Radiation Gauge
   HOME

TheInfoList



OR:

In the
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
of
gauge theories 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, t ...
, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
in
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
variables. By definition, a gauge theory represents each physically distinct configuration of the system as an
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a certain transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom. Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical configuration by a ''particular'' detailed configuration (or even a weighted distribution of them). Judicious gauge fixing can simplify calculations immensely, but becomes progressively harder as the physical model becomes more realistic; its application to
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
is fraught with complications related to
renormalization 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 the ...
, especially when the computation is continued to higher
orders Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * H ...
. Historically, the search for
logically consistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences o ...
and computationally tractable gauge fixing procedures, and efforts to demonstrate their equivalence in the face of a bewildering variety of technical difficulties, has been a major driver of
mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
from the late nineteenth century to the present.


Gauge freedom

The archetypical gauge theory is the
Heaviside Oliver Heaviside ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, a ...
Gibbs Gibbs or GIBBS is a surname and acronym. It may refer to: People * Gibbs (surname) Places * Gibbs (crater), on the Moon * Gibbs, Missouri, US * Gibbs, Tennessee, US * Gibbs Island (South Shetland Islands), Antarctica * 2937 Gibbs, an asteroid ...
formulation of continuum
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
in terms of an
electromagnetic four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
, which is presented here in space/time asymmetric Heaviside notation. The
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
E and
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
B of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
contain only "physical" degrees of freedom, in the sense that every ''mathematical'' degree of freedom in an electromagnetic field configuration has a separately measurable effect on the motions of test charges in the vicinity. These "field strength" variables can be expressed in terms of the
electric scalar potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work needed ...
\varphi and the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the ma ...
A through the relations: = -\nabla\varphi - \frac\,, \quad = \nabla\times. If the transformation is made, then B remains unchanged, since (with the identity \nabla \times \nabla \psi = 0) = \nabla\times (+ \nabla \psi) = \nabla\times. However, this transformation changes E according to \mathbf E = -\nabla\varphi - \frac - \nabla \frac = -\nabla \left( \varphi + \frac\right) - \frac. If another change is made then E also remains the same. Hence, the E and B fields are unchanged if one takes any function and simultaneously transforms A and ''φ'' via the transformations () and (). A particular choice of the scalar and vector potentials is a gauge (more precisely, gauge potential) and a scalar function ''ψ'' used to change the gauge is called a gauge function. The existence of arbitrary numbers of gauge functions corresponds to the
U(1) In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle g ...
gauge freedom of this theory. Gauge fixing can be done in many ways, some of which we exhibit below. Although classical electromagnetism is now often spoken of as a gauge theory, it was not originally conceived in these terms. The motion of a classical point charge is affected only by the electric and magnetic field strengths at that point, and the potentials can be treated as a mere mathematical device for simplifying some proofs and calculations. Not until the advent of quantum field theory could it be said that the potentials themselves are part of the physical configuration of a system. The earliest consequence to be accurately predicted and experimentally verified was the
Aharonov–Bohm effect The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanics, quantum-mechanical phenomenon in which an electric charge, electrically charged point particle, particle is affected by an elect ...
, which has no classical counterpart. Nevertheless, gauge freedom is still true in these theories. For example, the Aharonov–Bohm effect depends on a
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integr ...
of A around a closed loop, and this integral is not changed by \mathbf \rightarrow \mathbf + \nabla \psi\,. Gauge fixing in non-abelian gauge theories, such as
Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...
and
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, is a rather more complicated topic; for details see
Gribov ambiguity In gauge theory, especially in non-abelian gauge theories, global problems at gauge fixing are often encountered. Gauge fixing means choosing a representative from each gauge orbit, that is, choosing a section of a fiber bundle. The space of rep ...
,
Faddeev–Popov ghost In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral form ...
, and
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...
.


