formulasearchengine - User contributions [en]

Newman–Penrose formalism

2014-09-16T17:27:53Z

131.111.16.20: /* NP field equations */ Factor of lambda missing (the RHS is quadratic in the spin coefficients)

The title of the author is Jayson. Invoicing is what I do. For years he's been residing in Alaska and he doesn't strategy on altering it. The preferred hobby for him and his kids is to perform lacross and he'll be starting something else along with it. My web blog; psychic readings online ([http://chorokdeul.Co.kr/index.php?document_srl=324263&mid=customer21 My Home Page])

Dirichlet density

2013-12-06T01:17:04Z

131.111.16.20: /* Definition */

{{Use dmy dates|date=August 2013}}
{{Unreferenced stub|auto=yes|date=December 2009}}
'''Charge-carrier density''' denotes the number of [[charge carrier]]s per [[volume]]. It is measured in m−3. As any [[density]] it can depend on position.
It should not be confused with the [[charge density]], which is the number of charges per volume at a given energy.

The carrier density is obtained by [[integral|integrating]] the charge density over the energy that the charges are allowed to have.

Charge-carrier density is a [[particle density]], so [[integral|integrating]] it over a volume <math>V</math> gives the number of charge carriers <math>N</math> in that volume

<math>N=\int_V n(\mathbf r) \,\mathrm{d}V</math>.

where
:<math>n(\mathbf r)</math> is the position-dependent charge-carrier density.

If the density does not depend on position and is instead equal to a constant <math>n_0</math> this equation simplifies to

<math>N=V\cdot n_0</math>.

Charge-carrier densities involve equations concerning the [[electrical conductivity]] and related phenomena like the [[thermal conductivity]].

==Measurement==

The density of charge carriers can be determined in many cases using the [[Hall effect]], the voltage of which depends inversely on the density.

Such measurements show that the density for silver is around 5.8·1028 m−3 or around 1.2 per atom.

{{DEFAULTSORT:Charge Carrier Density}}
[[Category:Density]]
[[Category:Charge carriers]]

{{Phys-stub}}

Wick rotation

2013-11-26T16:50:43Z

131.111.16.20: /* Overview */ changed sign of Wick rotation. It is t->-i\tau

{{technical|date=May 2013}}
{{DISPLAYTITLE:E6 (mathematics)}}
{{Group theory sidebar |Topological}}
{{Lie groups |Simple}}

In [[mathematics]], '''E6''' is the name of some closely related [[Lie group]]s, linear [[algebraic group]]s or their [[Lie algebra]]s <math>\mathfrak{e}_6</math>, all of which have dimension 78; the same notation E6 is used for the corresponding [[root lattice]], which has [[Rank of a Lie group|rank]] 6. The designation E6 comes from the Cartan–Killing classification of the complex [[simple Lie algebra]]s, which fall into four infinite series labeled A''n'', B''n'', C''n'', D''n'', and [[Exceptional simple Lie group|five exceptional cases]] labeled E6, [[E7 (mathematics)|E7]], [[E8 (mathematics)|E8]], [[F4 (mathematics)|F4]], and [[G2 (mathematics)|G2]]. The E6 algebra is thus one of the five exceptional cases.

The fundamental group of the complex form, compact real form, or any algebraic version of E6 is the [[cyclic group]] '''Z'''/3'''Z''', and its [[outer automorphism group]] is the cyclic group '''Z'''/2'''Z'''. Its [[fundamental representation]] is 27-dimensional (complex), and a basis is given by the [[27 lines on a cubic surface]]. The [[dual representation]], which is inequivalent, is also 27-dimensional.

In [[particle physics]], E6 plays a role in some [[grand unified theory|grand unified theories]].

==Real and complex forms==
There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of [[complex dimension]] 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group '''Z'''/3'''Z''', has maximal [[Compact space|compact]] subgroup the compact form (see below) of E6, and has an outer automorphism group non-cyclic of order 4 generated by complex conjugation and by the outer automorphism which already exists as a complex automorphism.

As well as the complex Lie group of type E6, there are five real forms of the Lie algebra, and correspondingly five real forms of the group with trivial center (all of which have an algebraic double cover, and three of which have further non-algebraic covers, giving further real forms), all of real dimension 78, as follows:

*The compact form (which is usually the one meant if no other information is given), which has fundamental group '''Z'''/3'''Z''' and outer automorphism group '''Z'''/2'''Z'''.
*The split form, EI (or E6(6)), which has maximal compact subgroup Sp(4)/(±1), fundamental group of order 2 and outer automorphism group of order 2.
*The quasi-split form EII (or E6(2)), which has maximal compact subgroup SU(2) × SU(6)/(center), fundamental group cyclic of order 6 and outer automorphism group of order 2.
*EIII (or E6(-14)), which has maximal compact subgroup SO(2) × Spin(10)/(center), fundamental group '''Z''' and trivial outer automorphism group.
*EIV (or E6(-26)), which has maximal compact subgroup F4, trivial fundamental group cyclic and outer automorphism group of order 2.

The EIV form of E6 is the group of collineations (line-preserving transformations) of the [[octonionic projective plane]] '''OP'''2.<ref>{{Citation | last1=Rosenfeld | first1=Boris | title=Geometry of Lie Groups | year=1997 }} (theorem 7.4 on page 335, and following paragraph).</ref> It is also the group of determinant-preserving linear transformations of the exceptional [[Jordan algebra]]. The exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E6 has a 27-dimensional complex representation. The compact real form of E6 is the [[isometry group]] of a 32-dimensional [[Riemannian manifold]] known as the 'bioctonionic projective plane'; similar constructions for E7 and E8 are known as the [[Rosenfeld projective plane]]s, and are part of the [[Freudenthal magic square]].

