1. Finsler Geometrical Path Integral Erico Tanaka Palacký University Takayoshi Ootsuka Ochanomizu University 2009.5.27 @University of Debrecen WORKSHOP ON FINSLER GEOMETRY AND ITS APPLICATIONS hepth/arXiv:0904.2464

2. 27th May. 2009 @University of Debrecen Ideas of Feynman Path Integral Quantisation by Lagrangian formalism classical path Quantum Theory Least action principle There is a more fundamental theory behind. Wave optics Geometrical optics

3. 27th May. 2009 @University of Debrecen 1. The probability amplitude of a particle to take a path in a certain region of space-time is the sum of all contributions from the paths existing in this region. 2. The contributions from the paths are equal in magnitude, but the phase regards the classical action. Feynman’s Path Integral 3 Feynman’s path integral formula Rev.Mod.Phys 20, 367(1948) “Space-time approach to Non-relativistic Quantum mechanics” Problems One has to start from canonical quantisation to obtain a correct measure. (Lee-Yang term problem/constrained system) Time slicing and coordinate transformation are somewhat related. (Kleinert) Problems calculating centrifugal potentials. (Kleinert) What about singular or non- quadratic Lagrangians?

4. 27th May. 2009 @University of Debrecen The stage for Finsler path integral  Finsler manifold : Finsler function such that Reparametrisation invariant = Independent of time variable : n+1 dim. differentiable manifold with a foliation 4

5. 27th May. 2009 @University of Debrecen Measure induced from Finsler structure indicatrix indicatrix body Unit length Unit volume unit area Tamassy Lajos, Rep.Math.Phys 33, 233(1993) “ AREA AND CURVATURE IN FINSLER SPACES ” Indicatrix body ∩ ΔΣ x = φ

6. 27th May. 2009 @University of Debrecen 6 Measure induced from Finsler structure Assume a codimension 1 foliation such that: i) choose initial point and final point from two different leaves, such that these points are connected by curves(=path). On this curve is well-defined for all. ii) The leaves of foliation are transversal to these set of curves.

7. 27th May. 2009 @University of Debrecen 7 Finsler measure on leaf Measure induced from Finsler structure

8. 27th May. 2009 @University of Debrecen Lagrangian is a differentiable function + homogeneity condition Def. Finsler function as Lagrangian 8 Reparametrisation invariant

9. 27th May. 2009 @University of Debrecen 9 Finsler geometrical path integral Euclid measure only when special slicing (t=const.) We need more general slicing for relativity. has no geometrical structure in general. : configuration space Conventional Feynman path integral Finsler geometrical setting Much general choice of Foliation ← Time parameterisation free spacetime endowed with the Finsler function measure determined from

10. 27th May. 2009 @University of Debrecen Finsler geometrical path integral Feynman path integral The meaning of propagator on

11. 27th May. 2009 @University of Debrecen 11 For Classical Lagrange Mechanics ： Extended configuration space (n+1 dim smooth manifold) Finsler function determined by the Lagrangian Finsler manifold C. Lanczos, ” The Variational Principles of Mechanics” Example. Path Integral for non relativistic particle

12. 27th May. 2009 @University of Debrecen Summary  We created a new definition for the path integral by the usage of Finsler geometry.  The proposed method is a quantization by “Lagrangian formalism”, independent of canonical formalism (Hamiltonian formalism).  The proposed Finsler path integral is coordinate free, covariant frame work which does not depend on the choice of time variables.  With the proposed formalism, we could solve the problems conventional method suffered. 12 We greatly thank Prof. Tamassy for this work.

13. 27th May. 2009 @University of Debrecen Problems and further extensions Relativistic particles Application of foliation besides. First non quadratic application in a Lagrangian formalism. Irreversible systems ⇒ Measure depends on the orientation 13 Geometrical phase space path integral by the setting of Contact manifold areal metric Higher order Field theory Centrifugal potential etc etc etc …

14. 27th May. 2009 @University of Debrecen Are the problems in Feynman Path Integral solved?  One has to start from canonical quantization to obtain a correct measure. (Lee-Yang term problem/constrained system)  Time slicing and coordinate transformation are somewhat related. (Kleinert)  Problems calculating centrifugal potentials. (Kleinert)  What about singular or non-quadratic Lagrangians?

15. 27th May. 2009 @University of Debrecen 15 ex. non relativistic particle Finsler geometrical path integral Feynman path integral

16. 27th May. 2009 @University of Debrecen Finsler Path Integral ? top form on ∩ function on geodesic 16 geodesic connecting

17. 27th May. 2009 @University of Debrecen 17 ex. non relativistic particle

18. 27th May. 2009 @University of Debrecen chart associated to the foliation chart at Goldstein,”Classical Mechanics”

19. 27th May. 2009 @University of Debrecen 19 We can choose arbitrary “time”parameter dependant Trivial if Simple examples of Lagrange mechanics Particles in EM field ： Newtonian mechanics ： Equation of motion Randers metric

20. 27th May. 2009 @University of Debrecen 20 However, for most simple examples in physics… Assume existence of a foliation of M such that, ＝φ＝φ for i) choose initial point and final point from two different leaves, such that these points can be connected by curves and on this curve is well-defined. ii) The leaves of foliation are transversal to these set of curves.

21. 27th May. 2009 @University of Debrecen 21 Independent of the choice of Riemann metric F F :: = Finsler measure on Σ Finsler area of the infinitesimal domain of the submanifold ： constant

22. 27th May. 2009 @University of Debrecen 22 ex ： Free particle on Riemannian manifold Lee-Yang term

23. 27th May. 2009 @University of Debrecen ex ： particle constrained on all contributions from k winding