Symmetry in quantum mechanics

From Wikipedia, the free encyclopedia

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice; they are powerful methods for solving problems and predicting what could happen. While conservation laws do not always give the answer to the problem directly and alone, they form the correct constraints and the first steps to solving the problem.

This article outlines the connection between the classical form of continuous symmetries as well as their quantum operators, and relates them to the Lie groups, and relativistic transformations in the Lorentz group, and Poincaré group.

Notation

The notational conventions used in this article are as follows. Boldface indicates vectors, four vectors, matrices, and vectorial operators, while quantum states use bra–ket notation. Wide hats are for operators, narrow hats are for unit vectors (including their components in tensor index notation). The summation convention on the repeated tensor indices is used, unless stated otherwise. The Minkowski metric signature is (+−−−).

Symmetry transformations on the wavefunction in non-relativistic quantum mechanics

Continuous symmetries

Generally, the correspondence between continuous symmetries and conservation laws is given by Noether's theorem.

The form of the fundamental quantum operators, for example energy as a partial time derivative and momentum as a spatial gradient, becomes clear when one considers the initial state, then changes one parameter of it slightly. This can be done for displacements (lengths), durations (time), and angles (rotations). Additionally, the invariance of certain quantities can be seen by making such changes in lengths and angles, which illustrates conservation of these quantities.

In what follows, transformations on only one-particle wavefunctions in the form:

are considered, where denotes a unitary operator. Unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state (representing the total probability of finding the particle somewhere with some spin) must be invariant under these transformations. The inverse is the Hermitian conjugate . The results can be extended to many-particle wavefunctions. Written in Dirac notation as standard, the transformations on quantum state vectors are:

Now, the action of changes ψ(r, t) to ψ(r′, t′), so the inverse changes ψ(r′, t′) back to ψ(r, t), so an operator invariant under satisfies:

and thus:

for any state ψ. Quantum operators representing observables are also required to be Hermitian so that their eigenvalues are real numbers, i.e. the operator equals its Hermitian conjugate, .

Overview of Lie group theory

Following are the key points of group theory relevant to quantum theory, examples are given throughout the article. For an alternative approach using matrix groups, see the books of Hall^[1][2]
Let G be a Lie group, which is a group parameterized by a finite number N of real continuously varying parameters ξ₁, ξ₂, ... ξ_N.

• the dimension of the group, N, is the number of parameters it has.
• the group elements, g, in G are functions of the parameters:

and all parameters set to zero returns the identity element of the group:

Group elements are often matrices which act on vectors, or transformations acting on functions.

• The generators of the group are the partial derivatives of the group elements with respect to the group parameters with the result evaluated when the parameter is set to zero:

One aspect of generators in theoretical physics is they can be construed themselves as operators corresponding to symmetries, which may be written as matrices, or as differential operators. In quantum theory, for unitary representations of the group, the generators require a factor of i:

The generators of the group form a vector space, which means linear combinations of generators also form a generator.

• The generators (whether matrices or differential operators) satisfy the commutator:

where f_abc are the (basis dependent) structure constants of the group. This makes, together with the vector space property, the set of all generators of a group a Lie algebra. Due to the antisymmetry of the bracket, the structure constants of the group are antisymmetric in the first two indices.

• The representations of the group are denoted using a capital D and defined by:

without summation on the repeated index j. Representations are linear operators that take in group elements and preserve the composition rule:

A representation which cannot be decomposed into a direct sum of other representations, is called irreducible. It is conventional to label irreducible representations by a superscripted number n in brackets, as in D⁽ⁿ⁾, or if there is more than one number, we write D^{(n, m, ... )}.

Representations also exist for the generators and the same notation of a capital D is used in this context: D(X). The D in the representation of a generator D(X) is not the same mapping as the D in a representation of a group element, nevertheless this notational abuse of using the same letter to denote two different mappings is used in the literature. An example of this abuse is to be found in the defining equation above.

Momentum and energy as generators of translation and time evolution, and rotation

The space translation operator acts on a wavefunction to shift the space coordinates by an infinitesimal displacement Δr. The explicit expression can be quickly determined by a Taylor expansion of ψ(r + Δr, t) about r, then (keeping the first order term and neglecting second and higher order terms), replace the space derivatives by the momentum operator . Similarly for the time translation operator acting on the time parameter, the Taylor expansion of ψ(r, t + Δt) is about t, and the time derivative replaced by the energy operator .

Name	Translation operator	Time translation/evolution operator
Action on wavefunction
Infinitesimal operator
Finite operator
Generator	Momentum operator	Energy operator

The exponential functions arise by definition as those limits, due to Euler, and can be understood physically and mathematically as follows. A net translation can be composed of many small translations, so to obtain the translation operator for a finite increment, replace Δr by Δr/N and Δt by Δt/N, where N is a positive non-zero integer. Then as N increases, the magnitude of Δr and Δt become even smaller, while leaving the directions unchanged. Acting the infinitesimal operators on the wavefunction N times and taking the limit as N tends to infinity gives the finite operators.

Space and time translations commute, which means the operators and generators commute.

Commutators

Operators
Generators

For a time-independent Hamiltonian, energy is conserved in time and quantum states are stationary states: the eigenstates of the Hamiltonian are the energy eigenvalues E:

and all stationary states have the form

where t₀ is the initial time, usually set to zero since there is no loss of continuity when the initial time is set.
An alternative notation is .

Angular momentum as the generator of rotations

Orbital angular momentum

The rotation operator acts on a wavefunction to rotate the spatial coordinates of a particle by a constant angle Δθ:

where r′ are the rotated coordinates about an axis defined by a unit vector through an angular increment Δθ, given by:

where is a rotation matrix dependent on the axis and angle. In group theoretic language, the rotation matrices are group elements, and the angles and axis are the parameters, of the three-dimensional special orthogonal group, SO(3). The rotation matrices about the standard Cartesian basis vector through angle Δθ, and the corresponding generators of rotations J = (J_x, J_y, J_z), are:

More generally for rotations about an axis defined by , the rotation matrix elements are:^[3]

where δ_ij is the Kronecker delta, and ε_ijk is the Levi-Civita symbol.

It is not as obvious how to determine the rotational operator compared to space and time translations. We may consider a special case (rotations about the x, y, or z-axis) then infer the general result, or use the general rotation matrix directly and tensor index notation with δ_ij and ε_ijk. To derive the infinitesimal rotation operator, which corresponds to small Δθ, we use the small angle approximations sin(Δθ) ≈ Δθ and cos(Δθ) ≈ 1, then Taylor expand about r or r_i, keep the first order term, and substitute the angular momentum operator components.

	Rotation about	Rotation about
Action on wavefunction
Infinitesimal operator
Infinitesimal rotations		Same
Finite rotations		Same
Generator	z-component of the angular momentum operator	Full angular momentum operator .

The z-component of angular momentum can be replaced by the component along the axis defined by , using the dot product .

Again, a finite rotation can be made from lots of small rotations, replacing Δθ by Δθ/N and taking the limit as N tends to infinity gives the rotation operator for a finite rotation.

Rotations about the same axis do commute, for example a rotation through angles θ₁ and θ₂ about axis i can be written

However, rotations about different axes do not commute. The general commutation rules are summarized by

In this sense, orbital angular momentum has the common sense properties of rotations. Each of the above commutators can be easily demonstrated by holding an everyday object and rotating it through the same angle about any two different axes in both possible orderings; the final configurations are different.

In quantum mechanics, there is another form of rotation which mathematically appears similar to the orbital case, but has different properties, described next.

Spin angular momentum

All previous quantities have classical definitions. Spin is a quantity possessed by particles in quantum mechanics without any classical analogue, having the units of angular momentum. The spin vector operator is denoted . The eigenvalues of its components are the possible outcomes (in units of ) of a measurement of the spin projected onto one of the basis directions.

Rotations (of ordinary space) about an axis through angle θ about the unit vector in space acting on a multicomponent wave function (spinor) at a point in space is represented by:

Spin rotation operator (finite)

However, unlike orbital angular momentum in which the z-projection quantum number ℓ can only take positive or negative integer values (including zero), the z-projection spin quantum number s can take all positive and negative half-integer values. There are rotational matrices for each spin quantum number.

Evaluating the exponential for a given z-projection spin quantum number s gives a (2s + 1)-dimensional spin matrix. This can be used to define a spinor as a column vector of 2s + 1 components which transforms to a rotated coordinate system according to the spin matrix at a fixed point in space.

For the simplest non-trivial case of s = 1/2, the spin operator is given by

where the Pauli matrices in the standard representation are:

Total angular momentum

The total angular momentum operator is the sum of the orbital and spin

and is an important quantity for multi-particle systems, especially in nuclear physics and the quantum chemistry of multi-electron atoms and molecules.

We have a similar rotation matrix:

Lorentz group in relativistic quantum mechanics

Following is an overview of the Lorentz group; a treatment of boosts and rotations in spacetime. Throughout this section, see (for example) T. Ohlsson (2011)^[4] and E. Abers (2004).^[5]

Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector , and a rotation angle θ about a three-dimensional unit vector defining an axis, so and are together six parameters of the Lorentz group (three for rotations and three for boosts). The Lorentz group is 6-dimensional.

Pure rotations in spacetime

The rotation matrices and rotation generators considered above form the spacelike part of a four-dimensional matrix, representing pure-rotation Lorentz transformations. Three of the Lorentz group elements and generators J = (J₁, J₂, J₃) for pure rotations are:

The rotation matrices act on any four vector A = (A₀, A₁, A₂, A₃) and rotate the space-like components according to:

leaving the time-like coordinate unchanged. In matrix expressions, A is treated as a column vector.

Pure boosts in spacetime

A boost with velocity ctanhφ in the x, y, or z directions given by the standard Cartesian basis vector , are the boost transformation matrices. These matrices and the corresponding generators K = (K₁, K₂, K₃) are the remaining three group elements and generators of the Lorentz group:

The boost matrices act on any four vector A = (A₀, A₁, A₂, A₃) and mix the time-like and the space-like components, according to:

The term "boost" refers to the relative velocity between two frames, and is not to be conflated with momentum as the generator of translations, as explained below.

Combining boosts and rotations

Products of rotations give another rotation (a frequent exemplification of a subgroup), while products of boosts and boosts or of rotations and boosts cannot be expressed as pure boosts or pure rotations. In general, any Lorentz transformation can be expressed as a product of a pure rotation and a pure boost. For more background see (for example) B.R. Durney (2011)^[6] and H.L. Berk et al.^[7] and references therein.

The boost and rotation generators have representations denoted D(K) and D(J) respectively, the capital D in this context indicates a group representation.

For the Lorentz group, the representations D(K) and D(J) of the generators K and J fulfill the following commutation rules.

