Hot answers tagged classical-field-theory
10
Wu and Yang (1968) found a static solution to the sourceless SU(2) Yang-Mills equations, (please, see the following two relatively recent articles containing a rather detailed description of the solution: Marinho, Oliveira, Carlson, Frederico and Ngome
The solution constitutes of a generalization of the Abelian Dirac monopole. The vector potential is given ...
2
In my opinion it is better to work in an explicit covariant form. In my answer I will use two different definitions, the Greek indexes always run from $0$ to $3$ and Latin indexes from $1$ to $3$ and the metric $g_{\mu\nu}$ has signature $(-1,1,1,1)$.
To translate the expressions to a explicit covariant form we define some timelike vector field $v^\mu$. We ...
2
Consider the $4\times 4$ matrix $g_{\mu\nu}$ with zeroth row $g_{0\nu}$.
Now for $i=1,2,3$, add to the $i$'th row the zeroth row times $-g_{i0}/g_{00}$.
This produces the following matrix
$$\begin{bmatrix}
g_{00} & g_{01} & g_{02}& g_{03} \\
0 & -\gamma_{11} & -\gamma_{12}& -\gamma_{13} \\
0 & -\gamma_{21} & ...
1
I) Here is at least a partial answer. Assume the following set-up. Let there be given a classical Lagrangian field theory in $d+1$ spacetime dimensions, with dynamical field variables $\phi^{\alpha}(x,t)$, and with no explicit time dependence.
Action $S[\phi]:=\int \! dt~ L[\phi(t,\cdot)]$.
Lagrangian functional $L:=T-V$.
Energy functional $E=T+V$.
...
1
Exact solutions could not be the right way to understand infrared behavior of Yang-Mills theory. As we know from quantum field theory, we can start with some approximation (weak coupling). With this in mind, it can be proved that the following holds (see http://arxiv.org/abs/0903.2357) for a gauge coupling going formally to infinity
$$
...
1
This condition is due to the fact that for a free massless particle the Pauli-Lubanski vector $W=*(M\wedge P)$ must be proportional to the linear momentum (The proportionality factor being the helicity). Thus the condition must be valid to all free massless relativistic field theories.
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