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:
“The most perfect political community must be amongst those who are in the middle rank, and those states are best instituted wherein these are a larger and more respectable part, if possible, than both the other; or, if that cannot be, at least than either of them separate, so that being thrown into the balance it may prevent either scale from preponderating.”
—Aristotle (384–322 B.C.)
“Classical art, in a word, stands for form; romantic art for content. The romantic artist expects people to ask, What has he got to say? The classical artist expects them to ask, How does he say it?”
—R.G. (Robin George)
“We need a type of theatre which not only releases the feelings, insights and impulses possible within the particular historical field of human relations in which the action takes place, but employs and encourages those thoughts and feelings which help transform the field itself.”
—Bertolt Brecht (1898–1956)
“No one thinks anything silly is suitable when they are an adolescent. Such an enormous share of their own behavior is silly that they lose all proper perspective on silliness, like a baker who is nauseated by the sight of his own eclairs. This provides another good argument for the emerging theory that the best use of cryogenics is to freeze all human beings when they are between the ages of twelve and nineteen.”
—Anna Quindlen (20th century)