Ιωαννης Αντωνιου

1. ΟΙ ΦΥΣΙΚΕΣΑΛΓΕΒΡΕΣ ΧΩΡΟΧΡΟΝΙΚΩΝ ΣΥΜΜΕΤΡΙΩΝ ΚΑΙ ΤΟ ΣΧΕΤΙΚΙΣΤΙΚΟ XΑΟΣTHE NATURAL ALGEBRAS OF SPACETIME SYMMETRIES AND RELATIVISTIC CHAOS Ιωαννης Αντωνιου

2. ΠΕΡΙΛΗΨΗ HMIEYΘΕΑ ΑΘΡΟΙΣΜΑΤΑ ΑΛΓΕΒΡΩΝ LIE,ΣΥΜΜΕΤΡΙΕΣ ΚΑΙ ΧΑΟΣΤα Hμιευθεα Αθροισματα Αλγεβρων Lie περιγραφουν τις χωροχρονικες Συμμετριες. Οι αντιστοιχες Γεωμετριες προκυπτουν και ταξινομουνται από τις Συμμετριες που σεβονται (Lie-Klein).Οι καταστασεις των Φυσικων Συστηματων προκυπτουν επισης από την κατασκευη και ταξινομιση των Αναπαραστασεων των Αλγεβρων Lie των Συμμετριων τους που εμπεριεχονται στις Αλγεβρες των παρατηρισιμων τελεστων (Wigner-Von Neumann). H Aλγεβρα των Τα Σχετικιστικων Χαοτικων Συστηματων είναι μια απειροδιαστατη επεκταση της 10-παραμετρικης Lie Αλγεβρας Poincare της Ειδικης Σχετικοτητας που συμπεριλαμβανει τον Τελεστη του Χρονου ο οποιος περιγραφει τις διαδοχικες καινοτομιες, εγγενες χαρακτηριστικο του Χαους.

3. 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)

4. Euclidean Symmetries

6. The Structure of Euclidean Tranformations The composition of Euclidean Transformations S◦T(x)=S[T(x)] , x2³ is also a Euclidean Transformation The Euclidean Transformations are a Semi-Direct Product Lie Group

7. 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 • g=g, for all g2 G Inverse: for any g2 G, 9 gˉ¹2 G : gˉ¹ • g=u

8. 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

9. Lie’s motivation: Symmetries of Differential equations • Lie’s idea: Study the Group Action from the local properties in the neighbourhood of the unit • Lie Algebra is the Tangent space at the unit

10. Lie Algebra • A vector space G over R • With the operation- Lie bracket [,]: (X,Y) [X,Y] X,Y, [X,Y]2G Bilinearity over R: [iXi,jYj]=ij [X,Y], i,j2R Anti commutativity: [X,Y]=- [Y,X] Jacobi identity: [X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0

11. Semi-Direct Product of the Group (G,·) with the Group (X,+)

13. 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 Gwith the Lie Algebra X

21. Weyl Algebra:The Commutation Relations

24. (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

25. 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).

26. 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

27. (Θ)Innovation Process Time Operator formula for Age  Time Operator defined in terms of the Canonical Commutation Relation in the Weyl or Heisenberg form

28. Innovation Process

29. Time Operator formula for Age

30. Weyl CCR for the Time Operator

31. Weyl CCR:

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

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

34. Relativistic Chaos: Representation of the Lie Algebra generated by:

35. The simplest Example:Wave Equation

36. Wave Equation

37. Time Operator

38. Spectral Representation

39. The RIT Algebra

40. RIT

41. RIT

42. RIT Structure:

43. 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

44. 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

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

46. 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 Serre’s Problem Any finitely generated projective module over the ring of commutative polynomials over a field is free