https://en.formulasearchengine.com/api.php?action=feedcontributions&user=62.245.100.121&feedformat=atomformulasearchengine - User contributions [en]2024-03-28T21:06:58ZUser contributionsMediaWiki 1.42.0-wmf.5https://en.formulasearchengine.com/index.php?title=Talk:Lyapunov_stability&diff=296926Talk:Lyapunov stability2014-02-01T20:39:20Z<p>62.245.100.121: /* What is an equilibrium? */ new section</p>
<hr />
<div>Singapore has increased a tax on international property patrons as part of new momentary measures to chill its residential housing market which has seen continued sturdy demand regardless of previous efforts to curb prices.<br><br>Rising workplace sector leads actual estate market efficiency, while prime retail and enterprise park segments reasonable and residential sector continues in decline Latest information from Actual Capital Analytics (RCA) exhibits real estate advisory agency advised on the most deals by worth within the area in 2013 An alternative choice to property duties is an inheritance tax, which assesses beneficiaries slightly than the estate of the deceased, and could be easier to manage and gather, significantly when assets are held abroad. There is no reason for a company or particular person not to buy. Do not ask me why they pay rent. Some of them pay even lease to international homeowners or overseas owned companies. District 18, 999 yr LH apartment Residential items & fifty seven commercial models Posted by Edison Foo July 3, 2013<br><br>I am a firm believer that if we create a powerful, wholesome foundation with our lifestyle, we will age well and in a way that's a lot more healthy. After all all of us age, and there might come a time when medications are required irrespective of how nice our food plan and train habits are, however shouldn't all of us begin with a powerful foundation of wholesome lifestyle first? The actor is finest-recognized for his function as Michael Bluth in the TV sequence " Arrested Development" The cult traditional, which ran from 2003-2006, had a small but ardent fan base. Fox eventually canceled the present as a consequence of poor rankings however followers discovered methods to maintain the show alive The show was resurrected for a 2013 season, set to premiere in May exclusively on Netflix. 6.) Investing in New Technologies<br><br>We've lost depend of the number of rounds of cooling measures the Authorities has unleashed on the [http://dota.Thecollegegamer.com/groups/apple-is-working-on-two-versions-of-the-iphone-6-however-only-one-may-launch/ property search singapore] market to curb spiralling costs. The newest is the revised debt servicing framework introduced by the Monetary Authority of Singapore (MAS) in June 2013. This principally imposes stricter guidelines on debtors seeking a financial institution loan to fund their actual-estate purchases. Each property has an AV. This AV of a property is estimated based on market leases of comparable or comparable properties. What this means is that in the event you personal a 5-room flat in Toa Payoh, the Inland Income Authority of Singapore (IRAS) looks at comparable 5-room flats in Toa Payoh and how much they're rented out at, to find out the Annual Value. pending on the market licence Approval Apr, 2013<br><br>He mentioned the heightened ABSD and the tighter loan-to-valuation limits are "exceptional measures, imposed for cyclical causes, they don't seem to be permanent". On the other hand, measures affecting the PR owner-occupation of HDB flats and ECs are structural and for the long run. The adjustments for HDB flats are aimed toward moderating demand and ensuring consumers do not over-commit. Stricter mortgage eligibility akin to capping a mortgage service ratio at 30 per cent for loans from banks is one instance. Another rule bars PRs from sub-letting their whole HDB flat. Mr Tharman was additionally requested if the measures had been timed to coincide with the by-election. HDB restrictions to 'instil monetary prudence' by Sumita Sreedharan and Wong Wei Han, IMMEDIATELY, 12 Jan 2013<br><br>At the moment, the SSD rates are levied at the identical price as purchaser's stamp obligation, i.e. 1 per cent for the primary $one hundred eighty,000, 2 per cent for the following $one hundred eighty,000 and three% on the balance. The SSD charges are tiered according to the length of the holding interval, i.e. the vendor pays the total SSD charge if the residential property is bought within the first yr of buy; 2/3 the full SSD charge if the sale is in the second yr; 1/three the full SSD price if within the third 12 months.<br><br>German Shepard Buddy ran out of his proprietor Ben Heinrichs's Alaskan home when Heinrichs's workshop caught fireplace.The faithful canine spotted a state trooper, according to The Telegraph , and helped guide him to the burning property.Read the full story right here Be the FIRST proprietor of your desired challenge. Put it in the resale market in future as FIRST proprietor and you may reap probably the most harvest from your investments. Brand new unit means a model new way of life for your loved ones. But there are particular guidelines and restrictions for the ECs, including a 5-12 months minimal occupation interval from the proprietor before they're allowed to sell it within the resale market. And so they can solely sell them to foreigners after 10 years. Contact the agent to e-mail you the required documents.</div>62.245.100.121https://en.formulasearchengine.com/index.php?title=Uniqueness_quantification&diff=3587Uniqueness quantification2014-01-30T20:53:56Z<p>62.245.100.121: /* Proving uniqueness */ added missing verb</p>
<hr />
<div>{{about|root systems in mathematics|[[plant]]s' root systems|Root}}<br />
<br />
{{Lie groups |Semi-simple}}<br />
<br />
In [[mathematics]], a '''root system''' is a configuration of [[vector space|vector]]s in a [[Euclidean space]] satisfying certain geometrical properties. The concept is fundamental in the theory of [[Lie group]]s and [[Lie algebra]]s. Since Lie groups (and some analogues such as [[algebraic group]]s) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by [[Dynkin diagram]]s, occurs in parts of mathematics with no overt connection to Lie theory (such as [[singularity theory]]). Finally, root systems are important for their own sake, as in [[graph theory]] in the study of [[eigenvalue]]s.<br />
<br />
==Definitions and first examples==<br />
[[File:Root system A2 with labels.png|right|thumb|250px|The six vectors of the root system A<sub>2</sub>.]]<br />
As a first example, consider the six vectors in 2-dimensional [[Euclidean space]], '''R'''<sup>2</sub>, as shown in the image at the right; call them '''roots'''. These vectors [[Linear span|span]] the whole space. If you consider the line [[perpendicular]] to any root, say β, then the reflection of '''R'''<sup>2</sup> in that line sends any other root, say α, to another root. Moreover, the root to which it is sent equals β + n α, where n is an integer (in this case, n equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one is known as A<sub>2</sub>.<br />
<br />
===Definition===<br />
Let ''V'' be a finite-dimensional [[Euclidean space|Euclidean]] [[vector space]], with the standard [[Dot product|Euclidean inner product]] denoted by <math>(\cdot,\cdot)</math>. A '''root system''' in ''V'' is a finite set Φ of non-zero vectors (called '''roots''') that satisfy the following conditions:<ref>Bourbaki, Ch.VI, Section 1</ref><ref>Humphreys (1972), p.42</ref><br />
<br />
# The roots [[linear span|span]] ''V''.<br />
# The only scalar multiples of a root ''x''&nbsp;∈&nbsp;Φ that belong to Φ are ''x'' itself and&nbsp;–''x''.<br />
# For every root ''x''&nbsp;∈&nbsp;Φ, the set Φ is closed under [[Reflection (mathematics)|reflection]] through the [[hyperplane]] perpendicular to&nbsp;''x''.<br />
# ('''Integrality''') If ''x'' and ''y'' are roots in Φ, then the projection of ''y'' onto the line through ''x'' is a half-integral multiple of&nbsp;''x''.<br />
<br />
An equivalent way of writing conditions 3 and 4 is as follows:<br />
<br />
<ol><br />
<li value="3">For any two roots ''x'' and&nbsp;''y'', the set Φ contains the element <math>\sigma_x(y) =y-2\frac{(x,y)}{(x,x)}x \in \Phi.</math></li><br />
<li value="4">For any two roots ''x'' and&nbsp;''y'', the number <math> \langle y, x \rangle := 2 \frac{(x,y)}{(x,x)}</math> is an [[integer]].</li><br />
</ol> <br />
<br />
Some authors only include conditions 1&ndash;3 in the definition of a root system.<ref>Humphreys (1992), p.6</ref> In this context, a root system that also satisfies the integrality condition is known as a '''crystallographic root system'''.<ref>Humphreys (1992), p.39</ref> Other authors omit condition 2; then they call root systems satisfying condition 2 '''reduced'''.<ref>Humphreys (1992), p.41</ref> In this article, all root systems are assumed to be reduced and crystallographic.<br />
<br />
In view of property 3, the integrality condition is equivalent to stating that ''β'' and its reflection ''σ''<sub>''α''</sub>(''β'') differ by an integer multiple of&nbsp;''α''. Note that the operator<br />
<br />
:<math> \langle \cdot, \cdot \rangle \colon \Phi \times \Phi \to \mathbb{Z}</math><br />
<br />
defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument. <br />
{| class="wikitable" align="right" width=300<br />
|+'''Rank-2 root systems'''<br />
|- align=center<br />
| [[Image:Root system A1xA1.svg|150px|Root system A<sub>1</sub> + A<sub>1</sub>]]<br />
| [[Image:Root system D2.svg|150px|Root system D<sub>2</sub>]]<br />
|- align=center<br />
| Root system <math>A_1 \times A_1</math><BR>{{Dynkin|node_n1|2|node_n2}}<br />
| Root system <math>D_2</math><BR>{{Dynkin|nodes}}<br />
|- align=center<br />
| [[Image:Root system A2.svg|150px|Root system A<sub>2</sub>]]<br />
| [[Image:Root system G2.svg|150px|Root system G<sub>2</sub>]]<br />
|- align=center<br />
| Root system <math>A_2</math><BR>{{Dynkin|node_n1|3|node_n2}}<br />
| Root system <math>G_2</math><BR>{{Dynkin|node_n1|6a|node_n2}}<br />
|- align=center<br />
| [[Image:Root system B2.svg|150px|Root system B<sub>2</sub>]]<br />
| [[Image:Root system C2.svg|150px|Root system C<sub>2</sub>]]<br />
|- align=center<br />
| Root system <math>B_2</math><BR>{{Dynkin|node_n1|4a|node_n2}}<br />
| Root system <math>C_2</math><BR>{{Dynkin|node_n1|4b|node_n2}}<br />
|}<br />
The '''rank''' of a root system Φ is the dimension of ''V''. <br />
Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems ''A''<sub>2</sub>, ''B''<sub>2</sub>, and ''G''<sub>2</sub> pictured to the right, is said to be '''irreducible'''.<br />
<br />
Two root systems (''E''<sub>1</sub>,&nbsp;Φ<sub>1</sub>) and (''E''<sub>2</sub>,&nbsp;Φ<sub>2</sub>) are called '''isomorphic''' if there is an invertible linear transformation ''E''<sub>1</sub>&nbsp;→&nbsp;''E''<sub>2</sub> which sends&nbsp;Φ<sub>1</sub> to&nbsp;Φ<sub>2</sub> such that for each pair of roots, the number <math> \langle x, y \rangle</math> is preserved.<ref>Humphreys (1972), p.43</ref><br />
<br />
The [[group (mathematics)|group]] of [[isometry|isometries]] of&nbsp;''V'' generated by reflections through hyperplanes associated to the roots of&nbsp;Φ is called the [[Weyl group]] of&nbsp;Φ. As it [[Group action|acts faithfully]] on the finite set&nbsp;Φ, the Weyl group is always finite.<br />
<br />
The '''{{visible anchor|root lattice}}''' of a root system Φ is the '''Z'''-submodule of ''V'' generated by&nbsp;Φ. It is a [[lattice (discrete subgroup)|lattice]] in&nbsp;''V''.<br />
<br />
===Rank two examples===<br />
<br />
There is only one root system of rank 1, consisting of two nonzero vectors <math>\{\alpha, -\alpha\}</math>. This root system is called <math>A_1</math>.<br />
<br />
In rank 2 there are four possibilities, corresponding to <math>\sigma_\alpha(\beta) = \beta + n\alpha</math>, where <math>n = 0, 1, 2, 3</math>. Note that a root system that generates a lattice is not unique: <math>A_1 \times A_1</math> and <math>B_2</math> generate a [[square lattice]] while <math>A_2</math> and <math>G_2</math> generate a [[hexagonal lattice]], only two of the five possible types of [[Lattice (group)#Lattices in two dimensions: detailed discussion|lattices in two dimensions]].<br />
<br />
Whenever Φ is a root system in ''V'', and ''U'' is a [[Linear subspace|subspace]] of ''V'' spanned by Ψ&nbsp;=&nbsp;Φ&nbsp;∩&nbsp;''U'', then&nbsp;Ψ is a root system in&nbsp;''U''. Thus, the exhaustive list of four root systems of rank&nbsp;2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank. In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.<br />
<br />
==History==<br />
The concept of a root system was originally introduced by [[Wilhelm Killing]] around 1889 (in German, ''Wurzelsystem''<ref>Killing (1889)</ref>).<ref>Bourbaki (1998), p.270</ref> He used them in his attempt to classify all [[simple Lie algebra]]s over the [[field (mathematics)|field]] of [[complex number]]s. Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F<sub>4</sub>. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.<ref>Coleman, p.34</ref><br />
<br />
Killing investigated the structure of a Lie algebra <math>L</math>, by considering (what is now called) a [[Cartan subalgebra]] <math>\mathfrak{h}</math>. Then he studied the roots of the [[characteristic polynomial]] <math>\det (\mathrm{ad}_L x - t)</math>, where <math>x \in \mathfrak{h}</math>. Here a ''root'' is considered as a function of <math>\mathfrak{h}</math>, or indeed as an element of the dual vector space <math>\mathfrak{h}^*</math>. This set of roots form a root system inside <math>\mathfrak{h}^*</math>, as defined above, where the inner product is the [[Killing form]].<ref>Bourbaki (1998), p.270</ref><br />
<br />
==Elementary consequences of the root system axioms==<br />
[[Image:Integrality of root systems.svg|thumb|500px|right|The integrality condition for <''&beta;'',&nbsp;''&alpha;''> is fulfilled only for ''&beta;'' on one of the vertical lines, while the integrality condition for <''&alpha;'',&nbsp;''&beta;''> is fulfilled only for ''&beta;'' on one of the red circles. Any &beta; perpendicular to &alpha; (on the Y axis) trivially fulfills both with 0, but does not define an irreducible root system. <br>Modulo reflection, for a given &alpha; there are only 5 nontrivial possibilities for &beta;, and 3 possible angles between &alpha; and &beta; in a set of simple roots. Subscript letters correspond to the series of root systems for which the given &beta; can serve as the first root and &alpha; as the second root. (or in F<sub>4</sub> as the middle 2 roots)]]<br />
