Physical relation between quantum mechanics and solitons on a thin elastic rod

Abstract

The 2X2-matrix-valued first-order linear differential equation appears when we solve the modified Kortewegde Vries (MKdV) equation. This linear differential equation is regarded as a fictitious quanturn equation. On the other hand, it is known that the dynamics of an elastic rod is governed by the MKdV equation. In this paper, after we construct a Dirac equation on an elastic rod embedded into (2+1)-dimensional space-time, we show that this linear differential equation is naturally introduced through this Dirac equation. Then we can explain the, reason why the classical nonlinear differential equation is associated physically with quantum mechanics. In other words, we prove that fictitious quantum mechanics related to the soliton is real quantum mechanics on the soliton as a base space. We also argue that the Berry phase of the Dirac particle is related to the Lax pair L, =i [L,B ].

Exact Description of Rotational Waves in an Elastic Solid

“…Furthermore, the first-order Dirac equation may be regarded as a representation of a second-order wave equation with the bispinor wave function encoding the first derivatives [7,9,10,11,3]. Halfinteger spin is attributable to the fact that waves propagating in opposite directions are independent states separated by 180-degree rotation.…”

IMMERSION ANOMALY OF DIRAC OPERATOR ON SURFACE IN ℝ3

“…Since the Dirac operator should be regarded as a translator (a functor) between the analytical object and the geometrical object [8][9], in terms of the Dirac operator, I have been studying the physical and geometrical meanings of the abstract theorems in the soliton theory and quantum theory focusing on the elastica problem [28][29][30][31][32] and recently on the immersed surface [1].…”

“…In terms of the partition function of the Dirac field, I constructed the JimboMiwa theory of the MKdV hierarchy [32] and showed the physical meaning of the inverse scattering method and the monodromy preserving deformation [28,31]. Investigation on the Dirac operator of the KKWE equation might lead us to the Sato-type theory of the MNV hierarchy.…”

“…From the soliton theory, the MNV equation is a higher dimensional analogue of the MKdV equation [6]. The Dirac operator appearing in the auxiliary linear problems of the MKdV equation is realized as the operator for the Dirac field confined in the elastica [28][29][30][31][32] as the KKWE equation might be related to auxiliary linear problems of the MNV equation [2,6] and is realized as the equation of the Dirac field confined in the immersed surface [1].…”

“…In the series of works [28][29][30][31][32], I have been studying the elastica in terms of the quantized Dirac fields in the elastica. In terms of the partition function of the Dirac field, I constructed the JimboMiwa theory of the MKdV hierarchy [32] and showed the physical meaning of the inverse scattering method and the monodromy preserving deformation [28,31].…”