Commutators

	Pure rotation	Pure boost	Lorentz transformation
Generators
Representations

In all commutators, the boost entities mixed with those for rotations, although rotations alone simply give another rotation. Exponentiating the generators gives the boost and rotation operators which combine into the general Lorentz transformation, under which the spacetime coordinates transform from one rest frame to another boosted and/or rotating frame. Likewise, exponentiating the representations of the generators gives the representations of the boost and rotation operators, under which a particle's spinor field transforms.

Transformation laws

	Pure boost	Pure rotation	Lorentz transformation
Transformations
Representations

In the literature, the boost generators K and rotation generators J are sometimes combined into one generator for Lorentz transformations M, an antisymmetric four-dimensional matrix with entries:

and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:

The general Lorentz transformation is then:

with summation over repeated matrix indices α and β. The Λ matrices act on any four vector A = (A₀, A₁, A₂, A₃) and mix the time-like and the space-like components, according to:

Transformations of spinor wavefunctions in relativistic quantum mechanics

In relativistic quantum mechanics, wavefunctions are no longer single-component scalar fields, but now 2(2s + 1) component spinor fields, where s is the spin of the particle. The transformations of these functions in spacetime are given below.

Under a proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, all one-particle quantum states ψ_σ locally transform under some representation D of the Lorentz group:^{[8] [9]}

where D(Λ) is a finite-dimensional representation, in other words a (2s + 1)×(2s + 1) dimensional square matrix, and ψ is thought of as a column vector containing components with the (2s + 1) allowed values of σ:

Real irreducible representations and spin

The irreducible representations of D(K) and D(J), in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new operators:

so A and B are simply complex conjugates of each other, it follows they satisfy the symmetrically formed commutators:

and these are essentially the commutators the orbital and spin angular momentum operators satisfy. Therefore A and B form operator algebras analogous to angular momentum; same ladder operators, z-projections, etc., independently of each other as each of their components mutually commute. By the analogy to the spin quantum number, we can introduce positive integers or half integers, a, b, with corresponding sets of values m = a, a − 1, ... −a + 1, −a and n = b, b − 1, ... −b + 1, −b. The matrices satisfying the above commutation relations are the same as for spins a and b have components given by multiplying Kronecker delta values with angular momentum matrix elements:

where in each case the row number m′n′ and column number mn are separated by a comma, and in turn:

and similarly for J⁽ⁿ⁾.^{[note 1]} The three J^(m) matrices are each (2m + 1)×(2m + 1) square matrices, and the three J⁽ⁿ⁾ are each (2n + 1)×(2n + 1) square matrices. The integers or half-integers m and n numerate all the irreducible representations by, in equivalent notations used by authors: D^{(m, n)} ≡ (m, n) ≡ D^(m) ⊗ D⁽ⁿ⁾, which are each [(2m + 1)(2n + 1)]×[(2m + 1)(2n + 1)] square matrices.

Applying this to particles with spin s;

• left-handed (2s + 1)-component spinors transform under the real irreps D^{(s, 0)},
• right-handed (2s + 1)-component spinors transform under the real irreps D^{(0, s)},
• taking direct sums symbolized by ⊕ (see direct sum of matrices for the simpler matrix concept), one obtains the representations under which 2(2s + 1)-component spinors transform: D^{(m, n)} ⊕ D^{(n, m)} where m + n = s. These are also real irreps, but as shown above, they split into complex conjugates.

In these cases the D refers to any of D(J), D(K), or a full Lorentz transformation D(Λ).

Relativistic wave equations

In the context of the Dirac equation and Weyl equation, the Weyl spinors satisfying the Weyl equation transform under the simplest irreducible spin representations of the Lorentz group, since the spin quantum number in this case is the smallest non-zero number allowed: 1/2. The 2-component left-handed Weyl spinor transforms under D^{(1/2, 0)} and the 2-component right-handed Weyl spinor transforms under D^{(0, 1/2)}. Dirac spinors satisfying the Dirac equation transform under the representation D^{(1/2, 0)} ⊕ D^{(0, 1/2)}, the direct sum of the irreps for the Weyl spinors.

The Poincaré group in relativistic quantum mechanics and field theory

Space translations, time translations, rotations, and boosts, all taken together, constitute the Poincaré group. The group elements are the three rotation matrices and three boost matrices (as in the Lorentz group), and one for time translations and three for space translations in spacetime. There is a generator for each. Therefore the Poincaré group is 10-dimensional.

In special relativity, space and time can be collected into a four-position vector X = (ct, −r), and in parallel so can energy and momentum which combine into a four-momentum vector P = (E/c, −p). With relativistic quantum mechanics in mind, the time duration and spatial displacement parameters (four in total, one for time and three for space) combine into a spacetime displacement ΔX = (cΔt, −Δr), and the energy and momentum operators are inserted in the four-momentum to obtain a four-momentum operator,

which are the generators of spacetime translations (four in total, one time and three space):

There are commutation relations between the components four-momentum P (generators of spacetime translations), and angular momentum M (generators of Lorentz transformations), that define the Poincaré algebra:^[10][11]

• 
• 
• 

where η is the Minkowski metric tensor. (It is common to drop any hats for the four-momentum operators in the commutation relations). These equations are an expression of the fundamental properties of space and time as far as they are known today. They have a classical counterpart where the commutators are replaced by Poisson brackets.
To describe spin in relativistic quantum mechanics, the Pauli–Lubanski pseudovector

a Casimir operator, is the constant spin contribution to the total angular momentum, and there are commutation relations between P and W and between M and W:

Invariants constructed from W, instances of Casimir invariants can be used to classify irreducible representations of the Lorentz group.

