Effective field theory

Effective field theory

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In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, an effective field theory is, as any effective theory
Effective theory
-Effective Theory:An Effective Theory is a theory based on reasonable assumption, if not actual evidence, for the purpose of moving on in an area of study. An effective theory provides a framework to explain certain effects, without the claim that the theory correctly models the underlying ...

, an approximate theory, (usually a quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

) that includes appropriate degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

 to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances (or, equivalently, at higher energies).

The renormalization group


Presently, effective field theories are discussed in the context of the renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

 (RG) where the process of integrating out short distance degrees of freedom is made systematic. Although this method is not sufficiently concrete to allow the actual construction of effective field theories, the gross understanding of their usefulness becomes clear through a RG analysis. This method also lends credence to the main technique of constructing effective field theories, through the analysis of symmetries. If there is a single mass scale M in the microscopic theory, then the effective field theory can be seen as an expansion in 1/M. The construction of an effective field theory accurate to some power of 1/M requires a new set of free parameters at each order of the expansion in 1/M. This technique is useful for scattering or other processes where the maximum momentum scale k satisfies the condition k/M<<1. Since effective field theories are not valid at small length scales, they need not be renormalizable. Indeed, the ever expanding number of parameters at each order in 1/M required for an effective field theory means that they are generally not renormalizable in the same sense as quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

 which requires only the renormalization of three parameters.

Fermi theory of beta decay


The best-known example of an effective field theory is the Fermi theory of beta decay
Fermi's interaction
In particle physics, Fermi's interaction also known as Fermi coupling, is an old explanation of the weak force, proposed by Enrico Fermi, in which four fermions directly interact with one another at one vertex...

. This theory was developed during the early study of weak decays of nuclei when only the hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

s and lepton
Lepton
A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...

s undergoing weak decay were known. The typical reactions studied were:

This theory posited a pointlike interaction between the four fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s involved in these reactions. The theory had great phenomenological success and was eventually understood to arise from the gauge theory of electroweak interaction
Electroweak interaction
In particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different...

s, which forms a part of the standard model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

 of particle physics. In this more fundamental theory, the interactions are mediated by a flavour
Flavour (particle physics)
In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...

-changing gauge boson, the W±. The immense success of the Fermi theory was because the W particle has mass of about 80 GeV
GEV
GEV or GeV may stand for:*GeV or gigaelectronvolt, a unit of energy equal to billion electron volts*GEV or Grid Enabled Vehicle that is fully or partially powered by the electric grid, see plug-in electric vehicle...

, whereas the early experiments were all done at an energy scale of less than 10 MeV
MEV
MeV and meV are multiples and submultiples of the electron volt unit referring to 1,000,000 eV and 0.001 eV, respectively.Mev or MEV may refer to:In entertainment:* Musica Elettronica Viva, an Italian musical group...

. Such a separation of scales, by over 3 orders of magnitude, has not been met in any other situation as yet.

BCS theory of superconductivity


Another famous example is the BCS theory
BCS theory
BCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...

 of superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

. Here the underlying theory is of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s in a metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...

 interacting with lattice vibrations called phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

s. The phonons cause attractive interactions between some electrons, causing them to form Cooper pair
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair is two electrons that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper...

s. The length scale of these pairs is much larger than the wavelength of phonons, making it possible to neglect the dynamics of phonons and construct a theory in which two electrons effectively interact at a point. This theory has had remarkable success in describing and predicting the results of experiments.

Other examples


Presently, effective field theories are written for many situations.
  • One major branch of nuclear physics
    Nuclear physics
    Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...

     is quantum hadrodynamics, where the interactions of hadron
    Hadron
    In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

    s are treated as a field theory, which one hopes to derive from quantum chromodynamics
    Quantum chromodynamics
    In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

    , the true underlying theory, in the future. Due to the smaller separation of length scales here, this effective theory has some classificatory power, but not the spectacular success of the Fermi theory.
  • In particle physics
    Particle physics
    Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

     the effective field theory of QCD
    Quantum chromodynamics
    In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

     called chiral perturbation theory
    Chiral perturbation theory
    Chiral perturbation theory is an effective field theory constructed with a Lagrangian consistent with the chiral symmetry of quantum chromodynamics , as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD...

     has had better success. This theory deals with the interactions of hadron
    Hadron
    In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

    s with pion
    Pion
    In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....

    s or kaon
    Kaon
    In particle physics, a kaon is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness...

    s, which are the Goldstone boson
    Goldstone boson
    In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...

    s of spontaneous chiral symmetry breaking. The expansion parameter is the pion energy/momentum.
  • For hadron
    Hadron
    In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

    s containing one heavy quark
    Quark
    A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

     (such as the bottom
    Bottom quark
    The bottom quark, also known as the beauty quark, is a third-generation quark with a charge of − e. Although all quarks are described in a similar way by the quantum chromodynamics, the bottom quark's large bare mass , combined with low values of the CKM matrix elements Vub and Vcb, gives it a...

     or charm
    Charm quark
    The charm quark or c quark is the third most massive of all quarks, a type of elementary particle. Charm quarks are found in hadrons, which are subatomic particles made of quarks...

    ), an effective field theory which expands in powers of the quark mass, called the heavy-quark effective theory (HQET), has been found useful.
  • For hadron
    Hadron
    In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

    s containing two heavy quarks, an effective field theory which expands in powers of the relative velocity of the heavy quarks, called non-relativistic QCD (NRQCD), has been found useful, especially when used in conjunctions with lattice QCD
    Lattice QCD
    Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....

    .
  • For hadron
    Hadron
    In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

     reactions with light energetic (collinear) particles, the interactions with low-energetic (soft) degrees of freedom are described by the soft-collinear effective theory
    Soft-collinear effective theory
    In quantum field theory, soft-collinear effective theory is a theoretical framework for doing calculations in quantum chromodynamics that involve interacting particles carrying widely different energies....

     (SCET).
  • General relativity
    General relativity
    General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

     is expected to be the low energy effective field theory of a full theory of quantum gravity
    Quantum gravity
    Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

    , such as string theory
    String theory
    String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

    . The expansion scale is the Planck mass.
  • Much of condensed matter physics
    Condensed matter physics
    Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

     consists of writing effective field theories for the particular property of matter being studied.
  • Effective field theories have also been used to simplify problems in General Relativity (NRGR). In particular in calculating post-Newtonian corrections to the gravity wave signature of inspiralling finite-sized objects. http://arxiv.org/pdf/hep-th/0409156

See also

  • Form factor
  • Renormalization group
    Renormalization group
    In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

  • Quantum field theory
    Quantum field theory
    Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

  • Quantum triviality
    Quantum triviality
    In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. Ifthe only allowed value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting...