In the
physics
Physics is the natural science that studies matter, its 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 which r ...
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 (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant
degrees of freedom 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 of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a
gauge transformation
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 (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
, equivalent to a
shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
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 is fraught with complications related to
renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
, especially when the computation is continued to higher
orders
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
. Historically, the search for
logically consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
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 refers to the development of 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 developme ...
from the late nineteenth century to the present.
Gauge freedom
The archetypical gauge theory is the
Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
–
Gibbs formulation of continuum
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
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 E and
magnetic field 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, and electric circuits.
...
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
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and the
magnetic vector potential A through the relations:
If the transformation
is made, then B remains unchanged, since (with the identity
)
However, this transformation changes E according to
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 \mathbb S^1, 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 = \.
...
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 mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confine ...
, 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 to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, al ...
of A around a closed loop, and this integral is not changed by
Gauge fixing in
non-abelian gauge theories, such as
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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 re ...
,
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 formulat ...
, and
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to 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 nat ...
.
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 \mathbb S^1, 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 = \.
...
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 and
condensed matter physics and is defined by the gauge condition (more precisely, gauge fixing condition)
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
In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...
is given, in
SI units, by:
and in
Gaussian units
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
by:
This may be rewritten as:
where