Simple applications of Noether's first theorem in quantum mechanics and electromagnetism

Dirac-von Neumann 型经典电磁学公理化结构

Dirac-von Neumann Type Axiomatic Structure for Classical Electromagnetism

本文为经典电磁学建立了一个基于 Dirac-von Neumann 公理的希尔伯特空间结构，以经典势代替波函数并去除坍缩假设，推导出波对易关系、经典海森堡不确定性原理及相对论性 Feynman 证明，揭示经典与量子力学的本质差异在于 $\hbar$。

Noether conservation laws in quantum mechanics

Being quantized, conserved Noether symmetry functions are represented by Hermitian operators in the space of solutions of the Schrodinger equation, and their mean values are conserved.

Lipkin's conservation law, Noether's theorem, and the relation to optical helicity

A simple conserved quantity for electromagnetic fields in vacuum was discovered by Lipkin in 1964. In recent years this "zilch" has been used as a measure of the chirality of light. The conservation of optical zilch is here derived from a simple symmetry of the standard electromagnetic action. The symmetry transformation allows the identification of circularly polarized plane waves as zilch eigenstates. The same symmetry is present for electromagnetism in a homogeneous, dispersive medium, allowing the derivation of the zilch density and flux in such a medium. Optical helicity density and flux …

Casimir effect from macroscopic quantum electrodynamics

The canonical quantization of macroscopic electromagnetism was recently presented in New J. Phys. 12 (2010) 123008. This theory is here used to derive the Casimir effect, by considering the special case of thermal and zero-point fields. The stress-energy-momentum tensor of the canonical theory follows from Noether's theorem, and its electromagnetic part in thermal equilibrium gives the Casimir energy density and stress tensor. The results hold for arbitrary inhomogeneous magnetodielectrics and are obtained from a rigorous quantization of electromagnetism in dispersive, dissipative media. Conti…

Optical Activity of a Dilute Dielectric Medium: Pasteur and the Molecular Symmetries

In 1848 Pasteur conjectured that the rotation of the polarization plane of the light in a dilute dielectric medium is generated by the molecular symmetries of the medium where the light propagates. Our objective is to show that Pasteur hypothesis was correct using basic knowledge of electromagnetism and quantum mechanics. In Sections 2-5 we present a brief review of the fundamental concepts of the electromagnetism necessary to study the optical activity. In Sections 6-8 using the quantum mechanical perturbation theory and taking into account the molecular symmetries we calculate the optical ac…

On Galilean invariance and nonlinearity in electrodynamics and quantum mechanics

Recent experimental results on slow light heighten interest in nonlinear Maxwell theories. We obtain Galilei covariant equations for electromagnetism by allowing special nonlinearities in the constitutive equations only, keeping Maxwell's equations unchanged. Combining these with linear or nonlinear Schroedinger equations, e.g. as proposed by Doebner and Goldin, yields a Galilean quantum electrodynamics.

Non-Commutative Worlds

This paper shows how the forms of gauge theory, Hamiltonian mechanics and quantum mechanics arise from a non-commutative framework for calculus and differential geometry. Discrete calculus is seen to fit into this pattern by reformulating it in terms of commutators. Differential geometry begins here, not with the concept of parallel translation, but with the concept of a physical trajectory and the algebra related to the Jacobi identity that governs that trajectory. The paper discusses how Poisson brackets give rise to Jacobi identity, and how Jacobi identity arises in combinatorial contexts, …

Quantum mechanics of time-dependent systems. Construction of pure states

For time-dependent systems the wavefunction depends explicitly on time and it is not a pure state of the Hamiltonian. We construct operators for which the above wavefunction is a pure state. The method is based on the introduction of conserved quantities $Q$ and the pure states are defined by ${\hat Q}ψ=qψ$. The conserved quantities are constructed using parametrised mechanics and the Noether theorem.

Pseudo-chirality: a manifestation of Noether's theorem in non-Hermitian systems

Noether's theorem relates constants of motion to the symmetries of the system. Here we investigate a manifestation of Noether's theorem in non-Hermitian systems, where the inner product is defined differently from quantum mechanics. In this framework, a generalized symmetry which we term pseudo-chirality emerges naturally as the counterpart of symmetries defined by a commutation relation in quantum mechanics. Using this observation, we reveal previously unidentified constants of motion in non-Hermitian systems with parity-time and chiral symmetries. We further elaborate the disparate implicati…

From Lagrangian to Quantum Mechanics with Symmetries

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schrödinger equation is the key that takes classical mechanics into quantu…

深入探究诺特定理

Getting to the Bottom of Noether's Theorem

本文采用代数方法比较经典与量子诺特定理，揭示 Alfsen 和 Shultz 提出的动力学对应第二条件表达了量子与统计力学间的关系，与“逆温度是虚时间”原理密切相关。