An illustration

As an illustration of gauge fixing, one may look at a cylindrical rod and attempt to tell whether it is twisted. If the rod is perfectly cylindrical, then the circular symmetry of the cross section makes it impossible to tell whether or not it is twisted. However, if there were a straight line drawn along the length of the rod, then one could easily say whether or not there is a twist by looking at the state of the line. Drawing a line is gauge fixing. Drawing the line spoils the gauge symmetry, i.e., the circular symmetry
U(1) In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle g ...
of the cross section at each point of the rod. The line is the equivalent of a gauge function; it need not be straight. Almost any line is a valid gauge fixing, i.e., there is a large gauge freedom. In summary, to tell whether the rod is twisted, the gauge must be known. Physical quantities, such as the energy of the torsion, do not depend on the gauge, i.e., they are gauge invariant.


Coulomb gauge

The Coulomb gauge (also known as the transverse gauge) is used in
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
and
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
and is defined by the gauge condition (more precisely, gauge fixing condition) \nabla\cdot(\mathbf,t)=0\,. It is particularly useful for "semi-classical" calculations in quantum mechanics, in which the vector potential is quantized but the Coulomb interaction is not. The Coulomb gauge has a number of properties:


Lorenz gauge

The Lorenz gauge is given, in SI units, by: \nabla\cdot + \frac\frac=0 and in
Gaussian units Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit syst ...
by: \nabla\cdot + \frac\frac=0. This may be rewritten as: \partial_ A^ = 0. where A^\mu = \left ,\tfrac\varphi,\,\mathbf\,\right/math> is the
electromagnetic four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
, the
4-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and r ...
sing the metric signature (+, −, −, −)">metric_signature.html" ;"title="sing the metric signature">sing the metric signature (+, −, −, −) It is unique among the constraint gauges in retaining manifest Lorentz invariance. Note, however, that this gauge was originally named after the Danish physicist Ludvig Lorenz and not after Hendrik Lorentz; it is often misspelled "Lorentz gauge". (Neither was the first to use it in calculations; it was introduced in 1888 by
George Francis FitzGerald George Francis FitzGerald (3 August 1851 – 21 February 1901) was an Irish physicist known for hypothesising length contraction, which became an integral part of Albert Einstein's special theory of relativity. Life and work in physics FitzGer ...
.) The Lorenz gauge leads to the following inhomogeneous wave equations for the potentials: \frac\frac - \nabla^2 = \frac \frac\frac - \nabla^2 = \mu_0 \mathbf It can be seen from these equations that, in the absence of current and charge, the solutions are potentials which propagate at the speed of light. The Lorenz gauge is ''incomplete'' in some sense: there remains a subspace of gauge transformations which can also preserve the constraint. These remaining degrees of freedom correspond to gauge functions which satisfy the
wave equation The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...
\frac = c^2 \nabla^2\psi These remaining gauge degrees of freedom propagate at the speed of light. To obtain a fully fixed gauge, one must add boundary conditions along the
light cone In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single Event (relativity), event (localized to a single point in space and a single moment in time) and traveling in all direct ...
of the experimental region. Maxwell's equations in the Lorenz gauge simplify to \partial_\mu \partial^\mu A^\nu = \mu_0 j^\nu where j^\nu = \left ,c\,\rho,\,\mathbf\,\right/math> is the
four-current In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the current density, with units of charge per unit time per unit area. Also known as vector current, it is used in the ...
. Two solutions of these equations for the same current configuration differ by a solution of the vacuum wave equation \partial_\mu \partial^\mu A^\nu = 0. In this form it is clear that the components of the potential separately satisfy the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is named after Oskar Klein and Walter Gordon. It is second-order i ...
, and hence that the Lorenz gauge condition allows transversely, longitudinally, and "time-like" polarized waves in the four-potential. The transverse polarizations correspond to classical radiation, i.e., transversely polarized waves in the field strength. To suppress the "unphysical" longitudinal and time-like polarization states, which are not observed in experiments at classical distance scales, one must also employ auxiliary constraints known as
Ward identities Ward may refer to: Division or unit * Hospital ward, a hospital division, floor, or room set aside for a particular class or group of patients, for example the psychiatric ward * Prison ward, a division of a penal institution such as a pris ...
. Classically, these identities are equivalent to the
continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity ...
\partial_\mu j^\mu = 0. Many of the differences between classical and
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
can be accounted for by the role that the longitudinal and time-like polarizations play in interactions between charged particles at microscopic distances.


