aa r X i v : . [ phy s i c s . h i s t - ph ] S e p Kant and Hegel in Physics
Y. S. KimCenter for Fundamental Physics, University of Maryland,College Park, Maryland 20742, U.S.A.email: [email protected]
Abstract
Kant and Hegel are among the philosophers who are guiding the way in which wereason these days. It is thus of interest to see how physical theories have been developedalong the line of Kant and Hegel. Einstein became interested in how things appearto moving observers. Quantum mechanics is also an observer-dependent science. Thequestion then is whether quantum mechanics and relativity can be synthesized intoone science. The present form of quantum field theory is a case in point. This theoryhowever is based on the algorithm of the scattering matrix where all participatingparticles are free in the remote past and in the remote future. We thus need, inaddition, a Lorentz-covariant theory of bound state which will address the question ofhow the hydrogen atom would look to moving observers. The question is then whetherthis Lorentz-covariant theory of bound states can be synthesized with the field theoryinto a Lorentz-covariant quantum mechanics. This article reviews the progress madealong this line. This integrated Kant-Hegel process is illustrated in terms of the wayin which Americans practice their democracy.
Introduction
Let us look at a Coca-Cola can. It is a circle if we see it from its top, and its sideview is a rectangle, as is illustrated in Fig. 1. How would it then appear to a movingobserver. This is of course an Einsteinian problem. Einstein studied the philosophy ofKant during his high school years. It is thus natural for him to ask this question.Niels Bohr was interested in the electron orbit of the hydrogen atom. It is wellknown how his efforts led to the present form of quantum mechanics. The wave functionof the hydrogen atom consists of the rotation-invariant radial function and the angularfunction consisting of spherical harmonics and spinors. The angular function tells howthe orbit looks to observers looking at different angles. This is called the rotationalsymmetry in physics. The question is how this atom would appear to observers inmotion.
Figure 1: A Coca-Cola can appears differently to two observers from two different angles.Likewise, the electron orbit in the hydrogen atom should appear differently to two observersmoving with two different speeds. The elliptic deformation of the circular orbit is fromJ.S. Bells book [1]. This is deformation is only a speculation based on Einsteins lengthcontraction and does not have any scientific merits. The purpose of the present paper is toclarify this issue.
This is a Kantian question. The observer is in a different environment. Einsteinformulated this problem with a mathematics called the Lorentz group. His Lorentziansystem consists of three rotations around three different directions in the ( x, y, z )space, three Lorentz boosts in those directions, and three translations along thesedifferent directions, plus one translation along the time. There are thus ten operationsin the Lorentzian system. three translations along the three different directions. Themathematics governing these ten operations is called the inhomogeneous Lorentz group igner 1939]. The main purpose of this paper is to examine how the hydrogen appearsto moving observers in terms of the way Kant and Hegel suggested. We can then askwhether quantum mechanics and special relativity can be synthesized.In Sec. 2, we examine how the idea of Kant and that of Hegel can be integratedinto a single Kant-Hegel procedure in physics. In Sec. 4, we review the attempts madein the past to synthesize quantum mechanics and special relativity. It is noted thatthe present form of quantum field theory can only deal with scattering problems. Itis noted also that Paul A. M. Dirac made his life-long efforts to construct bound-statewave functions that can be Lorentz-boosted. By integrating those efforts. It is shownpossible to construct harmonic-oscillator wave functions that can be Lorentz-boosted.In Sec. 4, we examine whether this Lorentz-transformable wave function can explainwhat we see in the real world. Let us pick a proton which is a bound state of morefundamental particles called quarks [2]. When it moves with a speed close to that oflight, it appears as a collection of free particles called partons [4]. Why does the sameproton appear differently? This is precisely Einsteins Kantian question. After settlingthe issue of bound states in the Lorentzian system. We are led to the question ofwhether quantum mechanics and Einsteins special relativity can be derived from thesame set of mathematical formulas. It