Huygens^Principle as Rather Universal Model of

Huygens’Principle as Rather Universal Model of Propagation and Transport Peter Enders Senzig, Ahornallee 11, D-15712 Koenigs Wusterhausen, Germany; [email protected]

Abstract Huygens’ Principle (HP) contains both the principle of action-at-proximity and the superposition principle. Although the propagation of sharp, non-spreading wave fronts is included in Huygens’ (1690) original formulation, it can be left out without touching those principles. The formulation of HP by means of the Chapman-Kolmogorov equation (following Feynman 1948) comprises both versions and overcomes misunderstandings like "Huygens’principle is not exactly obeyed in Optics" (Feynman 1948) and "HP is incompatible with Green’s functions" (Johns 1974). This way, HP applies not only to the propagation of light, but also to heat and matter di¤usion, Schroedinger matter waves, ie, to virtually all propagation phenomena, which can be described through explicit linear di¤erential and di¤erence equations, respectively. HP for Maxwell’s equations is discussed in terms of the Helmholtz-decomposed …elds and currents. The appearances of HP in mechanics, in fractional Fourier transformation being exploited in optics and in linguistics are also mentioned.

1

Contents

3

I. Introduction

6

II. Huygens’principle in mechanics A. Free fall

6

B. Linear harmonic oscillator

6

1. Euler-Wignerian approach: Separation of internal and external parameters

6

2. Propagator and Huygens’construction

7

3. Separation of angular frequency and mass

7

4. New integrals of motion

8

5. Factorization of the Hamiltonian

8

6. First-order equations of motion

8

7. Appearance of i

8

C. Quantum mechanics

9

III. Kirchho¤’s formula and its di¢ culties of interpretation and application 9 IV. Hadamard’s notion of Huygens’principle

11

A. Hadamard’s syllogism

11

B. ’Special’Huygens’principle: Minor premise included

12

C. ’General’Huygens’principle: Minor premise not included

13

V. Green’s functions representing Huygens’principle A. Huygens propagators

13 13

B. The Chapman-Kolmogorov equation as generalization of Huygens’ construction

14

C. Treatment of di¤erential equations of higher-order in time

15

VI. The time derivative of the wave amplitude as independent dynamical 16

variable VII. Huygens’principle in electromagnetism A. The rationalized Maxwell equations 2

18 18

B. Helmholtz decomposition ~ B) ~ C. Propagation of (D;

18

D. Mie-Weyl set of independent dynamical variables

20

E. Other constitutive relations

21

19

23

VIII. Discrete models of propagation A. One-step Markov chains –discrete Huygens propagators

23

B. A two-step Markov chain (random walk with correlation)

25

C. Proper Huygens propagators

26

IX. Fractional Fourier Transformation and Huygens’Principle X. General Theses for Describing Propagation Processes

27 28 29

XI. Conclusions

32

Acknowledgments XII. References

32

References

39

I.

INTRODUCTION

No one doubts that physics is an exact science. Nevertheless, the notion ’exact science’ should not be interchanged with ’like mathematics’. As stressed by Huygens (1690, pp. IIIf.), within physics, "one will …nd proofs of a kind, which do not grant the same great certainness of that of geometry and which even are rather di¤erent from those, because here, the principles are veri…ed by the conclusions drawn from them, while the geometricians proof their theorems out of sure and unquestionable principles; the nature of the subjects dealt with conditions this". Huygens’ideas on how light propagates have become basic ingredients of our physical picture of the world. The notion ’Huygens’principle’(HP), however, is not uniquely used. This paper aims, on the one hand, at the clari…cation of some confusion existing in the literature, in particular, about the role of sharp, non-spreading wave fronts and about the 3

range of applicability. For instance, Feynman (1948) wrote, that HP holds exactly for wave mechanics, but only approximately for optics; and Scharf (1994) stated, that HP is a principle of geometrical optics, not of wave optics. On the contrary, the unifying power of HP will be demonstrated in this paper. Some of that confusion is related to Kirchhoff’s (1882/1883) formula and reaches up to doubts on the validity of HP at all (Miller 1991), or on the possibility to represent HP by means of Green’s functions (GF) (Johns 1974). Both doubts challenge every mind believing in the unity of physics; the second doubt contradicts also the widely exploited power of GF (Green 1828). Actually, Kirchhoff’s solution to the wave equation –while being mathematically exact –su¤ers from the drawback of requiring the knowledge of both the …eld amplitude and its gradient on the boundaries. I will trace the origin of these mathematical and physical di¢ culties to the notions of degrees of freedom of motion and of independent dynamical variables. For the sake of the unity of physics, a further goal of this paper is to generalize Huygens’ basic ideas. This means, that I will keep essentially the imagination, that, (i), each locus of a wave excites the local matter, which, in turn, reradiates a secondary wavelet, and that, (ii), all these wavelets superpose to a new, resulting wave (the envelope of those wavelets), and so on. Huygens’ad hoc omission of backward radiation as well as Fresnel’s (1826) and other auxiliary assumptions (cf Longhurst 1973, §10-2) will be included in a natural manner. Special attention will be paid to a simple, but general and exact description of virtually all transport and propagation processes, which obey the principle of action-by-proximity and can be described by explicit transport equations. Shortly, consider a complete set of independent dynamical variables of a given prob~ r; t) = (X1 (~r; t); : : : ; Xf (~r; t)), eg, X(~ ~ r; t) = (u(~r; t); @u(~r; t)=@t), u(~r; t) being the lem, X(~ amplitude of a scalar wave. I seek to represent its propagation in the most simple form. ZZZ ~ ^ r; t; ~r0 ; t0 ) X(~ ~ r0 ; t0 )d3~r0 ; t > t0 X(~r; t) = H(~ (1) ^ obeys the Chapman-Kolmogorov The ’Huygens propagator’(Enders 1996, 2009a), H, equation (Kolmogorov 1931, 1933, Chapman & Cowling 1939) that is known from, but not restricted to Markov processes (Markov 1906) and related problems of probability theory. ^ r; t; ~r0 ; t0 ) = H(~

ZZZ

^ r; t; ~r1 ; t1 ) H(~ ^ r1 ; t1 ; ~r0 ; t0 )d3~r0 ; H(~ 4

t > t1 > t0

(2)

Thus, following Feynman (1948), I will express HP through this equation. The rigorous treatment requires measure theory (Dynkin 1965), but this is much more than necessary for the understanding of ’common’ physical propagation processes. It may proven useful, however, for the fractal description of wave propagation in disordered media (West 1992) and the like. ~ r; t) obeys a set of partial di¤erential equation of …rst order in time, H(~ ^ r; t; ~r0 ; t0 ) If X(~ turns out to be the GF of that equation, and eq.(1) is the solution to the corresponding ~ r; t) obeys a set of partial di¤erential equainitial-boundary value problem. If, however, X(~ tion of second (or higher) order in time, no such simple equation exists. Often, the much more involved Kirchhoff formula (17) is used. Unfortunately, this has misled some authors to deny a relationship between GF, HP and wave propagation at all. The use of GF within such considerations is not new, of course (Courant et al 1928, Spitzer 1964, Keilson 1965). My goal, however, is the representation of HP through GF rather than a discussion of the probabilistic questions behind such approaches. These are interesting enough, but need (and deserve!) a separate treatment. For numerical calculations on digital computers, discrete forms are preferable. The mathematically natural formulation is in terms of Markov chains –the physically and technically natural formulation is in terms of transmission-line networks. On that basis, powerful algorithms have been developed not only for electromagnetic problems (Hoefer & So 1991, Christopoulos 1995), but also for di¤usion (de Cogan 1998) and even for mechanical problems (de Cogan et al 2005). Here, in contrast to other cellular automata algorithms (Chopard & Droz 1998), an (idealized) physical system is mapped. For this, it is not too surprizing, that HP applies to the TLM equations, too (Hoefer 1991, Enders 2001, Enders & Vanneste 2003). For historical and methodological reasons, I will start in Section 2 with HP in mechanics and will continue, in Section 3, with Kirchhoff’s formula and certain problems of its physical interpretation. Then, Hadamard’s rigorous de…nition of HP will be discussed. In section 5, the superposition of secondary wave(let)s will be represented and illustrated by means of general …eld propagators in the space-time domain. This will lead to a description of wave motion, which overcomes the di¢ culties in the interpretation and application of Kirchhoff’s formula mentioned above. Section 6 will stress the role of time-derivatives of dynamical variables as independent dynamical variables. When equations of 2nd order 5

in time, such as the wave eqation, are rewritten as systems of equations of 1st order in time, HP applies exactly to those and, consequently, to wave optics as well. Section 7 will consider Maxwell’s equations in the light of these results, where the …elds and currents will be Helmholtz-decomposed, in order to work with independent …eld variables only. Alternatively, Section 8 will treat Mie’s set of electromagnetic equations in that sense. Section 9 will apply these thoughts to di¤erence equations and discuss implications for practical computations. A relationship to the fractional Fourier transformation will be sketched in section 10. The application to linguistics (Enders 2011) will be outlined in Section 11. Section 12 will condense all those results into theses for general principles of the physics of propagation. Sections 13, …nally, will summarize and conclude this contribution.