<br />
<!--<br />
The integrality condition also means that the ratio of the lengths (magnitudes) of any two non-perpendicular roots cannot be 2 or greater, since otherwise either the projection of the shorter root onto the longer root will be less than half as long as the longer root, or the shorter root will be exactly half the longer root or its negative.<br />
--><br />
<br />
The cosine of the angle between two roots is constrained to be a [[Half-integer|half-integral]] multiple of a square root of an integer. This is because <math> \langle \beta, \alpha \rangle</math> and <math>\langle \alpha, \beta \rangle</math> are both integers, by assumption, and<br />
<br />
<math> \langle \beta, \alpha \rangle \langle \alpha, \beta \rangle = 2 \frac{(\alpha,\beta)}{(\alpha,\alpha)} \cdot 2 \frac{(\alpha,\beta)}{(\beta,\beta)} = 4 \frac{(\alpha,\beta)^2}{\vert \alpha \vert^2 \vert \beta \vert^2} = 4 \cos^2(\theta) = (2\cos(\theta))^2 \in \mathbb{Z}.</math><br />
<br />
Since <math>2\cos(\theta) \in [-2,2]</math>, the only possible values for <math>\cos(\theta)</math> are <math>0, \pm \tfrac{1}{2}, \pm\tfrac{\sqrt{2}}{2}, \pm\tfrac{\sqrt{3}}{2}, \pm\tfrac{\sqrt{4}}{2} = \pm 1</math>, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0 or 180°. Condition 2 says that no scalar multiples of ''α'' other than 1 and -1 can be roots, so 0 or 180°, which would correspond to ''2α'' or ''-2α'' are out.<br />
<br />
==Positive roots and simple roots==<br />
Given a root system Φ we can always choose (in many ways) a set of '''positive roots'''. This is a subset<br />
<math>\Phi^+</math> of Φ such that <br />
* For each root <math>\alpha\in\Phi</math> exactly one of the roots <math>\alpha</math>, –<math>\alpha</math> is contained in <math>\Phi^+</math>.<br />
* For any two distinct <math>\alpha, \beta\in \Phi^+</math> such that <math>\alpha+\beta</math> is a root, <math>\alpha+\beta\in\Phi^+</math>.<br />
<br />
If a set of positive roots <math>\Phi^+</math> is chosen, elements of <math>-\Phi^+</math> are called '''negative roots'''.<br />
<br />
An element of <math>\Phi^+</math> is called a '''simple root''' if it cannot be written as the sum of two elements of <math>\Phi^+</math>. The set <math>\Delta</math> of simple roots is a basis of <math>V</math> with the property that every vector in <math>\Phi</math> is a linear combination of elements of <math>\Delta</math> with all coefficients non-negative, or all coefficients non-positive. For each choice of positive roots, the corresponding set of simple roots is the unique set of roots such that the positive roots are exactly those that can be expressed as a combination of them with non-negative coefficients, and such that these combinations are unique.<br />
<br />
===The root poset===<br />
[[File:E6HassePoset.svg|thumb|300px|[[Hasse diagram]] of E6 [[Root_system#The_root_poset|root poset]] with edge labels identifying added simple root position]]<br />
The set of positive roots is naturally ordered by saying that <math>\alpha \leq \beta</math> if and only if <math>\beta-\alpha</math> is a nonnegative linear combination of simple roots. This [[Partially ordered set|poset]] is [[Graded poset|graded]] by <math>\operatorname{deg}\big(\sum_{\alpha \in \Delta} \lambda_\alpha \alpha\big) = \sum_{\alpha \in \Delta}\lambda_\alpha</math>, and has many remarkable combinatorial properties, one of them being that one can determine the degrees of the fundamental invariants of the corresponding Weyl group from this poset.<ref>Humphreys (1992), Theorem 3.20</ref> The Hasse graph is a visualization of the ordering of the root poset.<br />
<br />
==Dual root system and coroots==<br />
{{see also|Langlands dual group}}<br />
If Φ is a root system in ''V'', the '''coroot''' α<sup>V</sup> of a root α is defined by<br />
<br />
:<math>\alpha^\vee= {2\over (\alpha,\alpha)}\, \alpha.</math><br />
<br />
The set of coroots also forms a root system Φ<sup>V</sup> in ''V'', called the '''dual root system''' (or sometimes ''inverse root system'').<br />
By definition, α<sup>V V</sup> = α, so that Φ is the dual root system of Φ<sup>V</sup>. The lattice in ''V'' spanned by Φ<sup>V</sup> is called the ''coroot lattice''. Both Φ and Φ<sup>V</sup> have the same Weyl group ''W'' and, for ''s'' in ''W'',<br />
<br />
:<math> (s\alpha)^\vee= s(\alpha^\vee).</math><br />
<br />
If Δ is a set of simple roots for Φ, then Δ<sup>V</sup> is a set of simple roots for Φ<sup>V</sup>.<br />
<br />
==Classification of root systems by Dynkin diagrams==<br />
[[File:Finite Dynkin diagrams.svg|480px|thumb|Pictures of all the irreducible Dynkin diagrams]]<br />
<br />
Irreducible root systems [[bijection|correspond]] to certain [[graph (mathematics)|graphs]], the '''[[Dynkin diagram]]s''' named after [[Eugene Dynkin]]. The classification of these graphs is a simple matter of [[combinatorics]], and induces a classification of irreducible root systems.<br />
<br />
Given a root system, select a set Δ of [[root system#Positive roots and simple roots|simple roots]] as in the preceding section. The vertices of the associated Dynkin diagram correspond to vectors in Δ. An edge is drawn between each non-orthogonal pair of vectors; it is an undirected single edge if they make an angle of <math>2 \pi / 3</math> radians, a directed double edge if they make an angle of <math>3 \pi / 4</math> radians, and a directed triple edge if they make an angle of <math>5 \pi / 6</math> radians. The term "directed edge" means that double and triple edges are marked with an angle sign pointing toward the shorter vector.<br />
<br />
Although a given root system has more than one possible set of simple roots, the [[Weyl group]] acts transitively on such choices. Consequently, the Dynkin diagram is independent of the choice of simple roots; it is determined by the root system itself. Conversely, given two root systems with the same Dynkin diagram, one can match up roots, starting with the roots in the base, and show that the systems are in fact the same.<br />
<br />
Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams. Root systems are irreducible if and only if their Dynkin diagrams are connected. Dynkin diagrams encode the inner product on ''E'' in terms of the basis Δ, and the condition that this inner product must be [[positive definite bilinear form|positive definite]] turns out to be all that is needed to get the desired classification.<br />
<br />
The actual connected diagrams are as follows. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system).<br />
<br />
==Properties of the irreducible root systems==<br />
{|border=1 cellpadding=4 align="right" style="margin: 1em; text-align: center; border-collapse: collapse;"<br />
!<math>\Phi</math> || <math>|\Phi|</math> || <math>|\Phi^{<}|</math> || ''I'' || ''D'' || <math>|W|</math><br />
|-<br />
|A<sub>''n''</sub> (''n''&nbsp;≥&nbsp;1) || ''n''(''n''&nbsp;+&nbsp;1) || &nbsp; ||&nbsp; || ''n''&nbsp;+&nbsp;1 || (''n''&nbsp;+&nbsp;1)!<br />
|-<br />
|B<sub>''n''</sub> (''n''&nbsp;≥&nbsp;2) || 2''n''<sup>2</sup> || 2''n''|| 2 || 2 || 2<sup>''n''</sup> ''n''!<br />
|-<br />
|C<sub>''n''</sub> (''n''&nbsp;≥&nbsp;3) || 2''n''<sup>2</sup> || 2''n''(''n''&nbsp;&minus;&nbsp;1)|| 2 || 2 || 2<sup>''n''</sup> ''n''!<br />
|-<br />
|D<sub>''n''</sub> (''n''&nbsp;≥&nbsp;4) || 2''n''(''n''&nbsp;&minus;&nbsp;1) || &nbsp;||&nbsp; || 4 || 2<sup>''n''&nbsp;&minus;&nbsp;1</sup> ''n''!<br />
|-<br />
|[[E6 (mathematics)|E<sub>6</sub>]] || 72 || &nbsp; ||&nbsp;|| 3 || 51840 <br />
|-<br />
|[[E7 (mathematics)|E<sub>7</sub>]] || 126 || &nbsp; ||&nbsp;|| 2 || 2903040 <br />
|-<br />
|[[E8 (mathematics)|E<sub>8</sub>]] || 240 || &nbsp;||&nbsp; || 1 || 696729600<br />
|-<br />
|[[F4 (mathematics)|F<sub>4</sub>]] || 48 || 24|| 4 || 1 || 1152<br />
|-<br />
|[[G2 (mathematics)|G<sub>2</sub>]] || 12 || 6 || 3 || 1 || 12<br />
|}<br />
<br />
Irreducible root systems are named according to their corresponding connected Dynkin diagrams. There are four infinite families (A<sub>''n''</sub>, B<sub>''n''</sub>, C<sub>''n''</sub>, and D<sub>''n''</sub>, called the '''classical root systems''') and five exceptional cases (the '''exceptional root systems''').<ref>{{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction |publisher=Springer|year=2003|isbn=0-387-40122-9}}.</ref> The subscript indicates the rank of the root system. <br />
<br />
In an irreducible root system there can be at most two values for the length (''α'',&nbsp;''α'')<sup>1/2</sup>, corresponding to '''short''' and '''long''' roots. If all roots have the same length they are taken to be long by definition and the root system is said to be '''simply laced'''; this occurs in the cases A, D and E. Any two roots of the same length lie in the same orbit of the Weyl group. In the non-simply laced cases B, C, G and F, the root lattice is spanned by the short roots and the long roots span a sublattice, invariant under the Weyl group, equal to ''r''<sup>2</sup>/2 times the coroot lattice, where ''r'' is the length of a long root. <br />
<br />
In the table to the right, |Φ<sup>&nbsp;<&nbsp;</sup>| denotes the number of short roots, ''I'' denotes the index in the root lattice of the sublattice generated by long roots, ''D'' denotes the determinant of the [[Cartan matrix]], and |''W''| denotes the order of the [[Weyl group]].<br />
<br />
==Explicit construction of the irreducible root systems==<br />
<br />
===A<sub>''n''</sub>===<br />
{| align=right class=wikitable<br />
|+ '''A'''<sub>'''3'''</sub><br />
|-<br />
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub><br />
|- align=right<br />
! α<sub>1</sub><br />
|1||&minus;1||0||0<br />
|- align=right<br />
! α<sub>2</sub><br />
|0||1||&minus;1||0 <br />
|- align=right<br />
! α<sub>3</sub><br />
||0||0||1||&minus;1 <br />
|-<br />
|colspan=5 align=center|{{Dynkin2|node_n1|3|node_n2|3|node_n3}}<br />
|}<br />
Let ''V'' be the subspace of '''R'''<sup>''n''+1</sup> for which the coordinates sum to 0, and let Φ be the set of vectors in ''V'' of length √2 and which are ''integer vectors,'' i.e. have integer coordinates in '''R'''<sup>''n''+1</sup>. Such a vector must have all but two coordinates equal to 0, one coordinate equal to 1, and one equal to –1, so there are ''n''<sup>2</sup> + ''n'' roots in all. One choice of simple roots expressed in the [[standard basis]] is: '''α'''<sub>i</sub> = '''e'''<sub>i</sub> – '''e'''<sub>i+1</sub>, for 1 ≤ i ≤ n.<br />
<br />
The [[Reflection (mathematics)|reflection]] ''σ''<sub>''i''</sub> through the [[hyperplane]] perpendicular to '''α'''<sub>''i''</sub> is the same as [[permutation]] of the adjacent '''''i'''''-th and ('''''i''&nbsp;+&nbsp;1''')-th [[coordinates]]. Such <br />
[[Transposition (mathematics)|transpositions]] generate the full [[permutation group]].<br />
For adjacent simple roots, <br />
''σ''<sub>''i''</sub>('''α'''<sub>''i''+1</sub>) = '''α'''<sub>''i''+1</sub>&nbsp;+&nbsp;'''α'''<sub>i</sub> =&nbsp;''σ''<sub>''i''+1</sub>('''α'''<sub>''i''</sub>) =&nbsp;'''α'''<sub>''i''</sub>&nbsp;+&nbsp;'''α'''<sub>''i''+1</sub>, that is, reflection is equivalent to adding a multiple of&nbsp;1; but<br />
reflection of a simple root perpendicular to a nonadjacent simple root leaves it unchanged, differing by a multiple of&nbsp;0.<br />
<br />
The lattice generated by the A<sub>3</sub> root system is known to crystallographers as the '''face-centered cubic''' ('''fcc''') (or '''[[cubic crystal system|cubic]] close packed''') lattice.<ref>[[John Horton Conway|Conway, John Horton]]; [[Neil Sloane|Sloane, Neil James Alexander]]; & Bannai, Eiichi. ''Sphere packings, lattices, and groups''. Springer, 1999, Section 6.3.</ref><br />
<br />
===B<sub>''n''</sub>===<br />
{| align=right class=wikitable<br />
|+ '''B'''<sub>'''4'''</sub><br />
|-<br />
| &nbsp;1||-1||0||0<br />
|-<br />
|0|| &nbsp;&nbsp;1||-1||0 <br />
|-<br />
|0||0|| &nbsp;&nbsp;1||-1 <br />
|-<br />
|0||0|| 0||&nbsp;&nbsp;1<br />
|-<br />
|colspan=4 align=center|{{Dynkin2|node_n1|3|node_n2|3|node_n3|4b|node_n4}}<br />
|}<br />
Let ''V'' = '''R'''<sup>''n''</sup>, and let Φ consist of all integer vectors in ''V'' of length 1 or √2. The total number of roots is 2''n''<sup>2</sup>. One choice of simple roots is: '''α'''<sub>i</sub> = '''e'''<sub>i</sub> – '''e'''<sub>i+1</sub>, for 1 ≤ i ≤ n – 1 (the above choice of simple roots for '''A<sub>n-1</sub>'''), and the shorter root '''α'''<sub>n</sub> = '''e'''<sub>n</sub>.<br />
<br />
The reflection σ<sub>n</sub> through the hyperplane perpendicular to the short root '''α'''<sub>n</sub> is of course simply negation of the '''n'''th coordinate. <br />
For the long simple root '''α'''<sub>n-1</sub>, σ<sub>n-1</sub>('''α'''<sub>n</sub>) = '''α'''<sub>n</sub> + '''α'''<sub>n-1</sub>, but for reflection perpendicular to the short root, σ<sub>n</sub>('''α'''<sub>n-1</sub>) = '''α'''<sub>n-1</sub> + 2'''α'''<sub>n</sub>, a difference by a multiple of 2 instead of 1. <br />
<br />
B<sub>1</sub> is isomorphic to A<sub>1</sub> via scaling by √2, and is therefore not a distinct root system.<br />
<br />
===C<sub>''n''</sub>===<br />
{| align=right class=wikitable<br />
|+ '''C'''<sub>'''4'''</sub><br />
|-<br />
| &nbsp;1||-1||0||0<br />
|-<br />
|0|| &nbsp;&nbsp;1||-1||0 <br />
|-<br />
|0||0|| &nbsp;&nbsp;1||-1 <br />
|-<br />
|0||0|| 0||&nbsp;&nbsp;2<br />
|-<br />
|colspan=4 align=center|{{Dynkin2|node_n1|3|node_n2|3|node_n3|4a|node_n4}}<br />
|}<br />
Let ''V'' = '''R'''<sup>''n''</sup>, and let Φ consist of all integer vectors in ''V'' of length √2 together with all vectors of the form 2λ, where λ is an integer vector of length 1. The total number of roots is 2''n''<sup>2</sup>. One choice of simple roots is: '''α'''<sub>i</sub> = '''e'''<sub>i</sub> – '''e'''<sub>i+1</sub>, for 1 ≤ i ≤ n – 1 (the above choice of simple roots for '''A<sub>n-1</sub>'''), and the longer root '''α'''<sub>n</sub> = 2'''e'''<sub>n</sub>.<br />
The reflection σ<sub>n</sub>('''α'''<sub>n-1</sub>) = '''α'''<sub>n-1</sub> + '''α'''<sub>n</sub>, but σ<sub>n-1</sub>('''α'''<sub>n</sub>) = '''α'''<sub>n</sub> + 2'''α'''<sub>n-1</sub>.<br />
<br />
C<sub>2</sub> is isomorphic to B<sub>2</sub> via scaling by √2 and a 45 degree rotation, and is therefore not a distinct root system.<br />
<br />
[[File:Root vectors b3 c3-d3.png|400px]]<BR>Root system B<sub>3</sub>, C<sub>3</sub>, and A<sub>3</sub>=D<sub>3</sub> as points within a [[cube]] and [[octahedron]]<br />
<br />
===D<sub>''n''</sub>===<br />
{| align=right class=wikitable<br />
|+ '''D'''<sub>'''4'''</sub><br />
|- valign=top<br />
| &nbsp;1||-1||0||0<br />
|-<br />
|0|| &nbsp;1||-1||0 <br />