Symmetries in quantum field theory and particle physics

Unitary groups in quantum field theory

Group theory is an abstract way of mathematically analyzing symmetries. Unitary operators are paramount to quantum theory, so unitary groups are important in particle physics. The group of N dimensional unitary square matrices is denoted U(N). Unitary operators preserve inner products which means probabilities are also preserved, so the quantum mechanics of the system is invariant under unitary unitary transformations. Let be a unitary operator, so the inverse is the Hermitian adjoint , which commutes with the Hamiltonian:

then the observable corresponding to the operator is conserved, and the Hamiltonian is invariant under the transformation .

Since the predictions of quantum mechanics should be invariant under the action of a group, physicists look for unitary transformations to represent the group.

Important subgroups of each U(N) are those unitary matrices which have unit determinant (or are "unimodular"): these are called the special unitary groups and are denoted SU(N).

U(1) and SU(1)

The simplest unitary group is U(1), which is just a complex number of modulus 1. This one-dimensional matrix entry is of the form:

in which θ is the parameter of the group, and the group is Abelian since one-dimensional matrices always commute under matrix multiplication. Lagrangians in quantum field theory for complex scalar fields are often invariant under U(1) transformations. If there is a quantum number a associated with the U(1) symmetry, for example baryon and the three lepton numbers in electromagnetic interactions, we have:

U(2) and SU(2)

The general form of an element of a U(2) element is parametrized by two complex numbers a and b:

and for SU(2), the determinant is restricted to 1:

In group theoretic language, the Pauli matrices are the generators of the special unitary group in two dimensions, denoted SU(2). Their commutation relation is the same as for orbital angular momentum, aside from a factor of 2:

A group element of SU(2) can be written:

where σ_j is a Pauli matrix, and the group parameters are the angles turned through about an axis.

U(3) and SU(3)

The eight Gell-Mann matrices λ_n (see article for them and the structure constants) are important for quantum chromodynamics. They originally arose in the theory SU(3) of flavor which is still of practical importance in nuclear physics. They are the generators for the SU(3) group, so an element of SU(3) can be written analogously to an element of SU(2):

where θ_n are eight independent parameters. The λ_n matrices satisfy the commutator:

where the indices a, b, c take the values 1, 2, 3... 8. The structure constants f_abc are totally antisymmetric in all indices analogous to those of SU(2). In the standard colour charge basis (r for red, g for green, b for blue):

the colour states are eigenstates of the λ₃ and λ₈ matrices, while the other matrices mix colour states together.

The eight gluons states (8-dimensional column vectors) are simultaneous eigenstates of the adjoint representation of SU(3) , the 8-dimensional representation acting on its own Lie algebra su(3), for the λ₃ and λ₈ matrices. By forming tensor products of representations (the standard representation and its dual) and taking appropriate quotients, protons and neutrons, and other hadrons are eigenstates of various representations of SU(3) of color. The adjoint representation of above is isomorphic to the tensor product of the standard representation and its dual.

Matter and antimatter

In relativistic quantum mechanics, relativistic wave equations predict a remarkable symmetry of nature: that every particle has a corresponding antiparticle. This is mathematically contained in the spinor fields which are the solutions of the relativistic wave equations.

Charge conjugation switches particles and antiparticles. Physical laws and interactions unchanged by this operation have C symmetry.

Discrete spacetime symmetries

• Parity mirrors the orientation of the spatial coordinates from left-handed to right-handed. Informally, space is "reflected" into its mirror image. Physical laws and interactions unchanged by this operation have P symmetry.
• Time reversal negates the time coordinate, which amounts to time running from future to past. A curious property of time, which space does not have, is that it is unidirectional: particles traveling forwards in time are equivalent to antiparticles traveling back in time. Physical laws and interactions unchanged by this operation have T symmetry.

C, P, T symmetries

• CPT theorem
• CP violation
• Lorentz violation

Gauge theory

In quantum electrodynamics, the symmetry group is U(1) and is abelian. In quantum chromodynamics, the symmetry group is SU(3) and is non-abelian.
The electromagnetic interaction is mediated by photons, which have no electric charge. The electromagnetic tensor has an electromagnetic four-potential field possessing gauge symmetry.

The strong (color) interaction is mediated by gluons, which can have eight color charges. There are eight gluon field strength tensors with corresponding gluon four potentials field, each possessing gauge symmetry.

The strong (color) interaction

Color charge

Analogous to the spin operator, there are color charge operators in terms of the Gell-Mann matrices λ_j:

and since color charge is a conserved charge, all color charge operators must commute with the Hamiltonian:

Isospin

Isospin is conserved in strong interactions.

The weak and electromagnetic interactions

Duality transformation

Magnetic monopoles can be theoretically realized, although current observations and theory are consistent with them existing or not existing. Electric and magnetic charges can effectively be "rotated into one another" by a duality transformation.

Electroweak symmetry

• Electroweak symmetry
• Electroweak symmetry breaking

Supersymmetry

A Lie superalgebra is an algebra in which (suitable) basis elements either have a commutation relation or have an anticommutation relation. Symmetries have been proposed to the effect that all fermionic particles have bosonic analogues, and vice versa. These symmetry have theoretical appeal in that no extra assumptions (such as existence of strings) barring symmetries are made. In addition, by assuming supersymmetry, a number puzzling issues can be resolved. These symmetries, which are represented by Lie superalgebras, have not been confirmed experimentally. It is now believed that they are broken symmetries, if they exist. But it has been speculated that dark matter is constitutes gravitinos, a spin 3/2 particle with mass, its supersymmetric partner being the graviton.

