General relativity is the theory of gravitation published by Albert Einstein in 1915. According to it, the force of gravity is a manifestation of the local geometry of spacetime. Mathematically, the theory is modelled after Bernhard Riemann's metric geometry, but the Lorentz group of spacetime symmetries (an essential ingredient of Einstein's own theory of special relativity)
replaces the group of rotational symmetries of space. (Later, loop
quantum gravity inherited this geometric interpretation of gravity, and
posits that a quantum theory of gravity is fundamentally a quantum
theory of spacetime.)
In the 1920s, the French mathematician Élie Cartan formulated Einstein's theory in the language of bundles and connections, a generalization of Riemannian geometry to which Cartan made important contributions. The so-called Einstein–Cartan theory of gravity not only reformulated but also generalized general relativity, and allowed spacetimes with torsion as well as curvature. In Cartan's geometry of bundles, the concept of parallel transport is more fundamental than that of distance, the centerpiece of Riemannian geometry. A similar conceptual shift occurs between the invariant interval of Einstein's general relativity and the parallel transport of Einstein–Cartan theory.
Spin networks
In 1971, physicist Roger Penrose explored the idea of space arising from a quantum combinatorial structure. His investigations resulted in the development of spin networks. Because this was a quantum theory of the rotational group and not the Lorentz group, Penrose went on to develop twistors.
Loop quantum gravity
In 1982, Amitabha Sen tried to formulate a Hamiltonian formulation of general relativity based on spinorial
variables, where these variables are the left and right spinorial
component equivalents of Einstein–Cartan connection of general
relativity. Particularly, Sen discovered a new way to write down the two constraints of the ADM Hamiltonian formulation
of general relativity in terms of these spinorial connections. In his
form, the constraints are simply conditions that the spinorial Weyl curvature
is trace free and symmetric. He also discovered the presence of new
constraints which he suggested to be interpreted as the equivalent of
Gauss constraint of Yang–Mills field
theories. But Sen's work fell short of giving a full clear systematic
theory and particularly failed to clearly discuss the conjugate momenta
to the spinorial variables, its physical interpretation, and its
relation to the metric (in his work he indicated this as some lambda
variable).
In 1986–87, physicist Abhay Ashtekar
completed the project which Amitabha Sen began. He clearly identified
the fundamental conjugate variables of spinorial gravity: The
configuration variable is as a spinoral connection (a rule for parallel
transport; technically, a connection) and the conjugate momentum variable is a coordinate frame (called a vierbein) at each point. So these variable became what we know as Ashtekar variables,
a particular flavor of Einstein–Cartan theory with a complex
connection. General relativity theory expressed in this way, made
possible to pursue quantization of it using well-known techniques from quantum gauge field theory.
The quantization of gravity in the Ashtekar formulation was based on Wilson loops, a technique developed by Kenneth G. Wilson in 1974 to study the strong-interaction regime of quantum chromodynamics
(QCD). It is interesting in this connection that Wilson loops were
known to be ill-behaved in the case of standard quantum field theory on
(flat) Minkowski space, and so did not provide a nonperturbative
quantization of QCD. However, because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for nonperturbative quantization of gravity.
Due to efforts by Sen and Ashtekar, a setting in which the Wheeler–DeWitt equation was written in terms of a well-defined Hamiltonian operator on a well-defined Hilbert space was obtained. This led to the construction of the first known exact solution, the so-called Chern–Simons form or Kodama state. The physical interpretation of this state remains obscure.
In 1988–90, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labeled by Penrose's spin networks. In this context, spin networks arose as a generalization of Wilson
loops necessary to deal with mutually intersecting loops.
Mathematically, spin networks are related to group representation theory
and can be used to construct knot invariants such as the Jones polynomial. Loop quantum gravity (LQG) thus became related to topological quantum field theory and group representation theory.
In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum. Work on the semi-classical limit, the continuum limit, and dynamics was intense after this, but progress was slower.
LQG
was initially formulated as a quantization of the Hamiltonian ADM
formalism, according to which the Einstein equations are a collection of
constraints (Gauss, Diffeomorphism and Hamiltonian). The kinematics are
encoded in the Gauss and Diffeomorphism constraints, whose solution is
the space spanned by the spin network basis. The problem is to define
the Hamiltonian constraint as a self-adjoint operator on the kinematical state space. The most promising work in this direction is Thomas Thiemann's Phoenix Project.
Covariant dynamics
Much of the recent work in LQG has been done in the covariant formulation of the theory, called "spin foam
theory." The present version of the covariant dynamics is due to the
convergent work of different groups, but it is commonly named after a
paper by Jonathan Engle, Roberto Pereira and Carlo Rovelli in 2007–08. Heuristically, it would be expected that evolution between spin network
states might be described by discrete combinatorial operations on the
spin networks, which would then trace a two-dimensional skeleton of
spacetime. This approach is related to state-sum models of statistical mechanics and topological quantum field theory such as the Turaeev–Viro model of 3D quantum gravity, and also to the Regge calculus approach to calculate the Feynman path integral of general relativity by discretizing spacetime.
Babylonian clay tablet YBC 7289 (c. 1800–1600 BCE) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
It is the study of numerical methods that attempt to find approximate
solutions of problems rather than the exact ones. Numerical analysis
finds application in all fields of engineering and the physical
sciences, and in the 21st century also the life and social sciences like
economics, medicine, business and even the arts. Current growth in
computing power has enabled the use of more complex numerical analysis,
providing detailed and realistic mathematical models in science and
engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
Before modern computers, numerical methods often relied on hand interpolation
formulas, using data from large printed tables. Since the mid-20th
century, computers calculate the required functions instead, but many of
the same formulas continue to be used in software algorithms.
Numerical analysis continues this long tradition: rather than
giving exact symbolic answers translated into digits and applicable only
to real-world measurements, approximate solutions within specified
error bounds are used.
Applications
The
overall goal of the field of numerical analysis is the design and
analysis of techniques to give approximate but accurate solutions to a
wide variety of hard problems, many of which are infeasible to solve
symbolically:
Computing the trajectory of a spacecraft requires the accurate
numerical solution of a system of ordinary differential equations.
Car companies can improve the crash safety of their vehicles by
using computer simulations of car crashes. Such simulations essentially
consist of solving partial differential equations numerically.
In the financial field, (private investment funds) and other financial institutions use quantitative finance tools from numerical analysis to attempt to calculate the value of stocks and derivatives more precisely than other market participants.
Airlines use sophisticated optimization algorithms to decide ticket
prices, airplane and crew assignments and fuel needs. Historically, such
algorithms were developed within the overlapping field of operations research.
Insurance companies use numerical programs for actuarial analysis.
History
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method. The origins of modern numerical analysis are often linked to a 1947 paper by John von Neumann and Herman Goldstine, but others consider modern numerical analysis to go back to work by E. T. Whittaker in 1912.
NIST publication
To facilitate computations by hand, large books were produced with
formulas and tables of data such as interpolation points and function
coefficients. Using these tables, often calculated out to 16 decimal
places or more for some functions, one could look up values to plug into
the formulas given and achieve very good numerical estimates of some
functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun,
a 1000-plus page book of a very large number of commonly used formulas
and functions and their values at many points. The function values are
no longer very useful when a computer is available, but the large
listing of formulas can still be very handy.
The mechanical calculator
was also developed as a tool for hand computation. These calculators
evolved into electronic computers in the 1940s, and it was then found
that these computers were also useful for administrative purposes. But
the invention of the computer also influenced the field of numerical
analysis, since now longer and more complicated calculations could be done.
In contrast to direct methods, iterative methods
are not expected to terminate in a finite number of steps, even if
infinite precision were possible. Starting from an initial guess,
iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving the residual,
is specified in order to decide when a sufficiently accurate solution
has (hopefully) been found. Even using infinite precision arithmetic
these methods would not reach the solution within a finite number of
steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems.
Iterative methods are more common than direct methods in
numerical analysis. Some methods are direct in principle but are usually
used as though they were not, e.g. GMRES and the conjugate gradient method.
For these methods the number of steps needed to obtain the exact
solution is so large that an approximation is accepted in the same
manner as for an iterative method.
As an example, consider the problem of solving
3x3 + 4 = 28
for the unknown quantity x.
Direct method
3x3 + 4 = 28.
Subtract 4
3x3 = 24.
Divide by 3
x3 = 8.
Take cube roots
x = 2.
