Scale Invariance in Classical Field Theory
Classical field theory is generically described by a field, or set of fields, that depend on coordinates, x. Valid field configurations are then determined by solving differential equations for, and these equations are known as field equations.
For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields:
The parameter is known as the scaling dimension of the field, and its value depends on the theory under consideration. Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is not scale-invariant.
A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution, one always has other solutions of the form .
Read more about this topic: Scale Invariance
Famous quotes containing the words scale, classical, field and/or theory:
“I don’t pity any man who does hard work worth doing. I admire him. I pity the creature who does not work, at whichever end of the social scale he may regard himself as being.”
—Theodore Roosevelt (1858–1919)
“Several classical sayings that one likes to repeat had quite a different meaning from the ones later times attributed to them.”
—Johann Wolfgang Von Goethe (1749–1832)
“Hardly a book of human worth, be it heaven’s own secret, is honestly placed before the reader; it is either shunned, given a Periclean funeral oration in a hundred and fifty words, or interred in the potter’s field of the newspapers’ back pages.”
—Edward Dahlberg (1900–1977)
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)