New answers tagged field-theory
3
This is quite a standard textbook example. Good treatments can, for example, be found in Altland (Condensed Matter Field Theory) or Nagaosa (Quantum Field Theory in Condensed Matter Physics). However, the basic reasoning can be understood quite easily:
Consider the change in angle produced as we perform a closed loop around some point. Naively we obtain
...
0
Your intuition about the Helmholtz equation is correct in that it is the right way to handle the spatial dependence. If you are in free space, then you should be tackling this problem using a plane-wave basis by taking the Fourier transform of your equation:
$$
\frac{\partial^2 \tilde\phi(t,\mathbf k)}{\partial t^2} +(c^2k^2+\omega_0^2)\tilde\phi(t,\mathbf ...
2
You started off stating that the undeformed solution ($\lambda=1$) must be a stationary point. So, if you differentiate the energy of this one-parameter family wrt $\lambda$ you should get zero. So evaluate $\frac{dE}{d \lambda}$ @ $\lambda = 1$.
4
I didn't go through all of your equations. However, if you take (1), differentiate it w.r.t $\lambda$ and set $\lambda = 1$, then since $\lambda=1$ is the stationary point $E'(\lambda)|_{\lambda=1} = 0$. This is equation (2)
4
It's a bit hard to be sure without seeing the whole text, but it looks like they're discussing the problem of of obtaining finite minimum energy solutions of a gauge/Higgs system. In 3 space dimensions, for example, for the Georgi-Glashow model, $$ \mathcal{L}= \frac{1}{2}Tr(F^{\mu\nu}F_{\mu\nu})+Tr(D_{\mu}\phi D^{\mu}\phi)-\frac{\lambda}{4}(|\phi|^2-v^2)^2 ...
1
$\phi^4$ equation and the sine-Gordon equation are the partial differential equations (PDEs). Equations (1) and (2) are actually the solutions to these PDEs. (BTW, in Eq.(2), it should be trigonometric arctangent, not the hyperbolic one). One can obtain a $\phi^4$ model as an approximation of the sine-Gordon model, expanding the sine term in the equation or ...
0
First, what is a difference between linear and nonlinear physical processes? If a deviation of a system from an equilibrium is small, then the system is said to be linear. Formally, in this case, the system is described by a linear equation. A simple example of a linear system is a pendulum that performs small oscillations near the equilibrium (vertical) ...
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