|-<br />
|0||0|| &nbsp;1||-1 <br />
|-<br />
|0||0|| &nbsp;1||&nbsp;&nbsp;1<br />
|-<br />
|colspan=4 align=center|[[File:DynkinD4_labeled.png|80px]]<!--{{Dynkin2|node_n1|3|branch|3|node_n3}}--><br />
|}<br />
Let ''V'' = '''R'''<sup>''n''</sup>, and let Φ consist of all integer vectors in ''V'' of length √2. The total number of roots is 2''n''(''n'' – 1). One choice of simple roots is: '''α'''<sub>i</sub> = '''e'''<sub>i</sub> – '''e'''<sub>i+1</sub>, for 1 ≤ i &lt; n (the above choice of simple roots for '''A<sub>n-1</sub>''') plus '''α'''<sub>n</sub> = '''e'''<sub>n</sub> + '''e'''<sub>n-1</sub>.<br />
<br />
Reflection through the hyperplane perpendicular to '''α'''<sub>n</sub> is the same as [[Transposition (mathematics)|transposing]] and negating the adjacent '''n'''-th and ('''n – 1''')-th coordinates. Any simple root and its reflection perpendicular to another simple root differ by a multiple of 0 or 1 of the second root, not by any greater multiple.<br />
<br />
D<sub>3</sub> reduces to A<sub>3</sub>, and is therefore not a distinct root system.<br />
<br />
D<sub>4</sub> has additional symmetry called [[triality]].<br />
<br />
===E<sub>6</sub>, E<sub>7</sub>, E<sub>8</sub>===<br />
<br />
{| class=wikitable width=675 center<br />
|[[File:E6Coxeter.svg|200px]]<BR>72 vertices of [[1 22 polytope|1<sub>22</sub>]] represent the root vectors of [[E6 (mathematics)|E<sub>6</sub>]]<BR>(Green nodes are doubled in this E6 Coxeter plane projection)<br />
|[[File:E7Petrie.svg|225px]]<BR>126 vertices of [[2 31 polytope|2<sub>31</sub>]] represent the root vectors of [[E7 (mathematics)|E<sub>7</sub>]]<br />
|[[File:E8 graph.svg|250px]]<BR>240 vertices of [[4 21 polytope|4<sub>21</sub>]] represent the root vectors of [[E8 (mathematics)|E<sub>8</sub>]]<br />
|- align=center<br />
|[[File:DynkinE6AltOrder.svg|200px]]<br />
|[[File:DynkinE7AltOrder.svg|225px]]<br />
|[[File:DynkinE8AltOrder.svg|250px]]<br />
|}<br />
*The E<sub>8</sub> root system is any set of vectors in '''R'''<sup>8</sup> that is [[congruence (geometry)|congruent]] to the following set:<br />
: D<sub>8</sub> ∪ { ½( ∑<sub>i=1</sub><sup>8</sup> ε<sub>i</sub>'''e'''<sub>i</sub>) : ε<sub>i</sub> = &plusmn;1, ε<sub>1</sub>•••ε<sub>8</sub> = +1}.<br />
The root system has 240 roots. <br />
The set just listed is the set of vectors of length √2 in the [[E8 lattice]] Γ<sub>8</sub>, which is the set of points in '''R'''<sup>8</sup> such that:<br />
#all the coordinates are [[integer]]s or all the coordinates are [[half-integer]]s (a mixture of integers and half-integers is not allowed), and<br />
#the sum of the eight coordinates is an [[even integer]].<br />
Thus, <br />
:E<sub>8</sub> = {'''α''' ∈ '''Z'''<sup>8</sup> ∪ ('''Z'''+½)<sup>8</sup> : |'''α'''|<sup>2</sup> = ∑'''α'''<sub>i</sub><sup>2</sup> = 2, ∑'''α'''<sub>i</sub> ∈ 2'''Z'''}.<br />
<br />
* The root system E<sub>7</sub> is the set of vectors in E<sub>8</sub> that are perpendicular to a fixed root in E<sub>8</sub>. The root system E<sub>7</sub> has 126 roots.<br />
* The root system E<sub>6</sub> is not the set of vectors in E<sub>7</sub> that are perpendicular to a fixed root in E<sub>7</sub>, indeed, one obtains D<sub>6</sub> that way. However, E<sub>6</sub> is the subsystem of E<sub>8</sub> perpendicular to two suitably chosen roots of E<sub>8</sub>. The root system E<sub>6</sub> has 72 roots.<br />
<br />
{| align=right style="text-align: right; border: 1px gray solid" cellspacing=0<br />
|+ Simple roots in E<sub>8</sub> even coordinates:<br />
|-<br />
| 1||-1||0||0||0||0||0||0<br />
|-<br />
|0|| 1||-1||0||0||0||0||0 <br />
|-<br />
|0||0|| 1||-1||0||0||0||0 <br />
|-<br />
|0||0||0|| 1||-1||0||0||0 <br />
|-<br />
| 0||0||0||0|| 1||-1||0||0 <br />
|-<br />
|0||0||0||0||0|| 1||-1||0 <br />
|-<br />
|0||0||0||0||0||1|| 1||0 <br />
|- <br />
| -½||-½||-½||-½||-½||-½||-½||-½<br />
|}<br />
An alternative description of the E<sub>8</sub> lattice which is sometimes convenient is as the set Γ'<sub>8</sub> of all points in '''R'''<sup>8</sup> such that<br />
*all the coordinates are integers and the sum of the coordinates is even, or<br />
*all the coordinates are half-integers and the sum of the coordinates is odd.<br />
The lattices Γ<sub>8</sub> and Γ'<sub>8</sub> are [[isomorphic]]; one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ<sub>8</sub> is sometimes called the ''even coordinate system'' for E<sub>8</sub> while the lattice Γ'<sub>8</sub> is called the ''odd coordinate system''.<br />
<br />
One choice of simple roots for E<sub>8</sub> in the even coordinate system with rows ordered by node order in the alternate (non-canonical) Dynkin diagrams (above) is:<br />
:'''α'''<sub>i</sub> = '''e'''<sub>i</sub> – '''e'''<sub>i+1</sub>, for 1 ≤ i ≤ 6, and<br />
:'''α'''<sub>7</sub> = '''e'''<sub>7</sub> + '''e'''<sub>6</sub><br />
(the above choice of simple roots for D<sub>7</sub>) along with <br />
:'''α'''<sub>8</sub> = '''β'''<sub>0</sub> = <math>-\textstyle\frac{1}{2}(\textstyle \sum_{i=1}^8e_i)</math> = (-½,-½,-½,-½,-½,-½,-½,-½).<br />
{| align=right style="text-align: right; border: 1px gray solid" cellspacing=0<br />
|+ Simple roots in E<sub>8</sub>: odd coordinates<br />
|-<br />
| 1||-1||0||0||0||0||0||0<br />
|-<br />
|0|| 1||-1||0||0||0||0||0 <br />
|-<br />
|0||0|| 1||-1||0||0||0||0 <br />
|-<br />
|0||0||0|| 1||-1||0||0||0 <br />
|-<br />
| 0||0||0||0|| 1||-1||0||0 <br />
|-<br />
|0||0||0||0||0|| 1||-1||0 <br />
|-<br />
|0||0||0||0||0||0|| 1||-1 <br />
|- <br />
| -½||-½||-½||-½||-½||&nbsp;½||&nbsp;½||&nbsp;½<br />
|}<br />
One choice of simple roots for E<sub>8</sub> in the odd coordinate system with rows ordered by node order in alternate (non-canonical) Dynkin diagrams (above) is:<br />
:'''α'''<sub>i</sub> = '''e'''<sub>i</sub> – '''e'''<sub>i+1</sub>, for 1 ≤ i ≤ 7<br />
(the above choice of simple roots for A<sub>7</sub>) along with <br />
:'''α'''<sub>8</sub> = '''β'''<sub>5</sub>, where<br />
:'''β'''<sub>j</sub> = <math>\textstyle\frac{1}{2}(-\textstyle \sum_{i=1}^je_i+\textstyle \sum_{i=j+1}^8e_i)</math>.<br />
(Using '''β'''<sub>3</sub> would give an isomorphic result. Using '''β'''<sub>1,7</sub> or '''β'''<sub>2,6</sub> would simply give A<sub>8</sub> or D<sub>8</sub>. As for '''β'''<sub>4</sub>, its coordinates sum to 0, and the same is true for '''α'''<sub>1...7</sub>, so they span only the 7-dimensional subspace for which the coordinates sum to 0; in fact –2'''β'''<sub>4</sub> has coordinates (1,2,3,4,3,2,1) in the basis ('''α'''<sub>i</sub>).) <br />
<br />
Deleting '''α'''<sub>1</sub> and then '''α'''<sub>2</sub> gives sets of simple roots for E<sub>7</sub> and E<sub>6</sub>.<br />
Since perpendicularity to '''α'''<sub>1</sub> means that the first two coordinates are equal, E<sub>7</sub> is then the subset of E<sub>8</sub> where the first two coordinates are equal, and similarly E<sub>6</sub> is the subset of E<sub>8</sub> where the first three coordinates are equal. This facilitates explicit definitions of E<sub>7</sub> and E<sub>6</sub> as:<br />
<br />
:E<sub>''7''</sub> = {'''α''' ∈ '''Z'''<sup>7</sup> ∪ ('''Z'''+½)<sup>7</sup>''' : ''' ∑'''α'''<sub>i</sub><sup>2</sup> + '''α'''<sub>1</sub><sup>2</sup> = 2, ∑'''α'''<sub>i</sub> + '''α'''<sub>1</sub> ∈ 2'''Z'''},<br />
:E<sub>''6''</sub> = {'''α''' ∈ '''Z'''<sup>6</sup> ∪ ('''Z'''+½)<sup>6</sup>''' : ''' ∑'''α'''<sub>i</sub><sup>2</sup> + 2'''α'''<sub>1</sub><sup>2</sup> = 2, ∑'''α'''<sub>i</sub> + 2'''α'''<sub>1</sub> ∈ 2'''Z'''}<br />
<br />
===F<sub>4</sub>===<br />
{| align=right class=wikitable<br />
|+ Simple roots in F<sub>4</sub><br />
|-<br />
| 1||-1||0||0<br />
|-<br />
|0|| 1||-1||0 <br />
|-<br />
|0||0|| 1||0 <br />
|-<br />
| -½||-½||-½||-½<br />
|-<br />
|colspan=4 align=center|{{Dynkin2|node_n1|3|node_n2|4b|node_n3|3|node_n4}}<br />
|}<br />
[[File:F4 roots by 24-cell duals.svg|100px|left|thumb|48-root vectors of F4, defined by vertices of the [[24-cell]] and its dual, viewed in the [[Coxeter plane]]]]<br />
For F<sub>4</sub>, let ''V'' = '''R'''<sup>4</sup>, and let Φ denote the set of vectors α of length 1 or √2 such that the coordinates of 2α are all integers and are either all even or all odd. There are 48 roots in this system. One choice of simple roots is: the choice of simple roots given above for B<sub>3</sub>, plus '''α'''<sub>4</sub> = – <math>\textstyle\frac{1}{2} \sum_{i=1}^4 e_i</math>.<br />
<!--<br />
\left (<br />
\begin{smallmatrix}<br />
+1&-1&0&0 \\ 0&+1&-1&0 \\ 0&0&+1&0 \\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}<br />
\end{smallmatrix}<br />
\right )<br />
--><br />
<br />
===G<sub>2</sub>===<br />
{| align=right class=wikitable<br />
|+ Simple roots in G<sub>2</sub><br />
|-<br />
| 1||&nbsp;-1||&nbsp;&nbsp;0<br />
|-<br />
| -1||2||-1<br />
|-<br />
|colspan=3 align=center|{{Dynkin2|node_n1|6a|node_n2}}<br />
|}<br />
The root system G<sub>2</sub> has 12 roots, which form the vertices of a [[hexagram]]. See the picture [[Root system#Rank two examples|above]].<br />
<br />
One choice of simple roots is: ('''α'''<sub>1</sub>, <br />
'''β''' = '''α'''<sub>2</sub> – '''α'''<sub>1</sub>) where <br />
'''α'''<sub>i</sub> = '''e'''<sub>i</sub> – '''e'''<sub>i+1</sub> for i = 1, 2 is the above choice of simple roots for A<sub>2</sub>.<br />
<br />
==Root systems and Lie theory==<br />
Irreducible root systems classify a number of related objects in Lie theory, notably the<br />
*simple [[Lie group]]s (see the [[list of simple Lie groups]]), including the<br />
*[[Simple Lie group|simple complex Lie groups]];<br />
*their associated [[Simple Lie algebra|simple complex Lie algebras]]; and<br />
*[[simply connected]] complex Lie groups which are simple modulo centers.<br />
In each case, the roots are non-zero [[weight (representation theory)|weight]]s of the [[Adjoint representation of a Lie algebra|adjoint representation]]. <br />
<br />
In the case of a [[simply connected]] simple compact Lie group ''G'' with [[maximal torus]] ''T'', the root lattice can naturally be identified with Hom(''T'', '''T''') and the coroot lattice with Hom('''T''', ''T''); see {{harvtxt|Adams|1983}}.<br />
<br />
For connections between the exceptional root systems and their Lie groups and Lie algebras see [[E8 (mathematics)|E<sub>8</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], and [[G2 (mathematics)|G<sub>2</sub>]].<br />
<br />
{{Exceptional Lie groups}}<br />
<br />
==See also==<br />
*[[ADE classification]]<br />
*[[Affine root system]]<br />
*[[Coxeter–Dynkin diagram]]<br />
*[[Coxeter group]]<br />
*[[Coxeter matrix]]<br />
*[[Dynkin diagram]]<br />
*[[root datum]]<br />
*[[Root system of a semi-simple Lie algebra]]<br />
*[[Weyl group]]<br />
<br />
==Notes==<br />
{{multiple issues|<br />
{{Page numbers needed|date=January 2012}}<br />
{{Refimprove|date=January 2012}}<br />
}}<br />
{{Reflist}}<br />
<br />
==References==<br />
<br />
*{{citation|first=J.F.|last=Adams|authorlink=Frank Adams|title=Lectures on Lie groups|publisher= University of Chicago Press|year= 1983|isbn=0-226-00530-5}}<br />
*{{citation|first=Nicolas|last= Bourbaki|authorlink=Nicolas Bourbaki| title=Lie groups and Lie algebras, Chapters 4&ndash;6 (translated from the 1968 French original by Andrew Pressley)|series= Elements of Mathematics|publisher= Springer-Verlag|year= 2002|isbn=3-540-42650-7}}. The classic reference for root systems.<br />
*{{cite isbn|3540647678}}<br />
*{{citation|author=A.J. Coleman|title=The greatest mathematical paper of all time|journal=The Mathematical Intelligencer|volume=11|date=Summer 1989|issue=3|pages=29–38}}<br />
*{{cite isbn|0521436133}}<br />
*{{cite isbn|0387900535}}<br />
*Killing, ''Die Zusammensetzung der stetigen endlichen Transformationsgruppen'' [[Mathematische Annalen]], [http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0031&DMDID=DMDLOG_0026&L=1 Part 1]: Volume 31, Number 2 June 1888, Pages 252-290 {{DOI|10.1007/BF01211904}}; [http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0033&DMDID=DMDLOG_0007&L=1 Part 2]: Volume 33, Number 1 March 1888, Pages 1–48 {{DOI|10.1007/BF01444109}}; [http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0034&DMDID=DMDLOG_0009&L=1 Part3]: Volume 34, Number 1 March 1889, Pages 57–122 {{DOI|10.1007/BF01446792}}; [http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0036&DMDID=DMDLOG_0019&L=1 Part 4]: Volume 36, Number 2 June 1890,Pages 161-189 {{DOI|10.1007/BF01207837}}<br />
* {{citation|first=Victor G. |last= Kac |authorlink= Victor Kac |title= Infinite dimensional Lie algebras |year= 1994}}.<br />
*{{cite isbn|0817640215}}<br />
<br />
==Further reading==<br />
* Dynkin, E. B. ''The structure of semi-simple algebras.'' {{ru icon}} Uspehi Matem. Nauk (N.S.) 2, (1947). no. 4(20), 59&ndash;127.<br />
<br />
==External links==<br />
{{commons category|Root systems}}<br />
<br />
[[Category:Euclidean geometry]]<br />
[[Category:Lie groups]]<br />
[[Category:Lie algebras]]</div>62.245.100.121https://en.formulasearchengine.com/index.php?title=Nilpotent&diff=3435Nilpotent2013-12-16T18:58:17Z<p>62.245.100.121: /* Commutative rings */ grammar</p>
<hr />
<div>In [[proof theory]] and [[mathematical logic]], '''sequent calculus''' is a family of [[formal system]]s sharing a certain style of inference and certain formal properties. The first sequent calculi, systems '''LK''' and '''LJ''', were introduced by [[Gerhard Gentzen]] in 1934 as a tool for studying [[natural deduction]] in [[first-order logic]] (in [[Classical logic|classical]] and [[Intuitionistic logic|intuitionistic]] versions, respectively). Gentzen's so-called "Main Theorem" (''Hauptsatz'') about LK and LJ was the [[cut-elimination theorem]], a result with far-reaching [[Metatheory|meta-theoretic]] consequences, including [[consistency]]. Gentzen further demonstrated the power and flexibility of this technique a few years later, applying a cut-elimination argument to give a (transfinite) [[Gentzen's consistency proof|proof of the consistency of Peano arithmetic]], in surprising response to [[Gödel's incompleteness theorems]]. Since this early work, sequent calculi (also called '''Gentzen systems''') and the general concepts relating to them have been widely applied in the fields of proof theory, mathematical logic, and [[automated deduction]].<br />
<br />
==Introduction==<br />
<br />
One way to classify different styles of deduction systems is to look at the form of ''[[Judgment (mathematical logic)|judgments]]'' in the system, ''i.e.'', which things may appear as the conclusion of a (sub)proof. The simplest judgment form is used in [[Hilbert-style deduction system]]s, where a judgment has the form<br />