==E6 as an algebraic group==
By means of a [[Chevalley basis]] for the Lie algebra, one can define E6 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as “untwisted”) adjoint form of E6. Over an algebraically closed field, this and its triple cover are the only forms; however, over other fields, there are often many other forms, or “twists” of E6, which are classified in the general framework of [[Galois cohomology]] (over a [[perfect field]] ''k'') by the set ''H''1(''k'', Aut(E6)) which, because the Dynkin diagram of E6 (see [[#Dynkin diagram|below]]) has automorphism group '''Z'''/2'''Z''', maps to ''H''1(''k'', '''Z'''/2'''Z''') = Hom (Gal(''k''), '''Z'''/2'''Z''') with kernel ''H''1(''k'', E6,ad).<ref>{{cite book | last1=Платонов | first1=Владимир П. | last2=Рапинчук | first2=Андрей С. | title=Алгебраические группы и теория чисел | year=1991 | publisher=Наука | isbn=5-02-014191-7 }} (English translation: {{cite book | last1=Platonov | first1=Vladimir P. | last2=Rapinchuk | first2=Andrei S. | title=Algebraic groups and number theory | year=1994 | publisher=Academic Press | isbn=0-12-558180-7 }}), §2.2.4</ref>

Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E6 coincide with the three real Lie groups mentioned [[#Real and complex forms|above]], but with a subtlety concerning the fundamental group: all adjoint forms of E6 have fundamental group '''Z'''/3'''Z''' in the sense of algebraic geometry, with Galois action as on the third roots of unity; this means that they admit exactly one triple cover (which may be trivial on the real points); the further non-compact real Lie group forms of E6 are therefore not algebraic and admit no faithful finite-dimensional representations. The compact real form of E6 as well as the noncompact forms EI=E6(6) and EIV=E6(-26) are said to be ''inner'' or of type 1E6 meaning that their class lies in ''H''1(''k'', E6,ad) or that complex conjugation induces the trivial automorphism on the Dynkin diagram, whereas the other two real forms are said to be ''outer'' or of type 2E6.

Over finite fields, the [[Lang–Steinberg theorem]] implies that ''H''1(''k'', E6) = 0, meaning that E6 has exactly one twisted form, known as 2E6: see [[#Chevalley and Steinberg groups of type E6 and 2E6|below]].

== Algebra ==

===Dynkin diagram===
The [[Dynkin diagram]] for E6 is given by [[Image:Dynkin diagram E6.png|120px|Dynkin diagram of E 6]], {{Dynkin2|node|3|node|3|branch|3|node|3|node}} or sometimes positioned like this {{Dynkin|nodes|3s|nodes|loop2|3|node}}.

=== Roots of E6 ===
[[File:Gosset 1 22 polytope.svg|thumb|300px|The 72 vertices of the [[1 22 polytope|122]] polytope represent the root vectors of the E6, as shown in this [[Coxeter plane]] projection.[[Coxeter-Dynkin diagram]]: {{CDD|node_1|3|node|split1|nodes|3ab|nodes}}]]

Although they [[Linear span|span]] a six-dimensional space, it is much more symmetrical to consider them as [[Vector space|vectors]] in a six-dimensional subspace of a nine-dimensional space.
:(1,−1,0;0,0,0;0,0,0), (−1,1,0;0,0,0;0,0,0),
:(−1,0,1;0,0,0;0,0,0), (1,0,−1;0,0,0;0,0,0),
:(0,1,−1;0,0,0;0,0,0), (0,−1,1;0,0,0;0,0,0),
:(0,0,0;1,−1,0;0,0,0), (0,0,0;−1,1,0;0,0,0),
:(0,0,0;−1,0,1;0,0,0), (0,0,0;1,0,−1;0,0,0),
:(0,0,0;0,1,−1;0,0,0), (0,0,0;0,−1,1;0,0,0),
:(0,0,0;0,0,0;1,−1,0), (0,0,0;0,0,0;−1,1,0),
:(0,0,0;0,0,0;−1,0,1), (0,0,0;0,0,0;1,0,−1),
:(0,0,0;0,0,0;0,1,−1), (0,0,0;0,0,0;0,−1,1),

All 27 combinations of <math>(\bold{3};\bold{3};\bold{3})</math> where <math>\bold{3}</math> is one of <math>\left(\frac{2}{3}, -\frac{1}{3}, -\frac{1}{3}\right),\ \left( -\frac{1}{3}, \frac{2}{3}, -\frac{1}{3}\right),\ \left( -\frac{1}{3}, -\frac{1}{3}, \frac{2}{3} \right)</math>

All 27 combinations of <math>(\bar{\bold{3}};\bar{\bold{3}};\bar{\bold{3}})</math> where <math>\bar{\bold{3}}</math> is one of <math>\left(-\frac{2}{3},\frac{1}{3},\frac{1}{3}\right),\ \left(\frac{1}{3}, -\frac{2}{3}, \frac{1}{3} \right),\ \left( \frac{1}{3}, \frac{1}{3}, -\frac{2}{3} \right)</math>

'''Simple roots'''

:(0,0,0;0,0,0;0,1,−1)

:(0,0,0;0,0,0;1,−1,0)

:(0,0,0;0,1,−1;0,0,0)

:(0,0,0;1,−1,0;0,0,0)

:(0,1,−1;0,0,0;0,0,0)

:<math>\left(\frac{1}{3},-\frac{2}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3}\right)</math>

[[Image:E6Coxeter.svg|thumb|300px|Graph of E6 as a subgroup of E8 projected into the Coxeter plane]]
[[File:E6HassePoset.svg|thumb|300px|[[Hasse diagram]] of E6 [[Root system#The root poset|root poset]] with edge labels identifying added simple root position]]

==== An alternative description ====
An alternative (6-dimensional) description of the root system, which is useful in considering E6 × SU(3) as a [[E8 (mathematics)#Subgroups|subgroup of]] E8, is the following:

All <math>4\times\begin{pmatrix}5\\2\end{pmatrix}</math> permutations of
:<math>(\pm1,\pm1,0,0,0,0)</math> preserving the zero at the last entry,

and all of the following roots with an odd number of plus signs

:<math>\left(\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{\sqrt{3}\over 2}\right).</math>

Thus the 78 generators consist of the following subalgebras:
: A 45-dimensional SO(10) subalgebra, including the above <math>4\times\begin{pmatrix}5\\2\end{pmatrix}</math> generators plus the five [[Cartan subalgebra|Cartan generator]]s corresponding to the first five entries.
: Two 16-dimensional subalgebras that transform as a [[Weyl spinor]] of <math>\operatorname{spin}(10)</math> and its complex conjugate. These have a non-zero last entry.
: 1 generator which is their chirality generator, and is the sixth [[Cartan subalgebra|Cartan generator]].

One choice of [[Simple root (root system)|simple root]]s for E6, index as [[File:DynkinE6.svg|120px]], is given by the rows of the following matrix:

:<math>\left [\begin{smallmatrix}
1&-1&0&0&0&0 \\
0&1&-1&0&0&0 \\
0&0&1&-1&0&0 \\
0&0&0&1&1&0 \\
-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{\sqrt{3}}{2}\\
0&0&0&1&-1&0 \\
\end{smallmatrix}\right ]</math>

we have ordered them so that their corresponding nodes in the [[Dynkin diagram]] are ordered from left to right (in the diagram depicted above) with the side node last.

