equations-of-physics

"It has been known for some time that the Yang-Baxter equations can be solved using s. More recently it was discovered … that the YBE for the N state could be solved using special curves of (N – 1)2."

"At an early stage the Yang-Baxter equation (YBE) appeared in several different guises in the literature, and sometimes its solutions have preceded the equation. One can trace basically three streams of ideas from which YBE has emerged: the , commuting in statistical mechanics, and factorizable in field theory."

"About 40 years ago, in the study of quantum s … , in particular in the framework of the … , new algebraic structures arose, the generalizations of which were later called quantum groups … The Yang-Baxter equations became a unifying basis of all these investigations. The most important nontrivial examples of quantum groups are quantizations (or deformations) of ordinary classical s and algebras (more precisely, one considers the deformations of the algebra of functions of a Lie group and the universal enveloping of a Lie algebra). The quantization is accompanied by the introduction of an additional parameter q (the deformation parameter), which plays a role analogous to the role of in quantum mechanics. In the limit q → 1, the quantum groups and algebras go over into the classical ones."

"The origins of es is one of the biggest mysteries in modern physics since they are beyond the realm of the Standard Model. As massive particles, neutrinos undergo throughout their propagation. In this paper we show that when a neutrino oscillates from a flavor state α to a flavor state β, it follows three possible paths consistent with the Quantum Yang- Baxter Equations. These trajectories define the transition probabilities of the oscillations. Moreover, we define a probability matrix for flavor transitions consistent with the Quantum Yang-Baxter Equations, and estimate the values of the three neutrino mass eigenvalues within the framework of the triangular formulation."

"With the growing importance of models in statistical mechanics and in field theory, the path integral method of Feynman was soon recognized to offer frequently a more general procedure of enforcing the first quantization instead of the Schrödinger equation. To what extent the two methods are actually equivalent, has not always been understood. ... the Coulomb potential and the harmonic oscillator ... point the way: For scattering problems the path integral seems particularly convenient, whereas for the calculation of discrete security eigenvalues the Schrödinger equation."

"Interactions that look instantaneous are well suited to Schrödinger’s equation, which requires the potential between particles at equal times. It would be quite awkward to explicitly describe finite-velocity forces in the Schrödinger equation because the potential for one particle at a time t would depend on the positions of the others at the retarded times, and one would need the past histories of all the particles to propagate the system forward in time."

"Just four weeks after the first paper (Q1) the Annalen received on February 23 the second paper (Q2) in the series 'Quantization as an Eigenvalue Problem'. ... It consists of a detailed exploration of the Hamiltonian analogy between mechanics and optics leading to a new derivation of the wave equation, an analysis of the relations between security making geometrical and undulatory mechanics, and applications of the wave equation to the harmonic oscillator and the diatomic molecule."

"We have a wave which leaves the material source and goes outward at the velocity c, which is the speed of light. ... From a historical point of view, it wasn’t known that the coefficient c in Maxwell’s equations was also the speed of light propagation. There was just a constant in the equations. We have called it c from the beginning, because we knew what it would turn out to be. We didn’t think it would be sensible to make you learn the formulas with a different constant and then go back to substitute c wherever it belonged. ... just by experiments with charges and currents we find a number c2 which turns out to be the square of the velocity of propagation of electromagnetic influences. From static measurements—by measuring the forces between two unit charges and between two unit currents—we find that c = 3.00 × 108 meters/sec. When Maxwell first made this calculation with his equations, he said that bundles of electric and magnetic fields should be propagated at this speed. He also remarked on the mysterious coincidence that this was the same as the speed of light. “We can scarcely avoid the inference,” said Maxwell, “that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” Maxwell had made one of the great unifications of physics. Before his time, there was light, and there was electricity and magnetism. The latter two had been unified by the experimental work of Faraday, Oersted, and Ampère. Then, all of a sudden, light was no longer “something else,” but was only electricity and magnetism in this new form—little pieces of electric and magnetic fields which propagate through space on their own."

