A previous model for treating electromagnetic nonlinear wave systems is examined in the context o f wave mechanics. It is shown that nonlinear wave mechanics implies harmonic generation of new quasipartiele wave functions, which are absent in linear systems. The phenomenon is interpreted in terms of pail" (and higher order ensembles) coherence of the interacting particles. The implications are jar-reaching, and the present approach might contribute toward a common basis for diverse physical phenomena involving nonlinearity. An intimate relationship connecting coherence, nonlocal interaction, and nonlinearity has been previously noticed in the physics of superconductivity. It is shown here that all these ingredients are consistently contained in the present formalism. The present theory may contribute to elucidate a controversial theory proposed by Panarella, who claims to have measured high-energy photons due to high-intensity laser radiation, which cannot be predicted on the basis o f linear quantum theory. Panarella explains the new phenomena by stipulating a nonlinear intensity-dependent photon energy. It is argued here that nonlinearity, manifested in the presence of high intensity, may give rise to high- and low-energy photons, the so-called "effective" and "'t&ed" photons, respectively. However, the present explanation does not involve ad hoc assump- tions regarding the foundations o f quantum theory. In analogy with the electro- dynamic model, the present theory leads to particulate self-focusing in high- density streams of particles. Since such particulate beams are currently under consideration in connection with fusion reactions, this might be of future interest.
1. INTRODUCTORY REMARKS
Mutua l influence between field theories in various branches of physics is very common. This is due to the fact that investigators are using similar
1 Department of Electrical Engineering and Computer Science, University of California-- San Diego, La Jolla, California. On leave of absence from the Department of Electrical Engineering, Ben-Gurion Univers- ity of the Negev, Beer Sheva, Israel.
mathematical tools. Often this leads to new insight into the physics of the pertinent fields of interest. Thus, presently a model for nonlinear electro- dynamic wave systems is being investigated in connection with quantum mechanical systems. Recently a ray formalism was proposed for analyzing self-focusing of electromagnetic waves in lossless (1,2) and absorbing (3-~ media. The theory is based on an extended Fermat principle, (6) and the transition to wave mechanics is straightforward. For linear absorbing media, the wave mechanical analog has been recently examined, (7) showing the significance of complex potentials and giving a meaning to space- and time-independent probability density in dissipative systems. The similar procedure of investigating nonlinear wave mechanical systems, with reference to the electrodynamic model, pays even higher dividends, as shown below.
We start the next section with a formal representation of the weakly nonlinear wave mechanical system. A periodic solution is then assumed, facilitating the algebraization of the transformed equations. This yields a dispersion equation, similar to linear systems; however, here the amplitudes of the wave function are also involved in the dispersion equation.
The next section is concerned with the implications of the theory for various branches of physics. The relevance to the theory of superconductivity is pointed out. A somewhat tentative argument regarding "effective" and "tired" photons in high-intensity laser beams is given. This may contribute to the understanding of experiments cited by Panarella, (8,9) without invoking his ad hoc modification of quantum theory.
Finally the problem of particulate self-focusing is considered, in analogy with the electrodynamic case.
2. G E N E R A L F O R M A L I S M
A formalism is presented here for dealing with weakly nonlinear systems as defined below. The formalism is quite general, and no attempt is made to discuss special cases here. The method is the analog of the electromagnetic case discussed previously. (1,2,5)
Consider a general wave mechanical system represented by
LiAbj + aiAbj = 0 (1)
where Lij is a square matrix of operators involving space and time derivatives, alj is a square matrix whose entries may be space and time dependent, thus describing potentials, and ~ = (¢1 ,.-., CJ ,---) is a vector of probability wave functions. The dimensionality of q~ is determined by the quantum mechanical model at hand, e.g., for the scalar SchrSdinger equation q~ = (¢1), and for the Dirac wave equation (1°) t~ = (¢1 ,.:., ¢4).