=== Weyl group ===
The [[Weyl group]] of E6 is of order 51840: it is the [[automorphism]] group of the unique [[simple group]] of order 25920 (which can be described as any of: PSU4(2), PSΩ6−(2), PSp4(3) or PSΩ5(3)).<ref>{{cite book |last1=Conway |first1=John Horton |authorlink1=John Horton Conway |last2=Curtis |first2=Robert Turner |last3=Norton |first3=Simon Phillips |authorlink3=Simon P. Norton |last4=Parker |first4=Richard A |authorlink4=Richard A. Parker |last5=Wilson |first5=Robert Arnott |authorlink5=Robert Arnott Wilson |title=[[ATLAS of Finite Groups|Atlas of Finite Groups]]: Maximal Subgroups and Ordinary Characters for Simple Groups |year=1985 |month= |publisher=Oxford University Press |isbn=0-19-853199-0 |page=26 }}</ref>

=== Cartan matrix ===
:<math>\left [\begin{smallmatrix}
2&-1&0&0&0&0\\
-1&2&-1&0&0&0\\
0&-1&2&-1&0&-1\\
0&0&-1&2&-1&0\\
0&0&0&-1&2&0\\
0&0&-1&0&0&2
\end{smallmatrix}\right ]</math>

==Important subalgebras and representations==
The Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU(3) × SU(3) × SU(3) subalgebra. Other maximal subalgebras which have an importance in physics (see below) and can be read off the Dynkin diagram, are the algebras of SO(10) × U(1) and SU(6) × SU(2).

In addition to the 78-dimensional adjoint representation, there are two dual [[E8 (mathematics)#Subgroups|27-dimensional "vector" representation]]s.

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the [[Weyl character formula]]. The dimensions of the smallest irreducible representations are {{OEIS|id=A121737}}:

:1, 27 (twice), 78, 351 (four times), 650, 1728 (twice), 2430, 2925, 3003 (twice), 5824 (twice), 7371 (twice), 7722 (twice), 17550 (twice), 19305 (four times), 34398 (twice), 34749, 43758, 46332 (twice), 51975 (twice), 54054 (twice), 61425 (twice), 70070, 78975 (twice), 85293, 100386 (twice), 105600, 112320 (twice), 146432 (twice), 252252 (twice), 314496 (twice), 359424 (four times), 371800 (twice), 386100 (twice), 393822 (twice), 412776 (twice), 442442 (twice)…

The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by the adjoint form of E6 (equivalently, those whose weights belong to the root lattice of E6), whereas the full sequence gives the dimensions of the irreducible representations of the simply connected form of E6.

The symmetry of the Dynkin diagram of E6 explains why many dimensions occur twice, the corresponding representations being related by the non-trivial outer automorphism; however, there are sometimes even more representations than this, such as four of dimension 351, two of which are fundamental and two of which are not.

The [[fundamental representation]]s have dimensions 27, 351, 2925, 351, 27 and 78 (corresponding to the seven nodes in the [[#Dynkin diagram|Dynkin diagram]] in the order chosen for the [[#Cartan matrix|Cartan matrix]] above, i.e., the nodes are read in the five-node chain first, with the last node being connected to the middle one).

==E6 polytope==
The '''[[E6 polytope|E6 polytope]]''' is the [[convex hull]] of the roots of E6. It therefore exists in 6 dimensions; its [[symmetry group]] contains the [[Coxeter group]] for E6 as an [[Index of a subgroup|index]] 2 subgroup.

==Chevalley and Steinberg groups of type E6 and 2E6==

{{main|²E₆}}

The groups of type ''E''6 over arbitrary fields (in particular finite fields) were introduced by {{harvs|txt|last=Dickson|year1=1901|year2=1908}}.

The points over a [[finite field]] with ''q'' elements of the (split) algebraic group E6 (see [[#E6 as an algebraic group|above]]), whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a finite [[Group of Lie type|Chevalley group]]. This is closely connected to the group written E6(''q''), however there is ambiguity in this notation, which can stand for several things:
* the finite group consisting of the points over '''F'''''q'' of the simply connected form of E6 (for clarity, this can be written E6,sc(''q'') or more rarely <math>\tilde E_6(q)</math> and is known as the "universal" Chevalley group of type E6 over '''F'''''q''),
* (rarely) the finite group consisting of the points over '''F'''''q'' of the adjoint form of E6 (for clarity, this can be written E6,ad(''q''), and is known as the "adjoint" Chevalley group of type E6 over '''F'''''q''), or
* the finite group which is the image of the natural map from the former to the latter: this is what will be denoted by E6(''q'') in the following, as is most common in texts dealing with finite groups.

From the finite group perspective, the relation between these three groups, which is quite analogous to that between SL(''n,q''), PGL(''n,q'') and PSL(''n,q''), can be summarized as follows: E6(''q'') is simple for any ''q'', E6,sc(''q'') is its [[Schur multiplier|Schur cover]], and E6,ad(''q'') lies in its automorphism group; furthermore, when ''q''−1 is not divisible by 3, all three coincide, and otherwise (when ''q'' is congruent to 1 mod 3), the Schur multiplier of E6(''q'') is 3 and E6(''q'') is of index 3 in E6,ad(''q''), which explains why E6,sc(''q'') and E6,ad(''q'') are often written as 3·E6(''q'') and E6(''q'')·3. From the algebraic group perspective, it is less common for E6(''q'') to refer to the finite simple group, because the latter is not in a natural way the set of points of an algebraic group over '''F'''''q'' unlike E6,sc(''q'') and E6,ad(''q'').

Beyond this “split” (or “untwisted”) form of E6, there is also one other form of E6 over the finite field '''F'''''q'', known as 2E6, which is obtained by twisting by the non-trivial automorphism of the Dynkin diagram of E6. Concretely, 2E6(''q''), which is known as a Steinberg group, can be seen as the subgroup of E6(''q''2) fixed by the composition of the non-trivial diagram automorphism and the non-trivial field automorphism of '''F'''''q''2. Twisting does not change the fact that the algebraic fundamental group of 2E6,ad is '''Z'''/3'''Z''', but it does change those ''q'' for which the covering of 2E6,ad by 2E6,sc is non-trivial on the '''F'''''q''-points. Precisely: 2E6,sc(''q'') is a covering of 2E6(''q''), and 2E6,ad(''q'') lies in its automorphism group; when ''q''+1 is not divisible by 3, all three coincide, and otherwise (when ''q'' is congruent to 2 mod 3), the degree of 2E6,sc(''q'') over 2E6(''q'') is 3 and 2E6(''q'') is of index 3 in 2E6,ad(''q''), which explains why 2E6,sc(''q'') and 2E6,ad(''q'') are often written as 3·2E6(''q'') and 2E6(''q'')·3.

