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H1 Angles and Symmetry
H1 Angles and Symmetry

NMR in metals, metal particles and metal cluster compounds
NMR in metals, metal particles and metal cluster compounds

... the one-electron problem of the stationary states of the hydrogen atom, introducing the Aufbau principle. The physical viewpoint is an extension of the problem of a single particle in a box with rigid walls. In the first case, the prevailing symmetry is spherical, and the ‘natural’ quantum numbers o ...
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... closed because you can return to your starting point in both space and time. It is called “timelike” because the time changes from point to point along the curve. The statement that a closed timelike curve exists is just a fancy way of saying that you have a time machine. It would seem that time tra ...
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Classical and Quantum Trajectory-based Approaches to Electron

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Worked Examples from Introductory Physics Vol. IV: Electric Fields

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Theoretical Studies of Ultrafast Correlated Electron

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Lecture Notes on General Relativity

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6.013 Electromagnetics and Applications, Course Notes
6.013 Electromagnetics and Applications, Course Notes

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Introduction to gauge theory

A gauge theory is a type of theory in physics. Modern theories describe physical forces in terms of fields, e.g., the electromagnetic field, the gravitational field, and fields that describe forces between the elementary particles. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated quantities can be measured, such as charges, energies, and velocities. In field theories, different configurations of the unobservable fields can result in identical observable quantities. A transformation from one such field configuration to another is called a gauge transformation; the lack of change in the measurable quantities, despite the field being transformed, is a property called gauge invariance. Since any kind of invariance under a field transformation is considered a symmetry, gauge invariance is sometimes called gauge symmetry. Generally, any theory that has the property of gauge invariance is considered a gauge theory. For example, in electromagnetism the electric and magnetic fields, E and B, are observable, while the potentials V (""voltage"") and A (the vector potential) are not. Under a gauge transformation in which a constant is added to V, no observable change occurs in E or B.With the advent of quantum mechanics in the 1920s, and with successive advances in quantum field theory, the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, because all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. Over the course of the 20th century, physicists gradually realized that all forces (fundamental interactions) arise from the constraints imposed by local gauge symmetries, in which case the transformations vary from point to point in space and time. Perturbative quantum field theory (usually employed for scattering theory) describes forces in terms of force-mediating particles called gauge bosons. The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the Standard Model, a quantum field theory that accurately predicts all of the fundamental interactions except gravity.
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