PlanetPhysics/Spin and Mathematics of Spin Groups

Spin and spin group mathematics
The physics of spins and mathematics of spin groups are both important subjects respectively in Physics and 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 'spectrum' of spin clusters ranging from gravitons (as in spin networks and spin foams, for example) to 'up' ($$u$$) and 'down' ($$d$$) quark spins (fermions) coupled by gluons in nuclei (as treated in Quantum Chromodynamics or Theoretical nuclear physics), and electron spin Cooper pairs (regarded as bosons) in low-temperature superconductivity. On the other hand, in relation to quantum symmetry, spin groups are defined in quantum mechanics and quantum field theories (QFT) in a precise, mathematical (algebraic) sense as properly defined groups, as introduced next. (In a semi-classical approach, the related concept of a spinor has been introduced and studied in depth by É. Cartan, who found that with his definition of spinors the (special) relativistic Lorentz covariance properties were not recovered, or applicable.)

In the mathematical, precise sense of the term, a spin group --as for example the 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 groupoid) of the invertible elements in the Clifford algebra $$\mathbb{C}l(n)$$; (when defined as a double cover this should be $$Cl_{p, q}(R)$$, a Clifford algebra built up from an orthonormal basis of $$n = p + q$$ mutually orthogonal vectors under addition and multiplication, $$p$$ of which have norm +1 and $$q$$ of which have norm $$-1$$, as further explained in the spinor definition). Note also that other spin groups such as $$Spin ~ d$$ (ref. ) are mathematically defined, and also important, in QFT.

{\mathbf Important Examples of $$Spin(n)$$ and Quantum Symmetries} There exist the following 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 symmetry groups in the Standard Model (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, non-Abelian spin groups are also defined, as for example, spin quantum groups and spin 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 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 $$ p>2, q >2 $$