is noted that Dirac in 1963 started with twoharmonic oscillators satisfying the Heisenberg uncertainty brackets [3]. He then notedthat the symmetry from these two oscillators is like that of the Lorentz transformationsapplicable the five dimensional space with three space coordinate and two time-likecoordinates.In Sec. 5, it is shown that the second time variable in this five dimensional space canbe transformed to the translations along the three space-like directions and one timelike direction, just like Einsteins Lorentzian system. Indeed, quantum mechanics andspecial relativity can be derived from the same set of equations, namely the Heisenbergbrackets.Kant and Hegel developed their theories based on human societies and histories, noton physical theories. It is thus easier to illustrate the integrated Kant-Hegel mechanismin terms of history. The history is the United States is short and transparent. In theAppendix, we examine the role of Kant and Hegel while Americans practice theirdemocracy. As is indicated in Fig. 2, Immanuel Kant and Georg Wilhelm Friedrich Hegel are amongthe most respected philosophers. Yet, their books are very difficult to read. The bestway to understand their ways of reasoning is to construct illustrations.According to Kant, many things should become one, the ding-an-sich. They just E = mc . look differently depending on the observers environment and state of mind. Accordingto Hegel, we can create a new wonderful world by synthesizing two different traditions.His philosophy was based on the history. He realized that Christianity is a synthesisof Jewish ethics and Greek philosophy. How can we integrate Kant and Hegel? Kantwanted to derive one from many. Hegel wanted to derive one from two.Thus, we need a mechanism which will lead many to two, between Kant and Hegel.This way of thinking was developed by ancient Chinese. After the ice age, many peoplewith different backgrounds came to the banks to Chinas northern river. They drewpictures to communicate, and this led to Chinese characters. In order to express theirfeelings, they sang songs. This is the reason why there are tones in spoken Chinese.How about different ideas? They realized they cannot be united to one. They thusdivided them into two opposing groups, namely Yang (plus) and Ying (minus). Thisway is known as Taoism.Kant was born in East Prussia (now Kaliningrad, Russia), and spent 80 years of his entire life there. Thus, his way of thinking was framed by what he saw every day.His area was a maritime commercial hub of the Baltic Sea, just like Venice in theMediterranean world. Many people came to Kants place with many different points ofview for the same thing. Thus, Kantianism and Taoism were developed in the same way,as illustrated in Fig. 3. Kant wanted one, but Chinese had to settle with two. Thus,Taoism can stand between Kant and Hegel. We can thus integrate Kant and Hegel byplacing Taoism between them. We can illustrate this integrated Kant-Hegel system interms of the history of the United States. Europeans with different backgrounds cameto the new land and settled down in many different areas. They then set up their owngovernment. In order to develop their laws and national policies, they developed twodifferent political parties. These two parties produce the laws and policies applicableto all citizens. This American system is admired by many people of the world. This isan integrated Kant-Hegel system, as is illustrated in Fig. 4.In the past, the physical laws were developed according to this integrated Kant-Hegel process. Physicists like to unite many different events into one formula or one setof formulas. There are many heavenly bodies. They can be divided into two groups,namely comets (with open orbits) and planets with (localized orbits). Isaac Newtonsynthesised these two groups into one with his second-order differential equation. Thisis a Hegelian process. James Clerk Maxwell synthesised the equations governing elec-tricity and those for magnetism into a set of four equations. This created the present-day wireless civilization. As is well known, the present form of quantum mechanics isa synthesis of particle nature and wave nature of matter. Einsteins special relativitysynthesizes massive and massless particles. All these are the integrated processes ofKantianism and Hegelianism. The remaining problem is whether quantum mechanics and Einsteins relativity can be synthesised. In this paper, we restrict ourselves to hisspecial relativity, even though his general relativity receives more public attention thesedays. Einsteins nickname is still E = mc , which was a product of his special theoryof relativity. The present form