Finsler Geometrical Path Integral

Erico Tanaka 　 Palacký UniversityTakayoshi Ootsuka 　 Ochanomizu University2009.5.27 @University of DebrecenWORKSHOP ON FINSLER GEOMETRY AND ITS APPLICATIONS

hepth/arXiv:0904.2464　

27th May. 2009 @University of Debrecen

Ideas　of　Feynman　Path　IntegralQuantisation　by　Lagrangian　formalism

classical　path

Quantum　Theory

Least　action　principle

There　is　a　more　fundamental　theory　behind.

Wave　opticsGeometrical　optics

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

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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?　

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

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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　∩　ΔΣx　=　φ

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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.　　

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Finsler measure on　leaf

Measure　induced　from　Finsler　structure

27th May. 2009 @University of Debrecen

Lagrangian is　a　differentiable　function　+　homogeneity　condition

Def.

Finsler　function　as　Lagrangian

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Reparametrisation　invariant

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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　

27th May. 2009 @University of Debrecen

Finsler　geometrical　path　integral

Finsler geometrical path integral

Feynman　path　integral

The meaning of propagator

on on

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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

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.

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We greatly thank Prof. Tamassy for this work.

27th May. 2009 @University of Debrecen

Problems　and　further　extensionsRelativistic　particles

Application　of　foliation　besides　　　　　　　　　　　　　　　　　　　.First　non　quadratic　application　in　a　Lagrangian　formalism.　

Irreversible　systems　⇒ Measure　depends　on　the　orientation

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Geometrical　phase　space　path　integral　by　the　setting　of　Contact　manifold

areal　metric

Higher　order

Field　theory

Centrifugal　potential

etc etc etc …

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?　

27th May. 2009 @University of Debrecen 15

ex.　non　relativistic　particle

Finsler geometrical path integral

Feynman　path　integral

27th May. 2009 @University of Debrecen

Finsler　Path　Integral

?

top　form　on　　　　　

∩

function　on

geodesic

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geodesic　connecting

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ex.　non　relativistic　particle

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chart　associated　to　the　foliation　chart　at　

Goldstein,”Classical Mechanics”　

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We　can　choose　arbitrary　“time”parameter 　　　　　　　　　　

dependantTrivial　if

Simple　examples　of　Lagrange　mechanics

Particles　in　EM　field ：

Newtonian　mechanics ：

Equation　of　motion

Randers　metric

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However,　for　most　simple　examples　in　physics…

Assume　existence　of　a　foliation　of　M　such　that,　

＝ 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.　　

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Independent of the choice of Riemann metric

F F

: :=

Finsler measure on Σ

Finsler area of　the　infinitesimal　domainof　the　submanifold

： constant

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ex ： Free　particle　on　Riemannian　manifold

Lee-Yang　term

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ex ： particle　constrained　on　 　

all　contributions　from　k　winding