00:00 - 17:0017:00 - 23:00
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.
In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if i...
In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non-gravitational fields are present. These are distinct from the electrovacuum solutions, which take into account the electromagnetic field in addition to the gravitational field. Vacuum solutions are also distinct from the lambdavacuum solutions, where the only term in the stress–energy tensor is the cosmological constant term (and thus, the lambdavacuums...
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.For example, an electron theory may begin by postulating an electron with an initial mass...
The Pauli Hamiltonian is
$$H= \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))^2 + q \phi $$
For the 1-component Schrodinger equation in absence of applied field we have the Lagrangian
$$\mathcal{L} = i\hbar\psi^*\dot \psi - \frac{\hbar^2}{2m}\Big( \boldsymbol{\nabla}\psi^*\cdo...5
Consider the two components spinors (usually Weyl spinors): $\psi = (c_1, c_2)^{\top}$, where $c_i$ are complex numbers. The metric in spinor space is usually defined with the second Pauli matrix $\sigma_y$ (many authors are adding an $i$ factor):
\begin{equation}\tag{1}
\epsilon = i \, \sigma_y...
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
== Formulation ==
The association of a spinor with a 2×2 complex Hermitian matrix was formulated by Élie Cartan.In detail, given a vector x = (x1, x2, x3) of real (or complex) numbers, one can associate the complex matrix
x
→
→
X...
2
00:00 - 17:0017:00 - 23:00
