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Quantization of the Riemann zetafunction and cosmology
"... Quantization of the Riemann zetafunction is proposed. We treat the Riemann zetafunction as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of padic strings and by recent works on stringy cosmologi ..."
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Quantization of the Riemann zetafunction is proposed. We treat the Riemann zetafunction as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of padic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zetafunction field is equivalent to the sum of the KleinGordon Lagrangians with masses defined by the zeros of the Riemann zetafunction. Quantization of the mathematics of FermatWiles and the Langlands program is indicated. The Beilinson conjectures on the values of Lfunctions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zetafunction field theory are discussed. 1 1
A NonArchimedean Wave Equation
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"... Let K be a nonArchimedean local field with the normalized absolute value  · . It is shown that a “plane wave ” f(t + ω1x1 + · · · + ωnxn), where f is a BruhatSchwartz complexvalued test function on K, (t,x1,...,xn) ∈ Kn+1, max 1≤j≤n ωj  = 1, satisfies, for any f, a certain homogeneous ps ..."
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Let K be a nonArchimedean local field with the normalized absolute value  · . It is shown that a “plane wave ” f(t + ω1x1 + · · · + ωnxn), where f is a BruhatSchwartz complexvalued test function on K, (t,x1,...,xn) ∈ Kn+1, max 1≤j≤n ωj  = 1, satisfies, for any f, a certain homogeneous pseudodifferential equation, an analog of the classical wave equation. A theory of the Cauchy problem for this equation is developed.
NUMBER THEORY IN PHYSICS
"... always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation ..."
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always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The