PlanetPhysics/Spin and Mathematics of Spin Groups

Spin and spin group mathematics

The physics of [[../QuarkAntiquarkPair/|spins]] and mathematics of spin groups are both important subjects respectively in Physics and [[../PhysicalMathematics2/|mathematical physics]].

In Physics, the term spin 'groups' is often used with the broad meaning of a collection of coupled, or interacting spins, and thus covers the broad '[[../CohomologyTheoryOnCWComplexes/|spectrum]]' of spin clusters ranging from [[../BoseEinsteinStatistics/|gravitons]] (as in [[../SpinNetworksAndSpinFoams/|spin networks and spin foams]], for example) to 'up' ($u$) and 'down' ($d$) [[../QuarkAntiquarkPair/|quark]] spins ([[../QuarkAntiquarkPair/|fermions]]) coupled by [[../ExtendedQuantumSymmetries/|gluons]] in nuclei (as treated in Quantum Chromodynamics or Theoretical [[../Cyclotron/|nuclear physics]]), and electron spin [[../LongRangeCoupling/|Cooper pairs]] (regarded as [[../Boson/|bosons]]) in low-temperature [[../LongRangeCoupling/|superconductivity]]. On the other hand, in [[../Bijective/|relation]] to [[../HilbertBundle/|quantum symmetry]], spin groups are defined in [[../QuantumParadox/|quantum mechanics]] and [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theories]] ([[../CategoricalQuantumLMAlgebraicLogic2/|QFT]]) in a precise, mathematical ([[../CoIntersections/|algebraic]]) sense as properly defined [[../TrivialGroupoid/|groups]], as introduced next. (In a semi-classical approach, the related [[../PreciseIdea/|concept]] of a [[../ECartan/|spinor]] has been introduced and studied in depth by É. Cartan, who found that with his definition of spinors the (special) relativistic Lorentz [[../Covariance/|covariance]] properties were not recovered, or applicable.)

In the mathematical, precise sense of the term, a spin group --as for example the [[../BilinearMap/|Lie group]] $Spin(n)$-- is defined as a double cover of the special orthogonal (Lie) group $SO(n)$ satisfying the additional condition that there exists the short exact sequence of Lie groups:

$1\to {\mathbb{Z}}_2\to Spin(n)\to SO(n)\to 1$

Alternatively one can say that the above exact sequence of Lie groups defines the spin group $Spin(n)$. Furthermore, $Spin(n)$ can also be defined as the proper subgroup (or [[../EquivalenceRelation.html|groupoid]]) of the invertible elements in the Clifford algebra $ℂl(n)$; (when defined as a double cover this should be $C{l}_{p,q}(R)$, a Clifford algebra built up from an orthonormal basis of $n=p+q$ mutually orthogonal [[../Vectors/|vectors]] under addition and multiplication, $p$ of which have [[../NormInducedByInnerProduct/|norm]] +1 and $q$ of which have norm $-1$, as further explained in the [[../Spinor/|spinor definition]]). Note also that other spin groups such as $Spind$ (ref. ) are mathematically defined, and also important, in QFT.

{\mathbf Important Examples of $Spin(n)$ and Quantum Symmetries} There exist the following [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphisms]]:

1. $Spin(1)\cong O(1)$

1. $Spin(2)\cong U(1)\cong SO(2)$

1. $Spin(3)\cong Sp(1)\cong SU(2)$

1. $Spin(4)\cong Sp(1)\times Sp(1)$

1. $Spin(5)\cong Sp(2)$

1. $Spin(6)\cong SU(4)$

Thus, the [[../TopologicalOrder2/|symmetry groups]] in the Standard Model ([[../NewtonianMechanics/|SUSY]]) of current Physics can also be written as : $Spin(2)\times Spin(3)\times SU(3)$, where only $SU(3)$ does not have an isomorphic $Spin(n)$ group.

{\mathbf Remarks}

• In modern Physics, [[../CubicalHigherHomotopyGroupoid/|non-Abelian]] spin groups are also defined, as for example, spin [[../QuantumGroup4/|quantum groups]] and spin [[../WeakHopfAlgebra/|quantum groupoids]].
• An extension of the concepts of spin group and spinor, is the notion of a 'twistor', a mathematical concept introduced by Sir Roger Penrose, generally with distinct symmetry/mathematical properties from those of spin groups, such as those defined above. ===The Fundamental Groups of $Spin(p,q)$=== With the usual notation, the [[../HomotopyCategory/|fundamental groups]] ${\pi }_1(Spin(p,q))$ are as follows: #
• $\left\{0\right\}$ , for $(p,q)=(1,1)$ and $(p,q)=(1,0)$; #
• $\left\{0\right\}$ , if $p>2$ and $q=0,1$; #
• $\mathbb{Z}$ for $(p,q)=(2,0)$ and $(p,q)=(2,1)$; #
• $\mathbb{Z}\times \mathbb{Z}$ for $(p,q)=(2,2)$; #
• $\mathbb{Z}$ for $p>2,q=2$ #
• ${\mathbb{Z}}_2$ for $p>2,q>2$