Nonlinear Wave Mechanics and Particulate Self-Focusing 557
In the present discussion au are allowed to depend on de, i.e., nonlinear wave equations are introduced by allowing self-interaction. The physical implications of such a construct will be discussed below. It is therefore appropriate to rewrite (t) in the general form
Li j~ j + V~(q~) = 0 (2)
where Vi(q~) indicates that the potentials depend on the array q~ of wave functions. Part of the motivation for the present study is the fact that wave equations of the type (2) have found their use in physics, e.g., in the Ginzburg- Landau theory of superconductivity. (11)
Nonlinear field problems of physics are notoriously complicated, and there exists no general mathematical tool for dealing with them. The present approach is no exception, and cannot be expected to do better than to illuminate a special aspect of the general problem. We are dealing here with weakly nonlinear systems, for which V~ can be represented as a hierarchy in powers of q~,
where ~ ijk am, a(~), etc., are tensors of increasing rank and the braces denote a functional structure relating the field q~ to the tensors ui j ... v. In the scalar Schr6dinger equation V a) = a(X)~b and V (~) = 0 for n =~ 1, where a(X) is the potential, depending on spacetime. The latter are compactly denoted by the relativistic four-vector X = (x, ict). This simple form, on which a major part of our knowledge of quantum physics is based, seems to be too specialized for our present purposes. For reasons clarified below, we prefer to define the first-order potential by means of a four-dimensional convolution integral
p~) (~) . (o~ (i) = {~i~ (x ) , ~j} = d~xl a~ ( x , ) ~ ( x - x 0
d --co (4)
f ~ (1) = d~Xl aij (X -- X 0 4,5(Xl)
The integral (4) is subject to causality, and no contribution is assumed from (1) outside the light cone. If in (4) we assume ai~ (X X 1 ) (1) - = b~j ( X 0 a ( X - - X 0 ,
then V~a)= al~)(X)~bj(X) will be obtained, signifying a local interaction. However, many physical applications prescribe nonlocal interaction, and it seems advantageous to adopt the form (4), which is mandatory for spatially and temporally dispersive systems. (~2) A fourfold Fourier transform applied to (4) yields
V?)(K) a")(U) = ,~-, ~@(K) (5)
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where for simplicity of notation the functions and their transforms are denoted by the same symbols. The different functions are distinguished by means of their argument, which is explicitly displayed where necessary. The wavenumber vector and frequency are compactly denoted by the four-vector K (k, i~o/c). On the other hand, V/m (1) = = aij (X)@. in general implies a fourfold convolution integral in K space. For a general discussion (5) is preferable, leading to an algebraic dispersion equation.
The definition of the higher order nonlinear terms VI ") constitutes a crucial step in our discussion. We introduce the forms used in nonlinear optics by Akhamanov and Khokhlov, <1~) which are equivalent to those used by Schubert and Wilhelmi. <14) As done before, m a four-dimensional space- time representation is used. Thus we define
This definition, although it seems to be quite arbitrary, will be shown to provide a satisfacotry f ramework for our physical discussion. The reason for that will be the conservation of momentum and energy of quasiparticles produced by nonlinear interaction. The Fourier transform of (6), with a factor (27r) -4 for each integration, which is henceforth suppressed, is given by
Note that in (7) there are n - 1 fourfold integrations and an additional constraint on K. The latter, standing for k = kl + k2 + "'" + k , and oJ = o~ 1 + to z + ... + ~o,, provides for the momentum and energy con- servation of the quasiparticles created by the nonlinear interaction. To choose a somewhat oversimplified example, (7) states that a (multiplication) nonlinear interaction between two wave functions A1 exp(ikl • x - - i~olt ) and A2 exp(ik2 - - i¢o2t) yields A exp(ik • x i¢ot), such that k = kl + k2, and ¢o = to1 -t- o)2 • The intensity A 2 of the new quasiparticle depends on how many particles of species 1 and 2 interact. I f the plausibility of (7) is accepted, then the nonlocal nature of (6) must also be accepted, since they form a transform pair.
