Wave Motion In Terms of Field - Carrying Particles. Hisham B. Ghassib Royal Scientific Society Amman - Jordan

Transcription

1 Wave Motion In Terms of Field - Carrying Particles Hisham B. Ghassib Royal Scientific Society Amman - Jordan ABSTRACT In this paper, wave propagation is analyzed in terms of field-carrying particle kinematics. This analysis is used for kinematically dissecting classical field equations. The pedagogical value of this analysis is emphasized. It is also shown that, according to this analysis, proper mass could be interpreted as a measure of the deviation of a classical field from Huygens' propagation; a deviation which is externally generated and induced.

2 1- INTRODUCTION It is possible to assert, on purely theoretical grounds, that the concept of wave is pedagogically far more complicated and difficult to grasp than that other fundamental form of classical motion, the concept of particle(1). In fact, many a physics student has graduated from university with the uncomfortable feeling that, even though it is within his capacity to solve mathematically a number of wave equations under various conditions quite satisfactorily, he has not actually grasped the essence of the concept of wave. It would appear that this wide pedagogical gap between the two concepts is indeed an objective gap which has nothing to do with the specific psychology of the student as an individual, but is inherent in the very logico-empirical status of the two concepts. In the first place, wave motion is that fundamental form of classical motion which is characteristic of the transfer of energy in its pure field form. What is essentially transferred via wave motion is field energy, even though it may involve particle oscillation. Particle motion, on the other hand, is characteristic of the transfer of energy in its more tangible form, the mass form. The former is the fundamental building block of classical field dynamics, whereas the latter is the fundamental building block of classical matter dynamics. The intangibility and elusiveness of the classical concept of field, compared to the concept of mass, is one of the basic reasons for the aforementioned pedagogical gap().

3 In the second place, the correspondence between wave representations and physical phenomena involving wave motion is far less direct than that between the particle representation and its related phenomena. For example, the correspondence between the simple sinusoidal wave representation and light propagation is far less conspicuous than the correspondence between the particle representation and planetary motion. The concept of wave is usually introduced via a number of onedimensional and two-dimensional representations. The student usually grasps these representations as concrete systems and in their pure facticity without getting hold of the essence of the representation for which the latter has been designed in the first place. This is a danger latent in all representations, but especially so in wave representations. Accordingly, the student fails to make the leap, as it were, from the particular to the concrete general, and fails to grasp the physical significance of such general wave forms as electromagnetic and Schrodinger waves. In optics, the concept of wave is defined in terms of Huygens' Principle, the phenomena of interference and diffraction, and eventually the three-dimensional wave equation: 1 u t r,t

4 Interpreting wave propagation in terms of the continual generation of field sources a' la Huygens, one can account satisfactorily for the principal phenomena of wave motion. However, the exact mathematical and physical relationship between equation (1) and Huygens' Principle is never truly established. At least, it is not so established in a simple and straight-forward manner(3). The relationship is further confused when the Maxwell and Schrodinger equations are introduced in their capacity as exemplary wave equations(4). The student usually feels capable of grasping each one of these multifarious principles and representations on its own. However, when it comes to relating them dialectically in a comprehensive whole characteristic of the essence of wave motion, he feels utterly confused, and is at a loss. In this paper, we introduce the concept of field-carrying particle (point), and clarify wave propagation, including Huygens' principle, in terms of this concept. That is, we derive the wave equation from the kinematics of field-carrying particles. Then we use this method to dissect the Klein-Gordon equation, both in the absence and the presence of an electromagnetic field. This dissection leads us to view proper mass as a measure of the deviation from Huygens' propagation, and as an externally induced and generated interaction term. 4

5 - THE CONCEPT OF WAVE The concept of wave is rarely given a precise definition with clear boundaries and limits. It is used to describe all sorts of situations ranging from one-dimensional string waves to relativistic scalar fields. We shall therefore give a precise definition of wave motion which will assist us in classifying the various classical field equations. We shall specify the concept in terms of the answers to the following questions. Via what type of carrier the classical field travels in space and time? And, how do these field carriers propagate in space and time - i. e., what is the kinematics of waves? We start by viewing a scalar classical field as an infinite collection of field-carrying point particles moving in definite paths. The field,, carried by each particle is a function of both the position vector of the particle and the time. Thus, (r,t)...() We assume the following: - (I) changes with the position of the field particle and the time in such a way that it remains constant. This means that each particle carries a specific field value, just as classical Newtonian particles carry specific masses. It is in this sense that the particles are called fieldcarrying particles. Thus, 5

6 (ii) r,t for each particle. const...(3) That is, there is a level surface in four-dimensional (Minkowski) space for each particle. The field-carrying particles are continuously distributed in space such that ( r, t ) describes their space distribution at time t. It follows from (3) that: r. ( r,t) ( r, t) 0...(4) t where r(t) is the velocity of the field-carrying particle. Next, we interpret Huygens' principle to mean that each particle moves along a path which is always perpendicular to its level surface- i.e., r (t) is parallel to ( r, t ). Thus, r ( r,t) ( r,t)... (5) u t where u is the speed of the particles. Taking the divergence of each side of eq. (5), and then r,t, we obtain: r,t r,t u t t u t resubstituting for Thus, for constant speed fields, 1 r,t r, t 0...(7) u t...(6) 6