:<math>B\,</math><br />
where <math>B</math> is any formula of first-order-logic (or whatever logic the deduction system applies to, ''e.g.'', [[propositional calculus]] or a [[higher-order logic]] or a [[modal logic]]). The theorems are those formulae that appear as the concluding judgment in a valid proof. A Hilbert-style system needs no distinction between formulae and judgments; we make one here solely for comparison with the cases that follow.<br />
<br />
The price paid for the simple syntax of a Hilbert-style system is that complete formal proofs tend to get extremely long. Concrete arguments about proofs in such a system almost always appeal to the [[deduction theorem]]. This leads to the idea of including the deduction theorem as a formal rule in the system, which happens in [[natural deduction]]. In natural deduction, judgments have the shape<br />
:<math>A_1, A_2, \ldots, A_n \vdash B</math><br />
where the <math>A_i</math>'s and <math>B</math> are again formulae and <math>n\geq 0</math>. In words, a judgment consists of a list (possibly empty) of formulae on the left-hand side of a [[Turnstile (symbol)|turnstile]] symbol "<math>\vdash</math>", with a single formula on the right-hand side. The theorems are those formulae <math>B</math> such that <math>\vdash B</math> (with an empty left-hand side) is the conclusion of a valid proof.<br />
(In some presentations of natural deduction, the <math>A_i</math>s and the turnstile are not written down explicitly; instead a two-dimensional notation from which they can be inferred is used).<br />
<br />
The standard semantics of a judgment in natural deduction is that it asserts that whenever<ref>Here, "whenever" is used as an informal abbreviation "for every assignment of values to the free variables in the judgment"</ref> <math>A_1</math>, <math>A_2</math>, etc., are all true, <math>B</math> will also be true. The judgments<br />
:<math>A_1, \ldots, A_n \vdash B</math><br />
and<br />
:<math>\vdash (A_1 \land \cdots \land A_n) \rightarrow B</math><br />
are equivalent in the strong sense that a proof of either one may be extended to a proof of the other.<br />
<br />
Finally, ''sequent calculus'' generalizes the form of a natural deduction judgment to<br />
: <math>A_1, \ldots, A_n \vdash B_1, \ldots, B_k,</math><br />
a syntactic object called a '''[[sequent]]'''. The formulas on left-hand side of the [[Turnstile (symbol)|turnstile]] are called the ''antecedent'', and the formulas on right-hand side are called the ''succedent''; together they are called ''cedents''. Again, <math>A_i</math> and <math>B_i</math> are formulae, and <math>n</math> and <math>k</math> are nonnegative integers, that is, the left-hand-side or the right-hand-side (or neither or both) may be empty. As in natural deduction, theorems are those <math>B</math> where <math>\vdash B</math> is the conclusion of a valid proof. The empty sequent, having both cedents empty, is defined to be false.<br />
<br />
The standard semantics of a sequent is an assertion that whenever ''every'' <math> A_i</math> is true, ''at least one'' <math>B_i</math> will also be true. One way to express this is that a comma to the left of the turnstile should be thought of as an "and", and a comma to the right of the turnstile should be thought of as an (inclusive) "or". The sequents<br />
:<math>A_1, \ldots, A_n \vdash B_1, \ldots, B_k</math><br />
and<br />
:<math>\vdash (A_1 \land\cdots\land A_n)\rightarrow(B_1 \lor\cdots\lor B_k)</math><br />
are equivalent in the strong sense that a proof of either one may be extended to a proof of the other.<br />
<br />
At first sight, this extension of the judgment form may appear to be a strange complication — it is not motivated by an obvious shortcoming of natural deduction, and it is initially confusing that the comma seems to mean entirely different things on the two sides of the turnstile. However, in a [[Classical logic|classical context]] the semantics of the sequent can also (by propositional [[tautology (logic)|tautology]]) be expressed either as<br />
: <math>\vdash \neg A_1 \lor \neg A_2 \lor \cdots \lor \neg A_n \lor B_1 \lor B_2 \lor\cdots\lor B_k</math><br />
(at least one of the As is false, or one of the Bs is true) or as<br />
: <math>\vdash \neg(A_1 \land A_2 \land \cdots \land A_n \land \neg B_1 \land \neg B_2 \land\cdots\land \neg B_k)</math><br />
(it cannot be the case that all of the As are true and all of the Bs are false). In these formulations, the only difference between formulae on either side of the turnstile is that one side is negated. Thus, swapping left for right in a sequent corresponds to negating all of the constituent formulae. This means that a symmetry such as [[De Morgan's laws]], which manifests itself as logical negation on the semantic level, translates directly into a left-right symmetry of sequents — and indeed, the inference rules in sequent calculus for dealing with conjunction (∧) are mirror images of those dealing with disjunction (∨).<br />
<br />
Many logicians feel that this symmetric presentation offers a deeper insight in the structure of the logic than other styles of proof system, where the classical duality of negation is not as apparent in the rules.<br />
<br />
==The system LK==<br />
<br />
This section introduces the rules of the sequent calculus '''LK''' (which is short for “'''l'''ogistischer '''k'''lassischer Kalkül”), as introduced by Gentzen in 1934.<br />
<ref>{{cite journal | first=Gerhard | last=Gentzen | authorlink=Gerhard Gentzen | title=Untersuchungen über das logische Schließen. I | journal=Mathematische Zeitschrift | volume=39 | pages=176–210 [191] | year=1934/1935 | doi=10.1007/BF01201353 | issue=2}}</ref><br />
A (formal) proof in this calculus is a sequence of [[sequent]]s, where each of the sequents is derivable from sequents appearing earlier in the sequence by using one of the [[rule of inference|rules]] below.<br />
<br />
===Inference rules===<br />
<br />
The following notation will be used:<br />
* <math>\vdash</math> known as the [[Turnstile (symbol)|turnstile]], separates the ''assumptions'' on the left from the ''propositions'' on the right<br />
* <math>A</math> and <math>B</math> denote formulae of first-order predicate logic (one may also restrict this to propositional logic),<br />
* <math>\Gamma, \Delta, \Sigma</math>, and <math>\Pi</math> are finite (possibly empty) sequences of formulae (in fact, the order of formulae do not matter; see subsection '''Structural Rules'''), called contexts,<br />
** when on the ''left'' of the <math>\vdash</math>, the sequence of formulas is considered ''conjunctively'' (all assumed to hold at the same time),<br />
** while on the ''right'' of the <math>\vdash</math>, the sequence of formulas is considered ''disjunctively'' (at least one of the formulas must hold for any assignment of variables),<br />
* <math>t</math> denotes an arbitrary term,<br />
* <math>x</math> and <math>y</math> denote variables.<br />
* a variable is said to occur [[Free variables and bound variables|free]] within a formula if it occurs outside the scope of quantifiers <math>\forall</math> or <math>\exist</math>.<br />
* <math>A[t/x]</math> denotes the formula that is obtained by substituting the term <math>t</math> for every free occurrence of the variable <math>x</math> in formula <math>A</math> with the restriction that the term <math>t</math> must be free for the variable <math>x</math> in <math>A</math> (i.e., no occurrence of any variable in <math>t</math> becomes bound in <math>A[t/x]</math>).<br />
* <math>WL</math> and <math>WR</math> stand for ''Weakening Left/Right'', <math>CL</math> and <math>CR</math> for ''Contraction'', and <math>PL</math> and <math>PR</math> for ''Permutation''.<br />
<br />
<table border="0" cellpadding="5"><br />
<tr><td>Axiom:</td><td>Cut:</td></tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math> \cfrac{\qquad }{ A \vdash A} \quad (I) </math><br />
</td><br />
<td style="text-align: center;"><br />
<math> <br />
\cfrac{\Gamma \vdash \Delta, A \qquad A, \Sigma \vdash \Pi} {\Gamma, \Sigma \vdash \Delta, \Pi} \quad (\mathit{Cut})<br />
</math><br />
</td><br />
</tr><br />
<tr><td>Left logical rules:</td><td>Right logical rules:</td></tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math> \cfrac{\Gamma, A \vdash \Delta} {\Gamma, A \and B \vdash \Delta} \quad ({\and}L_1)<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math> \cfrac{\Gamma \vdash A, \Delta}{\Gamma \vdash A \or B, \Delta} \quad ({\or}R_1)<br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math> \cfrac{\Gamma, B \vdash \Delta}{\Gamma, A \and B \vdash \Delta} \quad ({\and}L_2)<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math> \cfrac{\Gamma \vdash B, \Delta}{\Gamma \vdash A \or B, \Delta} \quad ({\or}R_2)<br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math> \cfrac{\Gamma, A \vdash \Delta \qquad \Sigma, B \vdash \Pi}{\Gamma, \Sigma, A \or B \vdash \Delta, \Pi} \quad ({\or}L)<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math> \cfrac{\Gamma \vdash A, \Delta \qquad \Sigma \vdash B, \Pi}{\Gamma, \Sigma \vdash A \and B, \Delta, \Pi} \quad ({\and}R)<br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma \vdash A, \Delta \qquad \Sigma, B \vdash \Pi}{\Gamma, \Sigma, A\rightarrow B \vdash \Delta, \Pi} \quad ({\rightarrow }L)<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma, A \vdash B, \Delta}{\Gamma \vdash A \rightarrow B, \Delta} \quad ({\rightarrow}R)<br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma \vdash A, \Delta}{\Gamma, \lnot A \vdash \Delta} \quad ({\lnot}L)<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma, A \vdash \Delta}{\Gamma \vdash \lnot A, \Delta} \quad ({\lnot}R)<br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma, A[t/x] \vdash \Delta}{\Gamma, \forall x A \vdash \Delta} \quad ({\forall}L)<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma \vdash A[y/x], \Delta}{\Gamma \vdash \forall x A, \Delta} \quad ({\forall}R) <br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma, A[y/x] \vdash \Delta}{\Gamma, \exist x A \vdash \Delta} \quad ({\exist}L)<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma \vdash A[t/x], \Delta}{\Gamma \vdash \exist x A, \Delta} \quad ({\exist}R)<br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><td>Left structural rules:</td><td>Right structural rules:</td></tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma \vdash \Delta}{\Gamma, A \vdash \Delta} \quad (\mathit{WL})<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma \vdash \Delta}{\Gamma \vdash A, \Delta} \quad (\mathit{WR})<br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma, A, A \vdash \Delta}{\Gamma, A \vdash \Delta} \quad (\mathit{CL})<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma \vdash A, A, \Delta}{\Gamma \vdash A, \Delta} \quad (\mathit{CR})<br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma_1, A, B, \Gamma_2 \vdash \Delta}{\Gamma_1, B, A, \Gamma_2 \vdash \Delta} \quad (\mathit{PL})<br />
</math><br />
</td><br />
<td style="text-align: center;"><br />
<math><br />
\cfrac{\Gamma \vdash \Delta_1, A, B, \Delta_2}{\Gamma \vdash \Delta_1, B, A, \Delta_2} \quad (\mathit{PR})<br />
</math><br />
</td><br />
</tr><br />
</table><br />
<br />
''Restrictions: In the rules <math>({\forall}R)</math> and <math>({\exist}L)</math>, the variable <math>y</math> must not occur free within <math>\Gamma</math> and <math>\Delta</math>. Alternatively, the variable <math>y</math> must not appear anywhere in the respective lower sequents.''<br />
<br />
===An intuitive explanation===<br />
<br />
The above rules can be divided into two major groups: ''logical'' and ''structural'' ones. Each of the logical rules introduces a new logical formula either on the left or on the right of the [[Turnstile (symbol)|turnstile]] <math>\vdash</math>. In contrast, the structural rules operate on the structure of the sequents, ignoring the exact shape of the formulae. The two exceptions to this general scheme are the axiom of identity (I) and the rule of (Cut).<br />
<br />
Although stated in a formal way, the above rules allow for a very intuitive reading in terms of classical logic. Consider, for example, the rule <math>({\and}L_1)</math>. It says that, whenever one can prove that <math>\Delta</math> can be concluded from some sequence of formulae that contain A, then one can also conclude <math>\Delta</math> from the (stronger) assumption, that <math>A \and B</math> holds. Likewise, the rule <math>({\neg}R)</math> states that, if <math>\Gamma</math> and A suffice to conclude <math>\Delta</math>, then from Γ alone one can either still conclude <math>\Delta</math> or A must be false, i.e. <math>{\neg}A</math> holds. All the rules can be interpreted in this way.<br />
<br />
For an intuition about the quantifier rules, consider the rule <math>({\forall}R)</math>. Of course concluding that <math>\forall{x} A</math> holds just from the fact that <math>A[y/x]</math> is true is not in general possible. If, however, the variable y is not mentioned elsewhere (i.e. it can still be chosen freely, without influencing the other formulae), then one may assume, that <math>A[y/x]</math> holds for any value of y. The other rules should then be pretty straightforward.<br />
<br />
Instead of viewing the rules as descriptions for legal derivations in predicate logic, one may also consider them as instructions for the construction of a proof for a given statement. In this case the rules can be read bottom-up; for example, <math>({\and}R)</math> says that, to prove that <math>A \and B</math> follows from the assumptions <math>\Gamma</math> and <math>\Sigma</math>, it suffices to prove that A can be concluded from <math>\Gamma</math> and B can be concluded from <math>\Sigma</math>, respectively. Note that, given some antecedent, it is not clear how this is to be split into <math>\Gamma</math> and <math>\Sigma</math>. However, there are only finitely many possibilities to be checked since the antecedent by assumption is finite. This also illustrates how proof theory can be viewed as operating on proofs in a combinatorial fashion: given proofs for both A and B, one can construct a proof for A∧B.<br />
<br />
When looking for some proof, most of the rules offer more or less direct recipes of how to do this. The rule of cut is different: It states that, when a formula A can be concluded and this formula may also serve as a premise for concluding other statements, then the formula A can be "cut out" and the respective derivations are joined. When constructing a proof bottom-up, this creates the problem of guessing A (since it does not appear at all below). The [[cut-elimination theorem]] is thus crucial to the applications of sequent calculus in [[automated deduction]]: it states that all uses of the cut rule can be eliminated from a proof, implying that any provable sequent can be given a ''cut-free'' proof.<br />
