Quantum Field Theory
The standard model is a quantum field theory, meaning its fundamental objects are quantum fields which are defined at all points in spacetime:
- the fermion field, which accounts for "matter particles",
- the electroweak boson fields, and ,
- the gluon field, and
- the Higgs field .
That these are quantum rather than classical fields have the mathematical consequence that they are operator-valued. In particular, values of the fields generally don't commute. As operators, they act upon the quantum state (ket vector).
The dynamics of the quantum state and the fundamental fields are determined by the Lagrangian density (usually for short just called the Lagrangian). This plays a role similar to that of the Schrödinger equation in non-relativistic quantum mechanics, but a Lagrangian is not an equation — rather, it is an polynomial function of the fields and their derivatives. While it would be possible to derive a system of differential equations governing the fields from the Langrangian, it is more common to use other techniques to compute with quantum field theories.
The standard model is furthermore a gauge theory, which means there are degrees of freedom in the mathematical formalism which do not correspond to changes in the physical state. The gauge group of the standard model is, where U(1) acts on and, SU(2) acts on and, and SU(3) acts on . The fermion field also transforms under these symmetries, although all of them leave some parts of it unchanged.
Read more about this topic: Standard Model (mathematical Formulation)
Famous quotes containing the words quantum, field and/or theory:
“A personality is an indefinite quantum of traits which is subject to constant flux, change, and growth from the birth of the individual in the world to his death. A character, on the other hand, is a fixed and definite quantum of traits which, though it may be interpreted with slight differences from age to age and actor to actor, is nevertheless in its essentials forever fixed.”
—Hubert C. Heffner (1901–1985)
“How sweet I roam’d from field to field
And tasted all the summer’s pride,
Till I the Prince of Love beheld
Who in the sunny beams did glide!”
—William Blake (1757–1827)
“The struggle for existence holds as much in the intellectual as in the physical world. A theory is a species of thinking, and its right to exist is coextensive with its power of resisting extinction by its rivals.”
—Thomas Henry Huxley (1825–95)