For the iterative method, apply the bisection method to f(x) = 3x3 − 24. The initial values are a = 0, b = 3, f(a) = −24, f(b) = 57.
Iterative method
a
b
mid
f(mid)
0
3
1.5
−13.875
1.5
3
2.25
10.17...
1.5
2.25
1.875
−4.22...
1.875
2.25
2.0625
2.32...
From this table it can be concluded that the solution is between
1.875 and 2.0625. The algorithm might return any number in that range
with an error less than 0.2.
Conditioning
Ill-conditioned problem: Take the function f(x) = 1/(x − 1). Note that f(1.1) = 10 and f(1.001) = 1000: a change in x of less than 0.1 turns into a change in f(x) of nearly 1000. Evaluating f(x) near x = 1 is an ill-conditioned problem.
Well-conditioned problem: By contrast, evaluating the same function f(x) = 1/(x − 1) near x = 10 is a well-conditioned problem. For instance, f(10) = 1/9 ≈ 0.111 and f(11) = 0.1: a modest change in x leads to a modest change in f(x).
Discretization
Furthermore,
continuous problems must sometimes be replaced by a discrete problem
whose solution is known to approximate that of the continuous problem;
this process is called 'discretization'. For example, the solution of a differential equation is a function.
This function must be represented by a finite amount of data, for
instance by its value at a finite number of points at its domain, even
though this domain is a continuum.
The study of errors forms an important part of numerical analysis.
There are several ways in which error can be introduced in the solution
of the problem.
Truncation errors
are committed when an iterative method is terminated or a mathematical
procedure is approximated and the approximate solution differs from the
exact solution. Similarly, discretization induces a discretization error
because the solution of the discrete problem does not coincide with the
solution of the continuous problem. In the example above to compute the
solution of , after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01.
Once an error is generated, it propagates through the
calculation. For example, the operation + on a computer is inexact. A
calculation of the type is even more inexact.
A truncation error is created when a mathematical procedure is
approximated. To integrate a function exactly, an infinite sum of
regions must be found, but numerically only a finite sum of regions can
be found, and hence the approximation of the exact solution. Similarly,
to differentiate a function, the differential element approaches zero,
but numerically only a nonzero value of the differential element can be
chosen.
Numerical stability and well-posed problems
An algorithm is called numerically stable if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is well-conditioned, meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is 'ill-conditioned', then any small error in the data will grow to be a large error. Both the original problem and the algorithm used to solve that problem
can be well-conditioned or ill-conditioned, and any combination is
possible.
So an algorithm that solves a well-conditioned problem may be either
numerically stable or numerically unstable. An art of numerical analysis
is to find a stable algorithm for solving a well-posed mathematical
problem.
Areas of study
The field of numerical analysis includes many sub-disciplines. Some of the major ones are:
Computing values of functions
Interpolation: Observing that the temperature varies from 20 degrees
Celsius at 1:00 to 14 degrees at 3:00, a linear interpolation of this
data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at
1:30pm.
Extrapolation: If the gross domestic product
of a country has been growing an average of 5% per year and was 100
billion last year, it might be extrapolated that it will be 105 billion
this year.
Regression: In linear regression, given n points, a line is computed that passes as close as possible to those n points.
Optimization: Suppose lemonade is sold at a lemonade stand,
at $1.00 per glass, that 197 glasses of lemonade can be sold per day,
and that for each increase of $0.01, one less glass of lemonade will be
sold per day. If $1.485 could be charged, profit would be maximized, but
due to the constraint of having to charge a whole-cent amount, charging
$1.48 or $1.49 per glass will both yield the maximum income of $220.52
per day.
Differential equation: If 100 fans are set up to blow air from one
end of the room to the other and then a feather is dropped into the
wind, what happens? The feather will follow the air currents, which may
be very complex. One approximation is to measure the speed at which the
air is blowing near the feather every second, and advance the simulated
feather as if it were moving in a straight line at that same speed for
one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation.
One of the simplest problems is the evaluation of a function at a
given point. The most straightforward approach, of just plugging in the
number in the formula is sometimes not very efficient. For polynomials, a
better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errors arising from the use of floating-point arithmetic.
Interpolation, extrapolation, and regression
Interpolation
solves the following problem: given the value of some unknown function
at a number of points, what value does that function have at some other
point between the given points?