Two notational issues should be raised concerning the groups 2E6(''q''). One is that this is sometimes written 2E6(''q''2), a notation which has the advantage of transposing more easily to the Suzuki and Ree groups, but the disadvantage of deviating from the notation for the '''F'''''q''-points of an algebraic group. Another is that whereas 2E6,sc(''q'') and 2E6,ad(''q'') are the '''F'''''q''-points of an algebraic group, the group in question also depends on ''q'' (e.g., the points over '''F'''''q''2 of the same group are the untwisted E6,sc(''q''2) and E6,ad(''q''2)).

The groups E6(''q'') and 2E6(''q'') are simple for any ''q'',<ref>{{cite book | first=Roger W. | last=Carter | title=Simple Groups of Lie Type | authorlink=Roger Carter (mathematician) | publisher=John Wiley & Sons | series=Wiley Classics Library | isbn=0-471-50683-4 | year=1989 }}</ref><ref>{{cite book | first=Robert A. | last=Wilson | title=The Finite Simple Groups | authorlink=Robert Arnott Wilson | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics]] | volume=251 | isbn=1-84800-987-9 | year=2009 }}</ref> and constitute two of the infinite families in the [[classification of finite simple groups]]. Their order is given by the following formula {{OEIS|id=A008872}}:

:<math>|E_6 (q)| = \frac{1}{\mathrm{gcd}(3,q-1)}q^{36}(q^{12}-1)(q^9-1)(q^8-1)(q^6-1)(q^5-1)(q^2-1)</math>
:<math>|{}^2\!E_6 (q)| = \frac{1}{\mathrm{gcd}(3,q+1)}q^{36}(q^{12}-1)(q^9+1)(q^8-1)(q^6-1)(q^5+1)(q^2-1)</math>

{{OEIS|id=A008916}}. The order of E6,sc(''q'') or E6,ad(''q'') (both are equal) can be obtained by removing the dividing factor gcd(3,''q''−1) from the first formula {{OEIS|id=A008871}}, and the order of 2E6,sc(''q'') or 2E6,ad(''q'') (both are equal) can be obtained by removing the dividing factor gcd(3,''q''+1) from the second {{OEIS|id=A008915}}.

The Schur multiplier of E6(''q'') is always gcd(3,''q''−1) (i.e., E6,sc(''q'') is its Schur cover). The Schur multiplier of 2E6(''q'') is gcd(3,''q''+1) (i.e., 2E6,sc(''q'') is its Schur cover) outside of the exceptional case ''q''=2 where it is 22·3 (i.e., there is an additional 22-fold cover). The outer automorphism group of E6(''q'') is the product of the diagonal automorphism group '''Z'''/gcd(3,''q''−1)'''Z''' (given by the action of E6,ad(''q'')), the group '''Z'''/2'''Z''' of diagram automorphisms, and the group of field automorphisms (i.e., cyclic of order ''f'' if ''q''=''pf'' where ''p'' is prime). The outer automorphism group of 2E6(''q'') is the product of the diagonal automorphism group '''Z'''/gcd(3,''q''+1)'''Z''' (given by the action of 2E6,ad(''q'')) and the group of field automorphisms (i.e., cyclic of order ''f'' if ''q''=''p''''f'' where ''p'' is prime).

==Importance in physics==

[[File:E6GUT.svg|300px|right|thumb|The pattern of [[weak isospin]], W, weaker isospin, W', strong g3 and g8, and baryon minus lepton, B, charges for particles in the [[SO(10)]] [[Grand Unified Theory]], rotated to show the embedding in E6.]]

N=8 [[supergravity]] in five dimensions, which is a [[dimensional reduction]] from 11 dimensional supergravity, admits an E6 bosonic global symmetry and an Sp(8) bosonic [[gauge symmetry|local symmetry]]. The fermions are in representations of Sp(8), the gauge fields are in a representation of E6, and the scalars are in a representation of both (Gravitons are [[singlet]]s with respect to both). Physical states are in representations of the coset E6/Sp(8).

In [[Grand unification theory|grand unification theories]], E6 appears as a possible gauge group which, after its [[symmetry breaking|breaking]], gives rise to the SU(3) × SU(2) × U(1) [[gauge group]] of the [[standard model]] (also see [[E8 (mathematics)#Importance in physics|Importance in physics of E8]]). One way of achieving this is through breaking to SO(10) × U(1). The adjoint 78 representation breaks, as explained above, into an adjoint 45, spinor 16 and <math>\bar{16}</math> as well as a singlet of the SO(10) subalgebra. Including the U(1) charge we have

:<math>78 \rightarrow 45_0 \oplus 16_{-3} \oplus \bar{16}_3 + 1_0. </math>

Where the subscript denotes the U(1) charge.

==See also==
*[[En (Lie algebra)]]
*[[ADE classification]]
*[[Freudenthal magic square]]

==References==
*{{Citation | last1=Adams | first1=J. Frank | title=Lectures on exceptional Lie groups | url=http://books.google.com/books?isbn=0226005275 | publisher=[[University of Chicago Press]] | series=Chicago Lectures in Mathematics | isbn=978-0-226-00526-3 | mr=1428422 | year=1996}}
* {{cite journal|last=Baez|first=John|authorlink=John Baez|year=2002|title=The Octonions, Section 4.4: E6|url=http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html|journal=Bull. Amer. Math. Soc.|issn=0273-0979|volume=39|issue=2|pages=145–205|doi=10.1090/S0273-0979-01-00934-X}}. Online HTML version at [http://math.ucr.edu/home/baez/octonions/node17.html].
* {{cite journal|last=Cremmer|first=E.|coauthors=J. Scherk and J. H. Schwarz|year=1979|title=Spontaneously Broken N=8 Supergravity|url=|journal=Phys. Lett. B|volume=84|issue=1|pages=83–86|doi=10.1016/0370-2693(79)90654-3}}. Online scanned version at [http://ccdb4fs.kek.jp/cgi-bin/img_index?7904075].
*{{Citation | last1=Dickson | first1=Leonard Eugene | author1-link=Leonard Eugene Dickson | title=A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface | url=http://books.google.com/books?id=I_SWAAAAMAAJ&pg=PA145 | id=Reprinted in volume 5 of his collected works | year=1901 | journal=The quarterly journal of pure and applied mathematics | volume=33 | pages=145–173}}
*{{Citation | last1=Dickson | first1=Leonard Eugene | author1-link=Leonard Eugene Dickson | title=A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface (second paper) | url=http://books.google.com/books?id=16J7bgQU65oC&pg=PA145 | id=Reprinted in volume VI of his collected works | year=1908 | journal=The quarterly journal of pure and applied mathematics | volume=39 | pages=205–209}}
{{Reflist}}

{{Exceptional_Lie_groups}}

{{DEFAULTSORT:E6 (Mathematics)}}
[[Category:Algebraic groups]]
[[Category:Lie groups]]

Taub–NUT space

2013-10-28T11:01:07Z

131.111.16.20: It's not homogeneous.

