Seminars
完全WKB解析を用いたブラックホール準固有振動の解析について論じる。完全WKB解析は２階常微分方程式の解の大域的挙動を近似無しで与える手法であり、特に固有値の満たすべき条件式を出す際に有用である。本講演ではこの手法を球対称なブラックホールの摂動方程式に適用した、我々の研究arXiv:2503.17245 [hep-th]を紹介する。また、一般相対論を超えた重力理論による補正の効果を取り入れた場合の、高減衰率をもつブラックホール準固有振動数の評価についても論じる。
The identity of a black hole is still mysterious theoretically and observationally. As one candidate for the quantum definition of black holes, we propose that the black hole is the configuration that maximizes the thermodynamic entropy for a fixed surface area. As a first step, we explore this possibility in the 4D semi-classical Einstein equation, and reach a picture that the black hole is a self-gravity condensate of many excited quanta with the maximum entropy. The entropy of the interior quanta agrees with the entropy-area law exactly because of the self-gravity; quantum fluctuations induced by the high curvatures produce a strong pressure, resolving the classical singularity; and the configuration has no horizon but generates almost the same imaging as a classical black hole due to the strong time delay. Therefore, the gravitational condensate can be a candidate for the black hole in the universe. [Phys.Rev.D 111 (2025) 2, 026023]
We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix named as "anti-symmetrized adjacency matrix", specifically tailored for cycle digraphs ($T^1$ lattice) and simple directed paths ($B^1$ lattice).
The nontrivial relation between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the anti-symmetrized adjacency matrix, along with its extensions to higher dimensions, are equivalent to naive lattice Dirac operators.
Building upon these connections, we provide rigorous proofs for two key assertions:
(i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice).
(ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the $D$-dimensional graph results from a cartesian product of cycle digraphs ($T^1$ lattice) and simple directed paths ($B^1$ lattice).
Semi-classical approximation is one of the most commonly used techniques in quantum theories. As it produces factorially divergent series, they have to be analyzed in the context of the resurgence theory. In this talk, I will start with a general discussion on divergent series. After explaining how the non-perturbative quantities are hidden in the divergent behaviour and how they can be obtained via Borel summation, as an example, I will discuss 1 loop effective action in Schwinger's proper time formalism and the non-perturbative pair production in the presence of uniform electric field. Then, in order to handle general non-uniform fields, I will introduce a recursive method to compute a (proper) time dependent perturbative expansion from which the non-perturbative effects are obtained by a Borel-like summation. In the second part of the talk, I will discuss how the recursive expansion is connected to the semi-classical expansion in the WKB formalism and how it offers a generalization to higher dimensional problems which can not be handled by the standard WKB integrals. I will finish the talk with an example on the effective action of the time dependent periodic electric fields in connection with the exact WKB formalism.