Extrapolation
is very similar to interpolation, except that now the value of the
unknown function at a point which is outside the given points must be
found.
Regression
is also similar, but it takes into account that the data are imprecise.
Given some points, and a measurement of the value of some function at
these points (with an error), the unknown function can be found. The least squares-method is one way to achieve this.
Solving equations and systems of equations
Another
fundamental problem is computing the solution of some given equation.
Two cases are commonly distinguished, depending on whether the equation
is linear or not. For instance, the equation is linear while is not.
Root-finding algorithms
are used to solve nonlinear equations (they are so named since a root
of a function is an argument for which the function yields zero). If the
function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.
Optimization problems ask for the point at which a given function is
maximized (or minimized). Often, the point also has to satisfy some constraints.
The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programming
deals with the case that both the objective function and the
constraints are linear. A famous method in linear programming is the simplex method.
The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.
Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an
integral on a relatively large set is broken down into integrals on
smaller sets. In higher dimensions, where these methods become
prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasi-Monte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids.
Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.
Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free-software alternative is the GNU Scientific Library.
There are several popular numerical computing applications such as MATLAB, TK Solver, S-PLUS, and IDL as well as free and open-source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (similar to S-PLUS), Julia, and Python with libraries such as NumPy, SciPy and SymPy.
Performance varies widely: while vector and matrix operations are
usually fast, scalar loops may vary in speed by more than an order of
magnitude.
Also, any spreadsheetsoftware can be used to solve simple problems relating to numerical analysis.
Excel, for example, has hundreds of available functions, including for matrices, which may be used in conjunction with its built in "solver".
Figure 1: The pseudoscalar meson
nonet. Members of the original meson "octet" are shown in green, the
singlet in magenta. Although these mesons are now grouped into a nonet,
the Eightfold Way name derives from the patterns of eight for the mesons and baryons in the original classification scheme.
In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks that give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eightfold Way, the successful classification scheme organizing the large number of lighter hadrons that were being discovered starting in the 1950s and continuing through the 1960s. It received experimental verification
beginning in the late 1960s and is a valid and effective classification
of them to date. The model was independently proposed by physicists Murray Gell-Mann, who dubbed them "quarks" in a concise paper, and George Zweig, who suggested "aces" in a longer manuscript. André Petermann also touched upon the central ideas from 1963 to 1965, without as much quantitative substantiation. Today, the model has essentially been absorbed as a component of the established quantum field theory of strong and electroweak particle interactions, dubbed the Standard Model.
Hadrons are not really "elementary", and can be regarded as bound
states of their "valence quarks" and antiquarks, which give rise to the
quantum numbers of the hadrons. These quantum numbers are labels identifying the hadrons, and are of two kinds. One set comes from the Poincaré symmetry—JPC, where J, P and C stand for the total angular momentum, P-symmetry, and C-symmetry, respectively.
The other set is the flavor quantum numbers such as the isospin, strangeness, charm,
and so on. The strong interactions binding the quarks together are
insensitive to these quantum numbers, so variation of them leads to
systematic mass and coupling relationships among the hadrons in the same
flavor multiplet.
All quarks are assigned a baryon number of 1/3. Up, charm and top quarks have an electric charge of +2/3, while the down, strange, and bottom quarks have an electric charge of −1/3. Antiquarks have the opposite quantum numbers. Quarks are spin-1/2 particles, and thus fermions. Each quark or antiquark obeys the Gell-Mann–Nishijima formula individually, so any additive assembly of them will as well.
Mesons are made of a valence quark–antiquark pair (thus have a baryon number of 0), while baryons
are made of three quarks (thus have a baryon number of 1). This article
discusses the quark model for the up, down, and strange flavors of
quark (which form an approximate flavor SU(3) symmetry). There are generalizations to larger number of flavors.
History
Developing classification schemes for hadrons
became a timely question after new experimental techniques uncovered so
many of them that it became clear that they could not all be
elementary. These discoveries led Wolfgang Pauli to exclaim "Had I foreseen that, I would have gone into botany." and Enrico Fermi to advise his student Leon Lederman:
"Young man, if I could remember the names of these particles, I would
have been a botanist." These new schemes earned Nobel prizes for
experimental particle physicists, including Luis Alvarez,
who was at the forefront of many of these developments. Constructing
hadrons as bound states of fewer constituents would thus organize the
"zoo" at hand. Several early proposals, such as the ones by Enrico Fermi and Chen-Ning Yang (1949), and the Sakata model (1956), ended up satisfactorily covering the mesons, but failed with baryons, and so were unable to explain all the data.