''R''''ξ'' gauges

The ''R''''ξ'' gauges are a generalization of the Lorenz gauge applicable to theories expressed in terms of an
action principle In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action, an appr ...
with
Lagrangian density Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees ...
\mathcal. Instead of fixing the gauge by constraining the
gauge field 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, t ...
''a priori'', via an auxiliary equation, one adds a gauge ''breaking'' term to the "physical" (gauge invariant) Lagrangian \delta \mathcal = -\frac The choice of the parameter ''ξ'' determines the choice of gauge. The ''R''ξ Landau gauge is classically equivalent to Lorenz gauge: it is obtained in the limit ''ξ'' → 0 but postpones taking that limit until after the theory has been quantized. It improves the rigor of certain existence and equivalence proofs. Most
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
computations are simplest in the Feynman–'t Hooft gauge, in which ; a few are more tractable in other ''R''ξ gauges, such as the Yennie gauge (named afer Donald R. Yennie). An equivalent formulation of ''R''ξ gauge uses an
auxiliary field In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian (field theory), Lagrangian describing such a Field (physics), field A contains an algebraic quadr ...
, a scalar field ''B'' with no independent dynamics: \delta \mathcal = B\,\partial_ A^ + \frac B^2 The auxiliary field, sometimes called a Nakanishi–Lautrup field, can be eliminated by "completing the square" to obtain the previous form. From a mathematical perspective the auxiliary field is a variety of Goldstone boson, and its use has advantages when identifying the asymptotic states of the theory, and especially when generalizing beyond QED. Historically, the use of ''R''ξ gauges was a significant technical advance in extending
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
computations beyond one-loop order. In addition to retaining manifest Lorentz invariance, the ''Rξ'' prescription breaks the symmetry under local gauge ''transformations'' while preserving the ratio of functional measures of any two physically distinct gauge ''configurations''. This permits a change of variables in which infinitesimal perturbations along "physical" directions in configuration space are entirely uncoupled from those along "unphysical" directions, allowing the latter to be absorbed into the physically meaningless Normalizing constant, normalization of the functional integral. When ξ is finite, each physical configuration (orbit of the group of gauge transformations) is represented not by a single solution of a constraint equation but by a Gaussian distribution centered on the extremum of the gauge breaking term. In terms of the Feynman rules of the gauge-fixed theory, this appears as a contribution to the photon propagator for internal lines from virtual photons of unphysical Polarization (waves), polarization. The photon propagator, which is the multiplicative factor corresponding to an internal photon in the Feynman diagram expansion of a QED calculation, contains a factor ''g''μν corresponding to the Minkowski metric. An expansion of this factor as a sum over photon polarizations involves terms containing all four possible polarizations. Transversely polarized radiation can be expressed mathematically as a sum over either a linear polarization, linearly or circularly polarized basis. Similarly, one can combine the longitudinal and time-like gauge polarizations to obtain "forward" and "backward" polarizations; these are a form of light-cone coordinates in which the metric is off-diagonal. An expansion of the ''g''μν factor in terms of circularly polarized (spin ±1) and light-cone coordinates is called a spin sum. Spin sums can be very helpful both in simplifying expressions and in obtaining a physical understanding of the experimental effects associated with different terms in a theoretical calculation. Richard Feynman used arguments along approximately these lines largely to justify calculation procedures that produced consistent, finite, high precision results for important observable parameters such as the anomalous magnetic moment of the electron. Although his arguments sometimes lacked mathematical rigor even by physicists' standards and glossed over details such as the derivation of Ward–Takahashi identity, Ward–Takahashi identities of the quantum theory, his calculations worked, and Freeman Dyson soon demonstrated that his method was substantially equivalent to those of Julian Schwinger and Sin-Itiro Tomonaga, with whom Feynman shared the 1965 Nobel Prize in Physics. Forward and backward polarized radiation can be omitted in the asymptotic states of a quantum field theory (see Ward–Takahashi identity). For this reason, and because their appearance in spin sums can be seen as a mere mathematical device in QED (much like the electromagnetic four-potential in classical electrodynamics), they are often spoken of as "unphysical". But unlike the constraint-based gauge fixing procedures above, the ''Rξ'' gauge generalizes well to Non-abelian gauge theory, non-abelian gauge groups such as the SU(3) of quantum chromodynamics, QCD. The couplings between physical and unphysical perturbation axes do not entirely disappear under the corresponding change of variables; to obtain correct results, one must account for the non-trivial Jacobian matrix and determinant, Jacobian of the embedding of gauge freedom axes within the space of detailed configurations. This leads to the explicit appearance of forward and backward polarized gauge bosons in Feynman diagrams, along with
Faddeev–Popov ghost In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral form ...
s, which are even more "unphysical" in that they violate the spin–statistics theorem. The relationship between these entities, and the reasons why they do not appear as particles in the quantum mechanical sense, becomes more evident in the BRST formalism of quantization.