II.

HUYGENS’PRINCIPLE IN MECHANICS

Before discussing the fundamental issues, let me illustrate my view on HP by means of non-traditional examples (I fully agree with Huygens about optics).

A.

Free fall

As a matter of fact, the principle of superposition has …rst been formulated by Huygens for mechanical motions. Shortly, during free fall, the momentually achieved increments of speed add to the speed assumed just before (Horologium oscillatorium, 1673; after Simonyi 1990). This guarantees the di¤erentiability of the velocity: ~v (t + dt) = ~v (t) + d~v , and, thus, that the trajectories, ~r(t), are smooth.

B. 1.

Linear harmonic oscillator Euler-Wignerian approach: Separation of internal and external parameters

The trajectory, x(t), of a linear harmonic oscillator can be described as function of the initial values of position, x(0), and momentum, p(0), and of its mass, m, and angular velocity, !. x(t) = x(0) cos(!t) +

6

p(0) sin(!t) m!

(3)

Here, the internal (m, !) und external parameters (x(0), v(0)) occur in mixed form. Since, generally speaking, separations highlight the actual physical interrelations, it is desirable to separate internal und external parameters; in this case, the constants (laws of motion, system parameters) from the variable in‡uences (initial conditions, cf Wigner 1963). The separation makes it immediately, if one writes down the coupled solutions for both independent dynamical variables, x(t) and p(t): 10 1 0 1 0 1 0 1 sin(!t) x(0) x(0) x(t) cos(!t) m! ^ @ A@ A = D(t) A @ A=@ def m! sin(!t) cos(!t) p(0) p(0) p(t)

(4)

This form emerges, when one solves Hamilton’s equations of motion as a system of equations.

2.

Propagator and Huygens’ construction

^ The (matrix-valued) propagator, D(t); contains solely the internal parameters and the time. It describes rotations in phase space, fx(t); p(t)g, and exhibits the group property. ^ ^ D(t) = D(t

^ 0 ); t0 ) D(t

0

t0

(5)

t

This is an example for Huygens’ construction, and eq. (5) represents a discrete form of the Chapman-Kolmogorov equation.

3.

Separation of angular frequency and mass

Finally, one should separate angular frequency, !, and mass, m; for the oscillation is ^ determined by the former only. This is possible through the diagonalization of D(t) (what else?):

0 @

x~(t) p~(t)

1

0

A=@

ei!t

0

0 e

i!t

10 A@

x~(0) p~(0)

1

0

^~ @ A = D(t)

x~(0) p~(0)

1 A

The transformed variables are 0 10 1 0 1 0 1 i i 1 x(t) x(t) + p(t) x~(t) m! A @ m! A = p1 @ A @ A = p1 @ 2 2 im! 1 p(t) im!x(t) + p(t) p~(t)

(6)

(7)

They are most simple functions of time.

x~(t) = x~(0)ei!t ;

p~(t) = p~(0)e 7

i!t

(8)

4.

New integrals of motion

From eqs. (8), two …rst integrals of motion can be read o¤ immediately. I1 = e

i!t

x~(t) = x~(0);

I2 = ei!t p~(t) = p~(0)

(9)

Since there are no further independent …rst integrals, the total energy, E, can be expressed as function of I1;2 . Indeed, E =

5.

i!I1 I2 .

Factorization of the Hamiltonian

This indicates, that the variables (7) factorize the Hamilton function. H(x; p) =

1 2 m 2 2 ! x (t) + p (t) = 2 2m

i!~ x(t)~ p(t) =

i!~ x(0)~ p(0) = E

(10)

The physical relevance of this factorization emerges in quantum mechanics, where –despite of constant factors – x~(t) and p~(t) become creation and annihilation operators for energy quanta (Enders 2006).

6.

First-order equations of motion

Finally, the variables (7) obey equations of motion not of second order in time –as x(t) und p(t) do –, but of …rst order in time. d p~(t) = dt

d x~(t) = +i!~ x(t); dt

i! p~(t)

(11)

This corresponds to the factorization d2 + !2 = 2 dt

d + i! dt

d dt

i!

(12)

Di¤erential equations of …rst order in time are fundamental ones in that they determine, which variables are independent ones. (Recall, that independent variables are those, for which the initial values can be independently prescribed.)

7.

Appearance of i

It is noteworthy that this all was possible only through exploiting the imaginary unit, p i 1. This provides i with a physical (and not only mathematical-calculational) justi…8

cation already within classical mechanics (Schroedinger 1926d hesitated to use i for the formulation of his …rst-order time-dependent equation).

C.

Quantum mechanics

Being a probabilistic theory, quantum mechanics describes motion in terms of transition probabilities, Pba =< bja >, rather than trajectories from state a to state b. Usually, the state functions, fjc >g, form a complete set, so that they provide a decomposition of the P unit operator: ^1 = c jc >< cj. This enables one to write Pba =< bja >=

X

< bjc >< cja >=

c

X

(13)

Pbc Pca

c

This is the Chapman-Kolmogorov equation in state space. Referring to this equation, Feynman (1948) emphasized, that HP applies to Schroedinger’s wave mechanics. I fully agree with Feynman to consider the ChapmanKolmogorov equation (here, in state space) to be the most appropriate mathematical expression of HP.

III.

KIRCHHOFF’S FORMULA AND ITS DIFFICULTIES OF INTERPRETA-

TION AND APPLICATION

Within classical wave theory, the mathematical problem of wave propagation is usually reduced to the solution of the wave equation. In the simplest 3D case, u(~r; t)

1 @ 2 u(~r; t) = c2 @t2

u(~r; t)

(14)

q(~r; t)

where u(~r; t) is the scalar …eld amplitude and q(~r; t) the source density. The GF, g(~r; tj~r0 ; t0 ), to eq.(14) is its solution for the unit source density. g(~r; tj~r0 ; t0 ) =

(~r

~r0 ) (t

(15)

t0 )

A special solution to eq.(15) is the expanding (retarded) impulsive spherical wave. gr (~r; tj~r0 ; t0 ) =

( ) (R=c 4 R

)

9

;

R

j~r

~r0 j ;

t

t0

(16)

(DeSanto 1992, p.47). Using this GF, the retarded solution to eq.(14) reads

ur (~r; t) =

Zt Z Z Z

t0

1 c2

gr (~r; tj~r0 ; t0 )q(~r0 ; t0 )dV0 dt0 V

ZZZ

u(~r0 ; t0 )

V

+

Zt Z Z

@ gr (~r; tj~r0 ; t0 ) @t

[gr (~r; tj~r0 ; t0 )r0 u(~r0 ; t0 )

gr (~r; tj~r0 ; t0 )

@ u(~r0 ; t0 ) dV0 @t

~0 dt0 (17) u(~r0 ; t0 )r0 gr (~r; tj~r0 ; t0 )] dS

t0 @V0

The …rst term describes the propagation of that part of the …eld amplitude that stems from the external sources, while the second and third terms account for the initial and boundary conditions, respectively. The third term, Kirchhoff’s (1882/1883) formula, describes, in particular, the scattering at screens. Being mathematically correct, its physical interpretation is subject to various discussions. For Macke (1958), it is a "direct and quantitative formulation of Huygens’ principle" (cf also Naas & Schmid 1974). On the other hand, it has been criticized for calling in not only the …eld amplitude itself (which may be complex-valued), but also its spatial derivative, as if derivations had "to be performed by nature" (Johns 1974, Miller 1991). In other words, "two types of sources of varying strength occur, so that the simplicity of Huygens’approach is lost" (Miller 1991). More generally speaking, this makes this formula unable to exactly solve real di¤raction problems (Pockels 1906, p.1045). However, it represents a de…nite progress over Fresnel’s (1826) famous, though not exact formulation of Huygens’ principle and can be exploited for approximative solutions, if realistic assumptions can be made about the boundary values. It seems to us, however, that these problems originate not from HP itself, but from, (i), the use of the free-space GF (16), which does not account for the actual boundary conditions, such as screens, and, (ii), from the use of a GF which even does not represent HP by itself. Indeed, these problems disappear when rewriting the wave equation (14) as @u(~r; t) = w(~r; t) @t @w(~r; t) = c2 u(~r; t) + c2 q(~r; t) @t

(18a) (18b)

and using a GF for this system of equations which, additionally, obeys appropriate boundary conditions, as will be shown below. 10

IV.

