Zeta Function Regularization in Casimir Effect Calculations & J.S. Dowker's Contribution

Zeta Function Regularization in Casimir Effect Calculations & J.S. Dowker's Contribution

ICE Zeta Function Regularization in Casimir Effect Calculations & J.S. Dowker's Contribution EMILIO ELIZALDE ICE/CSIC & IEEC, UAB, Barcelona QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 1/27 Outline The Zero Point Energy Operator Zeta Functions in Math Physics: Origins J. S. Dowker & S. W. Hawking Contributions Pseudodifferential Operator (ΨDO) Existence of ζA for A a ΨDO ζ Determinant, Properties − Wodzicki Residue, Multiplicative Anomaly or Defect Extended CS Series Formulas (ECS) QFT in space-time with a non-commutative toroidal part Future Perspectives: Operator regularization & so on With THANKS to: S Carloni, G Cognola, J Haro, S Nojiri, S Odintsov, D Sáez-Gómez, A Saharian, P Silva, S Zerbini, . QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 2/27 Zero point energy QFT vacuum to vacuum transition: 0 H 0 h | | i QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 3/27 Zero point energy QFT vacuum to vacuum transition: 0 H 0 h | | i Spectrum, normal ordering (harm oscill): 1 † H = n + λn an an 2 QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 3/27 Zero point energy QFT vacuum to vacuum transition: 0 H 0 h | | i Spectrum, normal ordering (harm oscill): 1 † H = n + λn an an 2 ~ c 1 0 H 0 = λn = tr H h | | i 2 Xn 2 QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 3/27 Zero point energy QFT vacuum to vacuum transition: 0 H 0 h | | i Spectrum, normal ordering (harm oscill): 1 † H = n + λn an an 2 ~ c 1 0 H 0 = λn = tr H h | | i 2 Xn 2 gives physical meaning? ∞ QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 3/27 Zero point energy QFT vacuum to vacuum transition: 0 H 0 h | | i Spectrum, normal ordering (harm oscill): 1 † H = n + λn an an 2 ~ c 1 0 H 0 = λn = tr H h | | i 2 Xn 2 gives physical meaning? ∞ Regularization + Renormalization ( cut-off, dim, ζ ) QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 3/27 Zero point energy QFT vacuum to vacuum transition: 0 H 0 h | | i Spectrum, normal ordering (harm oscill): 1 † H = n + λn an an 2 ~ c 1 0 H 0 = λn = tr H h | | i 2 Xn 2 gives physical meaning? ∞ Regularization + Renormalization ( cut-off, dim, ζ ) Even then: Has the final value real sense ? QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 3/27 Operator Zeta F’s in MΦ: Origins The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one ∞ starts with the infinite series 1 ns nX=1 which converges for all complex values of s with real Re s> 1, and then defines ζ(s) as the analytic continuation, to the whole complex s plane, of the function given, Re s> 1, − by the sum of the preceding series. Leonhard Euler already considered the above series in 1740, but for positive integer values of s, and later Chebyshev extended the definition to Re s> 1. QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 4/27 Operator Zeta F’s in MΦ: Origins The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one ∞ starts with the infinite series 1 ns nX=1 which converges for all complex values of s with real Re s> 1, and then defines ζ(s) as the analytic continuation, to the whole complex s plane, of the function given, Re s> 1, − by the sum of the preceding series. Leonhard Euler already considered the above series in 1740, but for positive integer values of s, and later Chebyshev extended the definition to Re s> 1. Godfrey H Hardy and John E Littlewood, “Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Math 41, 119 (1916) Did much of the earlier work, by establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods G H Hardy, Divergent Series (Clarendon Press, Oxford, 1949) Srinivasa I Ramanujan had found for himself the functional equation of the zeta function QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 4/27 Operator Zeta F’s in MΦ: Origins The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one ∞ starts with the infinite series 1 ns nX=1 which converges for all complex values of s with real Re s> 1, and then defines ζ(s) as the analytic continuation, to the whole complex s plane, of the function given, Re s> 1, − by the sum of the preceding series. Leonhard Euler already considered the above series in 1740, but for positive integer values of s, and later Chebyshev extended the definition to Re s> 1. Godfrey H Hardy and John E Littlewood, “Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Math 41, 119 (1916) Did much of the earlier work, by establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods G H Hardy, Divergent Series (Clarendon Press, Oxford, 1949) Srinivasa I Ramanujan had found for himself the functional equation of the zeta function Torsten Carleman, “Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes" (French), 8. Skand Mat-Kongr, 34-44 (1935) QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 4/27 Operator Zeta F’s in MΦ: Origins The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one ∞ starts with the infinite series 1 ns nX=1 which converges for all complex values of s with real Re s> 1, and then defines ζ(s) as the analytic continuation, to the whole complex s plane, of the function given, Re s> 1, − by the sum of the preceding series. Leonhard Euler already considered the above series in 1740, but for positive integer values of s, and later Chebyshev extended the definition to Re s> 1. Godfrey H Hardy and John E Littlewood, “Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Math 41, 119 (1916) Did much of the earlier work, by establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods G H Hardy, Divergent Series (Clarendon Press, Oxford, 1949) Srinivasa I Ramanujan had found for himself the functional equation of the zeta function Torsten Carleman, “Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes" (French), 8. Skand Mat-Kongr, 34-44 (1935) Zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold for the case of a compact region of the plane QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 4/27 Robert T Seeley, “Complex powers of an elliptic operator. 1967 Singular Integrals" (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288-307, Amer. Math. Soc., Providence, R.I. QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 5/27 Robert T Seeley, “Complex powers of an elliptic operator. 1967 Singular Integrals" (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288-307, Amer. Math. Soc., Providence, R.I. Extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 5/27 Robert T Seeley, “Complex powers of an elliptic operator. 1967 Singular Integrals" (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288-307, Amer. Math. Soc., Providence, R.I. Extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization D B Ray, Isadore M Singer, “R-torsion and the Laplacian on Riemannian manifolds", Advances in Math 7, 145 (1971) QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 5/27 Robert T Seeley, “Complex powers of an elliptic operator. 1967 Singular Integrals" (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288-307, Amer. Math. Soc., Providence, R.I. Extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization D B Ray, Isadore M Singer, “R-torsion and the Laplacian on Riemannian manifolds", Advances in Math 7, 145 (1971) Used this to define the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifold in their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace −s ζA(s)= Tr (A) the method defines the possibly divergent infinite product ∞ a = exp[ ζ ′(0)] n − A nY=1 QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 5/27 J. Stuart Dowker, Raymond Critchley “Effective Lagrangian and energy-momentum tensor in de Sitter space", Phys. Rev. D13, 3224 (1976) QFEXT 2011, CC Pedro Pascual, Benasque, Sep 18-24, 2011 – p. 6/27 J. Stuart Dowker, Raymond Critchley “Effective Lagrangian and energy-momentum tensor in de Sitter space", Phys. Rev. D13, 3224 (1976) Abstract The effective Lagrangian and vacuum energy-momentum tensor < T µν > due to a scalar field in a de Sitter space background are calculated using the dimensional-regularization method.

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