?? F?S???S ??G????S ?O????????O? S???????O? ??? ?? S??????S???? X??S THE NATURAL ALGEBRAS OF SPACETIME SYMMETRIES AND RELATIVISTIC CHAOS

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?? F?S???S ??G????S ?O????????O? S???????O? ???
?? S??????S???? X??S THE NATURAL ALGEBRAS OF
SPACETIME SYMMETRIES AND RELATIVISTIC CHAOS

• ??a???? ??t?????

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????????

• HMIEYT?? ?T???S???? ??G???O?
  LIE,S????????S ??? ???S?a Hµ?e??ea ?????sµata
  ???eß??? Lie pe????af??? t?? ??????????e?
  S?µµet??e?. ?? a?t?st???e? Ge?µet??e? p????pt???
  ?a? ta????µ???ta? ap? t?? S?µµet??e? p?? seß??ta?
  (Lie-Klein).?? ?atastase?? t?? F?s???? S?st?µat??
  p????pt??? ep?s?? ap? t?? ?atas?e?? ?a?
  ta????µ?s? t?? ??apa?astase?? t?? ???eß??? Lie
  t?? S?µµet???? t??? p?? eµpe??e???ta? st??
  ???eß?e? t?? pa?at???s?µ?? te?est?? (Wigner-Von
  Neumann). H A??eß?a t?? ?a S?et???st???? ?a?t????
  S?st?µat?? e??a? µ?a ape???d?astat? epe?tas? t??
  10-pa?aµet????? Lie ???eß?a? Poincare t?? ??d????
  S?et???t?ta? p?? s?µpe???aµßa?e? t?? ?e?est? t??
  ?????? ? ?p???? pe????afe? t?? d?ad????e?
  ?a???t?µ?e?,
• e??e?e? ?a?a?t???st??? t?? ?a???.

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Symmetry

• Symmetry as wide or as narrow as you may define
  its meaning, is one idea by which man through the
  ages has tried to comprehend and create order,
  beauty and perfection
• H. Weyl Symmetry, Princeton (1952)

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Euclidean Symmetries
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The Structure of Euclidean Tranformations

• The composition of Euclidean Transformations
• S?T(x)ST(x) , x2 ?³
• is also a Euclidean Transformation
• The Euclidean Transformations are
• a Semi-Direct Product Lie Group

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Group

• A set G with an operation GxG?G
• (g,h) ? g h , for all g,h2 G
• with the properties
• Associative g (h q)(g h) q, for all
  g,h,q2 G
• Identity 9 u 2 G u gg, for all g2 G
• Inverse for any g2 G, 9 g¹2 G g¹ gu

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Lie Groups

• Lie Groups are Continuous Groups (G,) which are
  also Analytic Manifolds and
• the group operation GxG?G is Analytic
• (x,y)?x y
• x?x-1

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• Lies motivation
• Symmetries of Differential equations
• Lies idea
• Study the Group Action from
• the local properties in the neighbourhood of
  the unit
• Lie Algebra is the Tangent space at the unit

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Lie Algebra

• A vector space G over R
• With the operation- Lie bracket ,
• (X,Y)? X,Y X,Y, X,Y2 G
• Bilinearity over R ?iXi,?jYj?i?j X,Y, ?i,?j
  2 R
• Anti commutativity X,Y- Y,X
• Jacobi identity X,Y,ZZ,X,YY,Z,X0

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Semi-Direct Product of the Group (G,) with the
Group (X,)
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The Lie Algebra of the Semi-Direct Product of
the Lie Group (G,) with the Group (X,) isthe
Semi-Direct Sum of the corresponding Lie Algebras

• The Semi-Direct Sum of the Lie Algebra G with the
  Lie Algebra X

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Weyl AlgebraThe Commutation Relations
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• (O) A System has the Symmetry Group G?
• The Group of Automorphisms of the Algebra
  of Observables (Functions,Operators)
• 
  or the state space
• contains a Representation of G as a
  Subgroup
• Pythagoras, Plato,, Lie, Klein, Weyl
• Beings are manifestations of Symmetries
• Representations of the
  relevant Symmetry Group G
• Classifications of Beings ? Classifications of
  the Representations of G
• Composite Beings correspond to Reducible
  Representations of G
• Elementary Beings correspond to Irreducible
  Representations of G