In order to consider applications of the model given above, we look for a simple solution of the wave equation. It is clear that harmonic waves cannot be considered as a solution for a nonlinear wave equation such as (2). On the other hand, a periodic wave function can be used if it contains all the harmonics which will be created due to the nonlinearity of the system. This is
Nonlinear Wave Mechanics and Particulate Self-Focusing 559
characteristic of a system of identical particles which are noninteracting in the absence of nonlinearity, and this type of interaction due to nonlinear effects is termed self-interaction. Therefore we choose a solution
~ = ~ ~bj,~ exp(imK" X) (8)
where ~b~ is the amplitude of the nthe harmonic and K • X = k ' x -- oJt. It is presently assumed that ~b;,, are constants, due to the homogeneity of the system in space and time. Later, in order to discuss ray tracing and self- focusing, it will be necessary to replace (8) by an appropriate eikonal approxi- mation. Substitution of (8) into (6) and exchange of order of integration and summation yields
The integral in (9) defines a (4n)-fold Fourier transform, yielding a l j . . . ~ ( a K ..... ilK), depending on K, the summation indices ~ ..... fl, and the tensor indices i , j , . . . , v . Since no new harmonics are produced in this process of self- interaction, the series with a summation index 9,, such that y = a + ... + /3 , can be defined
V~ ~) = ~ V~ ") exp(iyK" X) (10) S t = - - ~
and compared to (9). By rearranging the series and noting that each term picked out from (9) involves n wave amplitudes with indices going f r o m j to v, it is possible to define new terms according to
V~,~) -(~) ,,. ". : ai¢. . .vY'sv "'" ~vv (11)
where all the symbols in (11) depend on yK. Note that the new 8 are different from a. At a first glance it might seem paradoxical that we succeeded in transforming (7) into an algebraic form involving only one value yK. However, it must be remembered, comparing (9) and (10), that this has been achieved in the presence of all spacetime harmonics, hence if one term in (10) is somehow filtered out, this will affect all other terms. This is the essential distinction between linear and nonlinear systems; in the former all harmonics, if present, are unrelated. Finally it is noted that the definition (11) is tanta-
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mount to a Taylor expansion of (") V~, (q~,) near ~ , 0, with a vanishing term
and by comparison of (11) and (12) the ~ tensors are identified. In view of the above discussion it is now possible to derive an algebraic
dispersion equation for self-interacting, weakly nonlinear systems. One must simply replace 0/0X operators in Li t , (2), by imK (i.e., O/Ox by imk, ~/Ot by -- imm), and thus corresponding to (2) we now have
Let us consider now only m = 1, i.e., the fundamental harmonic, which is the limiting case of the nonlinear case as it reduces to the linear case. All other m 4 = 1 will be considered as equations on the amplitudes ~ j ..... 4J~. Following the same argument as for linear systems of homogeneous algebraic equations, the determinant of (13) must vanish in order to have nontrivial solutions. Hence, if we rewrite (13) (suppressing m) in the form
= = 0 (14)
noting that ~b are now the amplitudes ~le, then
F = det(F~e) : 0 (15)
We have therefore derived a dispersion relation F(K, q~) = 0, i.e., an algebraic equation relating k and ~o, and involving the amplitudes of the fundamental harmonic wave functions. It is a characteristic of nonlinear systems that their dispersion equation involves the amplitudes of the fields. This appears also in other models for discussing nonlinear systems. (la)
3. DISCUSSIONS AND APPLICATIONS
The distortion of harmonic signals in nonlinear systems, conducive to harmonic generation, is an elementary phenomenon, whether we are dealing with time signals in lumped electric circuits or with waves in a distributed system. Still, it captures the imagination whenever it appears in a new and somewhat unexpected context. This is the case in nonlinear optical systems,
Nonlinear Wave Mechanics and Particulate Self-Focusing 561
where harmonic generation gives rise to new colors in the light beams. The emergence of probability wave harmonics in the presence of nonlinear self- interaction of streams of otherwise identical particles is therefore not surprising, but it requires physical interpretation. Thinking along classical mechanical lines, it seems incomprehensible that new particles will be created in a system of this kind. However, in wave mechanical terms it simply means that pairs and higher order ensembles become correlated, or coherent, acting as a new quasiparticle whose momentum and energy are n times those of a single particle. Obviously the overall momentum and energy of the system are conserved. According to this picture, harmonics in (8) describe the motion of the "center of gravity" of the new quasiparticles, comprised of several "real" particles. This bonding effect may involve particles situated at different locations. At a first glance this is somewhat puzzling, but actually this is a model accepted in the theory of superconductivity long ago. In fact, all the ingredients appearing in the present theory have already been included in the theory of superconductivity, albeit in a fragmentary form. The necessity for a nonlocal field theory, as displayed by (6), is intimately connected with the need for a nonlocal electrodynamic model as given by Pippard. (11) The nonlinear wave mechanical equations of the Ginzburg-Landau theory m) can be considered as a special case of (2). The celebrated BCS theory Im of superconductivity is founded on the idea that pairs of electrons in a bound state are producted due to mutual interaction mediated by phonons. This leads to explanations of various effects such as Josephson tunneling. The general ideas conform to the present model in which quasiparticles are created due to nonlinear self-interaction, although the present model, being very general, does not specify the nature and origin of this interaction. It is of course unrealistic to assume that the present formalism, without further specialization, could provide a general framework for the physics of super- conductors. However, it is still interesting to observe how all the seemingly disconnected aspects are consistently involved in the present theory. It is hoped that this will contribute to a better formulation and deeper under- standing of the subject in the future.