'''Hilbert C*-modules''' are [[mathematical object]]s which generalise the notion of a [[Hilbert space]] (which itself is a generalisation of [[Euclidean space]]), in that they endow a [[vector space|linear space]] with an "[[inner product]]" which takes values in a [[C*-algebra]]. Hilbert C*-modules were first introduced in the work of [[Irving Kaplansky]] in [[1953 in science|1953]], which developed the theory for [[commutative]], [[unital algebra]]s (though Kaplansky observed that the assumption of a unit element was not "vital").<ref>{{cite journal| last = Kaplansky| first = I.| authorlink = Irving Kaplansky| title = Modules over operator algebras| journal = [[American Journal of Mathematics]]| volume = 75| issue = 4| pages = 839–853| year = 1953| doi = 10.2307/2372552| jstor = 2372552}}</ref> In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke<ref>{{cite journal| last = Paschke| first = W. L.| title = Inner product modules over B*-algebras| journal = [[Transactions of the American Mathematical Society]]| volume = 182| pages = 443–468| year = 1973| doi = 10.2307/1996542| jstor = 1996542}}</ref> and Marc Aristide Rieffel, the latter in a paper which used Hilbert C*-modules to construct a theory of [[induced representation]]s of C*-algebras.<ref>{{cite journal| last = Rieffel| first = M. A.| title = Induced representations of C*-algebras| journal = Advances in Mathematics| volume = 13| pages = 176–257| publisher = [[Elsevier]]| year = 1974| doi = 10.1016/0001-8708(74)90068-1| issue = 2}}</ref> Hilbert C*-modules are crucial to Kasparov's formulation of [[KK-theory]],<ref>{{cite journal| last = Kasparov| first = G. G.| title = Hilbert C*-modules: Theorems of Stinespring and Voiculescu| journal = Journal of Operator Theory| volume = 4| pages = 133–150| publisher = Theta Foundation| year = 1980}}</ref> and provide the right framework to extend the notion of [[Morita equivalence]] to [[C*-algebras]].<ref>{{cite journal| last = Rieffel| first = M. A.| title = Morita equivalence for operator algebras| journal = Proceedings of Symposia in Pure Mathematics| volume = 38| pages = 176–257| publisher = American Mathematical Society| year = 1982}}</ref> They can be viewed as the generalization of [[vector bundles]] to noncommutative [[C*-algebras]] and as such play an important role in [[noncommutative geometry]], notably in [[C*-algebraic quantum group theory]],<ref>{{cite journal| last = Baaj| first = S.| coauthors = Skandalis, G.| title = Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres| journal = [[Annales Scientifiques de l'École Normale Supérieure]]| volume = 26| issue = 4| pages = 425–488| year = 1993}}</ref><ref>{{cite journal| last = Woronowicz| first = S. L.| authorlink = S. L. Woronowicz| title = Unbounded elements affiliated with C*-algebras and non-compact quantum groups| journal = Communications in Mathematical Physics| volume = 136| pages = 399–432| year = 1991| doi = 10.1007/BF02100032|bibcode = 1991CMaPh.136..399W| issue = 2 }}</ref> and [[groupoid]] [[C*-algebras]].

== Definitions ==

=== Inner-product ''A''-modules ===
Let ''A'' be a C*-algebra (not assumed to be commutative or unital), its [[Involution (mathematics)|involution]] denoted by *. An '''inner-product ''A''-module''' (or '''pre-Hilbert ''A''-module''') is a [[complex number|complex]] linear space ''E'' which is equipped with a compatible right [[Module (mathematics)|''A''-module]] structure, together with a map
:<math> \langle \cdot, \cdot \rangle : E \times E \rightarrow A </math>
which satisfies the following properties:

*For all ''x'', ''y'', ''z'' in ''E'', and α, β in '''C''':

::<math> \langle x, \alpha y + \beta z \rangle = \alpha \langle x, y \rangle + \beta \langle x, z \rangle</math>

:(''i.e.'' the inner product is linear in its second argument).

*For all ''x'', ''y'' in ''E'', and ''a'' in ''A'':

::<math> \langle x, ya \rangle = \langle x, y \rangle a</math>

*For all ''x'', ''y'' in ''E'':

::<math> \langle x, y \rangle = \langle y, x \rangle^*,</math>

:from which it follows that the inner product is [[conjugate linear]] in its first argument (''i.e.'' it is a [[sesquilinear form]]).

*For all ''x'' in ''E'':

::<math> \langle x, x \rangle \geq 0</math>

:and

::<math> \langle x, x \rangle = 0 \iff x = 0.</math>

:(An element of a C*-algebra ''A'' is said to be ''positive'' if it is [[self-adjoint]] with non-negative [[Spectrum (functional analysis)|spectrum]].)<ref>{{cite book| last = Arveson| first = William| authorlink = William Arveson|title = An Invitation to C*-Algebras| publisher = Springer-Verlag| year = 1976| page = 35}}</ref><ref>In the case when ''A'' is non-unital, the spectrum of an element is calculated in the C*-algebra generated by adjoining a unit to ''A''.</ref>

=== Hilbert ''A''-modules ===
An analogue to the [[Cauchy-Schwarz inequality]] holds for an inner-product ''A''-module ''E'':<ref>This result in fact holds for semi-inner-product ''A''-modules, which may have non-zero elements ''x'' such that <''x'',''x''> = 0, as the proof does not rely on the [[nondegeneracy]] property.</ref>

:<math>\langle x, y \rangle \langle y, x \rangle \leq \Vert \langle x, x \rangle \Vert \langle y, y \rangle</math>

for ''x'', ''y'' in ''E''.