of quantum mechanics was developed for the Galilean system. Whilethe Galilean system operates with three translations and three-rotations on the spaceof ( x, y, z ), the time variable does not interfere with the coordinate transformations.However, was stated in Sec. 1, the space and time of Einsteins special relativity isbased on the Lorentzian system. This system operates in the four-dimensional spaceof ( x, y, z ). In this Lorentzian system, there are rotations in the three-dimensionalspace of ( x, y, z ). In addition, when the observer moves with a constant speed, thetime variable comes in. We call the observers velocity change “Lorentz boost.” Theboost can be made in three different directions. We call this symmetry system Lorentzcovariance. In additional, there are four translations along the four-dimensional space.Let us call the system of these three rotations, three boosts, and four translation theLorentzian system.The difference between the Galilean system and the Lorentzian system is spelledout in Table 1. Quantum mechanics was originally developed in the Galilean system, J x , J y , J z . J x , J y , J z . J x , J y , J z . Boosts None K x , K y , K z . K x , K y , K z . Translations P x , P y , P z . P x , P y , P z , P t . Q x , Q y , Q z , S contracted to P x , P y , P z , P t . but it was a great challenge during the 20th Century to construct quantum mechanicsin the Lorentzian space and time.Quantum field theory is a case in point. The mathematical algorithm of this theoryis based on the scattering matrix where all participating particles are free in the remotepast and free in the remote future. For making computations of the scattering matrix,Feynman diagrams provide mathematical transparencies with excellent physical inter-pretations. How about bound-state particles? The particles are not free in the remotepast and remote future.As indicated in Fig. 5, our understanding of bound states and scattering statesdid not go together all the time. As Feynman suggested [5], while Feynman diagramsare useful for running waves, we can use harmonic oscillators to understand quantumbound states in the Lorentzian system. In their paper of 1971, however, Feynman et al. did not do a very good job in constructing the harmonic oscillators in the Lorentzianworld. Their oscillator wave functions increase as the time separation variable becomelarge. Thus, their wave functions are meaningless in quantum mechanics Indeed, beforeFeynman et al. , Dirac attempted to construct a representation of the Lorentz groupusing harmonic oscillator wave functions [6]. Before 1971, a number of authors wrote down Lorentz-covariant oscillator wave functions [7, 8, 9].Yet, Feynman et al. ignored them all. Since 1973, mostly with Marilyn Noz, thepresent author started publishing papers on this subject [10] and continued writingpapers and books along the same line [11, 12, 14, 15, 18, 19]. With those papers, itis now possible to integrate Diracs lifelong efforts to construct Lorentz-covariant os-cillator wave functions. Dirac published three papers toward the Lorentz-covariantoscillators. In 1927, Dirac said that the c-number time-energy uncertainty should beincluded in Einsteins Lorentzian world [22]. In 1945, he suggested harmonic oscillatorsfor a representation of the Lorentz group [6]. In 1949, he introduced the light-conecoordinate system for Lorentz boosts, saying that the Lorentz boost is a squeeze trans-formation dir49. Diracs papers are like poems, but they contain no diagrams. Thus,we can use diagrams to accomplish what Dirac did not do, that is to integrate his ownpapers. As is illustrated in Fig. 6, his three papers can be integrated into an ellipse asa squeezed circle tangent to Einsteins hyperbola.Let us go back to Fig. 5, it is important to note that both the Feynman diagramsand the oscillator formalism given in this section can be constructed from the sameset of commutation relations, which is known as the Lie algebra of the inhomoge-neous Lorentz group [11]. It is also important note that the Feynman diagrams andthe Lorentz-covariant oscillator wave functions are constructed from the same set ofphysical principles governing quantum mechanics and special relativity [20]. On hundred years ago, Bohr and Einstein met occasionally to discuss physics. Bohrwas worrying about the electron orbit of the hydrogen atom, while Einsteins maininterest was how things appear to moving observers. Thus, they could have talkedabout how the hydrogen atom looks to a moving observer. However, there are norecords indicating that they ever talked about this issue. If they did not, they areexcused. There were and still are no hydrogen atoms moving with