Due to the dual nature of light, the present model and the electrodynamic analog m become overlapping when photons are considered. The present discussion, based on a particulate wave mechanical model, may contribute to our understanding of the redistribution of photon energy and momentum in high-intensity laser pulses. Panarella ~s,9) discusses this effect and cites earlier studies. He claims to verify experimentally that high-energy photon are found in the presence of high-intensity light beams. His theory for explaining this effect, admittedly an ad hoc one, attributes to photons an intensity-dependent energy. Thus the fundamental quantum theoretical relation e = hv (where e is the energy and v is the frequency) is questioned.
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His discussion is very comprehensive, frankly pointing out the weak points of this theory. Although his ad hoc theory can account for his data, the modification of the fundamentals of quantum theory creates a bigger difficulty. It is always with great reluctance, and understandably so, that scientists are willing to modify fundamental physical theories. Witness the special theory of relativity and the related Michelson-Morely experiment, initiated by Michelson in 1881 and repeated in many ways by numerous physicists over a period of about fifty years. The present model of nonlinear quantum mechanics may contribute to the understanding of effects induced by high-intensity optical pulses without affecting the foundation of quantum mechanics. It is suggested that the appearance of "effective" and "tired" photons, as termed by Panarella, is the result of association and dissociation of quasiparticles produced in the presence of nonlinear self-interaction. A detailed mathematical picture for these processes is not available presently, and we have to take resort to a verbal argument which must be regarded as a conjecture, at best.
Due to the nonlinearity inherent in the environment in which the light travels, be it a crystal or a gas, particulate harmonic generation is induced, manifested by the creation of quasiparticles, as described by the harmonics in (8). These are created and annihilated in a random fashion and (8) describes only the average density of the process. It is therefore conceivable that because of the inhomogeneity of the system, e.g., due to the varying intensity within the laser light pulse, pairs and higher order ensembles, upon dissocia- tion, do not obtain the original energies of the single photons, although the total energy is conserved. Although e = hv is preserved for individual photons and bound aggregates, "effective" and "tired" photons of higher and lower energy can be created as a result of unequal redistribution of energy when dissociation occurs. The conjecture presented here can be tested experimentally. According to Panarella, the effective photons are created because of photon-photon interactions, while the present theory presupposes a mediating medium which brings about the nonlinearity. If an experiment is constructed showing the "effective" photons are produced in vacuum, this will disqualify the present theory and add credibility to Panarella's theory of intensity-dependent photon energy. Of course, negative results will unequivocally invalidate his theory.
4. PARTICULATE SELF-FOCUSING
Since it was first discussed by Askaryan] TM the self-focusing effect has been the subject of intensive investigation. Recently a ray-theoretic model for nonlinear media m has been discussed in connection with self-focusing
Nonlinear Wave Mechanics and Particulate Self-Focusing 563
lossless c2~ and dissipative (5) media. The present formalism for solving a system of equations of the type (2) is analogous, hence a similar effect of wave mechanical, or particulate, self-focusing effect is expected.
In order to take into account spatially and temporally varying systems, the phase K • X in (8) is replaced by fK • dX, a line integral in four-space, between the limits X1 and X. The slowly varying amplitudes Cj,,(X) now replace the constants @,~ in (8), taking into account slow variations in space and time. Accordingly we have to modify (14) and (15),
Gi(K, ,-k, X) = 0 (16)
F(K, g,, X) = 0 (17)
where X denotes the dependence of fi (13) (the analogs of the constitutive parameters in electrodynamics) on space and time, and da = ~bj~ are the amplitudes of the fundamental harmonic. The problem of ray tracing in nonlinear media has been discussed previously, (1,2,5J including real spacetime ray tracing in absorbing media. (3,4,7) It has been realized recently (~7) that F, (17), for the electromagnetic case must be augmented by the self-consistent requirement that the group velocity v be in the direction of the Poynting vector. Incorporating into F with a suitable Lagrange multiplier yields a new F = 0. Similarly, in the wave mechanical analog we require that the group velocity v be in the direction of the particle probability current (in the generalized sense, see Messiah, (~.s) p. 888), i.e., g = 9, where ~ and 9 are unit vectors in the direction of current density and group velocity, respectively. This again modifies F into F.
