The inverse scattering problem is applied to 2-dimensional partial differential equations called ... more The inverse scattering problem is applied to 2-dimensional partial differential equations called soliton equations such as the KdV equation and so on. It is also used to integrate the Einstein equations with axial symmetry. These inverse scattering problems look different. We show that they can be understood in a unified way. As an application to the Einstein equation, we find solutions of the Einstein-Maxwell equations with a magnetic charge.
Abstract For a wide class of solition equations in 1 + 1 dimensions an infinite set of symmetry t... more Abstract For a wide class of solition equations in 1 + 1 dimensions an infinite set of symmetry transformations obeying the Kac-Moody Lie algebra is obtained. These symmetry transformations, however, can be applied only within a limited domain of the x (space) variable. This restriction applies, in particular, to the Heisenberg spin chain and the nonlinear Schrodinger equation for which a Kac-Moody structure was recently derived by Eichenherr.
We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. The vac... more We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. The vacuums with and without using the normal ordering look different. The vacuum without the normal ordering is shown to be expressed using the Weyl ordering. The Weyl ordered vacuum is then compared with the normal ordered vacuum, and the implication of the difference between them is discussed.
It is known that the continuous soliton equations such as the KdV equation have hierarchy structu... more It is known that the continuous soliton equations such as the KdV equation have hierarchy structures. A higher order KdV equation constituting the hierarchy includes higher order derivative terms in the equation. The Toda equation is characterized by discreteness in the space dimension. In the hierarchy for the discrete soliton equation such as the Toda equation, neither spatial derivatives nor higher order spatial derivatives appear. Despite of this difference, we show both continuous and discrete soliton equation hierarchies starting with the noncommutative zero curvature equation. Especially, we show the explicit form of the higher order Toda equation.
We investigate the equation where the commutation relation in 2-dimensional zero-curvature equati... more We investigate the equation where the commutation relation in 2-dimensional zero-curvature equation composed of the algebra-valued potentials is replaced by the Moyal bracket and the algebra-valued potentials are replaced by the non-algebra-valued ones with two more new variables. We call the 4-dimensional equation the noncommutative zero-curvature equation. We show that various soliton equations are derived by the dimensional reduction of the equation.
The soliton technique is applied to the 5D static Einstein-Maxwell equations, and an infinite num... more The soliton technique is applied to the 5D static Einstein-Maxwell equations, and an infinite number of solutions are explicitly obtained. We study the rod structure of 2-soliton solutions and we show that the 5D Reissner-Nordstrom solution and the 5D Majumdar-Papapetrou solution are included as the 2-soliton solutions.
We connect the discrete and continuous Bogomolny equations. There exists one-parameter algebra re... more We connect the discrete and continuous Bogomolny equations. There exists one-parameter algebra relating two equations which is the deformation of the extended conformal algebra. This shows that the deformed algebra plays the role of the link between the matrix valued model and the model with one more space dimension higher.
We study the 5D static Einstein-Maxwell equations with a dilaton field. We construct an infinte n... more We study the 5D static Einstein-Maxwell equations with a dilaton field. We construct an infinte number of solutions by using a soliton technique. We study the rod structure of 2-soliton solution and show the 5D dilatonic black ring and black hole solutions are included.
We discuss the exact electrically charged BTZ black hole solutions to the Einstein-Maxwell equati... more We discuss the exact electrically charged BTZ black hole solutions to the Einstein-Maxwell equations with a negative cosmological constant in 2+1 spacetime dimensions assuming a (anti-)self dual condition between the electromagnetic fields. In a coordinate condition there appears a logarithmic divergence in the angular momentum at spatial infinity. We show how it is to be regularized by taking the contribution from the boundary into account. We show another coordinate condition which leads to a finite angular momentum though it brings about a peculiar spacetime topology.
We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vac... more We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vacua are defined, one with the normal ordering and the other with the Weyl ordering. Their equivalence is shown by using a differential equation satisfied by the normal ordered vacuum.
We give an infinite number of exact solutions to the 5-dimensional static Einstein equation with ... more We give an infinite number of exact solutions to the 5-dimensional static Einstein equation with axial symmetry by using the inverse scattering method. The solutions are characterized by two integers representing the soliton numbers. The first non trivial example of these solutions is the static black ring solution found recently.