HADAMARD’S NOTION OF HUYGENS’PRINCIPLE

Basing on his investigations on the Cauchy problem for partial di¤erential equations, Hadamard (1952, § 33) has given the most exact formulation of HP I am aware of.

A.

Hadamard’s syllogism

A syllogism is a form of logical conclusion; it has been developed already by Aristotle (Lukasiewicz, 1957, Wiener 1995). The conclusion is derived from two premises, a major and a minor ones. Major premise The action of phenomena produced at the instant t0 on the state of matter at the time t > t0 takes place by the mediation of every intermediate instant, t0 ; t0 < t0 < t. Minor premise The propagation of light pulses proceeds without deformation (spreading, tail building) of the pulse. Conclusion In order to calculate the e¤ect of our initial luminous phenomenon produced at t = t0 , one may replace it by a proper system of disturbances taking place at t = t0 and being distributed over the surface of the sphere with centre t0 and radius c(t0

t0 ).

The Major premise is the principle of action-by-proximity and, philosophically speaking, a "law of thought". (Unfortunately, it’s a still widely spread misunderstanding, that Newton’s theory of gravity is a theory of action-at-distance. Only the mathematical form of his force law describes action-at-distance.) The Minor premise postulates the propagation of none-spreading wave fronts. The Conclusion is essentially Huygens’construction. Now, as a matter of fact, in the Conclusion, the isotropy of re-irradiation can safely be replaced with the re-irradiation according to the actual local propagation conditions, without violating Huygens’imagination. This means, that the secondary sources take into account the local propagation properties of the material under consideration (or that of free space). For instance, at interfaces or in anisotropic media, the reaction of the secondary sources is anisotropic (Huygens 1690); in nonlinear media, the excitation and re-irradiation is not proportional to the amplitude of the exciting …eld (Guenther 1988). 11

Now, when compared with the Major premise, the Minor premise is rather special. It is necessary for geometrically constructing the wavefront, but not for the basic principle of action-by-proximity and not for the cycle of excitation and re-irradiation. One of the main points of this paper is to examine what happens, when it is relaxed.

B.

’Special’Huygens’principle: Minor premise included

Often, ferred

to

the as

existence HP

of

sharp,

(Courant

&

non-spreading Hilbert

1968,

wave

fronts

Naas

http://www.mathpages.com/home/kmath242/kmath242.htm).

&

is

already

Schmidt

re-

1974,

This phenomenon is

widely known for d’Alembert’s wave equation in 3+1 dimensions. Another classical example is the distortion-free pulse propagation along special 1D transmission lines (Heaviside, Pupin). Hadamard’s conjecture states that the wave front is not spreading in odd space dimensions (Hadamard 1953, Naas & Schmidt 1974). It should be clear, however, that, despite of its practical consequencies for signal transmission, the Minor premise (B) is a secondary attribute of propagation processes, while the Major premise (A) and the Conclusion (C) are primary ones. Thus, analogously to the solution of total-hyperbolic di¤erential equations (Naas & Schmidt 1974), one may de…ne also within optics and for general propagation phenomena, respectively, a ’Special’Huygens’Principle, where proposition (B) is an essential ingredient. It corresponds to a distortion-free signal transmission; the speed of propagation of the waves does not depend on the oscillation frequency of the source and the waves su¤er not any deformation through smearing or wake building (cf also Iwanenko & Sokolov 1953). In this case, for the validity of HP it is necessary and su¢ cient that the Green’s function of d’Alembert’s wave equation is proportional to the delta-function, (R

c ), or to its

derivatives (Naas & Schmidt 1974). The construction of equations the solutions to which are non-spreading sharp wave fronts has been developed to a special topic of its own (Guenther 1988). These results may proven to be useful for the design of dispersionless signal transmission systems. Is there a relationship to re‡ectionless potentials for the Schroedinger equation?

12

C.

’General’Huygens’principle: Minor premise not included

On the other hand, as mentioned above, action-at-proximity and superposition are not bounded to sharp wave fronts. Thus, Major premise (A) together with Conclusion (C) has also be termed HP (Johns 1974, Bickel & Hander 1988, Miller 1991). We will call HP the combination of action-at-proximity ("elastic waves in aether" in Huygens’pictural imagination) and superposition of secondary wavelets (Huygens’ construction, suitably generalized). The shape of the wave front may vary from case to case, without in‡uence on these basic ingredients of propagation, but the essentials of Huygens’(and Newton’s and Faraday’s, too) imagination of propagation are conserved. The advance of this notion of HP consists in that its applicability becomes extremely wide; in fact, in this form, HP quali…es to be a clue for unifying the physical and mathematical description of a huge variety of transport and propagation processes.

V.

GREEN’S FUNCTIONS REPRESENTING HUYGENS’PRINCIPLE

From the theoretician’s point of view, GF represent one of the most powerful and, at once, most beautiful and clear (propagator!) tools of mathematical physics at all (Dyson 1993). She simply expects that there are GF which can be used to represent HP.

A.

Huygens propagators

Without loss in generality, let us study the equation ZZZ u(~r; t) = H(~r; t; ~r0 ; t0 )u(~r0 ; t0 )d3~r0 ;

t > t0

(19)

describing the propagation of the scalar …eld u(~r; t) from the space-time point (~r0 ; t0 ) to the space-time point (~r; t). Which are the general properties of the integral kernel, H? 1. u(~r; t) ful…lls the partial di¤erential equation @u = L(~r)u(~r; t) @t

(20)

q(~r; t)

where L(~r) is a partial di¤erential expression in ~r and q(~r; t) the source density. Then, @ H(~r; t; ~r0 ; t0 ) = L(~r)H(~r; t; ~r0 ; t0 ) @t 13

(~r

~r0 ) (t

t0 )

(21)

H is the Green’s function of the di¤erential equation (20). If u(~r; t) ful…lls a partial di¤erential equation of higher order in time, there is no simple relationship like (19). 2. u(~r; t) ful…lls the initial condition (22)

u(~r; t = 0) = u0 (~r) if H obeys the initial condition lim H(~r; t; ~r0 ; t0 ) = (~r

t!t0 +0

and u(~r; t) =

ZZZ

(23)

~r0 )

H(~r; t; ~r0 ; 0)u0 (~r0 )d3~r0 ;

(24)

t>0

3. u(~r; t) ful…lls the boundary condition B(~r)u(~r; t) = 0;

~r 2 S

(25)

on the inner surface, S, of the domain under consideration, if H does so. B(~r)H(~r; t; ~r0 ; t0 ) = 0;

~r 2 S;

~r0 2 =S

(26)

This can be achieved by means of an eigenfunction expansion (DeSanto 1992). De…nition 1 The Huygens propagator is that GF of a di¤erential equation of …rst order in time, which, additionally, obeys the initial condition and the boundary conditions of the problem under consideration. Thus, it contains the propagation conditions both in the volume and on the spatiotemporal boundaries. Due to this, the di¢ culties with the boundary terms in Kirchhoff’s formula are overcome, because the 2nd and 3rd terms on the r.h.s. of formula (17) are absent in eq.(19).

B.

The Chapman-Kolmogorov equation as generalization of Huygens’ construc-

tion

Nesting the integral equation (17) into itself yields ZZZ ZZZ u(~r; t) = H(~r; t; ~r0 ; t0 ) H(~r0 ; t0 ; ~r1 ; t1 )u(~r1 ; t1 )d3~r1 d3~r0 ; 14

t > t0 > t1

(27)

Rearranging the indices and comparing this with the original equation (17) gives ZZZ H(~r; t; ~r0 ; t0 ) = H(~r; t; ~r1 ; t1 )H(~r1 ; t1 ; ~r0 ; t0 )d3~r1 ; t > t1 > t0

(28)

V1

This is the Chapman-Kolmogorov equation in the space-time domain. It generalizes Huygens’ construction to spreading wave fronts as the domain of sources of secondary wavelets is not necessarily a surface, but, in general, a certain …nite volume, V1 . Nonspreading wave fronts correspond to -functions in the GF reducing the volume integral to a surface integral. In such cases, di¤raction at screens is treated in a manner loosely resembling Kirchhoff’s formula, but without its di¢ culties mentioned above. Since the time interval t

t0 can be in…nitesimally small, the Chapman-Kolmogorov

equation is a mathematical formulation not only of the superposition of secondary wavelets, but also of the action-at-proximity. The validity of a relation like (28) is sometimes called a Markov property; it plays an important role for the path-integral representation of dynamical processes (Feynman 1948, Feynman & Hibbs 2005). Now, as a matter of fact, the GF (16) of the wave equation (14) does not obey the Chapman-Kolmogorov equation (28). Indeed, the latter is obeyed by functions being the solution to partial di¤erential equations of …rst order in time (this may be easily proven by means of the Fourier transformation w.r.t. the time variable). This was, perhaps, the reason for Feynman (1948) to state that, in optics, HP holds true only approximately. As mentioned above and outlined in the next Subsection, this misunderstanding is avoided when working with partial di¤erential equations of …rst order in time rather than of higher order in time like the wave equation.