Exchange symmetry

The concept of exchange symmetry is derived from a fundamental postulate of quantum statistics, which states that no observable physical quantity should change after exchanging two identical particles. It states that because all observables are proportional to for a system of identical particles, the wave function must either remain the same or change sign upon such an exchange.
Because the exchange of two identical particles is mathematically equivalent to the rotation of each particle by 180 degrees (and so to the rotation of one particle's frame by 360 degrees),^[12] the symmetric nature of the wave function depends on the particle's spin after the rotation operator is applied to it. Integer spin particles do not change the sign of their wave function upon a 360 degree rotation—therefore the sign of the wave function of the entire system does not change. Semi-integer spin particles change the sign of their wave function upon a 360 degree rotation (see more in spin–statistics theorem).

Particles for which the wave function does not change sign upon exchange are called bosons, or particles with a symmetric wave function. The particles for which the wave function of the system changes sign are called fermions, or particles with an antisymmetric wave function.

Fermions therefore obey different statistics (called Fermi–Dirac statistics) than bosons (which obey Bose–Einstein statistics). One of the consequences of Fermi–Dirac statistics is the exclusion principle for fermions—no two identical fermions can share the same quantum state (in other words, the wave function of two identical fermions in the same state is zero). This in turn results in degeneracy pressure for fermions—the strong resistance of fermions to compression into smaller volume. This resistance gives rise to the “stiffness” or “rigidity” of ordinary atomic matter (as atoms contain electrons which are fermions).

Quantum chromodynamics

From Wikipedia, the free encyclopedia

In theoretical physics, quantum chromodynamics (QCD) is the theory of strong interactions, a fundamental force describing the interactions between quarks and gluons which make up hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory with symmetry group SU(3). The QCD analog of electric charge is a property called color. Gluons are the force carrier of the theory, like photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A huge body of experimental evidence for QCD has been gathered over the years.
QCD enjoys two peculiar properties:

• Confinement, which means that the force between quarks does not diminish as they are separated. Because of this, when you do separate a quark from other quarks, the energy in the gluon field is enough to create another quark pair; they are thus forever bound into hadrons such as the proton and the neutron or the pion and kaon. Although analytically unproven, confinement is widely believed to be true because it explains the consistent failure of free quark searches, and it is easy to demonstrate in lattice QCD.
• Asymptotic freedom, which means that in very high-energy reactions, quarks and gluons interact very weakly creating a quark–gluon plasma. This prediction of QCD was first discovered in the early 1970s by David Politzer and by Frank Wilczek and David Gross. For this work they were awarded the 2004 Nobel Prize in Physics.

The phase transition temperature between these two properties has been measured by the ALICE experiment to be around 160 MeV^{[citation needed]}. Below this temperature, confinement is dominant, while above it, asymptotic freedom becomes dominant.

Terminology

The word quark was coined by American physicist Murray Gell-Mann (b. 1929) in its present sense. It originally comes from the phrase "Three quarks for Muster Mark" in Finnegans Wake by James Joyce. On June 27, 1978, Gell-Mann wrote a private letter to the editor of the Oxford English Dictionary, in which he related that he had been influenced by Joyce's words: "The allusion to three quarks seemed perfect." (Originally, only three quarks had been discovered.) Gell-Mann, however, wanted to pronounce the word to rhyme with "fork" rather than with "park", as Joyce seemed to indicate by rhyming words in the vicinity such as Mark. Gell-Mann got around that "by supposing that one ingredient of the line 'Three quarks for Muster Mark' was a cry of 'Three quarts for Mister ...' heard in H.C. Earwicker's pub", a plausible suggestion given the complex punning in Joyce's novel.^[1]

The three kinds of charge in QCD (as opposed to one in quantum electrodynamics or QED) are usually referred to as "color charge" by loose analogy to the three kinds of color (red, green and blue) perceived by humans. Other than this nomenclature, the quantum parameter "color" is completely unrelated to the everyday, familiar phenomenon of color.

Since the theory of electric charge is dubbed "electrodynamics", the Greek word "chroma" Χρώμα (meaning color) is applied to the theory of color charge, "chromodynamics".

History

With the invention of bubble chambers and spark chambers in the 1950s, experimental particle physics discovered a large and ever-growing number of particles called hadrons. It seemed that such a large number of particles could not all be fundamental. First, the particles were classified by charge and isospin by Eugene Wigner and Werner Heisenberg; then, in 1953, according to strangeness by Murray Gell-Mann and Kazuhiko Nishijima. To gain greater insight, the hadrons were sorted into groups having similar properties and masses using the eightfold way, invented in 1961 by Gell-Mann and Yuval Ne'eman. Gell-Mann and George Zweig, correcting an earlier approach of Shoichi Sakata, went on to propose in 1963 that the structure of the groups could be explained by the existence of three flavors of smaller particles inside the hadrons: the quarks.

Perhaps the first remark that quarks should possess an additional quantum number was made^[2] as a short footnote in the preprint of Boris Struminsky^[3] in connection with Ω⁻ hyperon composed of three strange quarks with parallel spins (this situation was peculiar, because since quarks are fermions, such combination is forbidden by the Pauli exclusion principle):

Three identical quarks cannot form an antisymmetric S-state. In order to realize an antisymmetric orbital S-state, it is necessary for the quark to have an additional quantum number.