By retracing the electromagnetic argument we derive the Hamilton equations pertinent to the present case of lossless wave mechanical systems:
dx F k dk F , do) Ft (18) v - - d t - - F~ ' d t - - f ~ ' d t - - F,~
where
F , - 0F OF [0Gj ] -1 0Gj 09)
for any varible I. For the linear case we replace F by F and ~ F / ~ ¢ i = 0
identically, since the dispersion equation is not dependent on field amplitudes. These are the familiar Hamilton equations of geometrical optics or geometrical mechanics. (6) For dissipative systems cv) we add the constraint Im v = 0 at the departure point of the ray and ensure that the ray stays in real spacetime by adding i[3 and iv . [~ to the second and third eqs. of (18), respectively, where
[Re(v~ + v~v)]-I Jm (v~ L F, v) (20) . . . . v=~ T vt + v~ • U~ F~
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Equations (18)-(20) are supplemented by
r°6 1-! r°a dk dco oc , , dt - - L-~-~J I Ok " d--t- + Oco dt @ TX-x" v ~- (21)
describing the evolution of the fields + along the ray.
5. DISCUSSION AND A SIMPLE EXAMPLE
The author is not aware of self-focusing phenomena that have been discussed in the context ot particulate systems. However, as particle beams, considered to date for fusion reaction, will achieve larger densities, the relevance may arise. The simple example considered here shows, as is expected, aG> that self-focusing depends on the gradients of the intensity in a beam. Consequently, if for some stochastic reasons ("noise") the intensity profile becomes a little corrugated, instability wilt set in, leading to filamenta- tion. This might be of interest for high-energy dense particulate beams.
As a simple example, consider the scalar Schr6dinger equation in which a nonlinear term is included. By assuming an instantaneous local plane wave (essentially the WKB approximation), the present form of (17) is
G = F~/J~ = [k 2 - - co + a + b~b~2]~b~ = 0 (22)
where k and co are proportional to the momentum and energy, respectively, ~b 1 is the amplitude of the fundamental harmoni c, and a and b are real constants. Since there is no explicit sPace and time dependence in (22), (18) prescribes that dco/dt = 0 and d k / d t = 0. Consequently (21) vanishes, prescribing ~bl = const along a ray path. Let us assume a steady state situation such that in
dco Oco Oco dk Ok cqk dt - - 8t + ~ " v O, dt -- 0t + - ~ - x ' v 0 (23)
we have &o/Ot = 0 and Ok/Ot 0. The latter prescribes &o/0x = 0 because of the uniqueness condition <19> Ok/Ot + &o/0x = 0. This leaves in (23) (0k/0x) • v = 0, i.e., k can change in space such that the changes are every- where perpendicular to the group Velocity, i.e., to the ray path. Since ~bl = const, all that is left of (18) is
v = - Fk/F,o = 2 k = 2 ( , . o a - - b ~ , . 2 ) 1 / ~ (24)
and g = ¢z = [~ is satisfied automatically, From (24) it is clear that for those parts of the beam where the amplitude is larger, the group velocity is smaller.
Nonlinear Wave Mechanics and Particulate Serf-Focusing 565
I f we assume, according to (23), that the changes in k (or v) are perpendicular to the ray path, then a family of orthogonal trajections is defined, with the rays perpendicular to the wavefronts. Since v depends on the intensity, the rays will converge toward :regions of locally higher intensity, which is in essence the manifestation of self-focusing. This also explains the instability with respect to undulations on the intensity profile, finally leading to filamentation. The electromagnetic analog of the present situation has been considered in greater detail previously. <~7~
6. CONCLUSIONS
A formalism for analyzing weakly nonlinear wave mechanical systems is presented. The analog to nonlinear optics facilitates the discussion of similar phenomena.
The concept of wave mechanical harmonic generation is discusseed and its relevance to the theory of superconductive materials is pointed out.
The present theory is used to explain, in a somewhat speculative manner, the possible existence of "effective" and "t i red" photons in high-intensity light beams.
Finally, it is pointed out that the theory implies particulate self-focusing, and a simple example is discussed.
R E F E R E N C E S
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Nat. Committee, Fed. Rep. Germany, October 1975). 4. D. Censor, J. Phys. A 10, 1781 (1977). 5. D. Censor, Phys. Rev. A 18, 2614 (1978). 6. J. L. Synge, Geometrical Mechanic's and de Broglie Waves (Cambridge University
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I0. L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1955). 11. M. Tinkham, Introduction to Superconductivity (McGraw-Hill, 1975). 12. V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas, 2rid ed.
(Pergamon Press, 1970). 13. S. A. Akhamanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and
Breach, 1972). 14. M. Schubert and B. Wilhetmi, Einj'dhrung in die nichttineare Optik (Teubner, 1971),
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16. G. A. Askaryan, Zh. Eksp. Teor. Fiz. 42, 1567 (1962) [Soy. Phys.--JETP 15, 1088 (1962)].
17. D. Censor, Wave packets and solitary waves--a dual approach. II: nonlinear systems, in preparation.
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