We give an infinite number of exact solutions to the 5-dimensional static Einstein equation with ... more We give an infinite number of exact solutions to the 5-dimensional static Einstein equation with axial symmetry by using the inverse scattering method. The solutions are characterized by two integers representing the soliton numbers. The first non trivial example of these solutions is the static black ring solution found recently. 1
We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vac... more We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vacua are defined, one with the normal ordering and the other with the Weyl ordering. Their equivalence is shown by using a differential equation satisfied by the normal ordered vacuum.
The Einstein-Maxwell equations with a negative cosmological constant Λ in 2 + 1 spacetime dimensi... more The Einstein-Maxwell equations with a negative cosmological constant Λ in 2 + 1 spacetime dimensions discussed by Bañados, Teitelboim and Zanelli are solved by assuming a self (anti-self) dual equation Eˆr = ±Bˆ, which is imposed on the orthonormal basis components of the electric field Eˆr and the magnetic field Bˆ. This solution describes an electrically charged extreme black hole with mass M = 8πGQ2 e, angular momentum J = ±8πGQ2 e /|Λ|1/2 and electric charge Qe. Although the coordinate components of the electric field Er and the magnetic field B have singularities on the horizon at r = (4πGQ2 e /|Λ|)1/2, the spacetime has the same value of constant negative curvature R = 6Λ as that of Bañados et al. MIRAMARE –TRIESTE
We study quantization via star products. We pursue a quantization scheme, where a quantum theory ... more We study quantization via star products. We pursue a quantization scheme, where a quantum theory is described entirely in terms of the function space without any reference to operators in the Hilbert space. For consistency of the theory, the associativity law plays the essential role in excluding the unwanted solutions to the stargen-value equation. This is exhibited in the D-dimensional harmonic oscillator explicitly. As the by-products, the interplay between the Laguerre polynomials and the creation and annihilation functions with the star product is demonstrated.
Proceedings of Workshop on Integrable Theories, Solitons and Duality — PoS(unesp2002), Dec 1, 2002
Defining the noncommutative zero-curvature equation, we show that many soliton equations can be w... more Defining the noncommutative zero-curvature equation, we show that many soliton equations can be written in this form. We demonstrate this by showing that various soliton equations are derived from this equation. The derived soliton equations differ according to different choices of manifolds in the reduction of the noncommutative zero-curvature equation.
The inverse scattering problem is applied to 2-dimensional partial differential equations called ... more The inverse scattering problem is applied to 2-dimensional partial differential equations called soliton equations such as the KdV equation and so on. It is also used to integrate the Einstein equations with axial symmetry. These inverse scattering problems look different. We show that they can be understood in a unified way. As an application to the Einstein equation, we find solutions of the Einstein-Maxwell equations with a magnetic charge.
Abstract For a wide class of solition equations in 1 + 1 dimensions an infinite set of symmetry t... more Abstract For a wide class of solition equations in 1 + 1 dimensions an infinite set of symmetry transformations obeying the Kac-Moody Lie algebra is obtained. These symmetry transformations, however, can be applied only within a limited domain of the x (space) variable. This restriction applies, in particular, to the Heisenberg spin chain and the nonlinear Schrodinger equation for which a Kac-Moody structure was recently derived by Eichenherr.
We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. The vac... more We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. The vacuums with and without using the normal ordering look different. The vacuum without the normal ordering is shown to be expressed using the Weyl ordering. The Weyl ordered vacuum is then compared with the normal ordered vacuum, and the implication of the difference between them is discussed.
It is known that the continuous soliton equations such as the KdV equation have hierarchy structu... more It is known that the continuous soliton equations such as the KdV equation have hierarchy structures. A higher order KdV equation constituting the hierarchy includes higher order derivative terms in the equation. The Toda equation is characterized by discreteness in the space dimension. In the hierarchy for the discrete soliton equation such as the Toda equation, neither spatial derivatives nor higher order spatial derivatives appear. Despite of this difference, we show both continuous and discrete soliton equation hierarchies starting with the noncommutative zero curvature equation. Especially, we show the explicit form of the higher order Toda equation.
We investigate the equation where the commutation relation in 2-dimensional zero-curvature equati... more We investigate the equation where the commutation relation in 2-dimensional zero-curvature equation composed of the algebra-valued potentials is replaced by the Moyal bracket and the algebra-valued potentials are replaced by the non-algebra-valued ones with two more new variables. We call the 4-dimensional equation the noncommutative zero-curvature equation. We show that various soliton equations are derived by the dimensional reduction of the equation.