— B. V. Struminsky, Magnetic moments of barions in the quark model, JINR-Preprint P-1939, Dubna, Submitted on January 7, 1965

Boris Struminsky was a PhD student of Nikolay Bogolyubov. The problem considered in this preprint was suggested by Nikolay Bogolyubov, who advised Boris Struminsky in this research.^[3] In the beginning of 1965, Nikolay Bogolyubov, Boris Struminsky and Albert Tavkhelidze wrote a preprint with a more detailed discussion of the additional quark quantum degree of freedom.^[4] This work was also presented by Albert Tavchelidze without obtaining consent of his collaborators for doing so at an international conference in Trieste (Italy), in May 1965.^[5][6]

A similar mysterious situation was with the Δ⁺⁺ baryon; in the quark model, it is composed of three up quarks with parallel spins. In 1965, Moo-Young Han with Yoichiro Nambu and Oscar W. Greenberg independently resolved the problem by proposing that quarks possess an additional SU(3) gauge degree of freedom, later called color charge. Han and Nambu noted that quarks might interact via an octet of vector gauge bosons: the gluons.

Since free quark searches consistently failed to turn up any evidence for the new particles, and because an elementary particle back then was defined as a particle which could be separated and isolated, Gell-Mann often said that quarks were merely convenient mathematical constructs, not real particles. The meaning of this statement was usually clear in context: He meant quarks are confined, but he also was implying that the strong interactions could probably not be fully described by quantum field theory.

Richard Feynman argued that high energy experiments showed quarks are real particles: he called them partons (since they were parts of hadrons). By particles, Feynman meant objects which travel along paths, elementary particles in a field theory.

The difference between Feynman's and Gell-Mann's approaches reflected a deep split in the theoretical physics community. Feynman thought the quarks have a distribution of position or momentum, like any other particle, and he (correctly) believed that the diffusion of parton momentum explained diffractive scattering. Although Gell-Mann believed that certain quark charges could be localized, he was open to the possibility that the quarks themselves could not be localized because space and time break down. This was the more radical approach of S-matrix theory.

James Bjorken proposed that pointlike partons would imply certain relations should hold in deep inelastic scattering of electrons and protons, which were spectacularly verified in experiments at SLAC in 1969. This led physicists to abandon the S-matrix approach for the strong interactions.

The discovery of asymptotic freedom in the strong interactions by David Gross, David Politzer and Frank Wilczek allowed physicists to make precise predictions of the results of many high energy experiments using the quantum field theory technique of perturbation theory. Evidence of gluons was discovered in three-jet events at PETRA in 1979. These experiments became more and more precise, culminating in the verification of perturbative QCD at the level of a few percent at the LEP in CERN.

The other side of asymptotic freedom is confinement. Since the force between color charges does not decrease with distance, it is believed that quarks and gluons can never be liberated from hadrons. This aspect of the theory is verified within lattice QCD computations, but is not mathematically proven. One of the Millennium Prize Problems announced by the Clay Mathematics Institute requires a claimant to produce such a proof. Other aspects of non-perturbative QCD are the exploration of phases of quark matter, including the quark–gluon plasma.

The relation between the short-distance particle limit and the confining long-distance limit is one of the topics recently explored using string theory, the modern form of S-matrix theory.^[7][8]

Theory

Some definitions

Every field theory of particle physics is based on certain symmetries of nature whose existence is deduced from observations. These can be

• local symmetries, that is the symmetry acts independently at each point in spacetime. Each such symmetry is the basis of a gauge theory and requires the introduction of its own gauge bosons.
• global symmetries, which are symmetries whose operations must be simultaneously applied to all points of spacetime.

QCD is a gauge theory of the SU(3) gauge group obtained by taking the color charge to define a local symmetry.
Since the strong interaction does not discriminate between different flavors of quark, QCD has approximate flavor symmetry, which is broken by the differing masses of the quarks.

There are additional global symmetries whose definitions require the notion of chirality, discrimination between left and right-handed. If the spin of a particle has a positive projection on its direction of motion then it is called left-handed; otherwise, it is right-handed. Chirality and handedness are not the same, but become approximately equivalent at high energies.

• Chiral symmetries involve independent transformations of these two types of particle.
• Vector symmetries (also called diagonal symmetries) mean the same transformation is applied on the two chiralities.
• Axial symmetries are those in which one transformation is applied on left-handed particles and the inverse on the right-handed particles.

Additional remarks: duality

As mentioned, asymptotic freedom means that at large energy – this corresponds also to short distances – there is practically no interaction between the particles. This is in contrast – more precisely one would say dual – to what one is used to, since usually one connects the absence of interactions with large distances. However, as already mentioned in the original paper of Franz Wegner,^[9] a solid state theorist who introduced 1971 simple gauge invariant lattice models, the high-temperature behaviour of the original model, e.g. the strong decay of correlations at large distances, corresponds to the low-temperature behaviour of the (usually ordered!) dual model, namely the asymptotic decay of non-trivial correlations, e.g. short-range deviations from almost perfect arrangements, for short distances. Here, in contrast to Wegner, we have only the dual model, which is that one described in this article.^[10]

Symmetry groups

The color group SU(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group U(1) which is gauged to give QED: this is an abelian group. If one considers a version of QCD with N_f flavors of massless quarks, then there is a global (chiral) flavor symmetry group SU_L(N_f) × SU_R(N_f) × U_B(1) × U_A(1). The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) SU_V(N_f) with the formation of a chiral condensate. The vector symmetry, U_B(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry U_A(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an anomaly. Gluon field configurations called instantons are closely related to this anomaly.

There are two different types of SU(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry which rotates different flavors of quarks to each other, or flavor SU(3). Flavor SU(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks.

In the QCD vacuum there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under SU(2) isospin rotations of up and down, and to a lesser extent under rotations of up, down and strange, or full flavor group SU(3), and the observed particles make isospin and SU(3) multiplets.