"... applications in gauge field theories and the physics of condensed matter. The starting point here is now quite well known: expressing the Maxwell equations for an electromagnetic field over Lorentz space as the Euler-Lagrange equations for a Lagrangian defined on the connections on a U(1) bundle, where the electromagnetic potential becomes the connection and the field tensor its curvature. The freedom of choice of "gauge" for the potential is a fundamental fact which stems, in the geometrical picture, from the lack of a preferred trivialisation of the bundle."

"But the mathematicians of the nineteenth century failed miserably to grasp the equally great opportunity offered to them in 1865 by Maxwell. If they had taken Maxwell's equations to heart as Euler took Newton's, they would have discovered, among other things, Einstein's theory of special relativity, the theory of s and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis. A great part of twentieth century physics and mathematics could have been created in the nineteenth century, simply by exploring to the end the mathematical concepts to which Maxwell's equations naturally lead."

"... the fullest statement Maxwell gave of his theory, his 1873 Treatise on Electricity and Magnetism, does not contain the four famous "Maxwell's equations," nor does it even hint at how electromagnetic waves might be produced or detected. These and many other aspects of the theory were quite thoroughly hidden in the version of it given by Maxwell himself; in the words of Oliver Heaviside, they were "latent" but hardly "patent." ... Maxwell was only forty-eight when he died of cancer in November 1879. ... the task of digging out the "latent" aspects of his theory and of exploring its wider implications was thus left to a group of younger physicists, most of them British."

"... the original field equations explicitly contain the magnetic vector potential, \overrightarrow{A} ... In Maxwell's original formulaton, Faraday's \overrightarrow{A} field was central and had physical meaning. The magnetic vector potential was not arbitrary, as defined by boundary conditions and choice of gauge as we will discuss; they were said to be gauge invariant. The original equations are thus often called the Faraday-Maxwell theory."

"The general equations are next applied to the case of a magnetic disturbance propagated through a non-conductive field, and it is shown that the only disturbances which can be so propagated are those which are transverse to the direction of propagation, and that the velocity of propagation is the velocity v, found from experiments such as those of Weber, which expresses the number of electrostatic units of electricity which are contained in one electromagnetic unit. This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws."

"Whenever I teach a course on electromagnetism, one of the first ... exam questions I will ask students is, "Why is the magnetic field called H in the textbook?" ... The reason it is called H is, of course, Maxwell did not use the vector notation. He writes out all the components. So the electric field he starts with E ... And so, the electric field is E, F, G, and the magnetic field is, obviously, H, I, J. ... So all six components are written out. And so, it's very misleading for a textbook nowadays to say there are four Maxwell's equations. There were actually, if you look at Maxwell's papers, something like twenty-four equations. And the whole thing looks incredibly complicated."

"The wave equation was quickly followed by remarkably similar equations for gravitation, electrostatics, elasticity, and heat flow. Many bore the names of their inventors: Laplace's equation, Poisson's equation. The equation for heat does not; it bears the unimaginative and not entirely accurate name 'heat equation'. It was introduced by Joseph Fourier, and his ideas led to the creation of a new area of mathematics whose ramifications were to spread far beyond its original source."

"The wave equation behaves nicely in one dimension and in three dimensions but not in two dimensions. In one dimension, waves on a uniform string propagate without distortion. In three dimensions, waves in a homogeneous isotropic medium propagate in an undistorted way except for a spherical correction factor. However, in two dimensions, wave propagation is complicated and distorted. By its very nature, 2D processing never can account for events originating outside of the plane. As a result, 2D processing is broken up into a large number of approximate partial steps in a sequence of operations. These steps are ingenious, but they can never give a true image."

"Dirac's holes, now called s, are no longer a marvel, but a tool. A notable use is... PET scans..."