C.

Treatment of di¤erential equations of higher-order in time

The way out of the tension between HP and the wave equation consists in the ’return’to systems of …rst-order equations. As a matter of fact, these are the fundamental relations as they comprise the constitutive relation(s) and the conservation law(s). For example, consider the system of equations of hyperbolic heat conduction theory

15

(Mueller 1967). 0

~ r; t) J(~

1

@ @ A= @t T (~r; t)

0 @

1=

( = )r

(1= Cp )r

0

(J~ –heat current density, T –temperature, unit volume at constant pressure,

10 A@

~ r; t) J(~ T (~r; t)

1 A

(29)

–heat conductivity, Cp –heat capacity per

–heat ‡ux relaxation time: Maxwell has introduced

it for making the propagation speed …nite). The GF for the system of equations (29) is the 4 4 matrix -valued function 0 1 ~ J~J~ ~ JT ^ G G ^=@ A G ~ T J~ GT T G

being de…ned as the solution to the matrix -valued equation 0 1 0 1 ^ ~ 1= ( = )grad 1 0 @ ^ ^+@ A G A (~r G= @ @t ~ (1= Cp )div 0 0 1

~r0 ) (t

t0 )

(30)

(31)

^ r; t; ~r0 ; t0 ) also accounts for the actual initial: J(~ ~ r; t = 0) = J~0 (~r), T (~r; t = 0) = When G(~ T0 (~r), and boundary conditions, it becomes the (matrix-valued) Huygens propagator, ^ r; t; ~r0 ; t0 ), of the problem under consideration, and the solution, (J; ~ T ), is given by a H(~ single integral, again. 0 1 10 1 0 ZZZ ~ 0 J~J~ ~ JT ~ ~ ^ J(~r; t) J (~r ) H H @ A= A@ 0 A d3~r0 ; @ ~ 0 TJ TT ~ T (~r; t) T0 (~r ) H H

t>0

(32)

Maxwell’s equations will be treated below.

VI.

THE TIME DERIVATIVE OF THE WAVE AMPLITUDE AS INDEPENDENT

DYNAMICAL VARIABLE

Huygens propagators of the wave equation in the form (18) describe the common propagation of the …eld (wave) amplitude, u(~r; t), and of its time-derivative, @u(~r; t)=@t = w(~r; t), as mutually independent dynamical variables, which are created simultaneously and propagates together and in mutual interaction. ~ T ) of eq.(29) or counter-propagating waves (d’Alembert’s When considering the pair (J; solution to his wave equation), the physical content of the time-derivative is even more 16

obvious. This is the fundamental di¤erence between our interpretation of HP and previous ones, but Hadamard (1953). Nowadays mechanical theories often concentrate on equations of motion, such as eq.(14) or Lagrange’s equation of motion, while the role of the velocity, ~v = d~r=dt, and of the momentum, p~ = md~r=dt, respectively, as dynamical variables on their own is explicitly considered only in Hamilton’s equations of motion (and in statistical mechanics). The independence of the initial values of location, ~r0 = ~r(t = 0), and of velocity, ~v0 = ~v (t = 0), implies the independence of the values of ~r(t) and ~v (t) for all later times. And exactly this is what quali…es ~r(t) and ~v (t) to be independent dynamical variables. Obviously, the same holds true for the (possibly complex-valued) wave amplitude, , and their time-derivative, @ =@t, as recognized within Lagrangean and Hamiltonian …eld theories. Within optics, this matter of fact is often hidden by the use of time-harmonic waves. The Schroedinger equation is one common parabolic equation for the two independent dynamical variables

and

. Indeed, there is a complete system of 1st-order equations in

time for Re( ) and Im( ). These results are generalized in the following Conjecture 2 The number of independent dynamical variables equals the number of timederivatives in the equation(s) of motion. In general, there are various complete sets of independent dynamical variables for a given problem. The number of independent dynamical variables can be reduced by symmetry. For instance, in a travelling electromagnetic wave in free space, all 12 …eld components in Maxwell’s macroscopic equations are proportional to only two …eld components (eg, Ex and Ey ) determining intensity and polarization. Scalar propagators obeying the Chapman-Kolmogorov equation are positive de…nite. Hence, the Huygens propagators for processes exhibiting interference are matrices (usually, classical waves) or complex-valued (eg, matter waves). The Hamilton-Jacobi equation (wave picture) converts Hamilton’s equations of motion (particle picture) into a non-linear ’wave’equation (cf also Einstein 1917). It would thus be interesting to explore the applicability of HP to the former. 17

VII. A.

HUYGENS’PRINCIPLE IN ELECTROMAGNETISM The rationalized Maxwell equations

From the point of view of initial-boundary value problems, the nowadays tought, so-called rationalized Maxwell equations (Heaviside, Poynting, Hertz) represent an incomplete ~ set of partial di¤erential equations of 1st order in time for the 4 3 = 12 …eld variables D, ~ E ~ and H. ~ B, ~ = r D

(33a)

~ =0 r B

(33b)

~ @B ~ = r E @t ~ @D ~ ~j =r H @t

(33c) (33d)

I complement them through the relaxational constitutive relations ~ ~ @E ~ = D +E @t "r "0 ~ ~ @H ~ = B + H ~ H @t r 0 ~ E

which account for …nite relaxation times (

~, E

~ ). H

(34a) (34b)

~ and H ~ to be The time-derivatives make E

independent dynamical variables, too, despite of the constraints represented by the source ~ and B ~ not to represent three independent eqs. (33a), (33b). They make the vectors D dynamical variables each. This de…ciency of 2 independent dynamical variables is usually ascribed to the implicitly imposed charge and energy conservations. However, this arguing provokes the question, why the conservations of momentum and angular momentum do not diminuish the number of independent dynamical variables? As a consequence, the Huygens propagator is degenerated, and so-called spurious modes may disturb numerical calculations.

B.

Helmholtz decomposition

One way out of this de…ciency consists in the Helmholtz decomposition (Helmholtz 1858, Keller 2005). 18

~ =B ~T + B ~ L is purely transverse: B ~ =B ~T , B ~ L = ~0, B ~ and E ~ enter the Maxwell equations (33), solely the transverse components of H ~ and ~j. the charge conservation is related solely to the longitudinal components of D @ r ~jL + = 0; @t

~L = r D

(35)

For this, one can rewrite eqs. (33) as ~L = r D

(36a)

~L = 0 r B

(36b)

~T @B ~T = r E @t ~T @D ~ T ~jT =r H @t

(36c) (36d)

The constitutive relations (34) can be kept. The Huygens propagator would not contain ~ L and B ~ L and thus would not be degenerated. D The Helmholtz decomposition genuinely relates the propagation of electromagnetic waves with the transverse …eld components. Its drawback – and, perhaps, the reason of low acceptance –consists in the fact that it is not Lorentz covariant, so that it has to be separately performed in each system of reference. The criterion of being compatible with special relativity is, however, not the Lorentz covariance, but the compatibility with the Poincare group (Dirac 1949). For more details, see (Enders 2009b)

C.

~ B) ~ Propagation of (D;

Eqs. (34) reveal that it depends on the properties of the matter in which the electromagnetic …eld under consideration exists, how many independent dynamical variable are ~ and H. ~ Without loss of generality, I con…ne myself to the simplest case, represented by E ~ = D=" ~ 0, H ~ = B= ~ 0 . Note, that in contrast to the common use viz, that of vacuum, E ~ B), ~ here, I keep the pair (D; ~ B) ~ in view of its position in the to work with the pair (E; Maxwell equations (see also Mie 1941 and Sommerfeld 2001).

