anomaly measure group gauge theory symmetry Lagrangian gauge symmetry U(1) path integral formulation Standard Model field SU(3) fields
The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly. For instance, in the path integral formulation, despite the invariance of the Lagrangian density 





L


[
ϕ
,

∂

μ


ϕ
]


{\displaystyle {\mathcal {L}}[\phi ,\partial _{\mu }\phi ]}

 under a certain local transformation of the fields, the measure 



∫


D


ϕ


{\textstyle \int {\mathcal {D}}\phi }

 of the path integral may change. For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group SU(3) × SU(2) × U(1), in which all anomalies exactly cancel.