<br />
The second rule that is somewhat special is the axiom of identity (I). The intuitive reading of this is obvious: every formula proves itself. Like the cut rule, the axiom of identity is somewhat redundant: the [[completeness of atomic initial sequents]] states that the rule can be restricted to [[atomic formula]]s without any loss of provability.<br />
<br />
Observe that all rules have mirror companions, except the ones for implication. This reflects the fact that the usual language of first-order logic does not include the "is not implied by" connective <math>\not\leftarrow</math> that would be the De Morgan dual of implication. Adding such a connective with its natural rules would make the calculus completely left-right symmetric.<br />
<br />
===Example derivations===<br />
<br />
Here is the derivation of "<math> \vdash A \or \lnot A </math>", known as<br />
the ''[[Law of excluded middle]]'' (''tertium non datur'' in Latin).<br />
<br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td>&nbsp;</td><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(I)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
A \vdash A<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(\lnot R)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\vdash \lnot A , A<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(\or R_2)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\vdash A \or \lnot A , A<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(PR)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\vdash A , A \or \lnot A<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(\or R_1)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\vdash A \or \lnot A , A \or \lnot A<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(CR)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\vdash A \or \lnot A<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td>&nbsp;</td><br />
</tr><br />
</table><br />
<br />
Next is the proof of a simple fact involving quantifiers. Note that the converse is not true, and its falsity can be seen when attempting to derive it bottom-up, because an existing free variable cannot be used in substitution in the rules <math>(\forall R)</math> and <math>(\exist L)</math>.<br />
<br />
<small><br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td>&nbsp;</td><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(I)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
p(x,y) \vdash p(x,y)<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(\forall L)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\forall x \left( p(x,y) \right) \vdash p(x,y)<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(\exists R)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\forall x \left( p(x,y) \right) \vdash \exists y \left( p(x,y) \right)<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(\exists L)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\exists y \left( \forall x \left( p(x,y) \right) \right) \vdash \exists y \left( p(x,y) \right)<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td rowspan=2><math><br />
(\forall R)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\exists y \left( \forall x \left( p(x,y) \right) \right) \vdash \forall x \left( \exists y \left( p(x,y) \right) \right)<br />
</math></td><br />
<td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td>&nbsp;</td><br />
</tr><br />
</table><br />
</small><br />
<br />
For something more interesting we shall prove <math> \left( \left( A \rightarrow \left( B \or C \right) \right) \rightarrow \left( \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) \rightarrow \lnot A \right) \right) </math>. It is straightforward to find the derivation, which exemplifies the usefulness of LK in automated proving.<br />
<br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td><br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td valign=bottom><br />
<br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td>&nbsp;</td> <td>&nbsp;</td> <td rowspan=2><math><br />
(I)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
A \vdash A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(\lnot R)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\vdash \lnot A , A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(PR)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\vdash A , \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td>&nbsp;</td><br />
</tr><br />
</table><br />
<br />
</td><br />
<td>&nbsp;&nbsp;</td><br />
<td valign=bottom><br />
<br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td><br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td valign=bottom><br />
<br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td><br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td valign=bottom><br />
<br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td><br />
<td>&nbsp;</td> <td>&nbsp;</td> <td rowspan=2><math><br />
(I)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
B \vdash B<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td>&nbsp;</td><br />
</tr><br />
</table><br />
<br />
<td>&nbsp;&nbsp;</td><br />
<td valign=bottom><br />
<br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td><br />
<td>&nbsp;</td> <td>&nbsp;</td> <td rowspan=2><math><br />
(I)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
C \vdash C<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td>&nbsp;</td><br />
</tr><br />
</table><br />
<br />
</td><br />
</tr><br />
</table><br />
</td><br />
<td>&nbsp;</td><br />
<td rowspan=2 valign=bottom><math><br />
(\or L)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
B \or C \vdash B , C<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(PR)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
B \or C \vdash C , B<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(\lnot L)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
B \or C , \lnot C \vdash B<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td>&nbsp;</td><br />
</tr><br />
</table><br />
<br />
</td><br />
<td>&nbsp;&nbsp;</td><br />
<td valign=bottom><br />
<br />
<table align=center border=0 cellspacing=0 cellpadding=0><br />
<tr><br />
<td>&nbsp;</td> <td>&nbsp;</td> <td rowspan=2><math><br />
(I)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\lnot A \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td>&nbsp;</td><br />
</tr><br />
</table><br />
<br />
</td><br />
</tr><br />
</table><br />
</td><br />
<td>&nbsp;</td><br />
<td rowspan=2 valign=bottom><math><br />
(\rightarrow L)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( B \or C \right) , \lnot C , \left( B \rightarrow \lnot A \right) \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(\and L_1)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( B \or C \right) , \lnot C , \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(PL)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( B \or C \right) , \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) , \lnot C \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(\and L_2)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( B \or C \right) , \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) , \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(CL)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( B \or C \right) , \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(PL)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) , \left( B \or C \right) \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td>&nbsp;</td><br />
</tr><br />
</table><br />
<br />
</td><br />
</tr><br />
</table><br />
</td><br />
<td>&nbsp;</td><br />
<td rowspan=2 valign=bottom><math><br />
(\rightarrow L)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) , \left( A \rightarrow \left( B \or C \right) \right) \vdash \lnot A , \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(CR)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) , \left( A \rightarrow \left( B \or C \right) \right) \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(PL)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( A \rightarrow \left( B \or C \right) \right) , \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) \vdash \lnot A<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(\rightarrow R)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\left( A \rightarrow \left( B \or C \right) \right) \vdash \left( \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) \rightarrow \lnot A \right)<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td> <td rowspan=2><math><br />
(\rightarrow R)<br />
</math></td><br />
</tr><br />
<tr><br />
<td align=center style='border-top:1px solid black;' rowspan=2><math><br />
\vdash \left( \left( A \rightarrow \left( B \or C \right) \right) \rightarrow \left( \left( \left( B \rightarrow \lnot A \right) \and \lnot C \right) \rightarrow \lnot A \right) \right)<br />
</math></td> <td>&nbsp;</td><br />
</tr><br />
<tr><br />
<td>&nbsp;</td><br />
<td>&nbsp;</td><br />
</tr><br />
</table><br />
<br />
These derivations also emphasize the strictly formal structure of the sequent calculus. For example, the logical rules as defined above always act on a formula immediately adjacent to the turnstile, such that the permutation rules are necessary. Note, however, that this is in part an artifact of the presentation, in the original style of Gentzen. A common simplification involves the use of [[multiset]]s of formulas in the interpretation of the sequent, rather than sequences, eliminating the need for an explicit permutation rule. This corresponds to shifting commutativity of assumptions and derivations outside the sequent calculus, whereas LK embeds it within the system itself.<br />
<br />
===Structural rules===<br />
<br />
The structural rules deserve some additional discussion.<br />
<br />
Weakening (W) allows the addition of arbitrary elements to a sequence. Intuitively, this is allowed in the antecedent because we can always restrict the scope of our proof (if all cars have wheels, then it's safe to say that all black cars have wheels); and in the succedent because we can always allow for alternative conclusions (if all cars have wheels, then it's safe to say that all cars have either wheels or wings).<br />
<br />
Contraction (C) and Permutation (P) assure that neither the order (P) nor the multiplicity of occurrences (C) of elements of the sequences matters. Thus, one could instead of [[sequence]]s also consider [[Set (mathematics)|sets]].<br />
<br />
The extra effort of using sequences, however, is justified since part or all of the structural rules may be omitted. Doing so, one obtains the so-called [[substructural logic]]s.<br />
<br />
===Properties of the system LK===<br />
<br />
This system of rules can be shown to be both [[soundness|sound]] and [[completeness|complete]] with respect to first-order logic, i.e. a statement <math>A\,</math> follows [[semantics|semantically]] from a set of premises <math>\Gamma\,</math> <math>(\Gamma \vDash A)</math> [[iff]] the sequent <math>\Gamma \vdash A</math> can be derived by the above rules.<br />
<br />
In the sequent calculus, the rule of [[cut-elimination|cut is admissible]]. This result is also referred to as Gentzen's ''Hauptsatz'' ("Main Theorem").<br />
<br />
==Variants==<br />
<br />
The above rules can be modified in various ways:<br />
<br />
===Minor structural alternatives===<br />
<br />
There is some freedom of choice regarding the technical details of how sequents and structural rules are formalized. As long as every derivation in LK can be effectively transformed to a derivation using the new rules and vice versa, the modified rules may still be called LK.<br />
<br />
First of all, as mentioned above, the sequents can be viewed to consist of sets or [[multiset]]s. In this case, the rules for permuting and (when using sets) contracting formulae are obsolete.<br />
<br />
The rule of weakening will become admissible, when the axiom (I) is changed, such that any sequent of the form <math>\Gamma , A \vdash A , \Delta</math> can be concluded. This means that <math>A</math> proves <math>A</math> in any context. Any weakening that appears in a derivation can then be performed right at the start. This may be a convenient change when constructing proofs bottom-up.<br />
<br />
Independent of these one may also change the way in which contexts are split within the rules: In the cases <math>({\and}R), ({\or}L)</math>, and <math>({\rightarrow}L)</math> the left context is somehow split into <math>\Gamma</math> and <math>\Sigma</math> when going upwards. Since contraction allows for the duplication of these, one may assume that the full context is used in both branches of the derivation. By doing this, one assures that no important premises are lost in the wrong branch. Using weakening, the irrelevant parts of the context can be eliminated later.<br />
<br />
===Substructural logics===<br />
{{main|Substructural logic}}<br />
<br />
Alternatively, one may restrict or forbid the use of some of the structural rules. This yields a variety of [[substructural logic]] systems. They are generally weaker than LK (''i.e.'', they have fewer theorems), and thus not complete with respect to the standard semantics of first-order logic. However, they have other interesting properties that have led to applications in theoretical [[computer science]] and [[artificial intelligence]].<br />
<br />
===Intuitionistic sequent calculus: System LJ===<br />
<br />
Surprisingly, some small changes in the rules of LK suffice to turn it into a proof system for [[intuitionistic logic]]. To this end, one has to restrict to sequents with exactly one formula on the right-hand side, and modify the rules to maintain this invariant. For example, <math>({\or}L)</math> is reformulated as follows (where C is an arbitrary formula):<br />
<br />
:<math><br />
\cfrac{\Gamma, A \vdash C \qquad \Sigma, B \vdash C }{\Gamma, \Sigma, A \or B \vdash C} \quad ({\or}L)<br />
</math><br />
<br />
The resulting system is called LJ. It is sound and complete with respect to intuitionistic logic and admits a similar cut-elimination proof.<br />
<br />
==See also==<br />
* [[Resolution (logic)]]<br />
<br />
==Notes==<br />
<references/><br />
<br />
==References==<br />
* {{cite book | first=Jean-Yves | last=Girard | authorlink=Jean-Yves Girard | coauthors=Paul Taylor, Yves Lafont | title=Proofs and Types | publisher=Cambridge University Press (Cambridge Tracts in Theoretical Computer Science, 7) | year=1990 | origyear=1989 | isbn=0-521-37181-3 | url= http://www.paultaylor.eu/stable/Proofs%2BTypes.html}}<br />
*{{cite book | author = Samuel R. Buss | chapter= An introduction to proof theory | editor = Samuel R. Buss | title=Handbook of proof theory | pages = 1–78 | url = http://math.ucsd.edu/~sbuss/ResearchWeb/handbookI/ | publisher = Elsevier | year = 1998 | isbn = 0-444-89840-9 }}<br />
<br />
==External links==<br />
* {{springer|title=Sequent calculus|id=p/s084580}}<br />
* [http://scienceblogs.com/goodmath/2006/07/17/a-brief-diversion-sequent-calc/ A Brief Diversion: Sequent Calculus]<br />
* [http://logitext.mit.edu/logitext.fcgi/tutorial Interactive tutorial of the Sequent Calculus]<br />
<br />
[[Category:Proof theory]]<br />
[[Category:Logical calculi]]<br />
[[Category:Automated theorem proving]]</div>62.245.100.121https://en.formulasearchengine.com/index.php?title=Levitzky%27s_theorem&diff=24583Levitzky's theorem2013-12-10T15:59:44Z<p>62.245.100.121: /* Proof */ this section is no longer empty</p>
<hr />
<div>In 1964, three teams wrote scientific papers which proposed related but different approaches to explain how mass could arise in local [[gauge theory|gauge theories]]. These three now famous papers were written by [[Robert Brout]] and [[François Englert]],<ref><br />
{{cite doi|10.1103/PhysRevLett.13.321<br />
}}</ref><ref><br />
{{cite arXiv<br />
|author=R. Brout, F. Englert<br />
|year=1998<br />
|title=Spontaneous Symmetry Breaking in Gauge Theories: A Historical Survey<br />
|class=hep-th<br />
|eprint=hep-th/9802142<br />
}}</ref> [[Peter Higgs]],<ref name="Peter W. Higgs 1964 508-509"><br />
{{cite doi|10.1103/PhysRevLett.13.508}}</ref> and [[Gerald Guralnik]], [[C. R. Hagen|C. Richard Hagen]], and [[Tom W. B. Kibble|Tom Kibble]],<ref><br />
{{cite doi|10.1103/PhysRevLett.13.585}}</ref><ref><br />
{{cite journal<br />
| author=G.S. Guralnik<br />
| year=2009<br />
| title=The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles<br />
| journal=[[International Journal of Modern Physics A]]<br />
| volume=24 | issue=14 | pages=2601–2627<br />
| arxiv=0907.3466<br />
| bibcode = 2009IJMPA..24.2601G<br />
| doi=10.1142/S0217751X09045431<br />
}}</ref> and are credited with the theory of the [[Higgs mechanism]] and the prediction of the [[Higgs boson|Higgs field and Higgs boson]]. Together, these provide a theoretical means by which [[Goldstone's theorem]] (a problematic limitation affecting early modern [[particle physics]] theories) can be avoided. They show how [[gauge bosons]] can acquire non-zero masses as a result of [[spontaneous symmetry breaking]] within [[gauge invariance|gauge invariant]] models of the universe.<ref><br />
{{cite doi|10.4249/scholarpedia.6441}}</ref><br />
<br />
As such, these form the key element of the [[electroweak theory]] that forms part of the [[Standard Model]] of [[particle physics]], and of many models, such as the [[Grand Unified Theory]], that go beyond it. The papers that introduce this mechanism were published in ''[[Physical Review Letters]]'' (PRL) and were each recognized as milestone papers by PRL's 50th anniversary celebration.<ref><br />
{{cite web<br />
|author=M. Blume, S. Brown, Y. Millev<br />
|year=2008<br />
|url=http://prl.aps.org/50years/milestones#1964<br />
|title=Letters from the past, a PRL retrospective (1964)<br />
|publisher=[[Physical Review Letters]]<br />
|accessdate=2010-01-30<br />
| archiveurl= http://web.archive.org/web/20100110134128/http://prl.aps.org/50years/milestones| archivedate= 10 January 2010 <!--DASHBot-->| deadurl= no}}</ref> All of the six physicists were awarded the 2010 [[Sakurai Prize|J. J. Sakurai Prize for Theoretical Particle Physics]] for this work,<ref><br />
{{cite web<br />
|author=<br />
|year=2010<br />
|url=http://www.aps.org/units/dpf/awards/sakurai.cfm<br />
|title=J. J. Sakurai Prize Winners<br />
|publisher=[[American Physical Society]]<br />
|accessdate=2010-01-30<br />
| archiveurl= http://web.archive.org/web/20100212025108/http://www.aps.org/units/dpf/awards/sakurai.cfm| archivedate= 12 February 2010 <!--DASHBot-->| deadurl= no}}</ref> and in 2013 Englert and Higgs received the Nobel Prize in Physics.<ref>http://www.nobelprize.org/nobel_prizes/physics/laureates/2013/</ref><br />
On 4 July 2012, the two main experiments at the LHC ([[ATLAS experiment|ATLAS]] and [[Compact Muon Solenoid|CMS]]) both reported independently the confirmed existence of a previously unknown particle with a mass of about {{val|125|ul=GeV/c2}} (about 133 proton masses, on the order of 10<sup>−25</sup>&nbsp;kg), which is "consistent with the Higgs boson" and widely believed to be the Higgs boson.<ref name=cern1207>{{cite news |url=http://press.web.cern.ch/press/PressReleases/Releases2012/PR17.12E.html |title=CERN experiments observe particle consistent with long-sought Higgs boson |publisher=CERN press release |date=4 July 2012 |accessdate=4 July 2012}}</ref><br />
<br />
== Introduction ==<br />
A [[gauge theory]] of [[elementary particles]] is a very attractive potential framework for constructing the [[grand unified theory|ultimate theory]]. Such a theory has the very desirable property of being potentially [[renormalization|renormalizable]]—shorthand for saying that all calculational infinities encountered can be consistently absorbed into a few parameters of the theory. However, as soon as one gives mass to the gauge fields, renormalizability is lost, and the theory rendered useless. [[Spontaneous symmetry breaking]] is a promising mechanism, which could be used to give mass to the vector gauge particles. A significant difficulty which one encounters, however, is [[Goldstone's theorem]], which states that in any [[quantum field theory]] which has a spontaneously broken symmetry there must occur a zero-mass particle. So the problem arises—how can one break a [[symmetry]] and at the same time not introduce unwanted zero-mass particles. The resolution of this dilemma lies in the observation that in the case of gauge theories, the Goldstone theorem can be avoided by working in the so-called [[radiation gauge]]. This is because the proof of Goldstone's theorem requires manifest [[Lorentz covariance]], a property not possessed by the radiation gauge.<br />
<br />
== History ==<br />
{| class="wikitable" style="float:right; margin:0 0 1em 1em; font-size:85%; width:354px;"<br />
|-<br />
| {{nowrap|[[File:AIP-Sakurai-best.JPG|x150px]]&nbsp;&nbsp;[[File:Higgs, Peter (1929) cropped.jpg|x150px]]}}<br /><br />
The six authors of the 1964 PRL papers, who received the 2010 [[Sakurai Prize|J. J. Sakurai Prize]] for their work. From left to right: [[T. W. B. Kibble|Kibble]], [[Gerald Guralnik|Guralnik]], [[C. R. Hagen|Hagen]], [[François Englert|Englert]], [[Robert Brout|Brout]]. ''Right:'' [[Peter Higgs|Higgs]].<br />
|}<br />
{{Wikinewshas|news related to|<br />
* [[n:2010 Sakurai Prize awarded for 1964 Higgs Boson theory work|2010 Sakurai Prize awarded for 1964 Higgs Boson theory work]]<br />
* [[n:Prospective Nobel Prize for Higgs boson work disputed|Prospective Nobel Prize for Higgs boson work disputed (2010)]]<br />
}}<br />
Particle physicists study [[matter]] made from [[fundamental particle]]s whose interactions are mediated by exchange particles known as [[force carrier]]s. At the beginning of the 1960s a number of these particles had been discovered or proposed, along with theories suggesting how they relate to each other, some of which had already been reformulated as [[quantum field theory|field theories]] in which the objects of study are not particles and forces, but [[quantum field]]s and their [[Symmetry (physics)|symmetries]].{{citation needed|date=August 2012}} However, [[Unified field theory|attempts to unify]] known [[fundamental forces]] such as the [[electromagnetic force]] and the [[weak nuclear force]] were known to be incomplete. One known omission was that [[gauge invariance|gauge invariant]] approaches, including [[non-abelian gauge theory|non-abelian]] models such as [[Yang–Mills theory]] (1954), which held great promise for unified theories, also seemed to predict known massive particles as massless.<ref name=woit>{{cite web|last=Woit|first=Peter|title=The Anderson–Higgs Mechanism|url=http://www.math.columbia.edu/~woit/wordpress/?p=3282|publisher=Dr. Peter Woit (Senior Lecturer in Mathematics [[Columbia University]] and Ph.D. particle physics)|accessdate=12 November 2012|date=13 November 2010}}</ref> [[Goldstone's theorem]], relating to [[continuous symmetry|continuous symmetries]] within some theories, also appeared to rule out many obvious solutions,<ref>{{cite journal | last =Goldstone | first = J | year = 1962 | title = Broken Symmetries | journal = Physical Review | volume = 127 | pages = 965–970 | doi =10.1103/PhysRev.127.965 | last2 =Salam | first2 =Abdus | last3 =Weinberg | first3 =Steven |bibcode = 1962PhRv..127..965G | issue =3 | ref =harv }}</ref> since it appeared to show that zero-mass particles would have to also exist that were "simply not seen".<ref name="Guralnik 2011">Guralnik – ''[http://arxiv.org/pdf/1110.2253 The Beginnings of Spontaneous Symmetry Breaking in Particle Physics]'' (2011, [[arXiv]]) pages 1 + 3</ref> According to [[Gerald Guralnik|Guralnik]], physicists had "no understanding" how these problems could be overcome.<ref name="Guralnik 2011" /><br />
<br />
Particle physicist and mathematician Peter Woit summarised the state of research at the time:<br />
<br />
: "Yang and Mills work on non-abelian gauge theory had one huge problem: in [[perturbation theory (quantum mechanics)|perturbation theory]] it has massless particles which don’t correspond to anything we see. One way of getting rid of this problem is now fairly well-understood, the phenomenon of [[color confinement|confinement]] realized in [[quantum chromodynamics|QCD]], where the strong interactions get rid of the massless “gluon” states at long distances. By the very early sixties, people had begun to understand another source of massless particles: spontaneous symmetry breaking of a continuous symmetry. What Philip Anderson realized and worked out in the summer of 1962 was that, when you have ''both'' gauge symmetry ''and'' spontaneous symmetry breaking, the Nambu–Goldstone massless mode can combine with the massless gauge field modes to produce a physical massive vector field. This is what happens in [[superconductivity]], a subject about which Anderson was (and is) one of the leading experts." ''[text condensed]'' <ref name="woit" /><br />
<br />
The Higgs mechanism is a process by which [[vector boson]]s can get [[rest mass]] ''without'' [[explicit symmetry breaking|explicitly breaking]] [[gauge invariance]], as a byproduct of [[spontaneous symmetry breaking]].<ref name="scholarpedia">{{cite web|url=http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism |title=Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia |publisher=Scholarpedia.org |date= |accessdate=2012-11-23}}</ref><ref name="scholarpedia_a">{{cite web|url=http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism_%28history%29 |title=History of Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia |publisher=Scholarpedia.org |date= |accessdate=2012-11-23}}</ref> The mathematical theory behind spontaneous symmetry breaking was initially conceived and published within particle physics by [[Yoichiro Nambu]] in 1960,<ref name="nambu nobel">[http://www.nobelprize.org/nobel_prizes/physics/laureates/2008 The Nobel Prize in Physics 2008] – official Nobel Prize website.</ref> the concept that such a mechanism could offer a possible solution for the "mass problem" was originally suggested in 1962 by [[Philip Warren Anderson|Philip Anderson]],<ref name="MyLifeAsABoson">{{cite web|last=Higgs|first=Peter|title=My Life as a Boson|url=http://www.kcl.ac.uk/nms/depts/physics/news/events/MyLifeasaBoson.pdf|publisher=Talk given by Peter Higgs at Kings College, London, Nov 24 2010, expanding on a paper originally presented in 2001|accessdate=17 January 2013|date=2010-11-24}} – the original 2001 paper can be found at: {{cite book|title=2001 A Spacetime Odyssey: Proceedings of the Inaugural Conference of the Michigan Center for Theoretical Physics, Michigan, USA, 21–25 May 2001|year=conference 2001, book of proceedings 2003|isbn=9812382313|publisher=World Scientific|editor=Duff and Liu|url=http://books.google.com/?id=ONhnbpq00xIC&pg=PR11&dq=2001:+A+Space+Time+Odyssey++%22life+as+a+boson%22#v=onepage&q=2001%3A%20A%20Space%20Time%20Odyssey%20%20%22life%20as%20a%20boson%22&f=false|pages=86–88|accessdate=17 January 2013}}</ref>{{rp|4–5}}<ref>{{cite journal|title=Plasmons, gauge invariance and mass|doi=10.1103/PhysRev.130.439|year=1963|last1=Anderson|first1=P.|journal=Physical Review|volume=130|page=439|bibcode=1963PhRv..130..439A|ref=harv}}</ref> and [[Abraham Klein (physicist)|Abraham Klein]] and [[Benjamin W. Lee|Benjamin Lee]] showed in March 1964 that Goldstone's theorem could be avoided this way in at least some non-relativistic cases and speculated it might be possible in truly relativistic cases.<ref>{{Cite journal <br />
| last1 = Klein | first1 = A. <br />
| last2 = Lee | first2 = B. <br />
| doi = 10.1103/PhysRevLett.12.266 <br />
| title = Does Spontaneous Breakdown of Symmetry Imply Zero-Mass Particles? <br />
| journal = Physical Review Letters <br />
| volume = 12 <br />
| issue = 10 <br />
| pages = 266 <br />
| year = 1964 <br />
| pmid = <br />
| pmc = <br />
|bibcode = 1964PhRvL..12..266K }}</ref><br />
<br />
These approaches were quickly developed into a full [[Special relativity|relativistic]] model, independently and almost simultaneously, by three groups of physicists: by [[François Englert]] and [[Robert Brout]] in August 1964;<ref name="eb64">{{Cite journal<br />
|last=Englert |first=François |authorlink=François Englert<br />
|last2=Brout |first2=Robert |authorlink2=Robert Brout<br />
|year=1964<br />
|title=Broken Symmetry and the Mass of Gauge Vector Mesons<br />
|journal=[[Physical Review Letters]]<br />
|volume=13 |issue=9 |pages=321–23<br />
|doi=10.1103/PhysRevLett.13.321 |bibcode=1964PhRvL..13..321E<br />
|ref=harv<br />
}}</ref> by [[Peter Higgs]] in October 1964;<ref name="higgs64">{{Cite journal<br />
|first=Peter |last=Higgs<br />
|year=1964<br />
|title=Broken Symmetries and the Masses of Gauge Bosons<br />
|journal=[[Physical Review Letters]]<br />
|volume=13 |issue=16 |pages=508–509<br />
|doi=10.1103/PhysRevLett.13.508<br />
|bibcode=1964PhRvL..13..508H<br />
|ref=harv<br />
}}</ref> and by [[Gerald Guralnik]], [[C. R. Hagen|Carl Hagen]], and [[T. W. B. Kibble|Tom Kibble]] (GHK) in November 1964.<ref name="ghk64">{{Cite journal<br />
|last=Guralnik |first=Gerald |authorlink=Gerald Guralnik<br />
|last2=Hagen |first2=C. R. |authorlink2=C. R. Hagen<br />
|last3=Kibble |first3=T. W. B. |authorlink3=T. W. B. Kibble<br />
|year=1964<br />
|title=Global Conservation Laws and Massless Particles<br />
|journal=[[Physical Review Letters]]<br />
|volume=13 |issue=20 |pages=585–587<br />
|doi=10.1103/PhysRevLett.13.585 |bibcode=1964PhRvL..13..585G<br />
|ref=harv<br />
}}</ref> Higgs also wrote a short but important<ref name="ScholarpediaKibble">{{cite web|last=Kibble|first=Tom|title=Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism (history)|url=http://www.scholarpedia.org/w/index.php?title=Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism|publisher=Scholarpedia|accessdate=17 January 2013|year=2009}}</ref> response published in September 1964 to an objection by [[Walter Gilbert|Gilbert]],<ref name="higgs64note">{{Cite journal<br />
|first=Peter |last=Higgs<br />
|year=1964<br />
|title=Broken symmetries, massless particles and gauge fields<br />
|journal=[[Physics Letters]]<br />
|volume=12 |issue=2 |pages=132–133<br />
|doi=10.1016/0031-9163(64)91136-9<br />
|ref=harv<br />
|bibcode = 1964PhL....12..132H }}</ref> which showed that if calculating within the radiation gauge, Goldstone's theorem and Gilbert's objection would become inapplicable.<ref group="Note" name="GoldstoneNote">[[Goldstone's theorem]] only applies to gauges having [[Lorentz covariance|manifest Lorentz covariance]], a condition that took time to become questioned. But the process of [[quantization (physics)|quantisation]]<!--BRITISH ENGLISH SPELLING--> requires a [[gauge fixing|gauge to be fixed]] and at this point it becomes possible to choose a gauge such as the 'radiation' gauge which is not invariant over time, so that these problems can be avoided.</ref> (Higgs later described Gilbert's objection as prompting his own paper.<ref>{{cite web|last=Higgs|first=Peter|title=My Life as a Boson|url=http://www.kcl.ac.uk/nms/depts/physics/news/events/MyLifeasaBoson.pdf|publisher=Talk given by Peter Higgs at Kings College, London, Nov 24 2010|accessdate=17 January 2013|date=2010-11-24|quote=Gilbert ... wrote a response to [Klein and Lee's paper] saying 'No, you cannot do that in a relativistic theory. You cannot have a preferred unit time-like vector like that.' This is where I came in, because the next month was when I responded to Gilbert’s paper by saying 'Yes, you can have such a thing' but only in a gauge theory with a gauge field coupled to the current.}}</ref>) Properties of the model were further considered by Guralnik in 1965,<ref>{{cite journal |author=G.S. Guralnik |year=2011 |title=GAUGE INVARIANCE AND THE GOLDSTONE THEOREM&nbsp;– 1965 Feldafing talk |journal=[[Modern Physics Letters A]] |volume=26 |issue=19 |pages=1381–1392 |doi=10.1142/S0217732311036188 |arxiv=1107.4592v1|bibcode = 2011MPLA...26.1381G |ref=harv }}</ref> by Higgs in 1966,<ref>{{Cite journal|first=Peter |last=Higgs |year=1966 |title=Spontaneous Symmetry Breakdown without Massless Bosons |journal=[[Physical Review]] |volume=145 |issue=4 |pages=1156–1163 |doi=10.1103/PhysRev.145.1156 |bibcode = 1966PhRv..145.1156H|ref=harv }}</ref> by Kibble in 1967,<ref>{{Cite journal|first=Tom |last=Kibble |year=1967 |title=Symmetry Breaking in Non-Abelian Gauge Theories |journal=[[Physical Review]] |volume=155 |issue=5 |pages=1554–1561 |doi=10.1103/PhysRev.155.1554|ref=harv|bibcode = 1967PhRv..155.1554K }}</ref> and further by GHK in 1967.<ref>{{cite web|url=http://www.datafilehost.com/download-7d512618.html |title=Guralnik, G S; Hagen, C R and Kibble, T W B (1967). Broken Symmetries and the Goldstone Theorem. Advances in Physics, vol. 2 |publisher=Datafilehost.com |date= |accessdate=2013-01-19}}</ref> The original three 1964 papers showed that when a [[gauge theory]] is combined with an additional field that spontaneously breaks the symmetry, the gauge bosons can consistently acquire a finite mass.<ref name="scholarpedia" /><ref name="scholarpedia_a" /><ref name="prl">{{Cite journal<br />
|url=http://prl.aps.org/50years/milestones#1964<br />
|title=Physical Review Letters&nbsp;– 50th Anniversary Milestone Papers<br />
|publisher=[[Physical Review Letters]]<br />
|ref=harv<br />
}}</ref> In 1967, [[Steven Weinberg]]<ref><br />
{{cite journal<br />
| author=S. Weinberg<br />
| year=1967<br />
| title=A Model of Leptons<br />
| journal=[[Physical Review Letters]]<br />
| volume=19 | pages=1264–1266<br />
| doi=10.1103/PhysRevLett.19.1264<br />
| bibcode=1967PhRvL..19.1264W<br />
| issue=21<br />
| ref=harv<br />
}}</ref> and [[Abdus Salam]]<ref><br />
{{cite conference<br />
| author=A. Salam<br />
| editor=N. Svartholm<br />
| year=1968<br />
| booktitle=Elementary Particle Physics: Relativistic Groups and Analyticity<br />
| pages=367<br />
| conference=Eighth Nobel Symposium<br />
| publisher=Almquvist and Wiksell<br />
| location=Stockholm<br />
}}</ref> independently showed how a Higgs mechanism could be used to break the electroweak symmetry of [[Sheldon Lee Glashow|Sheldon Glashow]]'s [[electroweak theory|unified model for the weak and electromagnetic interactions]]<ref><br />
{{cite journal<br />
| author=S.L. Glashow<br />
| year=1961<br />
| title=Partial-symmetries of weak interactions<br />
| journal=[[Nuclear Physics (journal)|Nuclear Physics]]<br />
| volume=22 | pages=579–588<br />
| doi=10.1016/0029-5582(61)90469-2<br />
|bibcode = 1961NucPh..22..579G<br />
| issue=4<br />
| ref=harv }}</ref> (itself an extension of work by [[Julian Schwinger|Schwinger]]), forming what became the [[Standard Model]] of particle physics. Weinberg was the first to observe that this would also provide mass terms for the fermions.<ref name="Ellis2012">{{cite arXiv<br />
|title=A Historical Profile of the Higgs Boson<br />
|first1=John |last1=Ellis<br />
|first2=Mary K. |last2=Gaillard<br />
|first3=Dimitri V. |last3=Nanopoulos<br />
|eprint=1201.6045<br />
|class=hep-ph<br />
|year=2012}}</ref>&nbsp;{{#tag:ref|A field with the "Mexican hat" potential <math>V(\phi)= \mu^2\phi^2 + \lambda\phi^4</math> and <math>\mu^2 < 0</math> has a minimum not at zero but at some non-zero value <math>\phi_0</math>. By expressing the action in terms of the field <math>\tilde \phi = \phi-\phi_0</math> (where <math>\phi_0</math> is a constant independent of position), we find the Yukawa term has a component <math>g\phi_0 \bar\psi\psi</math>. Since both ''g'' and <math>\phi_0</math> are constants, this looks exactly like the mass term for a fermion of mass <math>g\phi_0</math>. The field <math>\tilde\phi</math> is then the [[Higgs field]].|group=Note}}<br />
<br />
However, the seminal papers on spontaneous breaking of gauge symmetries were at first largely ignored, because it was widely believed that the (non-Abelian gauge) theories in question were a dead-end, and in particular that they could not be [[renormalizable|renormalised]]<!-- BRITISH ENGLISH SPELLING!-->. In 1971–72, [[Martinus Veltman]] and [[Gerard 't Hooft]] proved renormalisation of Yang–Mills was possible in two papers covering massless, and then massive, fields.<ref name="Ellis2012"/> Their contribution, and others' work on the [[renormalization group]], was eventually "enormously profound and influential",<ref name="Politzer 2004">{{cite web|last=Politzer|first=David|title=The Dilemma of Attribution|url=http://www.nobelprize.org/nobel_prizes/physics/laureates/2004/politzer-lecture.html|work=Nobel Prize lecture, 2004|publisher=Nobel Prize|accessdate=22 January 2013|quote=Sidney Coleman published in Science magazine in 1979 a citation search he did documenting that essentially no one paid any attention to Weinberg’s Nobel Prize winning paper until the work of ’t Hooft (as explicated by Ben Lee). In 1971 interest in Weinberg’s paper exploded. I had a parallel personal experience: I took a one-year course on weak interactions from Shelly Glashow in 1970, and he never even mentioned the Weinberg–Salam model or his own contributions.}}</ref> but even with all key elements of the eventual theory published there was still almost no wider interest. For example, [[Sidney Coleman|Coleman]] found in a study that "essentially no-one paid any attention" to Weinberg's paper prior to 1971{{#tag:ref| {{cite journal|last=[[Sidney Coleman{{!}}Coleman]]|first=Sidney|title=The 1979 Nobel Prize in Physics|journal=[[Science (magazine){{!}}Science]]|date=1979-12-14|volume=206|issue=4424|pages=1290–1292|doi=10.1126/science.206.4424.1290|url=http://www.sciencemag.org/content/206/4424/1290.extract|accessdate=22 January 2013|ref=harv|bibcode = 1979Sci...206.1290C }} – discussed by [[David Politzer]] in his 2004 Nobel speech.<ref name="Politzer 2004" />|name="Coleman 1979"}} – now the most cited in particle physics<ref name="PRL_50years">[http://prl.aps.org/50years/milestones#1967 Letters from the Past – A PRL Retrospective] (50 year celebration, 2008)</ref> – and even in 1970 according to [[David Politzer|Politzer]], Glashow's teaching of the weak interaction contained no mention of Weinberg's, Salem's, or Glashow's own work.<ref name="Politzer 2004" /> In practice, Politzer states, almost everyone learned of the theory due to physicist [[Benjamin W. Lee|Benjamin Lee]], who combined the work of Veltman and 't Hooft with insights by others, and popularised the completed theory.<ref name="Politzer 2004" /> In this way, from 1971, interest and acceptance "exploded"&nbsp;<ref name="Politzer 2004" /> and the ideas were quickly absorbed in the mainstream.<ref name="Ellis2012"/><ref name="Politzer 2004" /><br />
<br />
=== The significance of requiring manifest covariance ===<br />
Most students who have taken a course in [[electromagnetism]] have encountered the [[Coulomb potential]]. It basically states that two [[charged particles]] attract or repel each other by a force which varies according to the inverse square of their separation. This is fairly unambiguous for particles at rest, but if one or the other is following an arbitrary trajectory the question arises whether one should compute the force using the instantaneous positions of the particles or the so-called [[retarded potential|retarded position]]s. The latter recognizes that information cannot propagate instantaneously, rather it propagates at the [[speed of light]]. However, the [[radiation gauge]] says that one uses the instantaneous positions of the particles, but doesn't violate [[causality]] because there are compensating terms in the force equation. In contrast, the [[Lorenz gauge]] imposes [[manifest covariance]] (and thus causality) at all stages of a calculation. Predictions of observable quantities are identical in the two gauges, but the radiation gauge formulation of [[quantum field theory]] avoids [[Goldstone's theorem]].<ref><br />
{{cite book<br />
|author=G.S. Guralnik, C.R. Hagen, T.W.B. Kibble<br />
|chapter=Broken Symmetries and the Goldstone Theorem<br />
|chapterurl=http://www.datafilehost.com/download-7d512618.html<br />
|editor=R. L. Cool, R. E. Marshak<br />
|year=1968<br />
|title=Advances in Particle Physics<br />
|publisher=[[Interscience Publishers]]<br />
|volume=2 |issue= |pages=567–708<br />
|isbn=0-470-17057-3<br />
}}</ref><br />
<br />
=== Summary and impact of the PRL papers ===<br />
The three papers written in 1964 were each recognised as milestone papers during ''[[Physical Review Letters]]''{{'s}} 50th anniversary celebration.<ref name="prl" /> Their six authors were also awarded the 2010 [[Sakurai Prize|J. J. Sakurai Prize for Theoretical Particle Physics]] for this work.<ref name="sakuraiprize">American Physical Society – {{cite web|url=http://www.aps.org/units/dpf/awards/sakurai.cfm|title=J. J. Sakurai Prize for Theoretical Particle Physics}}</ref> (A controversy also arose the same year, because in the event of a [[Nobel Prize]] only up to three scientists could be recognised, with six being credited for the papers.<ref>{{cite news|last=Merali|first=Zeeya|title=Physicists get political over Higgs|url=http://www.nature.com/news/2010/100804/full/news.2010.390.html|accessdate=28 December 2011|newspaper=[[Nature Magazine]]|date=4 August 2010}}</ref> ) Two of the three PRL papers (by Higgs and by GHK) contained equations for the hypothetical [[quantum field theory|field]] that eventually would become known as the Higgs field and its hypothetical [[quantum]], the Higgs boson.<ref name="higgs64" /><ref name="ghk64" /> Higgs's subsequent 1966 paper showed the decay mechanism of the boson; only a massive boson can decay and the decays can prove the mechanism.{{citation needed|date=August 2012}}<br />
<br />
Each of these papers is unique and demonstrates different approaches to showing how mass arise in gauge particles. Over the years, the differences between these papers are no longer widely understood, due to the passage of time and acceptance of end-results by the [[particle physics]] community. A study of citation indices is interesting—more than 40 years after the 1964 publication in ''Physical Review Letters'' there is little noticeable pattern of preference among them, with the vast majority of researchers in the field mentioning all three milestone papers.{{citation needed|date=July 2012}}<br />
<br />