relativistic speeds.Since the total charge of the hydrogen atom is zero, it cannot be accelerated even thesedays.On the other hand, modern particle accelerators routinely produce many protonsmoving with the speed very close to that of light. These protons are not hydrogenatoms. However, they are also bound states within the same framework of quantummechanics. As indicated in Fig. 7, it is possible to study moving hydrogen atoms bylooking at moving protons. Indeed, according to the quark model [2], the proton is a quantum bound state of three quarks. Then the question is how the proton appearswhen it moves fast. In 1969, Feynman observed that the proton looks quite differentlywhen it moves with ultra-fast speed [4]. It appears like a collection of light-like particles.Feynman called them partons. These partons have the following peculiar properties.a. Feynmans parton picture is valid only for protons moving with velocity close tothat of light.b. The interaction time between the quarks becomes dilated, and partons behavelike free independent particles.c. The momentum distribution of partons becomes widespread as the proton speedincreases.d. The number of partons seems to be infinite or much larger than that of theconstituent quarks.The question is whether it is possible to explain Feynmans parton picture withinthe framework of quantum mechanics and special relativity. In order to answer thisquestion, we need a bound-state wave function which can be Lorentz-boosted. In Sec. ,we constructed the Gaussian function that can be Lorentz-boosted. The Gaussianfunction is different from the wave function for the hydrogen atom, but both wavefunctions share the same quantum mechanics. In 1964, Gell-Mann proposed the quark odel for hadrons. The hadrons are bound states of the quarks, and their mass spectraare like those of the harmonic oscillators [5]. The proton is a hadron and is a boundstate of three quarks. In the oscillator regime, the wave function for this three-bodysystem is a product of two wave functions [5]. It is thus sufficient to study the Lorentz-boost property of the two-quark system which was discussed in Sec. 4. In the oscillatorsystem, the momentum wave function of the Gaussian wave function is also Gaussian,and it becomes Lorentz-squeezed exactly in the same way as in the case of the space-time wave function. Thus, we can extend Fig. 6 to Fig. 8 [12, 15].Indeed, according to this figure, the quarks become light-like particles with a wide-spread momentum distribution, interacting with an external like free particles. Fur-thermore, since they are like light-like, the particle number is not constant as in thecase of black-body radiation. Thus, this figure provides the answers to all of the puzzlesraised on the parton picture listed earlier in this section. As was stated before, theproton wave function is a product of two oscillator wave functions. Using this wavefunction, Paul Hussar computed the parton distribution function for the proton, andit is in reasonable agreement with the observed parton distribution [24]. Let us go toTable 2. The Lorentz-covariant oscillator, which can be viewed as an integration ofDiracs papers, provides a one formula for the proton at rest (quark model) and theultra-fast proton (parton picture). The second row of this table is about the role ofWigners little group for internal space-time symmetries [25]. This row tells that Wign-ers little groups explain why the internal space-time symmetries appear differently tomoving observers. Again, this is a Kantian problem and was discussed detail in theliterature [12, 13, 15, 16, 17]. In Secs. 4 and 5, we observed that it is possible to construct harmonic oscillatorwave functions that can be Lorentz-boosted. Furthermore, it can settle the one of theKantian problems we observe in high-energy laboratories producing ultra-fast protons.In addition, we noted that the covariant oscillators and quantum field theory canbe constructed within the same Lorentzian system. They also share the same set ofphysical of physical principles.In that case, we are led to the question of whether these two scientific disciplinescan be derived from the same set of equations. For this purpose, Dirac consideredtwo harmonic oscillators. For the single-oscillator system, we can use the step-up andstep-down operators to write down Heisenbergs uncertainty brackets.For the two-oscillator system, there are four such operators. Dirac constructedten quadratic forms with those step-up and step-down operators. He then noted that E = mc . The first row of this table in well known.The second table is for internal space-time symmetries, the photon spin along the direction ofmomentum