Finsler Geometrical Path Integral

hepth/arXiv:0904.2464 Erico Tanaka Palacký University Takayoshi Ootsuka Ochanomizu University 2009.5.27 @University of Debrecen WORKSHOP ON FINSLER GEOMETRY AND ITS APPLICATIONS

Ideas of Feynman Path Integral

Quantisation by Lagrangian formalism Least action principle classical path There is a more fundamental theory behind.

Quantum Theory 27th May. 2009 @University of Debrecen

Feynman’s Path Integral

1.

2.

Problems

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.

The contributions from the paths are equal in magnitude, but the phase regards the classical action.

 Feynman’s path integral formula transformation are somewhat related. (Kleinert) •Problems calculating centrifugal potentials. (Kleinert) •What about singular or non quadratic Lagrangians? Rev.Mod.Phys 20, 367(1948) “Space-time approach to Non-relativistic Quantum mechanics” 27th May. 2009 @University of Debrecen 3

The stage for Finsler path integral



Finsler manifold

: n+1 dim. differentiable manifold with a foliation : Finsler function such that

Reparametrisation invariant = Independent of time variable

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Measure induced from Finsler structure

indicatrix Unit length indicatrix body Unit volume

Indicatrix body ∩ ΔΣ x = φ unit area Tamassy Lajos, Rep.Math.Phys 33, 233(1993) “ AREA AND CURVATURE IN FINSLER SPACES ”

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. 27th May. 2009 @University of Debrecen 6

Measure induced from Finsler structure

Finsler measure

on leaf 27th May. 2009 @University of Debrecen 7

Finsler function as Lagrangian

Def.

Lagrangian

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

Finsler geometrical setting : configuration space endowed with the Finsler function Euclid measure only when measure determined from special slicing (t=const.) ← Time parameterisation free We need more general slicing for relativity.

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Finsler geometrical path integral

Feynman path integral

Finsler geometrical path integral The meaning of propagator

on on 27th May. 2009 @University of Debrecen

For Classical Lagrange Mechanics

：

Extended configuration space (n+1 dim smooth manifold) Finsler function determined by the Lagrangian

C. Lanczos , ”

The Variational Principles of Mechanics

” Finsler manifold Example. Path Integral for non relativistic particle 27th May. 2009 @University of Debrecen 11

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.

We greatly thank Prof. Tamassy for this work.

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Problems and further extensions

Relativistic particles Application of foliation besides .

First non quadratic application in a Lagrangian formalism. Centrifugal potential Irreversible systems ⇒ Measure depends on the orientation Geometrical phase space path integral by the setting of Contact manifold areal metric Higher order Field theory

etc etc etc …

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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? 27th May. 2009 @University of Debrecen

Feynman path integral

Finsler geometrical path integral

ex. non relativistic particle 27th May. 2009 @University of Debrecen 15

Finsler Path Integral

?

top form on function on geodesic geodesic connecting 27th May. 2009 @University of Debrecen 16

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

chart associated to the foliation chart at Goldstein,”

Classical Mechanics

” 27th May. 2009 @University of Debrecen

Simple examples of Lagrange mechanics

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

However, for most simple examples in physics… for ＝

φ

Assume existence of a foliation of M such that, 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. 27th May. 2009 @University of Debrecen 20

:

=

F F

Finsler area

of the infinitesimal domain of the submanifold

: Finsler measure on Σ

： constant 27th May. 2009 @University of Debrecen

Independent of the choice of Riemann metric

21

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

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