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Matter, Nonlinear Theory Of

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Encyclopedia of Nonlinear Science

MATTER, NONLINEAR THEORY OF

A nonlinear theory of matter was first proposed in 1912 by Gustav Mie in a prescient series of papers that aimed to derive the elementary particles of matter as localized lumps of energy in a nonlinear field (Mie, 1912). To this end, Mie suggested a nonlinear augmentation of James Maxwell’s electromagnetic equations out of which the electron would arise in a natural way.

Specifically, he defined a Lagrangian density depending upon electric field intensity (E) and magnetic flux density (B) and the four components of the electromagnetic potential Requiring dependence on the parameters η≡(B²−E²) and ensured relativistic invariance, and the specific choice led to a static, spherically symmetric electric potential of the form

(1)

Setting (the electronic charge) yielded a spherically symmetric model for the electron with a radius of r₀ and electric potential

as r→∞. The Lorentz invariance that is built into the theory permits this solution to travel with any speed up to the limiting velocity of light with an appropriate Lorentz contraction.

Mie’s approach to a nonlinear theory of matter was supported by Albert Einstein, who offered the following opinion in the mid-1930s (Einstein, 1954).

In the foundation of any consistent field theory, the particle concept must not appear in addition to the field concept. The whole theory must be based solely on partial differential equations and their singularity-free solutions.

Empirical support for this perspective was provided in the early 1930s by Carl Anderson’s discovery of positron-electron creation from cosmic radiation. In other words, massive particles were observed to emerge from and collapse back into an electromagnetic field, which is not a property of the linear Maxwell equations.

Motivated by Anderson’s observation, Max Born revisited Mie’s nonlinear electromagnetics. Together with Leopold Infeld, he eliminated the χ-dependence in Mie’s functional formulation and chose instead the Lagrangian density (Born & Infeld, 1934)

(2)

where E₀ sets the field intensities at which nonlinearities arise. At low-field amplitudes, Equation (2) reduces to the classical Lagrangian density for Maxwell’s equations: (Even with the currently available, high-intensity lasers, these vacuum nonlinearities would be difficult to observe, as E₀ is estimated to be about 10²⁰ V/m in laboratory units.)

Among the solutions of this system, Born and Infeld found a spherically symmetric model electron with E finite everywhere, although the electric displacement (D) exhibits a singularity at the origin. Erwin Schrödinger became interested in Bern’s theory as early as 1935 and continued working on it through the 1940s when—as founding director of the Dublin Institute for Advanced Studies—he attempted to move research in physics toward key areas of nonlinear science (Schrödinger, 1935).

Plane waves derived from Equation (2) obey the equation where u is a component of the vector potential. Called the Born-Infeld equation, this system has been studied as an interesting nonlinear wave equation (Barbashov & Chernikov, 1966).

Einstein’s conviction that a consistent theory for particle physics must be based on localized solutions of nonlinear partial differential equations was shared by several of his colleagues. In addition to Mie, Born, Infeld, and Schrödinger, Werner Heisenberg (1966), Louis de Broglie (1960, 1963), and David Bohm (1957) have suggested nonlinear field theories that in their simplest representations, can be viewed as the augmentation of linear field equations by a nonlinear term of the form |u|²u, as in the nonlinear Schrödinger equation. This nonlinearity conserves the integral

(3)

which can be interpreted as a mass.

The ideas of de Broglie and Bohm are related to those of the inverse scattering method (ISM). In their “theory of the double solution,” the real particle is a localized solution of a nonlinear equation with the form

Associated with this localized nonlinear solution is the solution of a corresponding linear equation

ψ=Ψe^iθ

with

θ=θ′

(4)

except in a small region surrounding the real particle. The function ψ is taken to be a solution of Schrödinger’s quantum mechanical wave equation, and the phase condition of Equation (4) allows the particle to be guided by ψ. Similarly, in the context of the ISM, the nonlinear solution of a soliton equation is guided through space-time by the linear asymptotic solution of the associated linear operator. Although proposed almost a half century ago, the de Broglie-Bohm theory continues to offer possibilities for further studies (Holland, 1993).

During the 1960s, several investigators proposed the sine-Gordon (SG) equation as a field theory for elementary particles in one space dimension and time (Scott, 2003). This work gained momentum in the 1970s when it became known that special properties of the SG equation allow the corresponding quantum problem to be solved, showing that certain qualitative properties of the classical solution survive quantization (Dashen et al., 1974; Faddeev, 1975; Goldstone & Jakiw, 1975). In particular, the classical field energy was found to be a useful first approximation for the soliton mass, with quantum effects coming in as second-order corrections. More recently, this approach has been developed into the concept of a skyrmion, which is a generalization of the SG kink, carrying its topological stability into three space dimensions.

These examples suggest that it may be possible to develop a nonlinear theory of matter by proceeding as follows. First, guess a classical version of the correct nonlinear field. Second, solve this classical system for salient aspects of localized behavior. Third, analyze the corresponding quantum theory to obtain exact values for the mass spectrum. Finally, compare calculated values of mass with measured mass spectra. As Einstein was aware, however, this is a daunting program because there are no theoretical bounds on the range of conceivable nonlinear theories; thus, it is not surprising to find a proliferation of partially evaluated theories. In addition to the Born-Infeld, de Broglie, and skyrmion formulations, present candidates include string theory and the Yang-Mills equation. What others are out there?

ALWYN SCOTT

Matter, Nonlinear Theory Of

MATTER, NONLINEAR THEORY OF

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