解析力学とかのブクマしたツイート（'23年10月～12月分）

うべゆうと @uveyuto

Liouville方程式を正準量子化したらvon-Neumann方程式(Schrödinger描像)が出てきて、古典物理量のPoisson括弧による時間発展の式を正準量子化したらHeisenberg方程式(Heisenberg描像)が出てくるの不思議やなぁ

Haruki Watanabe @haruki_wtnb

ところで、講義中に生じた正準変換についての基本的な疑問で、まだ解決できていないものがあります。ご存知の方がいたら教えて下さい🙇‍♂️🙇‍♂️ 一言で言うと「全ての正準変換は無限小変換の合成で書けるか否か」という問題です。証明もできていなければ反例も見つけられていません（が流石に既知のはず）。

Physics In History @PhysInHistory

Noether’s theorem is a very important result in physics and mathematics that relates the symmetries of a physical system to the conservation laws that govern its behavior. It was discovered by Emmy Noether, a brilliant German mathematician, in 1915. To understand Noether’s theorem, we need to know some basic concepts of Lagrangian mechanics, which is a way of describing the dynamics of a system using a function called the Lagrangian. The Lagrangian depends on the coordinates and velocities of the system, and it encodes the information about the kinetic and potential energy of the system. The principle of least action states that the system will evolve in such a way that the action, which is the integral of the Lagrangian over time, is minimized. The equations of motion of the system can be derived from the Lagrangian using the Euler-Lagrange equations. A symmetry of a physical system is a transformation that does not change the physics of the system. For example, if we shift the origin of our coordinate system, or rotate it, or change the time origin, the system will behave the same way as before. Mathematically, a symmetry is a transformation that leaves the Lagrangian unchanged, or changes it by a total time derivative, which does not affect the action. Noether’s theorem states that for every continuous symmetry of the system, there is a quantity that is conserved, meaning that it does not change over time. This quantity is called the Noether charge, and it can be calculated from the symmetry transformation and the Lagrangian. Some examples of Noether’s theorem are: If the system is invariant under spatial translations (meaning that it does not depend on where we place the origin of our coordinate system), then the Noether charge is the linear momentum, which is conserved. If the system is invariant under rotations (meaning that it does not depend on how we orient our coordinate system), then the Noether charge is the angular momentum, which is conserved. If the system is invariant under time translations (meaning that it does not depend on when we start measuring time), then the Noether charge is the energy, which is conserved. Noether’s theorem can also be applied to more abstract symmetries, such as gauge symmetries, which are transformations that change the fields of the system in a specific way. For example, in electromagnetism, the electric and magnetic fields are invariant under a gauge transformation, which changes the scalar and vector potentials by a gradient of a function. The Noether charge associated with this symmetry is the electric charge, which is conserved. The theorem is very powerful and elegant, as it reveals the deep connection between the symmetries of nature and the conservation laws that we observe. It also helps us to construct new theories of physics, by imposing the symmetries that we want to have and finding the corresponding Lagrangians and conservation laws. Noether’s theorem is one of the most beautiful and profound discoveries in the history of science.

QmQ @gejiqmq

数学者と物理学者の最大の違いは、前者の興味の中心が数学的な現象で、後者は物理的な現象だ、という当たり前のところにあるんで、何も妙な区別をしなくても良いと思う。数学者じゃないからよくわからんけど、彼らもつまらない例外的な話よりも、より広い話題とつながりがある理論に興味があると思うが