7 which is the generalized 3-D Huygens-type equation. We believe that this derivation of the wave equation clarifies its kinematic meaning and gives a precise mathematical form to Huygens' principle. 3- THE KLEIN-GORDON EQUATION Eq. (4) admits a wide variety of solutions, each of which is characterized by a specific type of wave propagation. We have dealt in some detail with the Huygen's (orthogonal) solution. The question is what other solutions are physically interesting and significant? To answer the question, we shall enquire into the structure of the free classical relativistic scalar field equation (the Klein-Gordon equation)-namely, 1 m c c t h 0 r,t r,t r,t...(8) where m o is the proper mass of the field, and c is the speed of light in vacuo. It is evident that eq. (8) differs fundamentally from eq. (6). Therefore, it cannot possibly belong to the Huygens' field variety. The question is, what solution of eq. (4) would generate eq. (8)? Let us consider the following solution. r c t r, t r, t...(9) 7

8 (u = c, the speed of light in this case). where is a three - dimensional vector. Eq. (9) is equivalent to Eq. (4) if. r 0...(10) that is, if is perpendicular to r. Now, taking the divergence of both sides of eq. (9), as before, we arrive at the following equation. 1 r, t r, t r, t...(11) c t which is identical to eq. (8) if moc...(1) h In eq. (9), is a measure of the deviation from Huygens' propagation. When is zero, the propagation is of the Huygens' variety. Therefore eq. (1) implies that proper mass is a measure of the deviation from Huygens' propagation. This is a new wave - geometric interpretation of mass, which follows from dissecting the familiar equation (8) with our unfamiliar methodology. Next, we ask the question, how is our solution modified in the presence of an external electromagnetic field? What effect does the electromagnetic field have on our "generator" shown in eq. (9)? 8

9 The classical relativistic scalar field equation assumes the following form in the presence of an external electromagnetic field(5). t e i c A r, t) 1 h ( ih e m c...(13) o c where e is the field charge, A is the externally induced vector potential, and is the externally induced scalar potential. To simplify the treatment, we assume that the scalar potential is a vanishing term, and the vector potential is time-independent, in which case eq.(13) is reduced to the following: ih e c A r,t h c t r,t m c r,t...(14 o ) What "generator" in the spirit of eq. (9) will generate eq. (14)? If we assume: (i) u is time-independent, (ii) A, u, form an orthonormal system (i.e., the wave propagates in a direction perpendicular to both A and ), then we can generate eq. (14) with the following generator: u c ie r, t mo... (15) c t h ch 9

10 It can easily be verified that eq. (15) obeys eq. (4) and generates eq. (14) subject to the two assumptions stated. This means that eq. (14) is a wave equation in the strict sense given to it in eq. (14). We notice that the mass term is similar, in both form and function, to the electromagnetic field term on the right-hand side of eq. (15). They are both " - terms", both perpendicular to the fieldparticle velocity, and both measures of the deviation from Huygens' propagation. This suggests that the mass term is some sort of a vector potential of an external field, which is, however, quite different from the electromagnetic vector potential. The former is real, whilst the latter is imaginary. Nevertheless, eq. (15) lends itself to a Machian interpretation of mass as an externally induced interaction term (6). It could be worth pursuing this suggestion by inquiring into the type of gauge field that would induce the mass term. 01

11 References 1- This statement, which is based on theoretical grounds, is, nevertheless, corroborated by the author's physics teaching experience and conversations with physics graduates in England and Jordan. - See, in particular, the detailed discussions of this fundamental difference in: Hisham B. Ghassib, Excursions into Scientific Thought, RSS, Amman (1985); The Process of Unification in Field Physics, RSS and ISESCO, Amman (1988); A Simplified Approach to the Concept of Wave Motion, RSS and ISESCO, Amman (1991). 3- In textbooks on optics, the general 3 - D wave equation is never kinematically derived. Its organic connection with Huygens (orthogonal) propagation is never directly shown. For example, in: R.S. Longhurst, Geometrical and Physical Optics, nd. Edition, Longmans (1967), p. 96, the 3-D wave equation is introduced for the first time as follows: "An investigation into the propagation of sound waves leads to the general result that the velocity potential satisfies the equation, 00

12 x y z 1 c t This is known as the general equation of wave motion". Only much later does he discuss Kirchhoff's formulation of Huygens' Principle, which starts with the wave equation and ends up by showing that the disturbance at a point is the sum of all retarded disturbances coming from a closed surface. Kirchhoff's formulation is, however, basically a theory of diffraction, and does not show how Huygens (orthogonal) propagation forms a constituent part of the wave equation. A similar treatment occurs in a classic, Arnold Sommerfeld's "Optics" [(translated by O. Laporte and P. Moldauer, Academic Press, New York (1964)]. This treatment seems to be typical of most textbooks on optics. 4- There is a detailed attempt at clarifying this relationship in the context of accounting for the Schrodinger equation in: Hisham Ghassib, How Wave Mechanics was Constructed, RSS and Dar-al-Furqan, Amman (1983). 5- See, for example: A.S. Kompaneyets, Theoretical Physics, Mir Publishers, Moscow (1965), p See, in this regard: J. Rosen, Extended Mach Principle, Am. J. Phys., 49 (3), March 1981, pp ; M. Semon and G. Schmieg, Note on the Analogy between Intertial and Electromagnetic Forces, Am. J. Phys., 49 (7), July 1981, pp ; Y. Gingras, Comment on 0

13 "What the Electromagnetic Vector Potential Describes", Am. J. Phys., 48 (1), Jan. 1980, p,