The Gell-Mann–Nishijima formula, developed by Murray Gell-Mann and Kazuhiko Nishijima, led to the Eightfold Way classification, invented by Gell-Mann, with important independent contributions from Yuval Ne'eman,
in 1961. The hadrons were organized into SU(3) representation
multiplets, octets and decuplets, of roughly the same mass, due to the
strong interactions; and smaller mass differences linked to the flavor
quantum numbers, invisible to the strong interactions. The Gell-Mann–Okubo mass formula systematized the quantification of these small mass differences among members of a hadronic multiplet, controlled by the explicit symmetry breaking of SU(3).
The spin-3/2Ω− baryon,
a member of the ground-state decuplet, was a crucial prediction of that
classification. After it was discovered in an experiment at Brookhaven National Laboratory, Gell-Mann received a Nobel Prize in Physics for his work on the Eightfold Way, in 1969.
Finally, in 1964, Gell-Mann and George Zweig, discerned independently what the Eightfold Way picture encodes: They posited three elementary fermionic constituents—the "up", "down", and "strange"
quarks—which are unobserved, and possibly unobservable in a free form.
Simple pairwise or triplet combinations of these three constituents and
their antiparticles underlie and elegantly encode the Eightfold Way
classification, in an economical, tight structure, resulting in further
simplicity. Hadronic mass differences were now linked to the different
masses of the constituent quarks.
It would take about a decade for the unexpected nature—and physical reality—of these quarks to be appreciated more fully (See Quarks). Counter-intuitively, they cannot ever be observed in isolation (color confinement),
but instead always combine with other quarks to form full hadrons,
which then furnish ample indirect information on the trapped quarks
themselves. Conversely, the quarks serve in the definition of quantum chromodynamics,
the fundamental theory fully describing the strong interactions; and
the Eightfold Way is now understood to be a consequence of the flavor
symmetry structure of the lightest three of them.
The Eightfold Way classification is named after the following fact:
If we take three flavors of quarks, then the quarks lie in the fundamental representation, 3 (called the triplet) of flavorSU(3). The antiquarks lie in the complex conjugate representation 3. The nine states (nonet) made out of a pair can be decomposed into the trivial representation, 1 (called the singlet), and the adjoint representation, 8 (called the octet). The notation for this decomposition is
Figure 1 shows the application of this decomposition to the mesons.
If the flavor symmetry were exact (as in the limit that only the strong
interactions operate, but the electroweak interactions are notionally
switched off), then all nine mesons would have the same mass. However,
the physical content of the full theory includes consideration of the symmetry breaking induced by the quark
mass differences, and considerations of mixing between various
multiplets (such as the octet and the singlet).
N.B. Nevertheless, the mass splitting between the η and the η′ is larger than the quark model can accommodate, and this "η–η′ puzzle" has its origin in topological peculiarities of the strong interaction vacuum, such as instanton configurations.
If P = (−1)J, then it follows that S = 1, thus PC = 1. States with these quantum numbers are called natural parity states; while all other quantum numbers are thus called exotic (for example, the state JPC = 0−−).
Figure 4. The S = 1/2 ground state baryon octetFigure 5. The S = 3/2baryon decuplet
Since quarks are fermions, the spin–statistics theorem implies that the wavefunction
of a baryon must be antisymmetric under the exchange of any two quarks.
This antisymmetric wavefunction is obtained by making it fully
antisymmetric in color, discussed below, and symmetric in flavor, spin
and space put together. With three flavors, the decomposition in flavor
is
The decuplet is symmetric in flavor, the singlet antisymmetric and the
two octets have mixed symmetry. The space and spin parts of the states
are thereby fixed once the orbital angular momentum is given.
It is sometimes useful to think of the basis states of quarks as the six states of three flavors and two spins per flavor. This approximate symmetry is called spin-flavor SU(6). In terms of this, the decomposition is
The 56 states with symmetric combination of spin and flavour decompose under flavor SU(3) into
where the superscript denotes the spin, S, of the baryon. Since
these states are symmetric in spin and flavor, they should also be
symmetric in space—a condition that is easily satisfied by making the
orbital angular momentum L = 0. These are the ground-state baryons.