Maximal abelian gauge

In any non-Gauge theory, abelian gauge theory, any maximal abelian gauge is an ''incomplete'' gauge which fixes the gauge freedom outside of the maximal abelian subgroup. Examples are * For SU(2) gauge theory in D dimensions, the maximal abelian subgroup is a U(1) subgroup. If this is chosen to be the one generated by the Pauli matrix ''σ''3, then the maximal abelian gauge is that which maximizes the function \int d^Dx \left[\left(A_\mu^1\right)^2+\left(A_\mu^2\right)^2\right]\,, where _\mu = A_\mu^a \sigma_a\,. * For SU(3) gauge theory in D dimensions, the maximal abelian subgroup is a U(1)×U(1) subgroup. If this is chosen to be the one generated by the Gell-Mann matrices ''λ''3 and ''λ''8, then the maximal abelian gauge is that which maximizes the function \int d^Dx \left[\left(A_\mu^1\right)^2 + \left(A_\mu^2\right)^2 + \left(A_\mu^4\right)^2 + \left(A_\mu^5\right)^2 + \left(A_\mu^6\right)^2 + \left(A_\mu^7\right)^2\right]\,, where _\mu = A_\mu^a \lambda_a This applies regularly in higher algebras (of groups in the algebras), for example the Clifford Algebra and as it is regularly.


Less commonly used gauges

Various other gauges, which can be beneficial in specific situations have appeared in the literature.


Weyl gauge

The Weyl gauge (also known as the Hamiltonian or temporal gauge) is an ''incomplete'' gauge obtained by the choice \varphi=0 It is named after Hermann Weyl. It eliminates the negative-norm Ghost (physics), ghost, lacks manifest Lorentz invariance, and requires longitudinal photons and a constraint on states.


Multipolar gauge

The gauge condition of the multipolar gauge (also known as the line gauge, point gauge or Poincaré gauge (named after Henri Poincaré)) is: \mathbf\cdot\mathbf = 0. This is another gauge in which the potentials can be expressed in a simple way in terms of the instantaneous fields \mathbf(\mathbf,t) = -\mathbf \times\int_0^1 \mathbf(u \mathbf,t) u \, du \varphi(\mathbf,t) = -\mathbf \cdot \int_0^1 \mathbf(u \mathbf,t) du.


Fock–Schwinger gauge

The gauge condition of the Fock–Schwinger gauge (named after Vladimir Fock and Julian Schwinger; sometimes also called the relativistic Poincaré gauge) is: x^A_=0 where ''x''''μ'' is the position four-vector.


Dirac gauge

The nonlinear Dirac gauge condition (named after Paul Dirac) is: A_ A^ = k^2


References


Further reading

* * {{QED Electromagnetism Quantum field theory Quantum electrodynamics Gauge theories pl:Cechowanie (fizyka)#Wybór cechowania