19

Let us further assume that 0 1 D (z) B x C B ~ T = B D (z) C D C; @ y A 0

0

1

0

B (z) B x C B ~ = B B (z) C B C; @ y A 0

1

j (z) B x C B ~jT = B jy (z) C C @ A 0

(37)

Then, the four independent dynamical variables (Bx;y ; Dx;y ) obey the complete set of equations

0 B B B B B @

1 @ "0 @z

@ @t

0

0

0

@ @t

1 @ "0 @z

0

0

1 @ 0 @z

@ @t

0

0

0

@ @t

1 @ 0 @z

The corresponding GF, 0

10

Bx

1

CB C CB B C CB y C CB C= C B Dx C A@ A Dy

GBx Bx GBx By GBx Dx GBx Dy

B B GBy Bx GBy By GBy Dx GBy Dy ^=B G B B GDx Bx GDx By GDx Dx GDx Dy @ GDy Bx GDy By GDy Dx GDy Dy

0

1

B C B 0 C B C B C B jx C @ A jy

(38)

1 C C C C C A

is replaced with ^1 (z

(39)

z 0 ) (t

t0 ),

4 unit matrix. Since the matrix in eq.(38) is block diagonal, this 4

4-

obeys the same equation, where the vector ^1 denoting the 4

0

0 0 jx jy

matrix GF is reducible to a direct product of two 2

2-matrix GFs. These are irreducible,

but they can be diagonalized. One obtains the four GFs for the four wave equations. @2 Dy @t2

@2 Dy = "0 0 @z 2 1

@ jy ; @t

etc.

(40)

As in the 3D case, it is straightforward to show that the GF of the 1D wave equation (40) does not obey the Chapman-Kolmogorov equation, while the appropriate 2

2-matrix

GF does.

D.

Mie-Weyl set of independent dynamical variables

According to Weyl (1918, § 26), the principle of causality requires the basic equations to be in the form of partial di¤erential equations of …rst order in time. In order to circumvent the constraints (33a), (33b), he – referring to Mie – deals with the following subset of

20

Maxwell’s (1861, 1864, 1873) original set of equations.[1] @ + r ~j @t ~ @B ~ +r E @t ~ @A +r @t ~ @D ~ r H @t

=0

(W-1)

=0

(W-2) ~ E

=

(W-3)

= ~j

~ = 0, B ~ =r They comprise the constraints r B

(W-4)

~ and r D ~ = . A

Eqs. (W-1) and (W-3) are not coupled to eqs. (W-2) and (W-4). For this, it is not ~ into the Huygens propagator. The remaining eqs. necessary to include them, ie, and A (W-2) and (W-4) equal the rationalized Maxwell eqs. (33c) and (33d), resp. The bene…ts of this approach for constructing Huygens propagators for the electromagnetic …eld remain to be explored. Weyl has concentrated himself on the relationship between the Lagrangean and Hamiltonian descriptions of the electromagnetic …eld.

E.

Other constitutive relations

Let us examine the constitutive relations (34a) and (34b) in the light of, (i), Mie’s (1941) derivation of the rationalized Maxwell equations (cf Sommerfeld 2001). ~ describes the instantaneous (!) "electrical excitation" of its environment by an electrical D charge, q.

I

~ d ~r = D 2

Z

d3 r = q

(41)

V

@V

A …nite propagation speed of this excitation can be imposed by means of a Maxwellian relaxation time,

~ D

(which may be a tensor, ^D~ ). ! I Z ~ @D 2 ~ +D d ~r = ~ D @t

d3 r = q

(42)

V

@V

This would replace eq. (33a) with r

~ @D ~ +D ~ D @t

21

!

=

(43)

Replacing _ in the continuity equation accordingly yields ! 2~ ~ @ @ D D =0 r ~j + r + ~ D @t2 @t This means, that there is a sourceless …eld, r ~j +

(44)

~ such, that H,

~ ~ @2D @D =r + ~ D @t2 @t

~ H

(45)

~ = ~j + @ D=@t, ~ The l.h.s. is Maxwell’s (1861f.) "total current", C supplemented by a 2 ~ ~ describes the instantaneous (!) "magnetic excitation" relaxation term, ~ @ 2 D=@t . Hence, H D

~ 0 = ~j + @ D=@t ~ of its environment by the extended total current, C +

~@ D

2~

D=@t2 . A …nite

propagation speed of this excitation can be imposed by means of a Maxwellian relaxation time,

~ H

(which may be a tensor, ^H~ ), again. 2~ ~ ~ 0 = ~j + ~ @ D + @ D = r C D @t2 @t

~ ~ + ~ @H H H @t

!

(46)

Obviously, the problem of connecting the homogeneous and inhomogeneous rationalized Maxwell equations (33) still remains. ~ and H, ~ on their sources, Above, we have imposed a delayed reaction of the excitations, D ~ 0 , respectively. Similar, relaxational constitutive relations are expected to hold true and C ~ and E, ~ and H ~ and B ~ –but which are the causes and which the e¤ects? between D ~ as the cause of a dielectric Maxwell (1864, § 66) treats the electrical …eld strength, E, to be polarized. In isotropic substances, ~ ~ = D E "r "0

(Maxwell 1864 E)

~ causes the polarization, P~ = D ~ In other words, the force per unit charge, E,

~ The "0 E.

corresponding relaxational constitutive relation reads ~ P

~ @ P~ E + P~ = @t "0

(47)

This would introduce the electrical polarization, P~ , to be an independent dynamical variable and, hence, into the Huygens propagator. ~ and B ~ are always proportional each to another (for a critical In Maxwell’s work, H review, see Enders 2012). Analogues of the constitutive relation (47) are ~ @M ~ = +M ~ M @t 22

~

0H

(48)

and

~ @M ~ =B ~ +M ~ M @t

(49)

~ , to be an independent respectively. This would introduce the magnetic polarization, M dynamical variable and, hence, into the Huygens propagator. As the bene…ts of all these variants depend on the problem to be solved, their exploration is left to the reader.

VIII.

DISCRETE MODELS OF PROPAGATION

Some of the concepts proposed above can be illustrated in an even more obvious maner by means of simple discrete propagation models. Moreover, new concepts will be introduced here.

A.

One-step Markov chains –discrete Huygens propagators

In one-step Markov chains, each two subsequent states, ~uk

(uk;1 ; uk;2 ; :::) and ~uk+1 ,

where the second index may label spatial cells, are connected through a transition matrix, P^ , as ~uk+1 = P^ ~uk ;

(50)

k = 0; 1; :::

If P^ is independent of k, one obtains ~uk = P^ k ~u0 = P^ k

1

~u1 = P^ k

1

P^ ~u0

(51)

From this, the fundamental formula follows, which describes the evolution of such chains, viz, the Chapman-Kolmogorov equation(s) P^ k = P^ k

1

P^ = P^ k

l

P^ l ;

0

l

k

(52)

Obviously, eq.(52) is a discrete analogue to eq.(28) in describing the superposition of secondary ’wavelets’; and this, although one-step Markov chains with real-valued state variables describe di¤usion-like processes (overdamped waves). In discrete spaces, the principle of action-at-proximity means, that, during one time interval, only the next-neighbouring cells can be reached. A most important example of this class of Markov chains is constituted by the random walks (RW) (Spitzer 1964). 23

Consider the symmetric simple RW in 1D. An imaginary particle in an in…nite chain of cells is supposed to hop at each step to one of the two neighboring cells, where the probabilities of hopping forward and backward equal one-half each. The probability, pk;i , to …nd the particle at time step k in cell i is given by the recursion formula 1 pk;i = (pk 2

1;i 1

+ pk

1;i+1 );

k = 0; 1; 2; :::;

1 < i < +1

(53)

This is the well-known Euler forward scheme for the di¤usion equation, @T =@t = @ 2 T =@x2 . The f undamental solution to eq.(53) reads (the particle starts at k = 0 in cell i = 0) 8 9 > exp 2i (x2 + u2 ) cot > 2 < K (x; u) = (x u); = 2n > > > : (x + u); = (2n 1) 27

ixu csc

;

6= n (63)

This kernel is continuous in the generalized function sense. F 2n is the identity, F 2n the ordinary FT. Almeida (1993, 1994) has derived the group property F F = F K

+

Z+1 (x; z) = K (x; u)K (u; z)du

+ =2 +

is

, or (64)

1

The isomorphism with the Chapman-Kolmogorov equation (28) is obvious. In fact, up to a phase factor, K (x; u) is equivalent to the GF of the time-dependent Schroedinger equation for the harmonic oscillator (Agarwal & Simon 1994). Generally speaking, the kernel of any transformation satisfying a relation like (64) is equivalent to the GF of a parabolic di¤erential equation (such as the paraxial wave equation) in appropriate coordinates and, consequently, describes the propagation of some …eld. Therefore, the question arises, whether there are useful generalizations of the FracFT through chosing for the kernel other propagators, than that for the harmonic oscillator or for the parabolic index pro…le. In other words, are there further potential functions or index pro…les which yield integral transformations with similarly useful properties as those of the Fourier and fractional Fourier transformations? Furthermore, are there applications for the generalization of eq.(64) to matrix functions as kernels of integral transformations? From a computational point of view, it may be favourable to have got a discrete formulation of this theory. According to the foregoing section, this should be possible in terms of Markov chains or (transmission-line) networks. This could open a novel approach to wave-optical computations.

X.

GENERAL THESES FOR DESCRIBING PROPAGATION PROCESSES

The following theses are proposed to built a starting point for an extension of the de…nition and application of HP to all propagation phenomena, which can be described through linear explicit di¤erential and di¤erence equations, respectively. 1. Propagation via action-at-proximity proceeds such, that the …eld excites secondary sources, which re-irradiate the …eld accordingly to the actual boundary and continuity conditions. Topologically, this principle applies on structures with next-neighbour interaction (local theories; cellular automata; certain coupled maps).

28

2. The propagating …eld is represented by a set of f independent (but interacting, of course) dynamical variables, where f equals the number of time-derivatives in the governing equations. In general, there are several such sets. A complete set obeys a system of f di¤erential and di¤erence equations of …rst order, respectively. Examples are the right- and left-running waves in eq.(58), the wave amplitude and its "inner" speed of change, or …eld and ‡ux density [eqs. (29) to (32)]. The ‡ux density may play the role of the time-derivative of the …eld as independent dynamical variable, while its vector components are not independent of each other dynamical variables. A set of f one-step Markov chains provides the appropriate form for a discrete model of the propagation of f independent variables. 3. The (matrix-valued) GF of such a system contains the propagation of that complete set. It represents HP in the sense of action-at-proximity and superposition of secondary wavelets through the Chapman-Kolmogorov equation. In order to avoid perturbing boundary terms and to completely represent the propagation problem under consideration, the GF should ful…ll the boundary conditions for the …eld variables in appropriate form. For such GF the term Huygens propagator is proposed. 4. The elimination of backward motion and the conservation of sharp, non-spreading fronts during propagation are special cases, that emerge naturally from the governing equations and do not need additional assumptions.

XI.

CONCLUSIONS

For Feynman (1948), HP was – in geometrical-optical formulation – valid for matter waves; since the Schroedinger equation is of …rst order in time, the ChapmanKolmogorov equation holds true for the amplitudes of the quantum-mechanical transition probabilities. Schroedinger (1926b) has extensively quoted HP "in its old, naive form, not in the rigorous Kirchho¤ form", ie, in the same meaning as understood by Feynman. As its expression he has seen the Hamilton-Jacobi equation –an equation of …rst order in time. The representation of HP proposed in this paper uni…es the description of propagation processes modelled by parabolic and hyperbolic di¤erential equations. It is the same one for geometrical and for wave optics; the former being a limit case, but without ad hoc 29

assumptions. The mathematical formulation of HP in form of the Chapman-Kolmogorov equation (28) implies the following important conclusions. (i) Huygens’construction can be applied to spreading wave fronts as well. (ii) Wave propagation is a Markov process (speech recognition bases on this fact). (iii) HP in that sense holds true for Dirac and similar quantum …elds as well as for di¤usion processes. The theses of section 10 are deliberately formulated in such a general manner, that they apply, among others, also to the cases of spatial anisotropy (Huygens 1690), isotropy in the sense of the (local) line element (Schroedinger 1926a,b), nonlinear and ‡uctuating propagation conditions (Vanneste et al. 1992, Enders 1993), audio-holography (Illenyi & Jessel 1991), and the states in electrical power systems (Vasin 1990). HP needs no correction as proposed by Miller (1991), and the di¢ culties discussed by Johns (1974) are lifted as well. The mathematical representation of HP by means of propagators and the Chapman-Kolmogorov equation throws also new light upon the relation between the fractional Fourier transformation and wave propagation and suggests further generalizations and applications in this …eld. Schwartz (1987) wrote, "Physically this [HP] makes no sense at all. Light does not emit light; only accelerating charges emit light." Indeed, not the wavefront itself irradiates the secondary wavelets, but the matter it excites does so (including the so-called vacuum exhibiting …nite values of "0 and

0 ).

Since HP is not concerned with the mechanisms

of excitation and re-irradiation, the GF and, thus, the Chapman-Kolmogorov equation cope with this physical point. The following text resembles Feynman’s (1948) original statement and fosters the view on HP presented in this contribution. "Huygens principle follows formally from the fundamental postulate of quantum electrodynamics –that wavefunctions of every object propagate over any and all allowed (unobstructed) paths from the source to the given point. It is then the result of interference (addition) of all path integrals that de…nes the amplitude and phase of the wavefunction of the object at this given point, and thus de…nes the probability of …nding the object (say, a photon) at this point. Not only light quanta (photons), 30

but electrons, neutrons, protons, atoms, molecules, and all other objects obey this simple principle." (http://en.wikipedia.org/wiki/Huygens’_principle, Feb. 12, 2008) Di¤erence equations representing a discrete HP are directly suited for computing all propagation processes that can be modelled through explicit di¤erential equations. This should enable the simultaneous and self-consistent computation of interacting …elds of di¤erent type, eg, heat di¤usion and electromagnetic waves in lasers (Enders 1992), in microwave ovens or in lenses and mirrors for high-power beams. Within explicit schemes, self-consistency can be achieved at every (time) step, whereby convergency is considerably accelerated. One of such numerical algorithms is the Transmission-line Matrix Modeling Method (TLM), an explicit …nite-di¤erence scheme describing travelling voltage pulses on a mesh of lossless transmission lines and lumped resistors (Christopoulos 1995, de Cogan 1998, de Cogan et al 2005). These di¤erence equations trace a practically realizable physical process obeying HP, too. Due to that, a TLM routine exhibits excellent stability properties, which – among others – are exploited in commercial programm packages. The GF of the coupled one-step TLM equations is even a proper Huygens propagator exhibiting further computational advantages (cf Enders & Wlodarczyk 1993). Johns’(1987) symmetric condensed node for solving Maxwell’s equations in 3D obeys Hadamard’s Minor premise (Johns & Enders 1992). Delsanto and coworkers (1992) have stressed that a local interaction approach to simulation is favorized by three practical advantages: (i) extremalous speed due to immediate parallelizability; (ii) complex problems can be treated in a simple manner, since the local internodal connections are arbitrarily variable; (iii) the same code can be used for quite di¤erent problems, since the iterations (di¤erence equations) are principally, ie, up the values of the coe¢ cients, identical. Such algorithms belong to the class of cellular automata (Wolfram 1986), where there is no limitation for the state set of the nodes. Thus, if "the purpose of computing is insight, not numbers" (Hamming 1973), then an approach basing on a discrete HP is an ideal starting point for the development of codes, not at least due to its philosophy of modeling (Johns 1979, Toffioli 1984, Vichniac 31

1984, Hoefer 1991, Barr 1991). On the other side, the uni…ed treatment of propagation processes by means of clear division into elementary steps, which in turn display a large variety of behavior, may contribute to the unity of the treatment of propagation phenomena in di¤erent environmental conditions.

Acknowledgments

These investigations revise and extend the earlier reviews (Enders 1996) and (Enders 2009), the work on which had been started in 1991 at the University of East Anglia, Norwich, UK, with Dr. de Cogan and which had been inspired from various discussions with Prof. W. Brunner, Prof. W. Ebeling (Ebeling 1992), Prof. T. Elsaesser, Dr. H. Engel, Dr. R. Guether, D.P. Johns (Johns & Enders 1992), MSc, Dr. R. Mueller, Prof. H. Paul, Dr. A.J. Wlodarczyk (Enders & Wlodarczyk 1993) and Dr. M. Woerner. Later, I have learned from Dr. C. Vanneste how to account correctly for the incoming waves, so that HP for TLM could be established in general form, both locally (Enders & Vanneste 2003) and globally (Enders 2001). The new Section 2 emerged from my collaboration with Dr. Dr. D. Suisky on Euler’s representation of classical mechanics and its axiomatic generalization to quantum mechanics (Enders & Suisky 2005, Enders 2006). The new Section 7 has been added after I have learned about the Helmholtz decomposition, most of all from papers by Prof. O. Keller (for a review, see Keller 2005) and from various explanations by him personally. The section about HP and linguistics emerged from my lectures ’Science – Language – Society’ held in 2011 at the Faculty of Linguistics at the Al-Farabi Kazakh National University. The hospitality of the chair of Prof. G. Madiyeva is cordially acknowledged. Last but not least I feel indebted to Dr. C. Francis and other posters in the moderated Usenet group sci.physics.foundations for their clarifying remarks. The early work was partially supported by the Deutsche Akademie der Naturforscher Leopoldina (Enders 1996).

