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)
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