In the paper by Higgs the boson is massive, and in a closing sentence Higgs writes that "an essential feature" of the theory "is the prediction of incomplete multiplets of [[scalar boson|scalar]] and [[vector boson]]s".<ref name="higgs64"/> ([[Frank Close]] comments that 1960s gauge theorists were focused on the problem of massless ''vector'' bosons, and the implied existence of a massive ''scalar'' boson was not seen as important; only Higgs directly addressed it.<ref name="frank_close_infinity_puzzle">{{cite book|last=Close|first=Frank|title=The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe|year=2011|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-959350-7}}</ref>{{rp|154, 166, 175}}) In the paper by GHK the boson is massless and decoupled from the massive states.<ref name="ghk64" /> In reviews dated 2009 and 2011, Guralnik states that in the GHK model the boson is massless only in a lowest-order approximation, but it is not subject to any constraint and acquires mass at higher orders, and adds that the GHK paper was the only one to show that there are no massless [[Goldstone boson]]s in the model and to give a complete analysis of the general Higgs mechanism.<ref name="Guralnik 2011">{{cite arXiv |title=Guralnik, G.S. The Beginnings of Spontaneous Symmetry Breaking in Particle Physics. Proceedings of the DPF-2011 Conference, Providence, RI, 8–13 August 2011 |date=11 October 2011|eprint=1110.2253v1 |author1=Guralnik |class=physics.hist-ph}}</ref><ref name="Guralnik 2009">{{Cite journal | author=G.S. Guralnik | title=The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles | journal=[[International Journal of Modern Physics A]] | year=2009 | volume=24 | issue=14 | pages=2601–2627 | doi=10.1142/S0217751X09045431 | arxiv=0907.3466|bibcode = 2009IJMPA..24.2601G | ref=harv }}</ref> All three reached similar conclusions, despite their very different approaches: Higgs' paper essentially used classical techniques, Englert and Brout's involved calculating vacuum polarization in perturbation theory around an assumed symmetry-breaking vacuum state, and GHK used operator formalism and conservation laws to explore in depth the ways in which Goldstone's theorem explicitly fails.<ref name="ScholarpediaKibble" /><br />
<br />
In addition to explaining how mass is acquired by vector bosons, the Higgs mechanism also predicts the ratio between the [[W and Z bosons|W boson and Z boson]] masses as well as their [[coupling constant|couplings]] with each other and with the Standard Model quarks and leptons.{{citation needed|date=August 2012}} Subsequently, many of these predictions have been verified by precise measurements performed at the [[Large Electron–Positron Collider|LEP]] and the [[Stanford Linear Collider|SLC]] colliders, thus overwhelmingly confirming that some kind of Higgs mechanism does take place in nature,<ref name="EWWG">{{cite web |url=http://lepewwg.web.cern.ch/LEPEWWG/ |title=LEP Electroweak Working Group}}</ref> but the exact manner by which it happens has not yet been discovered.{{citation needed|date=August 2012}} The results of searching for the Higgs boson are expected to provide evidence about how this is realized in nature.{{citation needed|date=August 2012}}<br />
<br />
=== Consequences of the papers ===<br />
The resulting electroweak theory and Standard Model have [[Standard Model#Tests and predictions|correctly predicted]] (among other discoveries) [[weak neutral current]]s, [[W and Z bosons|three bosons]], the [[top quark|top]] and [[charm quark]]s, and with great precision, the mass and other properties of some of these.<ref name="predictions" group="Note">The success of the Higgs based electroweak theory and Standard Model is illustrated by their [[Standard Model#Tests and predictions|predictions]] of the mass of two particles later detected: the W boson (predicted mass: 80.390 ± 0.018 GeV, experimental measurement: 80.387 ± 0.019 GeV), and the Z boson (predicted mass: 91.1874 ± 0.0021, experimental measurement: 91.1876 ± 0.0021 GeV). The existence of the Z boson was itself another prediction. Other correct predictions included the [[weak neutral current]], the [[gluon]], and the [[top quark|top]] and [[charm quark]]s, all later proven to exist as the theory said.</ref> Many of those involved [[#Recognition and awards|eventually won]] [[Nobel Prize]]s or other renowned awards. A 1974 paper in ''[[Reviews of Modern Physics]]'' commented that "while no one doubted the [mathematical] correctness of these arguments, no one quite believed that nature was diabolically clever enough to take advantage of them".<ref>{{cite journal|last=Bernstein|first=Jeremy|title=Spontaneous symmetry breaking, gauge theories, the Higgs mechanism and all that|journal=Reviews of Modern Physics|date=January 1974|volume=7 46, No. 1|url=http://www.calstatela.edu/faculty/kaniol/p544/rmp46_p7_higgs_goldstone.pdf|accessdate=2012-12-10|ref=harv}}</ref> By 1986 and again in the 1990s it became possible to write that understanding and proving the Higgs sector of the Standard Model was "the central problem today in particle physics."&nbsp;<ref>{{cite book|last=José Luis Lucio and Arnulfo Zepeda|title=Proceedings of the II Mexican School of Particles and Fields, Cuernavaca-Morelos, 1986|year=1987|publisher=World Scientific|isbn=9971504340|pages=29|url=http://books.google.com/?id=jJ-yAAAAIAAJ&q=higgs+%22central+problem+today+in+particle+physics%22&dq=higgs+%22central+problem+today+in+particle+physics%22}}</ref><ref>{{cite book|last=Gunion, Dawson, Kane, and Haber|title=The Higgs Hunter's Guide (1st ed.)|year=199|pages=11 (?)|url=http://books.google.com/?id=M5moXN_SA-MC&pg=PA10&dq=higgs+hunter+crucial+central+prediction#v=snippet&q=central&f=false|isbn=9780786743186}} – quoted as being in the first (1990) edition of the book by Peter Higgs in his talk "My Life as a Boson", 2001, ref#25.</ref><br />
<br />
==See also==<br />
{{columns-list|2|<br />
*[[Higgs mechanism]]<br />
*[[Higgs boson]]<br />
*[[Standard Model]]<br />
*[[Symmetry breaking]]<br />
*[[Large Hadron Collider]]<br />
*[[Fermilab]]<br />
*[[Tevatron]]<br />
*[[Sakurai Prize|J. J. Sakurai Prize for Theoretical Particle Physics]]<br />
*''[[The God Particle: If the Universe Is the Answer, What Is the Question?|The God Particle]]'', a popular science book on the Higgs boson, written by [[Leon M. Lederman]]<br />
}}<br />
<br />
==Notes==<br />
{{reflist|group=Note}}<br />
<br />
==References==<br />
{{reflist|35em}}<br />
<br />
==Further reading==<br />
{{refbegin}}<br />
*{{cite doi|10.1016/0031-9163(64)91136-9}}<br />
*{{cite doi|10.1103/PhysRevLett.13.321}}<br />
*{{cite doi|10.1103/PhysRevLett.13.508}}<br />
*{{cite doi|10.1103/PhysRevLett.13.585}}<br />
*{{cite doi|10.1103/PhysRev.145.1156}}<br />
*{{cite doi|10.1103/PhysRev.122.345}}<br />
*{{cite doi|10.1103/PhysRev.127.965}}<br />
*{{cite doi|10.1103/PhysRev.130.439}}<br />
*{{cite doi|10.1103/PhysRevLett.12.266}}<br />
*{{cite doi|10.1103/PhysRevLett.12.713}}<br />
*{{Cite journal | first=Gerald| last=Guralnik | title=The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles | journal=[[International Journal of Modern Physics A]] | year=2009 | volume=24 | issue=14 | pages=2601–2627 | arxiv=0907.3466| ref=harv | bibcode=2009IJMPA..24.2601G | doi=10.1142/S0217751X09045431 }}, {{cite arXiv |first=Gerald| last=Guralnik |title=The Beginnings of Spontaneous Symmetry Breaking in Particle Physics. Proceedings of the DPF-2011 Conference, Providence, RI, 8–13 August 2011 |year=2011|eprint=1110.2253v1 |class=physics.hist-ph}}, and Guralnik, Gerald (2013). [http://www.sps.ch/en/articles/milestones_in_physics/heretical_ideas_that_provided_the_cornerstone_for_the_standard_model_of_particle_physics_1/ "Heretical Ideas that Provided the Cornerstone for the Standard Model of Particle Physics".] SPG MITTEILUNGEN March 2013, No. 39, (p.&nbsp;14)<br />
*{{cite doi|10.1143/PTP.49.652}}<br />
*{{cite doi|10.1016/0550-3213(72)90279-9}}<br />
*{{cite book<br />
|author=G.S. Guralnik, C.R. Hagen, T.W.B. Kibble<br />
|chapter=Broken Symmetries and the Goldstone Theorem<br />
|chapterurl=http://www.datafilehost.com/download-7d512618.html<br />
|editor=R. L. Cool, R. E. Marshak<br />
|year=1968<br />
|title=Advances in Particle Physics<br />
|publisher=[[Interscience Publishers]]<br />
|volume=2 |issue= |pages=567–708<br />
|isbn=0-470-17057-3<br />
}}<br />
{{refend}}<br />
<br />
==External links==<br />
{{external links|date=July 2012}}<br />
{{refbegin}}<br />
*[http://prl.aps.org/50years/milestones#1964 Physical Review Letters - 50th Anniversary Milestone Papers]<br />
*[http://www.aps.org/units/dpf/awards/sakurai.cfm American Physical Society - J. J. Sakurai Prize Winners]<br />
*[http://www.youtube.com/watch?v=WLZ78gwWQI0 Gerry Guralnik speaks at Brown University about the 1964 PRL papers]<br />
*[http://cerncourier.com/cws/article/cern/32522 In CERN Courier, Steven Weinberg reflects on spontaneous symmetry breaking]<br />
*[http://www.youtube.com/watch?v=Zl4W3DYTIKw Steven Weinberg on LHC]<br />
*[http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism Englert-Brout-Higgs-Guralnik-Hagen-Kibble Mechanism on Scholarpedia]<br />
*[http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism_%28history%29 History of Englert-Brout-Higgs-Guralnik-Hagen-Kibble Mechanism on Scholarpedia]<br />
*[http://arxiv.org/abs/0907.3466 The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles]<br />
*[http://www.worldscinet.com/ijmpa/24/2414/S0217751X09045431.html International Journal of Modern Physics A: The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles]<br />
*[http://arxiv.org/abs/1107.4592 G.S. Guralnik (2011) "Gauge Invariance and the Goldstone Theorem - 1965 Feldafing talk". International Journal of Modern Physics A]<br />
*[http://lanl.arxiv.org/abs/hep-th/9802142 Spontaneous Symmetry Breaking in Gauge Theories: a Historical Survey]<br />
*[http://cerncourier.com/cws/article/cern/36683 CERN Courier Letter from GHK – December 2008]<br />
*[http://www.godparticle.com/ God Particle]<br />
*[http://www.youtube.com/view_play_list?p=BDA16F52CA3C9B1D 2010 Sakurai Prize Videos]<br />
*[http://www.youtube.com/view_play_list?p=FD7DA4107D13E7B1 Brown University Celebration of 2010 Sakurai Prize - Videos]<br />
*[http://apps3.aps.org/aps/meetings/april10/roser.pdf The Hunt for the Higgs at Tevatron]<br />
*[http://www.nature.com/news/2010/100804/full/news.2010.390.html Physicists get political over Higgs]<br />
*[http://www.iansample.com/site/?q=content/higgs-row-and-nobel-reform Ian Sample on Controversy and Nobel Reform]<br />
*[http://www.amazon.com/dp/0465019471 Massive by Ian Sample]<br />
*[http://www.math.columbia.edu/~woit/wordpress/?p=3266 Blog Not Even Wrong, Review of Massive by Ian Sample]<br />
*[http://www.math.columbia.edu/~woit/wordpress/?p=3282 Blog Not Even Wrong, Anderson-Higgs Mechanism]<br />
{{refend}}<br />
<br />
{{DEFAULTSORT:1964 Prl Symmetry Breaking Papers}}<br />
[[Category:1964 in science|PRL symmetry breaking papers]]<br />
[[Category:Physics papers]]<br />
[[Category:Works originally published in American magazines]]<br />
[[Category:1964 works]]<br />
[[Category:Works originally published in science and technology magazines]]<br />
[[Category:Standard Model]]</div>62.245.100.121https://en.formulasearchengine.com/index.php?title=Talk:Computation_tree_logic&diff=295105Talk:Computation tree logic2013-01-10T12:38:11Z<p>62.245.100.121: /* Examples */</p>
<hr />
<div>This article discusses a few of the issues referring to a foreigner's acquisition of residential property in Singapore. For Singapore residents, there are no restrictions imposed in acquiring landed residential property. Nonetheless, for these other than Singapore citizens, sure restrictions apply and approval is required from the authorities.<br><br>Find your dream Singapore / JB Iskandar property or Choice Funding Properties on the market in London, Australia/Melbourne, Philippines and Cambodia at Singapore's leading property portal for brand new condominium launches and new business/industrial property launches We've the latest and latest condominiums / [http://www.thebestfriendfinder.com/index.php?dll=profile&sub=blogview&item_id=5369&item2_id=5519 New Project Launch Singapore] condos for sale in Singapore , landed homes on the market in Singapore , in addition to new outlets for sale, new offices on the market, and new eating places on the market in Singapore. Find out Why Buy New Apartment Launches , or learn on for a number of the scorching properties within the Featured Listings beneath.<br><br>In an image publish titled "Feeling sorry for her? She stays in a bigger house than you", Fabrications In regards to the PAP (FAP), a Fb page known for its support of the ruling Individuals's Motion Occasion, revealed an image of the entrance of Ms Yap's home, together with the street title of the property. The publish incited many to accuse her of making a sham of her poverty, alluding that Ms Yap might have offered her house to help herself instead of asking for the return of the money in her retirement account. Others, however, agreed that her money must be returned to her. Why was it even relevant for Mr Hri Kumar to mention that Ms Yap stays in a landed property, and has it triggered pointless lynching behaviour by chosen members of the online neighborhood?<br><br>Jun 18 NEWLY CONSTRUCTED CORNER TERRACE ON THE MARKET $6800000 / 6br - (1 Sembawang Hills Property) real estate - by proprietor Jun 14 Woodlands HDB for sale $400000 / 1076ft² - (787D woodlands Cres) real estate - by broker Jun 14 New Executive & Pte Condos For Sale~ pic real estate - by broker Jun 10 Property Investment in the Philippines pic real estate - by dealer Jun 9 Instant Property Portfolio - Bulk Purchase 7 Flats & Earn $39,255 p.a. (Florida) pic actual property - by owner PropGO helps you find Hong Kong Properties for Sale and Hong Kong Properties for Hire by providing real property listings and Hong Kong houses from multiple companies, agents, and home owners. We also provide Hong Kong Neighborhood data including Hong Kong Property Transactions KF Property Community Pte Ltd<br><br>We specialise in providing services for all your Singapore property needs. Whether or not you are a local searching for new Singapore houses, a foreigner searching for Singapore accomodation, or an established company taking a look at Singapore actual property, we have now the answer for you! Our property & real estate consultants have huge expertise and will see to it that your needs are met. Contact knowledgeable accredited Singapore Property Agent ( Real Estate Salesperson ) now to safe your splendid Singapore or overseas properties at the moment. Contact an accredited Singapore Actual Property Professional ( Property Professional ) now to safe your supreme overseas and Singapore properties today. HUDC Flat for Sale in Singapore Singapore Luxurious Properties Long run outlook on property in Singapore<br><br>As a part of JTC's regular evaluate of posted charges, we will likely be updating our posted rents and costs with impact from 1 July 2014. In keeping with the market trend and retaining enterprise prices competitive for industrialists, there will likely be a basic decrease in the posted rents and prices of most of our industrial land, while that of ready-constructed amenities will likely be stored at their present ranges. Total, the rents and prices for JTC industrial land and prepared-built services remain aggressive in contrast with the prevailing market rates. We will continue to observe the market closely and stay attentive to market tendencies.</div>62.245.100.121