On the pre-Hilbert module ''E'', define a norm by

:<math>\Vert x \Vert = \Vert \langle x, x \rangle \Vert^\frac{1}{2}.</math>

The norm-completion of ''E'', still denoted by ''E'', is said to be a '''Hilbert ''A''-module''' or a '''Hilbert C*-module over the C*-algebra ''A'''''.
The Cauchy-Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.

The action of ''A'' on ''E'' is continuous: for all ''x'' in ''E''

:<math>a_{\lambda} \rightarrow a \Rightarrow xa_{\lambda} \rightarrow xa.</math>

Similarly, if {''eλ''} is an [[Approximate identity|approximate unit]] for ''A'' (a [[Net (mathematics)|net]] of self-adjoint elements of ''A'' for which ''ae''λ and ''e''λ''a'' tend to ''a'' for each ''a'' in ''A''), then for ''x'' in ''E''

:<math> xe_\lambda \rightarrow x</math>

whence it follows that ''EA'' is [[Dense set|dense]] in ''E'', and ''x''1 = ''x'' when ''A'' is unital.

Let

:<math> \langle E, E \rangle = \operatorname{span} \{ \langle x, y \rangle | x, y \in E \},</math>

then the [[Closure (topology)|closure]] of <''E'',''E''> is a two-sided ideal in ''A''. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that ''E''<''E'',''E''> is dense in ''E''. In the case when <''E'',''E''> is dense in ''A'', ''E'' is said to be '''full'''. This does not generally hold.

== Examples ==

=== Hilbert spaces ===
A complex Hilbert space ''H'' is a Hilbert '''C'''-module under its inner product, the complex numbers being a C*-algebra with an involution given by [[complex conjugation]].

===Vector bundles===
If ''X'' is a [[locally compact Hausdorff space]] and ''E'' a [[vector bundle]] over ''X'' with a [[Riemannian metric]] ''g'', then the space of continuous sections of ''E'' is a Hilbert ''C(X)''-module. The inner product is given by
::<math> \langle f,h\rangle (x):=g(f(x),h(x)).</math>

The converse holds as well: Every countably generated Hilbert C*-module over a commutative C*-algebra ''A = C(X)'' is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over ''X''.

=== C*-algebras ===
Any C*-algebra ''A'' is a Hilbert ''A''-module under the inner product <''a'',''b''> = ''a''*''b''. By the C*-identity, the Hilbert module norm coincides with C*-norm on ''A''.

The (algebraic) [[direct sum of modules|direct sum]] of ''n'' copies of ''A''

:<math> A^n = \oplus_1^n A</math>

can be made into a Hilbert ''A''-module by defining

:<math>\langle (a_i), (b_i) \rangle = \sum a_i^* b_i.</math>

One may also consider the following elements in the countable direct product of ''A''

:<math>\mathcal{H}_A = \{ (a_i) | \sum a_i^{*}a_i\text{ converges in }A \}.</math>

Given an inner product analogous to that on ''An'', the resulting Hilbert ''A''-module is called the '''standard Hilbert module'''.

== See also ==
* [[Operator algebra]]

== Notes ==
{{Reflist}}

== References ==
* {{cite book |last=Lance |first=E. Christopher|title=Hilbert C*-modules: A toolkit for operator algebraists |series=London Mathematical Society Lecture Note Series|year=1995 |publisher=Cambridge University Press |location=Cambridge, England}}

== External links ==
* {{MathWorld |title=Hilbert C*-Module |urlname=HilbertC-Star-Module}}
* [http://www.imn.htwk-leipzig.de/~mfrank/hilmod.html Hilbert C*-Modules Home Page], a literature list

[[Category:C*-algebras]]
[[Category:Operator theory]]
[[Category:Theoretical physics]]

Inertial number

2013-08-20T13:48:16Z

131.111.16.74:

The '''Maxwell-Bloch equations''', also called the '''optical Bloch equations''', describe the dynamics of a [[two-state quantum system]] interacting with the electromagnetic mode of an optical resonator. They are analogous to (but not at all equivalent to) the [[Bloch equation]]s which describe the motion of the nuclear magnetic moment in an electromagnetic field. The equations can be derived either semiclassically or with the field fully quantized when certain approximations are made.

==Semi-classical formulation==
The derivation of the semi-classical optical Bloch equations is nearly identical to solving the [[two-state quantum system]] (see the discussion there). However, usually one casts these equations into a density matrix form. The system we are dealing with can be described by the wave function:
:<math> \psi = c_g\psi_g + c_e\psi_e </math>
:<math> \left|c_g\right|^2 + \left|c_e\right|^2 = 1 </math>

The [[density matrix]] is
:<math> \rho = \begin{bmatrix}\rho_{ee} & \rho_{eg} \\ \rho_{ge} & \rho_{gg}\end{bmatrix} = \begin{bmatrix}c_e c_{e}^* & c_e c_{g}^* \\ c_g c_{e}^* & c_g c_{g}^* \end{bmatrix}</math>

(other conventions are possible; this follows the derivation in Metcalf (1999)).<ref name="metcalf">Metcalf, Harold. ''Laser Cooling and Trapping'' Springer 1999 pg. 24-</ref> One can now solve the Heisenberg equation of motion, or translate the results from solving the Schrödinger equation into density matrix form. One arrives at the following equations, including spontaneous emission:
:<math>\frac{d \rho_{gg}}{dt} = \gamma \rho_{ee} + \frac{i}{2}(\Omega^* \bar \rho_{eg} - \Omega\bar \rho_{ge})</math>
:<math>\frac{d \rho_{ee}}{dt} = -\gamma \rho_{ee} + \frac{i}{2}(\Omega \bar \rho_{ge} - \Omega^*\bar \rho_{eg})</math>
:<math> \frac{d \bar \rho_{ge}}{dt} = -\left( \frac{\gamma}{2} + i\delta \right) \bar \rho_{ge} + \frac{i}{2}\Omega^*(\rho_{ee} - \rho_{gg})</math>
:<math>\frac{d \bar \rho_{eg}}{dt} = - \left( \frac{\gamma}{2} - i\delta \right) \bar \rho_{eg} + \frac{i}{2}\Omega^*(\rho_{gg} - \rho_{ee})</math>

In the derivation of these formulae it was explicitly assumed that spontaneous emission is described by an exponential decay of the coefficient <math>\rho_{eg}(t)</math> with decay constant <math>\frac{\gamma}{2}</math>. <math>\Omega</math> is the (generalized) [[Rabi frequency]], which is
:<math>\Omega = \sqrt{|\chi_{g,e}|^2 + \delta^2}</math>
where <math>\delta = \omega - \omega_{0}</math> is the detuning and measures how far the light frequency, <math>\omega</math>, is from the transition, <math>\omega_{0}</math>. <math> \chi_{g,e} = {\vec{d}_{g,e}\cdot\vec{E}_0 \over \hbar}</math> where <math>\scriptstyle{\vec{d}_{g,e}}</math> is the [[transition dipole moment]] for the <math>\scriptstyle{g \rightarrow e}</math> transition and <math>\scriptstyle{\vec{E}_0 = \hat{\epsilon}E_0}</math> is the [[vector (geometric)|vector]] [[electric field]] amplitude including the [[Polarization (waves)|polarization]].