The S = 1/2 octet baryons are the two nucleons (p+ , n0 ), the three Sigmas (Σ+ , Σ0 , Σ− ), the two Xis (Ξ0 , Ξ− ), and the Lambda (Λ0 ). The S = 3/2 decuplet baryons are the four Deltas (Δ++ , Δ+ , Δ0 , Δ− ), three Sigmas (Σ∗+ , Σ∗0 , Σ∗− ), two Xis (Ξ∗0 , Ξ∗− ), and the Omega (Ω− ).
For example, the constituent quark model wavefunction for the proton is
Mixing of baryons, mass splittings within and between multiplets,
and magnetic moments are some of the other quantities that the model
predicts successfully.
The group theory approach described above assumes that the quarks
are eight components of a single particle, so the anti-symmetrization
applies to all the quarks. A simpler approach is to consider the eight
flavored quarks as eight separate, distinguishable, non-identical
particles. Then the anti-symmetrization applies only to two identical
quarks (like uu, for instance).
Then, the proton wavefunction can be written in a simpler form:
and the
If quark–quark interactions are limited to two-body interactions,
then all the successful quark model predictions, including sum rules for
baryon masses and magnetic moments, can be derived.
Color quantum numbers are the characteristic charges of the strong
force, and are completely uninvolved in electroweak interactions. They
were discovered as a consequence of the quark model classification, when
it was appreciated that the spin S = 3/2 baryon, the Δ++ ,
required three up quarks with parallel spins and vanishing orbital
angular momentum. Therefore, it could not have an antisymmetric
wavefunction, (required by the Pauli exclusion principle). Oscar Greenberg noted this problem in 1964, suggesting that quarks should be para-fermions.
Instead, six months later, Moo-Young Han and Yoichiro Nambu
suggested the existence of a hidden degree of freedom, they labeled as
the group SU(3)' (but later called 'color). This led to three triplets
of quarks whose wavefunction was anti-symmetric in the color degree of
freedom.
Flavor and color were intertwined in that model: they did not commute.
The modern concept of color completely commuting with all other
charges and providing the strong force charge was articulated in 1973,
by William Bardeen, Harald Fritzsch, and Murray Gell-Mann.
States outside the quark model
While the quark model is derivable from the theory of quantum chromodynamics, the structure of hadrons is more complicated than this model allows. The full quantum mechanicalwavefunction of any hadron must include virtual quark pairs as well as virtual gluons, and allows for a variety of mixings. There may be hadrons which lie outside the quark model. Among these are the glueballs (which contain only valence gluons), hybrids (which contain valence quarks as well as gluons) and exotic hadrons (such as tetraquarks or pentaquarks).
![{\displaystyle |{\text{p}}_{\uparrow }\rangle ={\frac {1}{\sqrt {18}}}[2|{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }\rangle +2|{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }{\text{d}}_{\downarrow }\rangle +2|{\text{d}}_{\downarrow }{\text{u}}_{\uparrow }{\text{u}}_{\uparrow }\rangle -|{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }\rangle -|{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }\rangle -|{\text{u}}_{\downarrow }{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }\rangle -|{\text{d}}_{\uparrow }{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }\rangle -|{\text{d}}_{\uparrow }{\text{u}}_{\uparrow }{\text{u}}_{\downarrow }\rangle -|{\text{u}}_{\downarrow }{\text{u}}_{\uparrow }{\text{d}}_{\uparrow }\rangle ]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d84892af6c62784c12f0b691cd79d4d8584b3a)
![{\displaystyle {\text{p}}\left({\frac {1}{2}},{\frac {1}{2}}\right)={\frac {{\text{u}}{\text{u}}{\text{d}}}{\sqrt {6}}}[2\uparrow \uparrow \downarrow -\uparrow \downarrow \uparrow -\downarrow \uparrow \uparrow ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/530602aa73765eab71fa2e8847f9f2764325d62c)
![{\displaystyle \Delta ^{+}\left({\frac {3}{3}},{\frac {3}{2}}\right)={\text{u}}{\text{u}}{\text{d}}[\uparrow \uparrow \uparrow ]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c6b137d6e75df3c098948d8107f061ceafa1b8)
