Takayoshi Ootsuka

Finsler geometrical path integral

A new definition for the path integral is proposed in terms of Finsler geometry. The conventional Feynmans scheme for quantisation by Lagrangian formalism suffers problems due to the lack of geometrical structure of the configuration space where the path integral is defined. We propose that, by implementing the Feynmans path integral on an extended configuration space endowed with a Finsler structure, the formalism could be justified as a proper scheme for quantisation from Lagrangian only, that is, independent from Hamiltonian formalism. The scheme is coordinate free, and also a covariant framework which does not depend on the choice of time coordinates.

Energy-momentum conservation laws in Finsler/Kawaguchi Lagrangian formulation

We reformulate the standard Lagrangian formulation to a reparameterization invariant Lagrangian formulation by means of Finsler and Kawaguchi geometry. In our formulation, various types of symmetries that appear in theories of physics are expressed geometrically by symmetries of the Finsler (Kawaguchi) metric, and the conservation laws of energy momentum arise as a part of the Euler–Lagrange equations. The Euler–Lagrange equations are given geometrically in reparameterization invariant form, and the conserved energy-momentum currents can be obtained more easily than by the conventional Lagrangian formulation. The applications to scalar field, Dirac field, electromagnetic field and general relativity are introduced. In particular, we propose an alternative definition of the energy-momentum current of gravity, which satisfies gauge invariance under on-shell conditions.

Killing symmetry on the Finsler manifold

Symmetry and conservation law are discussed on the Finsler manifold M. We adopt the point Finsler approach, where we consider the geometry on a point manifold M not on TM. Generalized vector fields are defined on oriented curves on M, and Finsler non-linear connections are considered on M, not on the tangent space TM. Killing vector fields K are defined as generalized vector fields as , and the Killing symmetry is also reformulated simply as by using the Killing 1-form and the spray operator defined by using the non-linear connection. is related to the generalization of Killing tensors on the Finsler manifold, and our ansatz of and give an analytical method of finding higher derivative conserved quantities, which may be called hidden conserved quantities. We show two examples: the Carter constant on Kerr spacetime and the Runge–Lentz vectors in Newtonian gravity.

Variational principle of relativistic perfect fluid

We reformulate the relativistic perfect fluid system on curved space-time. Using standard variables, the velocity field

Chiral gravity in higher dimensions

u

Cusp singularity in mean field Ising model

,energy density

Finsler Connection for General Lagrangian Systems

\rho

Energy-momentum currents in Finsler/Kawaguchi Lagrangian formulation

and pressure

Non-associative Gauge Theory

p

Super Finsler Connection of Superparticle on Two Dimensional Curved Spacetime

, the covariant Euler-Lagrange equation is obtained from variational principle. This leads to the Euler equation and the equation of continuity in reparametrization invariant form.