==Derivation from Cavity Quantum Electrodynamics==
{{Unreferenced section|date=July 2011}}
Beginning with the [[Jaynes-Cummings model|Jaynes-Cummings Hamiltonian]] under [[coherent state|coherent drive]]

:<math>H=\omega_c a^\dagger a + \omega_a \sigma^\dagger\sigma+ig(a^\dagger\sigma-a\sigma^\dagger)+iJ(a^\dagger e^{-i\omega_l t}-a e^{i\omega_l t})</math>

where <math> a</math> is the [[lowering operator]] for the cavity field, and <math> \sigma=\frac{1}{2}\left(\sigma_x - i\sigma_y\right) </math> is the atomic lowering operator written as a combination of [[Pauli matrices]]. The time dependence can be removed by transforming the wavefunction according to <math> |\psi\rangle\rightarrow \operatorname{e}^{-i\omega_l t\left(a^\dagger a + \sigma^\dagger\sigma\right)}|\psi\rangle</math>, leading to a transformed Hamiltonian

:<math>H=\Delta_c a^\dagger a + \Delta_a \sigma^\dagger\sigma+ig(a^\dagger\sigma-a\sigma^\dagger)+iJ( a^\dagger-a)</math>

where <math> \Delta_i = \omega_i - \omega_l </math>. As it stands now, the Hamiltonian has four terms. The first two are the self energy of the atom (or other two level system) and field. The third term is an energy conserving interaction term allowing the cavity and atom to exchange population and coherence. These three terms alone give rise to the Jaynes-Cummings ladder of dressed states, and the associated anharmonicity in the energy spectrum. The last term models coupling between the cavity mode and a classical field, i.e. a laser. The drive strength <math> J </math> is given in terms of the power transmitted through the empty two-sided cavity as <math> J=\sqrt{2P(\Delta_c^2 + \kappa^2)/(\hbar\omega_c \kappa)} </math>, where <math>2\kappa</math> is the cavity linewidth. This brings to light a crucial point concerning the role of dissipation in the operation of a laser or other cqed device; dissipation is the means by which the system (coupled atom/cavity) interacts with its environment. To this end, dissipation is included by framing the problem in terms of the master equation, where the last two terms are in the [[Lindblad superoperator|Lindblad form]]

:<math>\dot{\rho}=-i[H,\rho] + 2\kappa\left(a\rho a^\dagger -\frac{1}{2}\left(a^\dagger a \rho + \rho a^\dagger a\right)\right) + 2\gamma\left(\sigma\rho
\sigma^\dagger -\frac{1}{2}\left(\sigma^\dagger \sigma\rho + \rho \sigma^\dagger \sigma\right)\right)</math>

The equations of motion for the expectation values of the operators can be derived from the master equation by the formula <math> \langle\dot{O}\rangle = \operatorname{tr}\left(O\rho\right) </math>. The equations of motion for <math> \langle a\rangle </math>, <math> \langle\sigma\rangle </math>, and <math> \langle\sigma_z\rangle </math>, the cavity field, atomic ground state population, and atomic inversion respectively, are

:<math>\frac{d}{dt}\langle a \rangle = i\left(-\Delta_c \langle a \rangle - ig\langle \sigma\rangle - iJ\right) -\kappa \langle a \rangle </math>
:<math>\frac{d}{dt}\langle \sigma \rangle = i\left(-\Delta_a \langle \sigma \rangle - ig\langle a \sigma_z \rangle\right) -\gamma \langle \sigma \rangle </math>
:<math>\frac{d}{dt}\langle \sigma_z \rangle = -2g\left(\langle a^\dagger \sigma \rangle+\langle a \sigma^\dagger \rangle\right) -2\gamma \langle \sigma_z\rangle-2\gamma </math>

At this point, we have produced three of an infinite ladder of coupled equations. As can be seen from the third equation, higher order correlations are necessary. The differential equation for the time evolution of <math>\langle a^\dagger \sigma \rangle </math> will contain expectation values of higher order products of operators, thus leading to an infinite set of coupled equations. We heuristically make the approximation that the expectation value of a product of operators is equal to the product of expectation values of the individual operators. This is akin to assuming that the operators are uncorrelated, and is a good approximation in the classical limit. It turns out that the resulting equations give the correct qualitative behavior even in the single excitation regime. Additionally, to simplify the equations we make the following replacements

:<math>\langle a \rangle = (\gamma/\sqrt{2} g)x </math>
:<math>\langle \sigma \rangle = -p/\sqrt{2}</math>
:<math>\langle \sigma_z\rangle = -D </math>
:<math>\Theta = \Delta_c/\kappa </math>
:<math> C = g^2/2\kappa\gamma </math>
:<math> y = \sqrt{2} g J/\kappa\gamma </math>
:<math> \Delta=\Delta_a/\gamma </math>

And the Maxwell-Bloch equations can be written in their final form

:<math>\dot{x}=\kappa\left(-2Cp+y-(i\Theta+1)x\right) </math>
:<math>\dot{p} = \gamma\left( -(1+i\Delta)p + xD\right) </math>
:<math>\dot{D}=\gamma\left(2(1-D)-(x^*p+xp^*)\right) </math>

==References==
{{reflist}}

[[Category:Quantum mechanics]]
[[Category:Theoretical physics]]

Catenoid

2013-03-12T08:38:57Z

131.111.16.20:

{{one source|date=December 2009}}
[[File:Paraboloid of Revolution.svg|thumb|Axially symmetrical paraboloid. The inside surface is concave]]

A '''rotating furnace''' is a device for making solid objects which have concave surfaces that are segments of [[Axial symmetry|axially symmetrical]] [[paraboloid]]s. Usually, the objects are made of [[glass]]. The [[furnace]] makes use of the fact, which was known already to [[Isaac Newton|Newton]], that the centrifugal-force-induced shape of the top surface of a spinning liquid is a concave paraboloid, identical to the shape of a reflecting telescope's primary focusing mirror.