The approximate flavor symmetries do have associated gauge bosons, observed particles like the rho and the omega, but these particles are nothing like the gluons and they are not massless. They are emergent gauge bosons in an approximate string description of QCD.

Lagrangian

The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The gauge invariant QCD Lagrangian is

where is the quark field, a dynamical function of spacetime, in the fundamental representation of the SU(3) gauge group, indexed by ; are the gluon fields, also dynamical functions of spacetime, in the adjoint representation of the SU(3) gauge group, indexed by a, b,... The γ^μ are Dirac matrices connecting the spinor representation to the vector representation of the Lorentz group.

The symbol represents the gauge invariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, F^μν, in quantum electrodynamics. It is given by:^[11]

where f_abc are the structure constants of SU(3). Note that the rules to move-up or pull-down the a, b, or c indexes are trivial, (+, ..., +), so that f^abc = f_abc = f^a_bc whereas for the μ or ν indexes one has the non-trivial relativistic rules, corresponding e.g. to the metric signature (+ − − −).

The constants m and g control the quark mass and coupling constants of the theory, subject to renormalization in the full quantum theory.

An important theoretical notion concerning the final term of the above Lagrangian is the Wilson loop variable. This loop variable plays a most important role in discretized forms of the QCD (see lattice QCD), and more generally, it distinguishes confined and deconfined states of a gauge theory. It was introduced by the Nobel prize winner Kenneth G. Wilson and is treated in a separate article.

Fields

The pattern of strong charges for the three colors of quark, three antiquarks, and eight gluons (with two of zero charge overlapping).

Quarks are massive spin-1/2 fermions which carry a color charge whose gauging is the content of QCD. Quarks are represented by Dirac fields in the fundamental representation 3 of the gauge group SU(3). They also carry electric charge (either −1/3 or 2/3) and participate in weak interactions as part of weak isospin doublets. They carry global quantum numbers including the baryon number, which is 1/3 for each quark, hypercharge and one of the flavor quantum numbers.

Gluons are spin-1 bosons which also carry color charges, since they lie in the adjoint representation 8 of SU(3). They have no electric charge, do not participate in the weak interactions, and have no flavor. They lie in the singlet representation 1 of all these symmetry groups.

Every quark has its own antiquark. The charge of each antiquark is exactly the opposite of the corresponding quark.

Dynamics

According to the rules of quantum field theory, and the associated Feynman diagrams, the above theory gives rise to three basic interactions: a quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. This contrasts with QED, in which only the first kind of interaction occurs, since photons have no charge. Diagrams involving Faddeev–Popov ghosts must be considered too (except in the unitarity gauge).

Area law and confinement

Detailed computations with the above-mentioned Lagrangian^[12] show that the effective potential between a quark and its anti-quark in a meson contains a term , which represents some kind of "stiffness" of the interaction between the particle and its anti-particle at large distances, similar to the entropic elasticity of a rubber band (see below). This leads to confinement ^[13] of the quarks to the interior of hadrons, i.e. mesons and nucleons, with typical radii R_c, corresponding to former "Bag models" of the hadrons^[14] . The order of magnitude of the "bag radius" is 1 fm (= 10⁻¹⁵ m). Moreover, the above-mentioned stiffness is quantitatively related to the so-called "area law" behaviour of the expectation value of the Wilson loop product P_W of the ordered coupling constants around a closed loop W; i.e. is proportional to the area enclosed by the loop. For this behaviour the non-abelian behaviour of the gauge group is essential.

Methods

Further analysis of the content of the theory is complicated. Various techniques have been developed to work with QCD. Some of them are discussed briefly below.

Perturbative QCD

This approach is based on asymptotic freedom, which allows perturbation theory to be used accurately in experiments performed at very high energies. Although limited in scope, this approach has resulted in the most precise tests of QCD to date.

Lattice QCD

A quark and an antiquark (red color) are glued together (green color) to form a meson (result of a lattice QCD simulation by M. Cardoso et al.^[15])

Among non-perturbative approaches to QCD, the most well established one is lattice QCD. This approach uses a discrete set of spacetime points (called the lattice) to reduce the analytically intractable path integrals of the continuum theory to a very difficult numerical computation which is then carried out on supercomputers like the QCDOC which was constructed for precisely this purpose. While it is a slow and resource-intensive approach, it has wide applicability, giving insight into parts of the theory inaccessible by other means, in particular into the explicit forces acting between quarks and antiquarks in a meson. However, the numerical sign problem makes it difficult to use lattice methods to study QCD at high density and low temperature (e.g. nuclear matter or the interior of neutron stars).

1/N expansion

A well-known approximation scheme, the 1/N expansion, starts from the premise that the number of colors is infinite, and makes a series of corrections to account for the fact that it is not. Until now, it has been the source of qualitative insight rather than a method for quantitative predictions. Modern variants include the AdS/CFT approach.

Effective theories

For specific problems effective theories may be written down which give qualitatively correct results in certain limits. In the best of cases, these may then be obtained as systematic expansions in some parameter of the QCD Lagrangian. One such effective field theory is chiral perturbation theory or ChiPT, which is the QCD effective theory at low energies. More precisely, it is a low energy expansion based on the spontaneous chiral symmetry breaking of QCD, which is an exact symmetry when quark masses are equal to zero, but for the u,d and s quark, which have small mass, it is still a good approximate symmetry. Depending on the number of quarks which are treated as light, one uses either SU(2) ChiPT or SU(3) ChiPT . Other effective theories are heavy quark effective theory (which expands around heavy quark mass near infinity), and soft-collinear effective theory (which expands around large ratios of energy scales). In addition to effective theories, models like the Nambu–Jona-Lasinio model and the chiral model are often used when discussing general features.