remains invariant, but the perpendicular components become one gauge degree offreedom et al. E = mc E = p / m E = q ( cp ) + m c E = cp Wigner’sLittle Groups S ,S , S InternalSymmetry HelicityGauge Trans.Integration of Gell-Mann’s Covariant Feyman’sDirac 1927,45,49. Qaurks Model Oscillators Parton Picture13able 3: Diracs ten quadratic forms which satisfy a closed set of commutation relationsidentical to that of the Lorentz group applicable to three space-like coordinates and twotime-like coordinates. The J operators are for the rotations in the three-dimensional space.The three K operators generate Lorentz boosts along three different directions. This tablecontains only six of the ten generators.Dirac Oscillators Differential J = (cid:16) a † a + a † a (cid:17) − i (cid:16) y ∂∂z − z ∂∂y (cid:17) J = i (cid:16) a † a − a † a (cid:17) − i (cid:16) z ∂∂x − x ∂∂z (cid:17) J = (cid:16) a † a − a † a (cid:17) − i (cid:16) x ∂∂y − y ∂∂x (cid:17) K = − (cid:16) a † a † + a a − a † a † − a a (cid:17) − i (cid:16) x ∂∂t + t ∂∂x (cid:17) K = + i (cid:16) a † a † − a a + a † a † − a a (cid:17) − i (cid:16) y ∂∂t + t ∂∂y (cid:17) K = (cid:16) a † a † + a a (cid:17) − i (cid:16) z ∂∂t + t ∂∂z (cid:17) Q = − i (cid:16) a † a † − a a − a † a † + a a (cid:17) − i (cid:16) x ∂∂s + s ∂∂x (cid:17) Q = − (cid:16) a † a † + a a + a † a † + a a (cid:17) − i (cid:16) y ∂∂s + s ∂∂y (cid:17) Q = i (cid:16) a † a † − a a (cid:17) − i (cid:16) z ∂∂s + s ∂∂z (cid:17) S = (cid:16) a † a + a a † (cid:17) − i (cid:16) t ∂∂s − s ∂∂t (cid:17) they satisfy the closed set of commutation relations which is the same as that for thegenerators of the Lorentz group applicable to the five-dimensional space consisting ofthree space coordinates ( x, y, z ) and two time coordinates t and s [3].Table 3 gives three rotation generators and three boost generators with respectto the time variable t, along with corresponding Diracs two-oscillator forms. Theoperators of this table do not depend on the second time variable s. The J operatorsthere generate rotations in the three-dimensional space ( x, y, z ), and the K operatorsgenerate Lorentz boosts along those three different directions. Thus, the six operatorsgiven in Table generate the Lorentz group familiar to us.In addition to the six quadratic forms given in Table 4, Dirac constructed fouradditional quadratic forms. They correspond to the differential operators given inTable 4. The differential operators in Table 4 do not depend on the time variable t ,but depend only on the second time variable s . We are now interested in convertingthe differential forms in this table into four translation generators, using the groupcontraction procedure introduced first by In¨on¨u and Wigner inonu53. In their originalpaper, In¨on¨u and Wigner obtained the Galilean system from the Lorentzian system.Since then, this contraction procedure was used for the unification of Wigners littlegroups for massive and massless particles [13, 15, 16]. This procedure unifies theinternal space-time symmetries of massive and massless particles, as Einsteins E = mc Q − i (cid:16) x ∂∂s + s ∂∂x (cid:17) − i ∂∂x Q − i (cid:16) y ∂∂s + s ∂∂y (cid:17) − i ∂∂y Q − i (cid:16) z ∂∂s + s ∂∂z (cid:17) − i ∂∂z S − i (cid:16) t ∂∂ − s ∂∂t (cid:17) i ∂∂t does for the energy-momentum relation, as indicated in Table 2. This contractionprocedure has been employed for the present purpose of converting all four operatorsin Table 3 into four translation generators [18, 19]. The result is shown in Table 5.This contraction procedure tells us to fix the s variable and set s = 1 for thegenerators given in Table 4 [18, 19]. They then become contracted to the translationgenerators given in Table 5. Indeed, the six generators of the Lorentz group given inTable 1 together with the four translation generators constitute the generators of theinhomogeneous Lorentz group or Einsteins Lorentzian system of space and time. Thisprocess is compared with the traditional Galilean and Lorentz systems in Table 1.Let us go back to Fig. 5. This figure asks whether it is possible to derive Ein-steins Lorentzian world from the principles of quantum mechanics. The answer to thisquestion is YES. While the Diracs oscillator algebra is derivable from the Heisenbergbrackets, the Heisenberg brackets are also derivable from the oscillator algebra. Thus,both quantum mechanics and special relativity are derivable from the same set ofequations. The synthesis of quantum mechanics and special relativity is now complete. Concluding Remarks
