A wavy line represents the propagation of a photon. If the following transformation to the fields is performed at every spacetime point x a local transformation , then the QED Lagrangian remains unchanged, or invariant:. If a theory's Lagrangian or more precisely the action is invariant under a certain local transformation, then the transformation is referred to as a gauge symmetry of the theory.
U 1 is an Abelian group , meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on non-Abelian groups , giving rise to non-Abelian gauge theories also known as Yang—Mills theories. Repeated indices i , j , a are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:. The preceding discussion of symmetries is on the level of the Lagrangian.
In other words, these are "classical" symmetries. After quantisation, some theories will no longer exhibit their classical symmetries, a phenomenon called anomaly. The theoretical foundation of general relativity , the equivalence principle , can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the Lorentz group. Noether's theorem states that every continuous symmetry, i. Gauge transformations do not relate distinct quantum states.
Rather, it relates two equivalent mathematical descriptions of the same quantum state. In this sense, gauge invariance is not a "real" symmetry, but are a reflection of the "redundancy" of the chosen mathematical description. To account for the gauge redundancy in the path integral formulation, one must perform the so-called Faddeev—Popov gauge fixing procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts".
Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally. Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it. To illustrate the mechanism, consider a linear sigma model containing N real scalar fields, described by the Lagrangian density:. The theory admits an O N global symmetry:. Without loss of generality, let the ground state be in the N -th direction:. The original O N global symmetry is no longer manifest, leaving only the subgroup O N The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.
Goldstone's theorem states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. On the other hand, when a gauge as opposed to global symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarised massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.
In the QFT of ferromagnetism , spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures. All experimentally known symmetries in nature relate bosons to bosons and fermions to fermions. Theorists have hypothesised the existence of a type of symmetry, called supersymmetry , that relates bosons and fermions.
In a supersymmetric theory, every fermion has a bosonic superpartner and vice versa.
If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called supergravity. Supersymmetry is a potential solution to many current problems in physics. For example, the hierarchy problem of the Standard Model — why the mass of the Higgs boson is not radiatively corrected under renormalisation to a very high scale such as the grand unified scale or the Planck scale — can be resolved by relating the Higgs field and its superpartner, the Higgsino.
Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of dark matter. Nevertheless, as of [update] , experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.
However, QFT a priori imposes no restriction on the number of dimensions nor the geometry of spacetime. For QFTs in curved spacetime on the other hand, a general metric such as the Schwarzschild metric describing a black hole is used:.
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For a real scalar field, the Lagrangian density in a general spacetime background is. For a special class of QFTs called topological quantum field theories TQFTs , all correlation functions are independent of continuous changes in the spacetime metric. This means that all calculational results of TQFTs are topological invariants of the underlying spacetime.
Using perturbation theory , the total effect of a small interaction term can be approximated order by order by a series expansion in the number of virtual particles participating in the interaction. Every term in the expansion may be understood as one possible way for physical particles to interact with each other via virtual particles, expressed visually using a Feynman diagram.
The electromagnetic force between two electrons in QED is represented to first order in perturbation theory by the propagation of a virtual photon. In a similar manner, the W and Z bosons carry the weak interaction, while gluons carry the strong interaction. The interpretation of an interaction as a sum of intermediate states involving the exchange of various virtual particles only makes sense in the framework of perturbation theory.
In contrast, non-perturbative methods in QFT treat the interacting Lagrangian as a whole without any series expansion. Instead of particles that carry interactions, these methods have spawned such concepts as 't Hooft—Polyakov monopole , domain wall , flux tube , and instanton. In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to Haag's theorem , there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally not rigorous.
Quantum field theory | Definition of Quantum field theory at tusriesaretma.ga
Since the s,  theoretical physicists and mathematicians have attempted to organise all QFTs into a set of axioms , in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called constructive quantum field theory , a subfield of mathematical physics ,  : 2 which has led to such results as CPT theorem , spin-statistics theorem , and Goldstone's theorem. Compared to ordinary QFT, topological quantum field theory and conformal field theory are better supported mathematically — both can be classified in the framework of representations of cobordisms.
Algebraic quantum field theory is another approach to the axiomatisation of QFT, in which the fundamental objects are local operators and the algebraic relations between them.
Axiomatic systems following this approach include Wightman axioms and Haag-Kastler axioms. Yang-Mills existence and mass gap , one of the Millenium Prize Problems , concerns the well-defined existence of Yang-Mills theories as set out by the above axioms. The full problem statement is as follows. From Wikipedia, the free encyclopedia. Quantum field theory Feynman diagram. Standard Model. Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism. Incomplete theories.
Anderson P. See also: Classical field theory. Main article: Canonical quantisation. Main article: Path integral formulation. Main article: Correlation function quantum field theory. Main article: Feynman diagram. Main article: Renormalisation. Main article: Renormalization group. Main article: Gauge theory.
Main article: Spontaneous symmetry breaking. Main article: Supersymmetry. Main article: Topological quantum field theory.
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Mathematics portal Physics portal. An Introduction to Quantum Field Theory. Westview Press. American Journal of Physics. Bibcode : AmJPh.. Heilbron 14 February Oxford University Press.