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:
“It is in vain that we would circumscribe the power of one half of our race, and that half by far the most important and influential. If they exert it not for good, they will for evil; if they advance not knowledge, they will perpetuate ignorance. Let women stand where they may in the scale of improvement, their position decides that of the race.”
—Frances Wright (1795–1852)
“Against classical philosophy: thinking about eternity or the immensity of the universe does not lessen my unhappiness.”
—Mason Cooley (b. 1927)
“When it had long since outgrown his purely medical implications and become a world movement which penetrated into every field of science and every domain of the intellect: literature, the history of art, religion and prehistory; mythology, folklore, pedagogy, and what not.”
—Thomas Mann (1875–1955)
“If my theory of relativity is proven correct, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.”
—Albert Einstein (1879–1955)