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 perch swallows the grub-worm, the pickerel swallows the perch, and the fisherman swallows the pickerel; and so all the chinks in the scale of being are filled.”
—Henry David Thoreau (1817–1862)
“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)
“My prime of youth is but a frost of cares,
My feast of joy is but a dish of pain,
My crop of corn is but a field of tares,
And all my good is but vain hope of gain:
The day is past, and yet I saw no sun,
And now I live, and now my life is done.”
—Chidiock Tichborne (1558–1586)
“Psychotherapy—The theory that the patient will probably get well anyway, and is certainly a damned ijjit.”
—H.L. (Henry Lewis)