The soliton technique is applied to the 5D static Einstein-Maxwell equations, and an infinite num... more The soliton technique is applied to the 5D static Einstein-Maxwell equations, and an infinite number of solutions are explicitly obtained. We study the rod structure of 2-soliton solutions and we show that the 5D Reissner-Nordstrom solution and the 5D Majumdar-Papapetrou solution are included as the 2-soliton solutions.
We connect the discrete and continuous Bogomolny equations. There exists one-parameter algebra re... more We connect the discrete and continuous Bogomolny equations. There exists one-parameter algebra relating two equations which is the deformation of the extended conformal algebra. This shows that the deformed algebra plays the role of the link between the matrix valued model and the model with one more space dimension higher.
We study the 5D static Einstein-Maxwell equations with a dilaton field. We construct an infinte n... more We study the 5D static Einstein-Maxwell equations with a dilaton field. We construct an infinte number of solutions by using a soliton technique. We study the rod structure of 2-soliton solution and show the 5D dilatonic black ring and black hole solutions are included.
We discuss the exact electrically charged BTZ black hole solutions to the Einstein-Maxwell equati... more We discuss the exact electrically charged BTZ black hole solutions to the Einstein-Maxwell equations with a negative cosmological constant in 2+1 spacetime dimensions assuming a (anti-)self dual condition between the electromagnetic fields. In a coordinate condition there appears a logarithmic divergence in the angular momentum at spatial infinity. We show how it is to be regularized by taking the contribution from the boundary into account. We show another coordinate condition which leads to a finite angular momentum though it brings about a peculiar spacetime topology.
We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vac... more We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vacua are defined, one with the normal ordering and the other with the Weyl ordering. Their equivalence is shown by using a differential equation satisfied by the normal ordered vacuum.
We give an infinite number of exact solutions to the 5-dimensional static Einstein equation with ... more We give an infinite number of exact solutions to the 5-dimensional static Einstein equation with axial symmetry by using the inverse scattering method. The solutions are characterized by two integers representing the soliton numbers. The first non trivial example of these solutions is the static black ring solution found recently.
We give an infinite number of exact solutions to the 5-dimensional static Einstein equation with ... more We give an infinite number of exact solutions to the 5-dimensional static Einstein equation with axial symmetry by using the inverse scattering method. The solutions are characterized by two integers representing the soliton numbers. The first non trivial example of these solutions is the static black ring solution found recently. 1
We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vac... more We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vacua are defined, one with the normal ordering and the other with the Weyl ordering. Their equivalence is shown by using a differential equation satisfied by the normal ordered vacuum.
The Einstein-Maxwell equations with a negative cosmological constant Λ in 2 + 1 spacetime dimensi... more The Einstein-Maxwell equations with a negative cosmological constant Λ in 2 + 1 spacetime dimensions discussed by Bañados, Teitelboim and Zanelli are solved by assuming a self (anti-self) dual equation Eˆr = ±Bˆ, which is imposed on the orthonormal basis components of the electric field Eˆr and the magnetic field Bˆ. This solution describes an electrically charged extreme black hole with mass M = 8πGQ2 e, angular momentum J = ±8πGQ2 e /|Λ|1/2 and electric charge Qe. Although the coordinate components of the electric field Er and the magnetic field B have singularities on the horizon at r = (4πGQ2 e /|Λ|)1/2, the spacetime has the same value of constant negative curvature R = 6Λ as that of Bañados et al. MIRAMARE –TRIESTE
We study quantization via star products. We pursue a quantization scheme, where a quantum theory ... more We study quantization via star products. We pursue a quantization scheme, where a quantum theory is described entirely in terms of the function space without any reference to operators in the Hilbert space. For consistency of the theory, the associativity law plays the essential role in excluding the unwanted solutions to the stargen-value equation. This is exhibited in the D-dimensional harmonic oscillator explicitly. As the by-products, the interplay between the Laguerre polynomials and the creation and annihilation functions with the star product is demonstrated.
Proceedings of Workshop on Integrable Theories, Solitons and Duality — PoS(unesp2002), Dec 1, 2002
Defining the noncommutative zero-curvature equation, we show that many soliton equations can be w... more Defining the noncommutative zero-curvature equation, we show that many soliton equations can be written in this form. We demonstrate this by showing that various soliton equations are derived from this equation. The derived soliton equations differ according to different choices of manifolds in the reduction of the noncommutative zero-curvature equation.
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Papers by Takao Koikawa