Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of SO(N), roughly speaking) similar to vectorss, but which change sign under a rotation of 2π radians. Spinors are often described as "square roots of vectors" because the vector representation appears in the tensor product of two copies of the spinor representation. Spinors were invented by Wolfgang Pauli and Paul Dirac to describe the physical properties of spin, especially the properties of fermions whose spin numerically equals one half. The word "spinor" was coined by Paul Ehrenfest. The mathematics of spinors is said to have been anticipated by Elie Cartan as early as 1913.

An n-dimensional spinor of a certain type is an element of a specific projective representation of the rotation group SO(n,R), or more generally of the group SO(p,q,R), where p + q = n for spinors in a space of nontrivial signature. This is equivalent to an ordinary (non-projective) representation of the double cover of SO(p,q,R), which is a real Lie group called the spinor group Spin(p,q).

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra C(p,q), into which Spin(p,q) may be embedded. In even dimensions, this representation is reducible when taken as a representation of Spin(p,q) and may be decomposed into two: the left-handed and right-handed Weyl spinor representations. These may be distinguished only by the action of parity transformations (not part of Spin(p,q), but present in C(p,q)). In addition, sometimes the non-complexified version of C(p,q) has a smaller real representation, the Majorana spinor representation. If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana-Weyl spinor representations. Of all these, only the Dirac representation exists in all dimensions. Dirac spinors are complex representations while Majorana spinors are real representations.

A 2n- or 2n+1-dimensional Dirac spinor may be represented as a vector of 2ⁿ complex numbers. (See Special unitary group.)

In the early 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute created games such as Tangloids to teach and model the calculus of spinors.

Examples in low dimensions

• In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a real 1-dimensional representation that does not transform

• In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-component complex representations, i.e. complex numbers that get multiplied by under a rotation by angle

• In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and pseudoreal. The existence of spinors in 3 dimensions follows from the isomorphism of the groups which allows us to define the action of , up to the ambiguous sign, on a complex 2-component column (a spinor); the generators of can be written as Pauli matrices

• In 4 Euclidean dimensions, the corresponding isomorphism is . There are two inequivalent pseudoreal 2-component Weyl spinors and each of them transforms under one of the factors only

• In 5 Euclidean dimensions, the relevant isomorphism is which implies that the single spinor representation is 4-dimensional and pseudoreal

• In 6 Euclidean dimensions, the isomorphism guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another

• In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no simple isomorphisms exist from this dimension on

• In 8 Euclidean dimensions, there are two Weyl-Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of SO(8) called triality

• In dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases

• In spacetimes with spatial and time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the -dimensional Euclidean space, but the reality projections mimic the structure in Euclidean dimensions. For example, in 3+1 dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism .

• spinor bundle
• anyon