QCD sum rules

Based on an Operator product expansion one can derive sets of relations that connect different observables with each other.

Nambu–Jona-Lasinio model

In one of his recent works, Kei-Ichi Kondo derived as a low-energy limit of QCD, a theory linked to the Nambu–Jona-Lasinio model since it is basically a particular non-local version of the Polyakov–Nambu–Jona-Lasinio model.^[16] The later being in its local version, nothing but the Nambu–Jona-Lasinio model in which one has included the Polyakov loop effect, in order to describe a 'certain confinement'.

The Nambu–Jona-Lasinio model in itself is, among many other things, used because it is a 'relatively simple' model of chiral symmetry breaking, phenomenon present up to certain conditions (Chiral limit i.e. massless fermions) in QCD itself. In this model, however, there is no confinement. In particular, the energy of an isolated quark in the physical vacuum turns out well defined and finite.

Experimental tests

The notion of quark flavors was prompted by the necessity of explaining the properties of hadrons during the development of the quark model. The notion of color was necessitated by the puzzle of the Δ++. This has been dealt with in the section on the history of QCD.

The first evidence for quarks as real constituent elements of hadrons was obtained in deep inelastic scattering experiments at SLAC. The first evidence for gluons came in three jet events at PETRA.

Several good quantitative tests of perturbative QCD exist:

• The running of the QCD coupling as deduced from many observations
• Scaling violation in polarized and unpolarized deep inelastic scattering
• Vector boson production at colliders (this includes the Drell-Yan process)
• Jet cross sections in colliders
• Event shape observables at the LEP
• Heavy-quark production in colliders

Quantitative tests of non-perturbative QCD are fewer, because the predictions are harder to make. The best is probably the running of the QCD coupling as probed through lattice computations of heavy-quarkonium spectra.
There is a recent claim about the mass of the heavy meson B_c [4]. Other non-perturbative tests are currently at the level of 5% at best. Continuing work on masses and form factors of hadrons and their weak matrix elements are promising candidates for future quantitative tests. The whole subject of quark matter and the quark–gluon plasma is a non-perturbative test bed for QCD which still remains to be properly exploited.

One qualitative prediction of QCD is that there exist composite particles made solely of gluons called glueballs that have not yet been definitively observed experimentally. A definitive observation of a glueball with the properties predicted by QCD would strongly confirm the theory. In principle, if glueballs could be definitively ruled out, this would be a serious experimental blow to QCD. But, as of 2013, scientists are unable to confirm or deny the existence of glueballs definitively, despite the fact that particle accelerators have sufficient energy to generate them.

Cross-relations to solid state physics

There are unexpected cross-relations to solid state physics. For example, the notion of gauge invariance forms the basis of the well-known Mattis spin glasses,^[17] which are systems with the usual spin degrees of freedom for i =1,...,N, with the special fixed "random" couplings Here the ε_i and ε_k quantities can independently and "randomly" take the values ±1, which corresponds to a most-simple gauge transformation This means that thermodynamic expectation values of measurable quantities, e.g. of the energy are invariant.
However, here the coupling degrees of freedom , which in the QCD correspond to the gluons, are "frozen" to fixed values (quenching). In contrast, in the QCD they "fluctuate" (annealing), and through the large number of gauge degrees of freedom the entropy plays an important role (see below).

For positive J₀ the thermodynamics of the Mattis spin glass corresponds in fact simply to a "ferromagnet in disguise", just because these systems have no "frustration" at all. This term is a basic measure in spin glass theory.^[18] Quantitatively it is identical with the loop product along a closed loop W. However, for a Mattis spin glass – in contrast to "genuine" spin glasses – the quantity P_W never becomes negative.

The basic notion "frustration" of the spin-glass is actually similar to the Wilson loop quantity of the QCD. The only difference is again that in the QCD one is dealing with SU(3) matrices, and that one is dealing with a "fluctuating" quantity. Energetically, perfect absence of frustration should be non-favorable and atypical for a spin glass, which means that one should add the loop product to the Hamiltonian, by some kind of term representing a "punishment". In the QCD the Wilson loop is essential for the Lagrangian rightaway.

The relation between the QCD and "disordered magnetic systems" (the spin glasses belong to them) were additionally stressed in a paper by Fradkin, Huberman und Shenker,^[19] which also stresses the notion of duality.

A further analogy consists in the already mentioned similarity to polymer physics, where, analogously to Wilson Loops, so-called "entangled nets" appear, which are important for the formation of the entropy-elasticity (force proportional to the length) of a rubber band. The non-abelian character of the SU(3) corresponds thereby to the non-trivial "chemical links", which glue different loop segments together, and "asymptotic freedom" means in the polymer analogy simply the fact that in the short-wave limit, i.e. for (where R_c is a characteristic correlation length for the glued loops, corresponding to the above-mentioned "bag radius", while λ_w is the wavelength of an excitation) any non-trivial correlation vanishes totally, as if the system had crystallized.^[20]

There is also a correspondence between confinement in QCD – the fact that the color field is only different from zero in the interior of hadrons – and the behaviour of the usual magnetic field in the theory of type-II superconductors: there the magnetism is confined to the interiour of the Abrikosov flux-line lattice,^[21]   i.e., the London penetration depth λ of that theory is analogous to the confinement radius R_c of quantum chromodynamics. Mathematically, this correspondendence is supported by the second term, on the r.h.s. of the Lagrangian.