From Information Theory to Quantum Mechanics

  • K. R. Parthasarathy
Part of the Applied Probability book series (APPLIEDPROB, volume 1)

Abstract

The first 20 years of my life were spent in the intensely religious and ritualistic atmosphere of a south Indian Brahmin family confined to the towns of Thanjavur and Madras and the paddy fields of the village of Thalanayar in Thanjavur district. Emphasis was laid at home on the philosophy and teachings of the Vaishnavite saints Ramanuja and Desika, the Hindu epics Ramayana and Mahabharata, the devotional songs of Tamil saints known as the Alwars and finally the Bhagavad Gita. To this day, I believe in their basic idea that personal and global peace are possible only through renunciation and action with detachment.

Keywords

Entropy Income Convolution Kato Univer 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Publications and References

  1. [1]
    Applebaum, D. (1984) Fermion Stochastic Calculus. Ph.D. Thesis, University of Nottingham.Google Scholar
  2. [2]
    Araki, H. (1970) Factorisable representations of current algebra. Publ. Res. Inst. Math. Sci. Kyoto Univ. A 5, 361–422.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Araki, H. and Woods, E. J. (1966) Complete boolean algebras of type I factors. Publ. Res. Inst. Math. Sci. Kyoto Univ. 2, 157–242.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Barnett, C., Streater, R. F. and Wilde, I. F. (1982) The Itô—Clifford integral. J. Functional Anal. 48, 172–212.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Bhattacharya, R. N. and Ranga Rao, R. (1976) Normal Approximation and Asymptotic Expansions. Wiley, New York.MATHGoogle Scholar
  6. [6]
    Erven, J. and Falkowski, B. J. (1981) Low Order Cohomology and Applications. Lecture Notes in Mathematics 877, Springer-Verlag, Berlin.Google Scholar
  7. [7]
    Frigerio, A. (1984) Covariant Markov dilations of quantum dynamical semi-groups. Preprint.Google Scholar
  8. [8]
    Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA.MATHGoogle Scholar
  9. [9]
    Gray, R. M. and Davisson, L. D. (EDS.) (1977) Ergodic and Information Theory. Dowden, Hutchinson and Ross, Stroudsburg, PA.Google Scholar
  10. [10]
    Halmos, P. R. (1953) Lectures on Ergodic Theory. Chelsea, New York.Google Scholar
  11. [11]
    Hudson, R. L. and Ion, P. D. F. (1981) The Feynman-Kac formula for a canonical Wiener process. Proc. Colloq. Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory. North-Holland, Amsterdam.Google Scholar
  12. [12]
    Hudson, R. L. and Parthasarathy, K. R. (1983) Quantum diffusions. In Theory and Applications of Random Fields ed. G. Kallianpur et al., Lecture Notes in Control and Information Sciences 49, Springer-Verlag, Berlin, 111–121.Google Scholar
  13. [13]
    Hudson, R. L. and Parthasarathy, K. R. (1984) Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Hudson, R. L. and Parthasarathy, K. R. (1984) Stochastic dilations of uniformly continuous, completely positive semigroups. Acta Applicandae Math.Google Scholar
  15. [15]
    Hudson, R. L. and Streater, R. F. (1981) Itô’s formula is the chain rule with Wick ordering. Phys. Letters 86 A, 277–279.Google Scholar
  16. [16]
    Hudson, R. L., Karandikar, R. L. and Parthasarathy, K. R. (1983) Towards a theory of non-commutative semi-martingales adapted to Brownian motion and a quantum It?’s formula. In Theory and Applications of Random Fields, ed. G. Kallianpur et al., Lecture Notes in Control and Information Sciences 49, Springer-Verlag, Berlin, 96–110.Google Scholar
  17. [17]
    Kato, T. (1976) Perturbation of Linear Operators. Springer-Verlag, Berlin.Google Scholar
  18. [18]
    Khinchine, A. I. (1957) Mathematical Foundations of Information Theory. Dover, New York.Google Scholar
  19. [19]
    Lévy, P. (1952) Sur une classe de lois de probabilité indécomposables. C. R. Acad. Sci. Paris 235, 489–492.MathSciNetMATHGoogle Scholar
  20. [20]
    Mackey, G. W. (1963) The Mathematical Foundations of Quantum Mechanics. Benjamin, New York.MATHGoogle Scholar
  21. [21]
    Oxtoby, J. C. (1961) On two theorems of Parthasarathy and Kakutani concerning the shift transformation. In Ergodic Theory, Proc. Internat. Symp. Tulane University,Academic Press, New York, 203–215.Google Scholar
  22. [22]
    Parthasarathy, K. R. (1961) On the category of ergodic measures. Illinois J. Math. 5, 648–656.MathSciNetMATHGoogle Scholar
  23. [23]
    Parthasarathy, K. R. (1961) On the integral representation of the rate of transmission of a stationary channel Illinois J. Math. 5, 299–305.MathSciNetMATHGoogle Scholar
  24. [24]
    Parthasarathy, K. R. (1963) Effective entropy rate and transmission of information through channels with additive random noise. Sankhyii A 25, 75–84.MATHGoogle Scholar
  25. [25]
    Parthasarathy, K. R. (1964) The central limit theorem for rotation groups. Theory Prob. Appl. 9, 248–257.CrossRefGoogle Scholar
  26. [26]
    Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.MATHGoogle Scholar
  27. [27]