XII.

REFERENCES

Agarwal, G.S. & Simon, R. 1994 A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function. Opt. Commun. 110, 23-26

32

Almeida, L.B. 1993 Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, Minneapolis, Minnesota, April 1993 (New York: IEEE) Almeida, L.B. 1994 An introduction to the angular Fourier transform. IEEE Trans. Signal Process. 33, 6182-6187 Alieva, T., Lopez, V., Agullo-Lopez, F. & Almeida, L.B. 1994 The fractional Fourier transform in optical propagation problems. J. Mod. Optics 41, 1037-1044 Barr, A.H. 1991 Physically-Based Modeling in Computer Graphics and Animation. Addison-Wesley, Reading (Mass.) Chapman, S. & Cowling, T. G. 1939 The Mathematical Theory of Non-Uniform Gases. An account of the kinetic theory of viscosity, thermal conduction, and di¤usion in gases, Cambridge: Cambridge Univ. Press Chopard, B. & Droz, M. 1998 Cellular Automata Modeling of Physical Systems. Cambridge, Cambridge Univ. Press Christopoulos, C. 1995 The Transmission-Line Matrix Method – TLM. Oxford, Oxford Univ. Press Courant, R., Friedrichs, K. & Lewy, H. 1928 Über die partiellen Di¤erenzengleichungen der Mathematischen Physik. Math. Annalen 100, 32-74; Engl. transl.: IBM J. Res. Develop. 11 (1967) 215-234 Courant, R. & Hilbert, D. 1968 Methoden der Mathematischen Physik II. Springer, Berlin/Heidelberg/New York (2nd ed.) de Cogan, D. 1998 Transmission Line Matrix (TLM) Techniques for Di¤usion Applications. Gordon & Breach de Cogan, D., O’Connor, W.J. & Pulko, S. 2005 Transmission Line Matrix in Computational Mechanics. Routledge, New York de Cogan, D. & Enders, P. 1991 Microscopic e¤ects in TLM heat ‡ow modelling. IEE Digest No. 1991/157, pp. 8/1-8/11 DeSanto, J.A. 1992 Scalar Wave Theory. Green’s Functions and Applications. Springer, New York etc. Delsanto, P.P., Chaskelis, H.H., Mignognia, R.B., Whitcombe, T.V. & Schechter, R.S. 1992 Connection Machine Simulation of Ultrasonic Wave Propagation: Two Dimensional Case. Rev. Progr. Quant. Nondestr. Eval. 11 (D.O. Thompson & D.E. Chimenti, Eds.), Plenum Press, New York, pp.113-120 33

Dirac, P.A.M. 1949 Forms of Relativistic Dynamics, Rev. Mod. Phys. 21, 392-399 Du Fort, E.C. & Frankel, S.P. 1953 Stability Conditions in the Numerical Treatment of Parabolic Di¤erential Equations. Math. Tables and other Aids to Computing 7, 135-152 Dynkin, E.B. 1965 Markov Processes. Springer, Berlin/Göttingen/Heidelberg (2 Vols.) Dyson, F. J. 1993 George Green and physics. Physics World, Aug., 33-38 Ebeling, W. 1992, Talk at the Landsberg Symposium, Berlin, Aug. 8 (unpubl.), and private communication Einstein, A. 1917 Eine Ableitung des Theorems von Jacobi. Sitz.ber. Kgl. Preuss. Akad. d. Wiss., phys.-math. Kl. XLVI, 606-608 Enders, P. 1992 The Chapman-Kolmogorov Equation as Representation of Huygens’ Principle and the Monolithic Self-Consistent Numerical Modelling of Lasers. Proc. 4th Int. Conf. Physics Computing ’92, Prague, Aug. 24-28 (R.A. de Groot & J. Nadrchal, Eds.), World Scienti…c, Singapore etc., pp.328-329 Enders, P. 1993 Comment on “A wave automaton for time-dependent wave propagation in random media“ by C. Vanneste et al. europhys. lett. 21, 791-793 Enders, P. 1996a A new view on Huygens principle, OSA Annual Meeting, Rochester 1996, Paper WGG50 Enders, P. 1996b Huygens’ Principle and the Modelling of Propagation. Eur. J. Phys. 17 (1996) 226-235 Enders, P. 2001 Huygens’ principle in the transmission line matrix method (TLM). Global theory. Int. J. Num. Modell.: Electronic Networks, Dev. and Fields 14, 451-456 Enders, P. 2006 Von der klassischen Physik zur Quantenphysik. Eine historisch-kritische Ableitung mit Anwendungsbeispielen aus der Festkörperphysik. Berlin Heidelberg: Springer Enders, P. 2009a Huygens principle as universal model of propagation, Latin Am. J. Phys. Educ. 3, 19-32; http://dialnet.unirioja.es/servlet/articulo?codigo=3688899 Enders, tions.

P.

2009b

Underdeterminacy

and

Redundance

in

Maxwell’s

Equa-

Origin of Gauge Freedom – Transversality of Free Electromagnetic Waves

– Gaugefree Canonical Treatment without Constraints, EJTP 6 (2009) 135-166; http://www.ejtp.com/articles/ejtpv6i22p135.pdf Enders, P. 2011 Huygens’Principle for Linguistics, Abai’Inst. Khab. 6 (12), 70-74 Enders, P. 2012 Electromagnetic Momentum Balance in Maxwell’s and Hertz’s Works, Galilean Electrodynamics 23 (2012) 5, 83-94 34

Enders, P. & de Cogan, D. 1992 Correlated random walk and discrete modelling of propagation through inhomogeneous media. Proc. 4th Int. Conf. Physics Computing ’92, Prague, Aug. 24-28 (R.A. de Groot & J. Nadrchal, Eds.), World Scienti…c, Singapore etc., pp.307-308 Enders, P. & de Cogan, D. 1993 The e¢ ciency of transmission-line matrix modelling – a rigorous viewpoint. Int. J. Num. Modell.: Electron. Networks, Dev. and Fields 6, 109-126 Enders, P. & Suisky, D. 2005 Quantization as selection problem, Int. J. Theor. Phys. 44 (2005) 161-194 Enders, P. & Vanneste, C. 2003 Huygens’ principle in the transmission line matrix method (TLM). Local theory. Int. J. Num. Modell.: Electronic Networks, Dev. and Fields 16, 175-178 Enders, P. & Wlodarczyk, A.J. 1993 Towards Exactly Modelling Open/Absorbing Boundaries. 2nd Int. Workshop Discrete Time Domain Modelling of Electromagnetic Fields and Networks, Berlin, Oct. 28-29 (unpubl.) Feynman, R.P. 1948 Space-Time Approach to Non-Relativistic Quantum Mechanics. Rev. Mod. Phys. 20, 367-387; reprint in: J. Schwinger (Ed.), Selected Papers on Quantum Electrodynamics, New York: Dover 1958, No.27 Feynman, R.P. & Hibbs, A.R. 2005 Quantum Mechanics and Path Integrals (emended by D. F. Styer), New York: McGraw-Hill 2005, Dover 2010 Fresnel, A. 1826 Mémoire sur la di¤raction de la lumière, Mém. de l’Acad. 5, 339¤. Goldstein, S. 1951 On di¤usion by discontinuous movements, and on the telegraph equation, Quart. Journ. Mech. and Appl. Math. IV, 129-156 Green, G. 1828 An Essay on the Applicability of Mathematical Analysis on the Theories of Electricity and Magnetism, Nottingham Guenther, P. 1988 Huygens’Principle and Hyperbolic Equations. Academic Press, New York Hadamard, J. 1953 The Cauchy Problem and the Linear Hyperbolic Partial Di¤erential Equations. Dover, New York 1953 Hamming, R.W. 1973 Numerical methods for scientists and engineers. McGraw-Hill, New York, 2nd ed. Helmholtz, H. Über Integrale der hydrodynamischen Gleichungen, welche den Wirbel35

