A Pedagogical Example: The Mexican Hat Potential
In the simplest idealized relativistic model, the spontaneously broken symmetry is summarized through an illustrative scalar field theory. The relevant Lagrangian, which essentially dictates how a system behaves, can be split up into kinetic and potential terms,
It is in this potential term V(Φ) that the symmetry breaking is triggered. An example of a potential, due to Jeffrey Goldstone is illustrated in the graph at the right.
This potential has an infinite number of possible minima (vacuum states) given by
for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to Φ = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (amounting to a choice of θ), this symmetry will appear to be lost, or "spontaneously broken".
In fact, any other choice of θ would have exactly the same energy, implying the existence of a massless Nambu–Goldstone boson, the mode running around the circle at the minimum of this potential, and indicating there is some memory of the original symmetry in the Lagrangian.
Read more about this topic: Spontaneous Symmetry Breaking
Famous quotes containing the words mexican, hat and/or potential:
“The germ of violence is laid bare in the child abuser by the sheer accident of his individual experience ... in a word, to a greater degree than we like to admit, we are all potential child abusers.”
—F. Gonzalez-Crussi, Mexican professor of pathology, author. “Reflections on Child Abuse,” Notes of an Anatomist (1985)
“The agent never receipts his bill, puts his hat on and bows himself out. He stays around forever, not only for as long as you can write anything that anyone will buy, but as long as anyone will buy any portion of any right to anything that you ever did write. He just takes ten per cent of your life.”
—Raymond Chandler (1888–1959)
“And what is the potential man, after all? Is he not the sum of all that is human? Divine, in other words?”
—Henry Miller (1891–1980)