Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, ...
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Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.Less
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.
Simon Donaldson
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0001
- Subject:
- Mathematics, Geometry / Topology, Analysis
This chapter reviews some examples of holomorphic functions in complex analysis. It emphasizes the idea of ‘analytic continuation’, which is a fundamental motivation for Riemann surface theory.
This chapter reviews some examples of holomorphic functions in complex analysis. It emphasizes the idea of ‘analytic continuation’, which is a fundamental motivation for Riemann surface theory.
Norman J. Morgenstern Horing
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198791942
- eISBN:
- 9780191834165
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198791942.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the ...
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Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.Less
Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
Ercüment H. Ortaçgil
- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198821656
- eISBN:
- 9780191860959
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821656.003.0004
- Subject:
- Mathematics, Pure Mathematics
If the nonlinear curvature of a splitting vanishes, then the local solutions form a pseudogroup, and we the manifold is called a local Lie group. This is a global concept!
If the nonlinear curvature of a splitting vanishes, then the local solutions form a pseudogroup, and we the manifold is called a local Lie group. This is a global concept!
Norman J. Morgenstern Horing
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198791942
- eISBN:
- 9780191834165
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198791942.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator ...
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Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.Less
Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.
Jean Zinn-Justin
- Published in print:
- 2021
- Published Online:
- June 2021
- ISBN:
- 9780198834625
- eISBN:
- 9780191872723
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198834625.003.0042
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
In general, a linear combination of instanton solutions is not a solution of the imaginary-time equations of motion, because the equations not linear. Moreover, in quantum mechanics (QM), all ...
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In general, a linear combination of instanton solutions is not a solution of the imaginary-time equations of motion, because the equations not linear. Moreover, in quantum mechanics (QM), all solutions of the classical equations can depend only on one time collective coordinate (in this respect, in field theory, the situation is different). However, a linear combination of largely separated instantons (a multi-instanton configuration) renders the action almost stationary, because each instanton solution differs, at large distances, from a constant solution by only exponentially small corrections (in field theory this is only true if the theory is massive). A situation where multi-instantons play a role is provided by large order behaviour estimates of perturbation theory for potentials with degenerate minima. When one starts from a situation in which the minima are almost degenerate, one obtains, in the degenerate limit, a contribution of the superposition of two, infinitely separated, instantons, but with an infinite multiplicative coefficient. Indeed, in this limit, the fluctuations which tend to change the distance between the instanton and the anti-instanton induce a vanishingly small variation of the action. To correctly determine the limit, one has to introduce a second collective coordinate which describes these fluctuations. The determination, at leading order, of all many-instanton contributions has led to conjecture the exact form of the semi-classical expansion for potentials with degenerate minima, generalizing the exact Bohr-Sommerfeld quantization condition.Less
In general, a linear combination of instanton solutions is not a solution of the imaginary-time equations of motion, because the equations not linear. Moreover, in quantum mechanics (QM), all solutions of the classical equations can depend only on one time collective coordinate (in this respect, in field theory, the situation is different). However, a linear combination of largely separated instantons (a multi-instanton configuration) renders the action almost stationary, because each instanton solution differs, at large distances, from a constant solution by only exponentially small corrections (in field theory this is only true if the theory is massive). A situation where multi-instantons play a role is provided by large order behaviour estimates of perturbation theory for potentials with degenerate minima. When one starts from a situation in which the minima are almost degenerate, one obtains, in the degenerate limit, a contribution of the superposition of two, infinitely separated, instantons, but with an infinite multiplicative coefficient. Indeed, in this limit, the fluctuations which tend to change the distance between the instanton and the anti-instanton induce a vanishingly small variation of the action. To correctly determine the limit, one has to introduce a second collective coordinate which describes these fluctuations. The determination, at leading order, of all many-instanton contributions has led to conjecture the exact form of the semi-classical expansion for potentials with degenerate minima, generalizing the exact Bohr-Sommerfeld quantization condition.