bewegungen entsprechen, J. Reine Angew. Math.55 (1858) 25-55 Hoefer, W.J.R. 1991 Huygens and the Computer – A Powerful Allianz in Numerical Electromagnetics. Proc. IEEE 79, 1459-1471 Hoefer, W.J.R. & So, P.P.M. 1991 The Electromagnetic Wave Simulator. A Dynamical Visula Electromagnetics Laboratory based on the Two-Dimanesional TLM Method. Wiley: Chichester etc. Huygens, Chr. 1669 De motu corporum ex percussione, London: Royal society, Talk Jan. 4, 1669, publ. 1703 Huygens, Chr. 1673 Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae, Paris 1673; in: The complete Works of Christiaan Huygens, Vol. XVII, Den Haag 1934 Huygens, Chr. 1690 Traité de la lumière. Pierre van der Aa, Leiden Illenyi, A. & Jessel, M. 1991 Holographics, a spread-out of the basic ideas on holography into audio-acoustics. In: Holography, Commemorating the 90th Anniversary of the Birth of Dennis Gabor, Tatabanya, H, 2-5 June 1990; SPIE Institutes for Advanced Optical Technologies, Vol. IS 8 (1991), Bellingham: SPIE Optical Engineering Press, pp. 39-52 Iwanenko, D. & Sokolow, A. 1953 Klassische Feldtheorie. Akademie-Verlag, Berlin, p.78 Johns, D.P. & Enders, P. 1992 Wave Optics Computing by a Network-Based Vector Wave Automaton. Proc. 4th Int. Conf. Physics Computing ’92, Prague, Aug. 24-28, R.A. de Groot & J. Nadrchal (Eds.), World Scienti…c, Singapore etc., pp.359-361 Johns, P.B. 1974 A new mathematical model to describe the physics of propagation. The Radio and Electronic Engin. 44, 657-666 Johns, P.B. 1977 A simple explicit and unconditionally stable numerical routine for the solution of the di¤usion equation. Int. J. Numer. Meth. Engin. 11, 1307-1328 Johns, P.B. 1979 The art of modelling. Electronics and Power 25, 565-569 Johns, P.B. 1987 A Symmetrical Condensed Node for the TLM Method. IEEE Trans. Microwave Theory Tech. MTT-35, 370-377 Keilson, J. 1965 Green’s Function Methods in Probability Theory. Gri¢ n & Co., London Keller, O. 2005 On the theory of localization of photons, Phys. Rep. 411, 1-232 Kirchhoff, G. 1882/1883 Zur Theorie der Lichtstrahlen, Berl. Sitzungsber. 1882, p. 641; Wied. Ann. 18 (1883) 663¤. 36

Kolmogorov, A.N. 1931 Über die analytischen Methoden der Wahrscheinlichkeitsrechnung, Math. Ann. 104, 415¤. Kolmogorov, A.N. 1933 Grundbegri¤e der Wahrscheinlichkeitsrechnung, Springer, Berlin Lax, P.D. 1954 Week solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation. Comm. Pure Appl. Math. 7, 159-193 Lohmann, A. 1993 Image rotation, Wigner rotation, and the fractional Fourier transform. JOSA A 10, 2181-2186 Longhurst, R.S. 1973 Geometrical and Physical Optics. Longman, London and New York (3rd ed.) Lukasiewicz, J. 1957 Aristotle’s Syllogistic From the Standpoint of Modern Formal Logic. Clarendon, Oxford Macke, W. 1958 Wellen. Geest & Portig, Leipzig, Section 443 Markov, A. A. 1906 Rasprostranenie zakona bol’shih chisel na velichiny, zavisyaschie drug ot druga, Izv. Fiz.-mat. obsch. Kazan. univ. [2] 15, 135–156; Engl.: Extension of the limit theorems of probability theory to a sum of variables connected in a chain, in: R. Howard, Dynamic Probabilistic Systems, vol. 1: Markov Models, New York: Wiley 1971 (Series in Decision and Control 1), Appendix B; reprint: Dover 2007 Markov, A. A. 1913 An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains, Science in Context 19 (2006) 591600;

http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=637500

(lecture at the Royal Academy of Sciences, St. Petersburg, 23 January 1913) Maxwell, J. C. 1861 On physical lines of force, Phil. Mag. [4] 21 (1861) 161¤., 281¤., 338¤.; 23 (1862) 12¤., 85¤.; Scient. Papers, I, 451¤. Maxwell, J. C. 1864 Dynamical Theory of the electromagnetic …eld, Trans. Roy. Soc. CLV (1964), Pt. III Maxwell, ford:

J.

C.

Clarendon 1873;

1873

A

Treatise

on

Electricity

reprint of the 1891 ed.:

&

Magnetism,

New York:

Ox-

Dover 1954;

https://archive.org/details/electricandmagne01maxwrich Mie, G. 1941 Lehrbuch der Elektrizität und des Magnetismus. Eine Experimentalphysik des Weltäthers für Physiker, Chemiker, Elektrotechniker. Enke, Stuttgart (2nd ed.) Miller, D.A.B. 1991 Huygens’s wave propagation principle corrected. Optics Lett. 16, 37

1370-1372 Mueller, I. 1967 Zum Paradoxen der Wärmeleitungstheorie. Z. Physik 198, 329-344 Naas, J. & Schmid, H.L. 1974 Mathematisches Wörterbuch. Akademie-Verlag, Berlin, and Teubner, Leipzig (3rd ed.), Vol. I Namias, V. 1980 The fractional Fourier transform and its application in quantum mechanics. J. Inst. Math. Appl. 39, 241-265 Pockels, F. 1906 Beugung des Lichtes, in: Winkelmann 1906, Art. XXXI, pp.10321119 Scharf, G. 1994 From Electrostatics to Optics. Springer, Berlin etc. Schroedinger, E. 1926a Quantisierung als Eigenwertproblem. Erste Mitteilung. Ann. Physik [4] 79, 361-376; 1926b Zweite Mitteilung. Ibid. 489-527; 1926c Dritte Mitteilung: Störungstheorie, mit Anwendung auf den Starke¤ekt der Balmerlinien. Ibid. 80, 437-490; 1926d Vierte Mitteilung. Ibid. 81, 109-139 Schwartz, M. 1987 Principles of Electrodynamics. Dover, New York (quoted after http://www.mathpages.com/home/kmath242/kmath242.htm) Simonyi, K. 1990 Kulturgeschichte der Physik. Urania, Leipzig etc., pp.241f. Sommerfeld, A. 2001 Vorlesungen über theoretische Physik, Bd. III Elektrodynamik, Frankfurt a. Main: Deutsch (4th ed.) Spitzer, F. 1964 Principles of Random Walk. Van Nostrand, Princeton (N.J.) Toffioli, T. 1984 Cellular Automata as an Alternative to (Rather Than an Approximation of) Di¤erential Equations in Modeling Physics. Physica 10D, 117-127 Vanneste, C., Sebbah, P. & Sornette, D. 1992 A Wave Automaton for Time- Dependent Wave Propagation in Random Media. Europhys. Lett. 17, 715-720 Vasin, V. P. 1990 Methods of global analysis of electric power systems’states of operation. Power Engineering (New York) 28, 22-34 Vichniac, G.Y. 1984 Simulating Physics with Cellular Automata. Physica 10D, 96-116 West, B.C. 1992 The use of fractals in optical wave propagation through random media. Proc. Conf. Lasers and Electro-Optic CLEO’92, 10.-15.05.1992, Anaheim, CA, p.506 Weyl, H. Raum – Zeit – Materie. Vorlesungen über allgemeine Relativitätstheorie, Berlin: Springer 4 1922; Engl.: Space – Time – Matter, New York: Dover 1952 Wiener,

J.

1995

Aristotle’s

Syllogism:

Logic

Takes

http://www.perseus.tufts.edu/GreekScience/Students/Jordana/LOGIC.html 38

Form,

Wigner, E.P. 1963 Events, laws of nature, and invariance principles. In: G. Ekspong (Ed.) 1998 Nobel Lectures in Physics Vol 4 1963-1980. World Scienti…c, Singapore, pp.6-17 Winkelmann, A. (Ed.) 1906 Handbuch der Physik, 2nd ed., Vol. 6. Optik, Leipzig: Barth Wolfram, S. 1986 Theory and Applications of Cellular Automata, World Scienti…c, Singapur Zauderer, E. 1989 Partial Di¤erential Equations of Applied Mathematics, Wiley, New York

[1] I replace d=dt with @=@t and use the nowadays symbols: s ! ~j; f !

39

~ A;

!

.