Conservation of the spin and orbital angular momenta in electromagnetism

We review and re-examine the description and separation of the spin and orbital angular momenta (AM) of an electromagnetic field in free space. While the spin and orbital AM of light are not separately-meaningful physical quantities in orthodox quantum mechanics or classical field theory, these quantities are routinely measured and used for applications in optics. A meaningful quantum description of the spin and orbital AM of light was recently provided by several authors, which describes separately conserved and measurable integral values of these quantities. However, the electromagnetic fiel…

Constrained Mechanics and Noiseless Subsystems

Many theories are formulated as constrained systems. We provide a mechanism that explains the origin of physical states of a constrained system by a process of selection of noiseless subsystems when the system is coupled to an external environment. Effectively, physical states that solve all the constraints are selected by a passive error correction scheme which has been developed in the context of quantum information processing. We apply this mechanism to several constrained theories including the relativistic particle, electromagnetism, and quantum gravity, and discuss some interesting (and …

Extension of Noether's theorem in PT-symmetric systems and its experimental demonstration in an optical setup

Noether's theorem is one of the fundamental laws in physics, relating the symmetry of a physical system to its constant of motion and conservation law. On the other hand, there exist a variety of non-Hermitian parity-time (PT)-symmetric systems, which exhibit novel quantum properties and have attracted increasing interest. In this work, we extend Noether's theorem to a class of significant PT -symmetric systems for which the eigenvalues of the PT-symmetric Hamiltonian H change from purely real numbers to purely imaginary numbers,and introduce a generalized expectation value of an operator base…

Stochastic Fractal and Noether's Theorem

We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability $p$ or disappears with probability $1\!-\!p$. It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter $α$, which also determines the fragmentation rate. For a fractal dimension $d_f$, we find that the $d_f$-th mo…

A Geometric Approach to the Quantum Mechanics of de Broglie and Vigier

Following de Broglie and Vigier, a fully relativistic causal interpretation of quantum mechanics is given within the context of a geometric theory of gravitation and electromagnetism. While the geometric model shares the essential principles of the causal interpretation initiated by de Broglie and advanced by Vigier, the particle and wave components of the theory are derived from the Einstein equations rather than a nonlinear wave equation. This geometric approach leads to several new features, including a solution to the de Broglie variable mass problem.

Quantum Mechanical Black Holes: Towards a Unification of Quantum Mechanics and General Relativity

In this paper, starting from vortices we are finally lead to a treatment of Fermions as Kerr-Newman type Black Holes wherein we identify the horizon at the particle's Compton wavelength periphery. A naked singularity is avoided and the singular processes inside the horizon of the Black Hole are identified with Quantum Mechanical effects within the Compton wavelength. Inertial mass, gravitation, electromagnetism and even QCD type interactions emerge from such a description including relative strengths and also other features like the anomalous gyromagnetic ratio, the discreteness of the charge,…

Noether Symmetries, Dynamical Constants of Motion, and Spectrum Generating Algebras

When discussing consequences of symmetries of dynamical systems based on Noether's first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether's first theorem. Rather a much less restrictive statement applies,…

Dynamics-dependent symmetries in Newtonian mechanics

We exhibit two symmetries of one-dimensional Newtonian mechanics whereby a solution is built from the history of another solution via a generally nonlinear and complex potential-dependent transformation of the time. One symmetry intertwines the square roots of the kinetic and potential energies and connects solutions of the same dynamical problem (the potential is an invariant function). The other symmetry connects solutions of different dynamical problems (the potential is a scalar function). The existence of corresponding conserved quantities is examined using Noethers theorem and it is show…

The quantization of gravity - an introduction

This is an introduction to quantum gravity, aimed at a fairly general audience and concentrating on what have historically two main approaches to quantum gravity: the covariant and canonical programs (string theory is not covered). The quantization of gravity is discussed by analogy with the quantization of the electromagnetic field. The conceptual and technical problems of both approaches are discussed, and the paper concludes with a discussion of evidence for quantum gravity from the rest of physics. The paper assumes some familiarity with non-relativistic quantum mechanics, special relativi…

Emergent quantum mechanics as a thermal ensemble

It has been argued that gravity acts dissipatively on quantum-mechanical systems, inducing thermal fluctuations that become indistinguishable from quantum fluctuations. This has led some authors to demand that some form of time irreversibility be incorporated into the formalism of quantum mechanics. As a tool towards this goal we propose a thermodynamical approach to quantum mechanics, based on Onsager's classical theory of irreversible processes and on Prigogine's nonunitary transformation theory. An entropy operator replaces the Hamiltonian as the generator of evolution. The canonically conj…