Discrete determinants and the Gel’fand–Yaglom formula

Dowker, J. S.

doi:10.1088/1751-8113/45/21/215203

Cited by 7 publications

(13 citation statements)

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“…In particular, if ∆ N is the graph Laplacian of a graph G,∆ 2N is the graph Laplacian of the graph Cartesian product G × P 2 . 12 Then, the use of the formula on deteminants…”

Section: The Graph Cartesian Products G × Pmentioning

confidence: 99%

“…Several years ago, the discrete Gel'fand-Yaglom method for difference operators was reviewed and studied by Dowker [12]. We generalize the discrete Gel'fand-Yaglom method for studying one-loop vacuum energies in extended deconstructed theories and models with discrete dimensions in the present paper.…”

Section: Introductionmentioning

confidence: 99%

“…The method to obtain the determinant of tridiagonal matrices in this section is substantially equivalent to the method for difference operators described by Dowker [12], except for a specific choice for an Hermitian matrix in the present section.…”

Section: Introductionmentioning

confidence: 99%

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Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

Kan

Shiraishi

2019

Advances in Mathematical Physics

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The discrete Gel'fand-Yaglom theorem was studied several years ago. In the present paper, we generalize the discrete Gel'fand-Yaglom method to obtain the determinants of mass matrices which appear current works in particle physics, such as dimensional deconstruction and clockwork theory. Using the results, we show the expressions for vacuum energies in such various models. PACS: 02.10.Ox, 04.60.Nc, 11.10.Kk, 11.25.Mj.After completion of the first version of the manuscript of the present paper (arXiv:1711.06806), a paper which treats the determinants of discrete Laplace operators appeared [13]. Their method is substantially the same as ours, because the author also relies on the recurrence relation among three variables on a lattice (see Sec. 3 in the present paper and below). We recently become aware of another similar paper on the determinants of matrix differential operators [14]. They studied generalization of Gel'fand-Yaglom method to obtain the functional determinants. Their work differs essentially from ours because they considered differential operators while we treat matrices as operators. We also point out that they did not consider the matrices of large size which have certain continuum limits.The organization of this paper is as follows. In order to make the present paper selfcontained, we show a short review of the Gel'fand-Yaglom method for a differential operator, along with the Dunne's review [2], in Sec. 2. In Sec. 3, we give the method to obtain determinants of tridiagonal matrices with repeated structure. This is a straightforward generalization of description in Ref. [12]. In Sec. 4, we give the method to obtain determinants of periodic tridiagonal matrices. Determinants of extended periodic tridiagonal matrices are obtained in Sec. 5. The rest of the present paper is devoted to applications to deconstructed theories and discrete systems. In Sec. 6, free energy on a graph is discussed by using the results of previous sections. In Sec. 7, we show the method of calculation for evaluating one-loop vacuum energy in deconstructed models from the determinants of mass matrices. In Sec. 8, we show a few more examples of one-loop vacuum energies for slightly complicated theory spaces. We give conclusions in the last section, Sec. 9.

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“…In particular, if ∆ N is the graph Laplacian of a graph G,∆ 2N is the graph Laplacian of the graph Cartesian product G × P 2 . 12 Then, the use of the formula on deteminants…”

Section: The Graph Cartesian Products G × Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

Kan

Shiraishi

2019

Advances in Mathematical Physics

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“…Fifty years ago Gel'fand and Yaglom [16] showed that the functional determinant of a one-dimensional second order differential (Schrödinger) operator O with the homogeneous Dirichlet boundary conditions (BCs) can be expressed explicitly using only the solutions of the homogeneous equation Oh = 0. This remarkable result has been extended to higher order [17,18], discrete [19] and Sturm-Liouville operators [20], as well as to more abstract objects, such as vector bundles [21,22] and non-compact Riemannian manifolds [23]. The Gel'fand and Yaglom formula for the homogeneous Dirichlet BCs has been also extended by Forman [24] to more general (elliptic) BCs, as well as to partial differential operators.…”

Section: Introductionmentioning

confidence: 99%

On functional determinants of matrix differential operators with multiple zero modes

Falco¹,

Fedorenko²,

Gruzberg³

2017

J. Phys. A: Math. Theor.

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We generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by McKane and Tarlie to differential operators with multiple zero eigenvalues. We derive general formulas for such functional determinants of r × r matrix second order differential operators O with 0 < n 2r linearly independent zero modes. We separately discuss the cases of the homogeneous Dirichlet boundary conditions, when the number of zero modes cannot exceed r, and the case of twisted boundary conditions, including the periodic and anti-periodic ones, when the number of zero modes is bounded above by 2r. In all cases the determinants with excluded zero eigenvalues can be expressed only in terms of the n zero modes and other r − n or 2r − n (depending on the boundary conditions) solutions of the homogeneous equation Oh = 0, in the spirit of Gel'fand-Yaglom approach. In instanton calculations, the contribution of the zero modes is taken into account by introducing the so-called collective coordinates. We show that there is a remarkable cancellation of a factor (involving scalar products of zero modes) between the Jacobian of the transformation to the collective coordinates and the functional fluctuation determinant with excluded zero eigenvalues. This cancellation drastically simplifies instanton calculations when one uses our formulas.

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“…As the functional determinants usually diverge, this formula should be understood either in the discrete setting [14] or using some regularisation, for example, by considering a ratio of two functional determinants with and without V (x) [2,10,12,13,14]. Thus in one dimension, functional determinants can be computed without knowing the eigenvalues explicitly.…”

Section: Introductionmentioning

confidence: 99%

Gelfand-Yaglom formula for functional determinants in higher dimensions

Ossipov

2018

Preprint

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The Gelfand-Yaglom formula relates functional determinants of the onedimensional second order differential operators to the solutions of the corresponding initial value problem. In this work we generalise the Gelfand-Yaglom method by considering discrete and continuum partial second order differential operators in higher dimensions. To illustrate our main result we apply the generalised formula to the twodimensional massive and massless discrete Laplace operators and calculate asymptotic expressions for their determinants.

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Discrete determinants and the Gel’fand–Yaglom formula

Cited by 7 publications

References 30 publications

Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

On functional determinants of matrix differential operators with multiple zero modes

Gelfand-Yaglom formula for functional determinants in higher dimensions

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