Kant and Hegel are very familiar names to us. They formulated their ideas basedon what they observed and what they learned. It is interesting to note that Einsteinstarted as a Kantianist but become a Hegelianist while doing physics. Indeed, physicsdevelops along the integrated Kant-Hegel line. The most pressing task of our time inphysics is a Hegelian synthesis of quantum mechanics and theories of relativity. Forthe single-oscillator system, we can use step-up and step-down operators to write downHeisenbergs uncertainty brackets. For the two-oscillator system, there are four suchoperators. Dirac constructed ten quadratic forms with those step-up and step-down op-erators. He then noted that they satisfy the closed set of commutation relations whichis the same as that generators of the Lorentz group applicable to the five-dimensionalspace consisting of three space coordinates ( x, y, z ) and two time coordinates t and s [3]. In this paper, it is pointed out that those ten generators can be transformedto the ten generators of Einsteins Lorentzian system. Thus, quantum mechanics andspecial relativity come from the same set of equations. nowledgnents I came to the University of Maryland in 1962 as an assistant professor one year afterI received my PhD degree from Princeton in 1961. At that time, the chairman of thephysics department was John S. Tall. He invited Paul A. M. Dirac to his departmentfor ten days in October of 1962, and he assigned me to be a personal assistant toDirac. I asked Dirac many questions, but Diracs answer was very consistent. Americanphysicists should study more about Lorentz covariance and its difference from theLorentz invariance. Why was he saying this?Three months earlier (in July 1962), Dirac met Feynman in Poland during a rel-ativity conference organized by Leopold Infeld. The photo of their meeting was laterpublished on the cover of the Physics Today, as shown in Fig. 5. After reading the1971 paper by Feynman et al. [5], it became clear to me that Feynman and his youngerco-authors did not understand the difference between the covariance and invariancein the Lorentzian world. This is the reason why they ignored the Lorentz-covariantoscillator wave functions which existed in the literature before 1971. When Dirac wastelling me about the weakness of American physicists in 1962, he was talking aboutFeynman he met three months earlier in Poland.Yet, the names of Feynman and Dirac are prominently displayed in the historytable given in Fig. 5. They have been and still are my great inspirational figures to,and I have been eager to place them into one box according to the Hegelian process ofsynthesis. In order to show my gratitude toward them, I visited in 2013 the JabllonaPalace north of Warsaw where they met in 1962, as shown in Fig. 9. In had thepleasure of my photo taken at the spot where they spoke to each other. Their photo alack of communication between them. It has been a great challenge for me to fix thegap between them. Finally, I am indebted to John S. Toll who provided my meetingwith Dirac in 1962. He was always helpful to me whenever I whenever needed helpthroughout my academic career.
Appendix
Traditionally, philosophers wrote their theories based on the religion, history, culturalconflicts. Their theories are quite separate from physical phenomena. Thus, it is mucheasier to illustrate their philosophies using historical developments.The history of the United States is short and transparent. After the first journey ofChristopher Columbus (1492-93), many Europeans moved to the New Land. In 1776,the Declaration of Independence was ratified. This document was written before Kantand Hegel became prominent, and it does not say anything about political parties.These days, the democratic system of the United States is functioning with twopolitical parties. Americans did not construct this system based on any theories of overnment written before. They developed this two-party system while practicingtheir democracy. How did they construct? The country consisted of many differentethnic groups with different cultural backgrounds. They were spread over many differ-ent areas in the North American continent. How it is possible to construct a nationalpolicy satisfactory to all those citizens?While practicing democracy, it is necessary to construct one national policy basedon all different opinions. Thus, the Kantian process of ”many-to-one” is desirable.However, it is not practical. Therefore, a more practical solution was to place thosemany opinions into two different groups. It is then possible to ”synthesize” two opinionsinto one, according to the Hegelian synthesis of ”two-to-one.” This process is illustratedin Fig. 4, which illustrates how physical theories are developed. References [1] Bell, J. S. (2004),
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