They can be used in various ways, including (after being [[silvered]]) as [[primary mirror]]s in [[reflecting telescope]]s; they have also been used for [[solar cooker]]s.<ref>http://solarcooking.org/research/SpinningParaboicConcentrators.htm</ref>

== Design ==
[[File:Coriolis effect11.jpg|thumb|Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of plexiglass. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre]]

The furnace includes a mechanism that rotates an open-topped container at constant speed around a vertical axis. A quantity of glass sufficient to make the mirror is placed in the container, heated until it is completely molten, and then allowed to cool while continuing to rotate until it has completely solidified. When the rotation is stopped, the glass is solid, so the paraboloidal shape of its top surface is preserved. This process is called ''spin casting''.

The same process can be used to make a [[lens (optics)|lens]] with a concave paraboloidal surface. The other surface is shaped by the container that holds the molten glass acting as a mould. Lenses made this way are sometimes used as [[Objective (optics)|objectives]] in [[refracting telescope]]s.

The word ''spin'' is frequently used in this context without implying that the rotation is rapid. Making a mirror with a focal length of five metres, for example, requires a rotation speed less than ten revolutions per minute. Since the speed is low, accurate [[Balancing of rotating masses|dynamic balancing]] of the rotating components is not needed.

The axis of rotation becomes the axis of the paraboloid. It is not necessary for this axis to be in the centre of the container of glass, or even for it to pass through the container. By placing the container away from the axis, off-axis paraboloidal segments can be cast. This is done in the making of very large telescopes which have mirrors consisting of several segments.

== Mathematical model ==
[[File:Forces in a Parabolic Dish.svg|thumb|The force of gravity (red), the [[normal force]] (green), and the resultant centripetal force (blue)]]

In fluid mechanics, the state when no part of the fluid has motion relative to any other part of the fluid is called 'solid body rotation'. When the liquid has reached a state of solid body rotation, then the dynamic equilibrium can be understood as a balance of two energies: gravitational [[potential energy]], and [[rotational kinetic energy]]. When a fluid is in solid body rotation it is the lowest state of energy that is available, because in a state of solid body rotation there is no friction to [[dissipation|dissipate]] any of the energy.

In an inertial reference frame, the dynamic equilibrium cannot be understood in terms of an equilibrium of forces. This is because when the liquid is rotating, there is an unbalanced force acting on the liquid – the force of gravity is acting in a vertical direction on the liquid, and the surface of the parabolic dish exerts a [[normal force]] on the liquid resting on it. The resultant force is a net centripetal force toward the axis of rotation.

The kinetic energy of a parcel of liquid given by the formula: <math>E_{kin} = \frac{1}{2} m v^2</math>

In the case of circular motion the relation <math>v = \omega r</math> holds (<math>\omega</math> is in radians per second), hence <math>E_{kin} = \frac{1}{2} m \omega^2 r^2</math>

The gravitational potential energy is given by <math>E_{pot} = m g h</math> where <math>g</math> is the acceleration of gravity and <math>h</math> is the height of the liquid's surface above some arbitrary elevation, for instance, we can set <math>h=0</math> to be the lowest liquid surface. We set the potential energy equal to the kinetic energy to find the liquid's shape: <math>h = \frac{1}{2 g} \omega^2 r^2</math> This is of the form <math>h=kr^2</math>, where <math>k</math> is a constant, which is, by definition, a [[paraboloid]].

It is not necessary to invoke the equality of rotational kinetic energy and gravitational potential energy. With reference to the force diagram above, the vertical component of the normal force (green arrow) must equal the weight of the parcel (red arrow), which is <math>mg</math>, and the horizontal component of the normal force must equal the [[centripetal force]] (blue arrow) that keeps the parcel in circular motion, which is <math>m \omega^2 r</math>. Since the green arrow is perpendicular to the surface of the liquid, the slope of the surface must equal the quotient of these forces: <math>\frac{d h}{d r} = \frac{m \omega^2 r}{m g}</math> Cancelling the <math>m</math>'s, integrating, and setting <math>h=0</math> when <math>r=0</math> leads to <math>h = \frac{1}{2 g} \omega^2 r^2</math> which is identical to the result obtained by the previous method, and likewise shows that the liquid surface is paraboloidal.

The focal length of the paraboloid is related to the angular speed at which the liquid is rotated by the equation: <math>2f \omega^2=g</math>, where <math>f</math> is the focal length, <math>\omega</math> is the rotation speed, and <math>g</math> is the [[Gravitational acceleration|acceleration due to gravity]]. They must be in compatible units so, for example, <math>f</math> can be in metres, <math>\omega</math> in radians per second, and <math>g</math> in metres per second-squared. The angle unit in <math>\omega</math> must be [[radian]]s. 1 radian per second is about 9.55 rotations per minute ([[RPM]]). On the Earth's surface, <math>g</math> is about 9.81 metres per second-squared. Putting these numbers into the equation produces the approximation: <math>fs^2 \approx 447</math>, where <math>f</math> is the focal length in metres, and <math>s</math> is the rotation speed in RPM.

== Uses ==

Generally, a spin-cast paraboloid is not sufficiently accurate to permit its immediate use as a telescope mirror or lens, so it is corrected by computer-controlled grinding machines. The amount of grinding done, and the mass of glass material wasted, are much less than would have been required without spinning.

Spin casting can also be used, often with materials other than glass, to produce prototype paraboloids, such as spotlight reflectors or solar-energy concentrators, which do not need to be as exactly paraboloidal as telescope mirrors. Spin casting every paraboloid that is made would be too slow and costly, so the prototype is simply copied relatively quickly and cheaply and with adequate accuracy.

[[Liquid mirror telescope]]s have rotating mirrors that consist of a liquid metal such as [[mercury (element)|mercury]] or a [[low-melting alloy]] of [[gallium]]. The mirrors do not solidify, but are used while liquid and rotating. The rotation shapes them into paraboloids that are accurate enough to be used as primary reflectors in telescopes. No correction of the shape is necessary. Spin-cast glass mirrors need correction because of distortions that arise during and after solidification.

== See also ==

* [[List of largest optical reflecting telescopes]]

== References ==

{{reflist}}

[[Category:Telescopes]]
[[Category:Optics]]