In
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 ...
, a charge is any of many different quantities, such as the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
in
electromagnetism
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 ...
or the
color charge in
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
. Charges correspond to the
time-invariant generators of a
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
, and specifically, to the generators that
commute with the
Hamiltonian. Charges are often denoted by the letter ''Q'', and so the invariance of the charge corresponds to the vanishing
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
, where H is the Hamiltonian. Thus, charges are associated with conserved
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can ...
s; these are the eigenvalues ''q'' of the generator ''Q''.
Abstract definition
Abstractly, a charge is any generator of a
continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to ano ...
of the physical system under study. When a physical system has a symmetry of some sort,
Noether's theorem implies the existence of a
conserved current. The thing that "flows" in the current is the "charge", the charge is the
generator of the (local) symmetry group. This charge is sometimes called the Noether charge.
Thus, for example, the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
is the generator of the
U(1) symmetry of
electromagnetism
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 ...
. The conserved current is the
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movi ...
.
In the case of local, dynamical symmetries, associated with every charge is a
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 (is invariant) under local transformations according to certain smooth families of operations (Lie grou ...
; when quantized, the gauge field becomes a
gauge boson
In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of ga ...
. The charges of the theory "radiate" the gauge field. Thus, for example, the gauge field of electromagnetism is the
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
; and the gauge boson is the
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
.
The word "charge" is often used as a synonym for both the generator of a symmetry, and the conserved quantum number (eigenvalue) of the generator. Thus, letting the upper-case letter ''Q'' refer to the generator, one has that the generator
commutes with the
Hamiltonian 'Q'', ''H''= 0.
Commutation
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
implies that the eigenvalues (lower-case) ''q'' are time-invariant: = 0.
So, for example, when the symmetry group is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
, then the charge operators correspond to the simple roots of the
root system of the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
; the
discreteness of the root system accounting for the quantization of the charge. The simple roots are used, as all the other roots can be obtained as linear combinations of these. The general roots are often called raising and lowering operators, or
ladder operators.
The charge quantum numbers then correspond to the weights of the
highest-weight modules of a given
representation of the Lie algebra. So, for example, when a particle in a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
belongs to a symmetry, then it transforms according to a particular representation of that symmetry; the charge quantum number is then the weight of the representation.
Examples
Various charge quantum numbers have been introduced by theories of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. These include the charges of the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
:
* The
color charge of
quark
A quark () is a type of 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. All commonly ...
s. The color charge generates the
SU(3) color symmetry of
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
.
* The
weak isospin quantum numbers of the
electroweak interaction. It generates the
SU(2) part of the electroweak SU(2) × U(1) symmetry. Weak isospin is a local symmetry, whose
gauge boson
In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of ga ...
s are the
W and Z bosons
In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
.
* The
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
for electromagnetic interactions. In mathematics texts, this is sometimes referred to as the
-charge of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
.
Charges of approximate symmetries:
* The
strong isospin
In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
charges. The symmetry groups is
SU(2) flavor symmetry; the gauge bosons are the
pions. The pions are not
elementary particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
s, and the symmetry is only approximate. It is a special case of flavor symmetry.
* Other
quark
A quark () is a type of 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. All commonly ...
-flavor charges, such as
strangeness
In particle physics, strangeness ("''S''") is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic interactions which occur in a short period of time. The strangeness of a parti ...
or
charm
Charm may refer to:
Social science
* Charisma, a person or thing's pronounced ability to attract others
* Superficial charm, flattery, telling people what they want to hear
Science and technology
* Charm quark, a type of elementary particle
* Ch ...
. Together with the – isospin mentioned above, these generate the global
SU(6)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
flavor symmetry of the fundamental particles; this symmetry is
badly broken by the masses of the heavy quarks. Charges include the
hypercharge, the
X-charge and the
weak hypercharge.
Hypothetical charges of extensions to the Standard Model:
* The hypothetical
magnetic charge
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
is another charge in the theory of electromagnetism. Magnetic charges are not seen experimentally in laboratory experiments, but would be present for theories including
magnetic monopoles.
In
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
:
* The
supercharge refers to the generator that rotates the fermions into bosons, and vice versa, in the supersymmetry.
In
conformal field theory:
* The
central charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other elemen ...
of the
Virasoro algebra, sometimes referred to as the ''conformal central charge'' or the
conformal anomaly
A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.
A classically conformal theory is a theory which, when placed on a surface ...
. Here, the term 'central' is used in the sense of the
center in group theory: it is an operator that commutes with all the other operators in the algebra. The central charge is the eigenvalue of the
central generator of the algebra; here, it is the
energy–momentum tensor Energy–momentum may refer to:
* Four-momentum
* Stress–energy tensor
* Energy–momentum relation
{{dab ...
of the two-dimensional conformal field theory.
In
gravitation
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
:
* Eigenvalues of the energy–momentum tensor correspond to physical
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
.
Charge conjugation
In the formalism of particle theories, charge-like quantum numbers can sometimes be inverted by means of a
charge conjugation
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
operator called C. Charge conjugation simply means that a given symmetry group occurs in two inequivalent (but still
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
)
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s. It is usually the case that the two charge-conjugate representations are
complex conjugate fundamental representations of the Lie group. Their product then forms the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
of the group.
Thus, a common example is that the
product of two charge-conjugate fundamental representations of
SL(2,C) (the
spinors) forms the adjoint rep of the
Lorentz group SO(3,1); abstractly, one writes
:
That is, the product of two (Lorentz) spinors is a (Lorentz) vector and a (Lorentz) scalar. Note that the complex Lie algebra sl(2,C) has a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
real form su(2) (in fact, all Lie algebras have a unique compact real form). The same decomposition holds for the compact form as well: the product of two spinors in
su(2) being a vector in the
rotation group O(3)
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
and a singlet. The decomposition is given by the
Clebsch–Gordan coefficients.
A similar phenomenon occurs in the compact group
SU(3), where there are two charge-conjugate but inequivalent fundamental representations, dubbed
and
, the number 3 denoting the dimension of the representation, and with the quarks transforming under
and the antiquarks transforming under
. The Kronecker product of the two gives
:
That is, an eight-dimensional representation, the octet of the
eight-fold way, and a
singlet. The decomposition of such products of representations into direct sums of irreducible representations can in general be written as
:
for representations
. The dimensions of the representations obey the "dimension sum rule":
:
Here,
is the dimension of the representation
, and the integers
being the
Littlewood–Richardson coefficients. The decomposition of the representations is again given by the Clebsch–Gordan coefficients, this time in the general Lie-algebra setting.
See also
*
Casimir operator
References
{{reflist
Electromagnetism
Quantum chromodynamics
Physical quantities