Finsler Geometrical Path Integral

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Finsler Geometrical Path Integral
hepth/arXiv0904.2464

• Erico Tanaka Palacký University
• Takayoshi Ootsuka Ochanomizu University
• 2009.5.27 _at_University of Debrecen
• WORKSHOP ON FINSLER GEOMETRY AND ITS APPLICATIONS

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Ideas of Feynman Path Integral
Quantisation by Lagrangian formalism
Least action principle
classical path
There is a more fundamental theory behind.
Quantum Theory
Geometrical optics
Wave optics
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Feynmans Path Integral

• 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?

• 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.

• Feynmans path integral formula

Rev.Mod.Phys 20, 367(1948) Space-time approach
to Non-relativistic Quantum mechanics
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The stage for Finsler path integral

• Finsler manifold

n1 dim. differentiable manifold with a
foliation
Finsler function such that
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Measure induced from Finsler structure
indicatrix
Unit length
indicatrix body
Unit volume
Indicatrix body n ?Sx f
Tamassy Lajos, Rep.Math.Phys 33, 233(1993) AREA
AND CURVATURE IN FINSLER SPACES
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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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Measure induced from Finsler structure
Finsler measure on leaf
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Finsler function as Lagrangian
Lagrangian is a differentiable function
homogeneity condition
Def.
Reparametrisation invariant
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Finsler geometrical path integral
Conventional Feynman path integral
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Finsler geometrical path integral
Feynman path integral
Finsler geometrical path integral
The meaning of propagator
on
on
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For Classical Lagrange Mechanics
Extended configuration space (n1 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
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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?
  

?
?
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Feynman path integral
Finsler geometrical path integral
ex. non relativistic particle
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Finsler Path Integral
?
top form on
function on
geodesic
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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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Simple examples of Lagrange mechanics
Newtonian mechanics
We can choose arbitrary timeparameter
Trivial if
dependant
Equation of motion
Randers metric
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However, for most simple examples in physics
for
f
Assume existence of a foliation of M such that,
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Finsler measure on S
Independent of the choice of Riemann metric
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exFree particle on Riemannian manifold
Lee-Yang term
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ex particle constrained on
all contributions from k winding