"Dirac's equation consists of... four separate s to describe electrons. Two components have an... immediately successful interpretation... describing the two possible directions of an electron's spin. ...The extra ...equations contain solutions with negative energy... Assuming Dirac's equation, if you start with an electron in one of the positive-energy solutions, you can calculate the rate for it to emit a photon and move into one of the negative-energy solutions. Energy must be conserved, but that... means... the emitted photon has higher energy than the electron that emitted it! ...Dirac was well aware of this problem. ...He proposed ...empty' space ...contains electrons obeying all the negative-energy solutions. ... A positive-energy electron can't go to a negative energy solution, because there's always another electron already there, and the won't allow a second... [T]he idea... the ordinary state of 'empty' space is far from empty... a different word for it... is 'vacuum'... a medium, with dynamical properties... [S]hine light [photons with enough energy] on the vacuum... then a negative-energy electron can absorb... [a] photon... and go into a positive-energy solution... an ordinary electron... But in the final state there is... a hole... originally occupied by the negative-energy electron... [I]f there is a pre-existing hole... a positive-energy electron can emit a photon and occupy the vacant negative-energy solution. ...Dirac's first hole-theory paper was... 'A theory of electrons and protons'."

"Possibly... [Dirac's] biggest achievement was... the equation for the electron... a sort of extension to Schrödinger's equation, which allowed quantum theory to be combined with Einstein's special theory of relativity. ...Dirac's equation is very ...similar to Schrödinger's equation, the difference being that these ...are not numbers, they are matrices, and this is one of the ...great equations of all time ...Farmelo wrote a book about the eighteen greatest formulae of all time, and that's one of them. ...This equation ...explained other things about the electron. These have been experimentally verified to huge precision.{{center|1=\left(\beta mc^2 + c\; (\sum_{n = 1}^{3}\alpha_n p_n)\right) \psi (z,t) = i \hbar \frac{\partial\psi(z,t) }{\partial t} }}"

"[E]quations developed by Heisenberg and Schrodinger did not take... from Einstein's relativistic mechanics, but from the old mechanics of Newton. ...[E]xperimental data on atomic spectra... were so accurate that small deviations from the Heisenberg-Schrodinger predictions could be observed. So there was a strong 'practical' motivation... [A]ncient and fundamental dichotomies were in play: ...light versus matter; ...continuous versus discrete ...tremendous barriers to ...achieving a unified description of nature ...[[Special relativity|[R]elativity]] was the child of light and the continuum ...quantum theory the child of matter and the discrete. After Dirac's revolution... all were reconciled, in the... conceptual amalgam... a quantum field. Maxwell's electrodynamics is a continuum theory of electric and magnetic fields, and of light, that makes no mention of . Newton's mechanics is a theory of discrete particles, whose only mandatory properties are mass and ."

"There is one topic I was not sorry to skip: the relativistic wave equation of Dirac. It seems to me that the way this is usually presented in books on quantum mechanics is profoundly misleading. Dirac thought that his equation was a relativistic generalization of the non-relativistic time-dependent Schrödinger equation that governs the probability amplitude for a point particle in an external electromagnetic field. For some time after, it was considered to be a good thing that Dirac’s approach works only for particles of spin one half, in agreement with the known spin of the electron, and that it entails negative energy states, states that when empty can be identified with the electron’s antiparticle. Today we know that there are particles like the W± that are every bit as elementary as the electron, and that have distinct antiparticles, and yet have spin one, not spin one half. The right way to combine relativity and quantum mechanics is through the quantum theory of fields, in which the Dirac wave function appears as the matrix element of a quantum field between a one-particle state and the vacuum, and not as a probability amplitude."

"In... 1928... Dirac... a twenty-five-year-old recent convert from electrical engineering to theoretical physics, produced... the [remarkable] Dirac equation. ...He wanted to ...describe the behaviour of electrons more accurately ...[Previous] equations had either incorporated special relativity or quantum mechanics, but not both. ...Unlike other physicists, and ...Newton and Maxwell, Dirac did not proceed from a minute study of experimental facts. ...By 'playing with equations' he hit upon...[the] uniquely simple, elegant ...Dirac equation ...[which] predicts ...electrons are ...spinning and ...act as ...bar magnets, and [predicts] the rate of the spin and ... strength of ...magnetism."

"The saddest chapter of modem physics is and remains the Dirac theory... In order not to be irritated with Dirac I have decided to do something else for a change..."