Lagrangian (field theory) in the context of Riemann–Roch theorem

In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, the Lagrangian is invariant under these transformations.

The term "gauge" refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory.

Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.

For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, mathematically replaces the initially postulated mass and charge of an electron with the experimentally observed mass and charge. Mathematics and experiments prove that positrons and more massive particles such as protons exhibit precisely the same observed charge as the electron – even in the presence of much stronger interactions and more intense clouds of virtual particles.

In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two static bodies to the "charges" of the bodies (i.e. the electric charge for electrostatic and the mass for Newtonian gravity) divided by the distance squared, ${\displaystyle r^{2}}$, between the bodies; thus: ${\displaystyle G}$ in ${\displaystyle F=Gm_{1}m_{2}/r^{2}}$ for Newtonian gravity and ${\displaystyle k_{\text{e}}}$ in ${\displaystyle F=k_{\text{e}}q_{1}q_{2}/r^{2}}$ for electrostatic. This description remains valid in modern physics for linear theories with static bodies and massless force carriers.${\displaystyle Q^{2}}$.

A modern and more general definition uses the Lagrangian ${\displaystyle {\mathcal {L}}}$ (or equivalently the Hamiltonian ${\displaystyle {\mathcal {H}}}$) of a system. Usually, ${\displaystyle {\mathcal {L}}}$ (or ${\displaystyle {\mathcal {H}}}$) of a system describing an interaction can be separated into a kinetic part ${\displaystyle T}$ and an interaction part ${\displaystyle V}$: ${\displaystyle {\mathcal {L}}=T-V}$ (or ${\displaystyle {\mathcal {H}}=T+V}$).In field theory, ${\displaystyle V}$ always contains 3 fields terms or more, expressing for example that an initial electron (field 1) interacts with a photon (field 2) producing the final state of the electron (field 3). In contrast, the kinetic part ${\displaystyle T}$ always contains only two fields, expressing the free propagation of an initial particle (field 1) into a later state (field 2).The coupling constant determines the magnitude of the ${\displaystyle T}$ part with respect to the ${\displaystyle V}$ part (or between two sectors of the interaction part if several fields that couple differently are present). For example, the electric charge of a particle is a coupling constant that characterizes an interaction with two charge-carrying fields and one photon field (hence the common Feynman diagram with two arrows and one wavy line). Since photons mediate the electromagnetic force, this coupling determines how strongly electrons feel such a force, and has its value fixed by experiment. By looking at the QED Lagrangian, one sees that indeed, the coupling sets the proportionality between the kinetic term ${\displaystyle T={\bar {\psi }}(i\hbar c\gamma ^{\sigma }\partial _{\sigma }-mc^{2})\psi -{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }}$ and the interaction term ${\displaystyle V=-e{\bar {\psi }}(\hbar c\gamma ^{\sigma }A_{\sigma })\psi }$.

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.