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• Aristotle Mechanics - Euclidean Group
• Newton Mechanics - Galilei Group
• Relativistic Mechanics - Poincare Group
• Construction , Classification and Significance of
  
• the Representations of the Symmetry Groups
• Barut A.O. and Raczka R., The Theory of Group
  Representations and Applications, Polish Sci.
  Publishers, Warsaw (1977).
• Wigner E.P., Unitary representations of the
  Inhomogenous Lorenz group, Ann. Math. 40, 149-204
  (1939).
• Naimark M.A. Linear Representations of the
  Lorentz Group, Pergamon Press (1964).
• Wigner E.P. Group Theory and its Application to
  the Quantum Mechanics of Atomic Spectra, Academic
  Press, New York (1959).
• Wigner E.P., Houtappel R. and Van Dam H., The
  conceptual basis and use of the geometric
  invariance principles, Rev. Mod. Phys. 37,
  595-632 (1965).
• Mackey G.W. Unitary Group Representations in
  Physics, Probability and Number Theory,
  Benjamin/Cummings, London (1978).
• Mackey G.W. Harmonic Analysis as the Exploitation
  of Symmetry-A Historical Survey, Bull. Am. Math.
  Soc. 3, 543-698 (1980).
• Wigner E., The unreasonable effectiveness of
  mathematics in the natural sciences, Comm. Pure
  Appl. Math. 13, 1-14 (1960).

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• What is the Algebra of Observables of
  Relativistic Chaotic Systems?
• It should include the Poincare Algebra
• qualifying Relativistic Symmetry
• and the Internal Time qualifying Chaos

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• (T)Innovation Process?
• Time Operator formula for Age ?
• Time Operator defined in terms of
• the Canonical Commutation Relation
• in the Weyl or Heisenberg form

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• Innovation Process

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• Time Operator formula for Age

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• Weyl CCR for the Time Operator

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• Weyl CCR

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• Stone-Von Neuman-Mackey Theorem
• Representations of CCR
• Translation Representation
• Spectral Representation

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• Construction of a Time Operator for Relativistic
  Systems
• Work within the Rational envelopping Algebra of
  the Relativistic System

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• Relativistic Chaos
• Representation of the Lie Algebra generated by

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• The simplest ExampleWave Equation

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• Wave Equation

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• Time Operator

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• Spectral Representation

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• The RIT Algebra

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• RIT

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• RIT

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• RIT Structure

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• Antoniou I. and Misra B., The relativistic
  internal time algebra, Nuclear Physics, Proceed.
  Suppl. Sect. 6, 240-242 (1989).
• Antoniou I. and Misra B., Relativistic Internal
  Time Operator, Int. J. Theor.Phys. 31, 119-136
  (1992).
• RIT is not in the known / classified Infinite
  Dimensional Lie Algebras
• Kac V.G., Infinite Dimensional Lie
  Algebras, Cambridge University Press (1985).
• How to characterise RIT
• So that we can compare RIT with other known
  Infinite Dimensional Lie Algebras ?
• ex. The Bondi-Metzner-Sachs Algebra of
  Asymptotic SpaceTime

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• Antoniou I. and and Misra B., Characterization of
  semidirect sum Lie algebras,
• J. Math. Phys. 32, 864-868 (1991).
• If we know the Representations of the Lie Algebra
  G
• then we may characterise the semi-direct sum of G
  with X
• by characterising the representation of G
  provided by X

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• The Representations of the Lorentz Algebra are
  known
• Naimark M.A. Linear Representations of the
  Lorentz Group, Pergamon Press (1964).

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• Antoniou I., Iyudu N. Poincare-Hilbert Series,pi
  and Noetherianity of the Enveloping of the
  Relativistic Internal Time Algebra, Comm. in
  Algebra 29, 4183-4196 (2001).
• Study of the Envelopping Algebra of RIT,
  Groebner basis technique
• Antoniou I., Iyudu N. Wisbauer R., On Serre's
  Problem for the RIT Algebra, Comm. in Algebra 31,
  6037-6050 (2003).
• RIT provides a counterexample to the
  Serres Problem
• Any finitely generated projective module
  over the ring of commutative polynomials over a
  field is free