    Parthasarathy, K. R. (1972) A probabilistic approach to the Pontrjagin duality theorem. Period. Math. Hungar. 2, 21–26.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Parthasarathy, K. R. (1974) The central limit theorem for positive definite functions on a locally compact group. J. Multivariate Anal. 4, 123–149.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    Parthasarathy, K. R. (1977) Introduction to Probability and Measure. Macmillan India, New Delhi (Russian translation: MIR, Moscow, 1983 ).Google Scholar
  30. [30]
    Parthasarathy, K. R. (1977) The central limit theorem for positive definite functions on Lie groups. Symposia Mathematica, Pub. Ist. Naz. Alt. Mat. XXI, Academic Press, New York, 245–256.Google Scholar
  31. [31]
    Parthasarathy, K. R. (1979) Lectures on Functional Analysis II. ISI Lecture Notes 6, Macmillan India, New Delhi.Google Scholar
  32. [32]
    Parthasarathy, K. R. (1982) On a class of time inhomogeneous nonsingular flows and Schrödinger operators. Math. Z. 179, 123–133.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Parthasarathy, K. R. (1984) A remark on the integration of Schrödinger equation using quantum Itô’s formula. Lett. Math. Phys.Google Scholar
  34. [34]
    Parthasarathy, K. R. and Bhatia, R. (1978) Lectures on Functional Analysis I. ISI Lecture Notes 3, Macmillan India, New Delhi.Google Scholar
  35. [35]
    Parthasarathy, K. R. and Bingham, M. S. (1968) A probabilistic proof of Bochner’s theorem on positive definite functions. J. London Math. Soc. 43, 626–632.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    Parthasarathy, K. R. and Sazonov, V. V. (1964) On the representation of infinitely divisible distribution on a locally compact abelian group. Theory Prob. Appl. 9, 118–122.MathSciNetMATHGoogle Scholar
  37. [37]
    Parthasarathy, K. R. and Schmidt, K. (1972) Factorisable representations of current groups and the Araki—Woods imbedding theorem. Acta Math. 128, 53–71.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    Parthasarathy, K. R. and Schmidt, K. (1972) Positive Definite Kernels, Continuous Tensor Products and Central Limit Theorems of Probability Theory. Lecture Notes in Mathematics 272, Springer-Verlag, Berlin.Google Scholar
  39. [39]
    Parthasarathy, K. R. and Schmidt, K. (1975) Stable positive definite functions. Trans. Amer. Math. Soc. 203, 161–174.MathSciNetMATHGoogle Scholar
  40. [40]
    Parthasarathy, K. R. and Schmidt, K. (1976) A new method for constructing factorisable representations of current groups and current algebras. Commun. Math. Phys. 50, 167–175.MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    Parthasarathy, K. R. and Schmidt, K. (1977) On the cohomology of a hyperfinite action. Monatsh. Math. 84, 37–48.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    Parthasarathy, K. R. and Sinha, K. B. (1982) A random Trotter?Kato product formula. In Statistics and Probability, Essays in Honour of C. R. Rao, ed. G. Kallianpur et al., North-Holland, Amsterdam, 553–565.Google Scholar
  43. [43]
    Parthasarathy, K. R., Ranga Rao, R. and Varadhan, S. R. S. (1962) On the category of indecomposable distributions on topological groups. Trans. Amer. Math. Soc. 102, 200–217.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    Parthasarathy, K. R., Ranga Rao, R. and Varadhan, S. R. S. (1963) Probability distributions on locally compact abelian groups. Illinois J. Math. 7, 337–369.MathSciNetMATHGoogle Scholar
  45. [45]
    Parthasarathy, K. R., Ranga Rao, R. and Varadarajan, V. S. (1967) Representations of complex semi-simple Lie groups and Lie algebras. Ann. Math. 25, 383–420.CrossRefGoogle Scholar
  46. [46]
    Ranga Rao, R. (1962) Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33, 659–680.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    Rohlin, V. A. (1967) Lectures on the entropy theory of transformations with invariant measure. Uspehi Mat. Nauk 22, 3–26 (Russian Math. Surveys 22, 1–52 ).CrossRefGoogle Scholar
  48. [48]
    Schmidt, K. (1971) Limits of uniformly infinitesimal families of projective representations of locally compact groups. Math. Ann. 192, 107–118.MathSciNetMATHCrossRefGoogle Scholar
  49. [49]
    Shohat, J. A. and Tamarkin, J. D. (1943) The Problem of Moments. Math. Surveys 1, American Mathematical Society, New York.Google Scholar
  50. [50]
    Streater, R. F. (1969) Current commutation relations, continuous tensor products and infinitely divisible group representations. Rend. Soc. Int. Fisica E. Fermi XI, 247–263.Google Scholar
  51. [51]
    Varadarajan, V. S. (1961) Measures on topological spaces. Mat. Sb. 55, 35–100 (Amer. Math. Soc. Transi. (2) 48, 161–228).Google Scholar
  52. [52]
    Varadhan, S. R. S. (1962) Convolution Properties of Distributions on Topological Groups. Ph.D. Thesis, ISI, Calcutta.Google Scholar

Copyright information

© Applied Probability Trust 1986

Authors and Affiliations

  • K. R. Parthasarathy

There are no affiliations available

Personalised recommendations