London. 1928, pp. 610-624. given place at a given time. Further research is necessary to [10] I. Barukčić, “Anti Heisenberg-refutation of Heisenberg's uncertainty make the proof whether the relationship relation,” in Proc. American Institute of Physics Conference, 2011, pp. 322-325. [11] I. Barukčić, “Anti-Bell-Refutation of Bell's theorem,” in Proc. E Em² ² ²p 00 (39) American Institute of Physics Conference, 2012, pp. 354-358. E E² m ² [12] I. Barukčić, “Anti-Bell-Refutation of Bell's theorem,” in Proc. r r r American Institute of Physics Conference, 2012, pp. 354-358. or a similar one can be regarded as being correct.
st Ilija Barukčić was born October 1 , 1961 in Novo Selo (Bosina and Hercegovina, former Yugoslavia). He V. CONCLUSION studies at the University of Hamburg from 1982 to 1989), Germany. He ends of studies with the degree: Today the Copenhagen interpretation dominated quantum State exam. His basic field of research is the mechanics, a joint name for the ideas of Bohr, Heisenberg, relationship between cause and effect under conditions Born and other physicists, is repeatedly regarded as the most of quantum and relativity theory, in biomedical sciences, in philosophy et cetera. Thus far, among his successful scientific theory in the history of mankind. From a main publications in physics, published by the American Institute of Physics, different but equivalent point of view, it is the same quantum are the refutation of Heisenberg’s uncertainty principle (Ilija Barukčić, “Anti theory which challenges not only our imagination but Heisenberg - Refutation Of Heisenberg's Uncertainty Relation,” American Institute of Physics - Conference Proceedings, Volume 1327, pp. 322-325, violates some fundamental principles (principle of causality) 2011), the refutation of Bell’s theorem and the CHSH inequality (Ilija of modern science too. Barukčić, “Anti-Bell - Refutation of Bell's theorem,” American Institute of Meanwhile, the historical dominance of the Copenhagen Physics - Conference Proceedings, Volume 1508, pp. 354-358, 2012.). In another publication, he was able to provide the proof that time and the interpretation of quantum mechanics is no longer justified. In gravitational field are equivalent (Ilija Barukčić, “The Equivalence of Time fact, the main pillars (principles) of today's Copenhagen and Gravitational Field,” Physics Procedia, Volume 22, pp. 56-62, 2011).
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