In physics, an anomaly is a classical symmetry — a symmetry of the Lagrangian — that is broken in quantum field theories. The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

Any anomaly can be considered to be a short-distance effect (an ultraviolet effect, in the language of quantum field theory), since it arises during the process of renormalization, when some divergent integrals cannot be regularized in such a way that all the symmetries are preserved simultaneously. However, their effects are often felt at long distances (ie, by the infra-red physics, in the language of quantum field theory) since the lack of this classical symmetry is often manifested in the physics of massless (or nearly massless) particles described by the theory. In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy approaching infinity, or, equivalently, because of physical phenomena at very short distances. ... The mathematical term regularization has two main meanings, both associated with making a function more `regular or smooth. ...

Anomalies in global symmetries pose no problems in a quantum field theory, and are often encountered (see the example of the chiral anomaly). Anomalies in gauge theories pose problems, since a gauge symmetry is usually required by the phenomenon under study. The attempt to remove them, ie, to build theories consistent with the gauge symmetries often leads to extra constraints on the theories (such is the case of the gauge anomaly in the standard model of particle physics). Anomalies in gauge theories have important connections to the topology and geometry of the gauge group. Chiral anomaly is the anomalous nonconservation of charge in a quantized theory of chiral fermions coupled to a background gauge field. ... In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics - usually a one-loop diagram - that invalidates the gauge symmetry of a quantum field theory i. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

Examples

• consistent anomaly
• covariant anomaly
• conformal anomaly
• chiral anomaly
• gravitational anomaly
• gauge anomaly
• mixed anomaly
• global anomaly
• parity anomaly
• diffeomorphism anomaly

Conformal anomaly is an anomaly a quantum phenomenon that breaks the conformal symmetry of the classical theory. ... Chiral anomaly is the anomalous nonconservation of charge in a quantized theory of chiral fermions coupled to a background gauge field. ... Anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams In theoretical physics, a gravitational anomaly is an example of an anomaly: it is an effect of quantum mechanics - usually a one-loop diagram - that invalidates the general covariance of a theory of general relativity combined with some... In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics - usually a one-loop diagram - that invalidates the gauge symmetry of a quantum field theory i. ... In theoretical physics, a mixed anomaly is an example of an anomaly: it is an effect of quantum mechanics - usually a one-loop diagram - that implies that the classically valid general covariance and gauge symmetry of a theory of general relativity combined with gauge fields and fermionic fields cannot be... In theoretical physics, a global anomaly is a type of anomaly: in this particular case, it is a quantum effect that invalidates a discrete symmetry that would otherwise be preserved in the classical theory. ... In a quantum field theory over an odd dimensional spacetime with fermions with a classical action which is invariant under parity, we may sometimes have parity anomalies, which is an anomaly associated with parity, which is broken. ...