natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra: So also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose For the complex representations of the real Clifford algebras, the associated reality structure on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate of the representation Δ, and the following isomorphism is seen to hold: In particular, note that the representation Δ of the orthochronous spin group is a unitary representation. In general, there are Clebsch–Gordan decompositions In metric signature , the following isomorphisms hold for the conjugate half-spin representations If q is even, then and If q is odd, then and Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations . Odd dimensions If is odd, then In the real case, once again the isomorphism holds Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by Consequences There are many far-reaching consequences of the Clebsch–Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are A manner of regarding the product of two spinors ψ as a scalar. In physical terms, a spinor should determine a probability amplitude for the quantum state. A manner of regarding the product ψ as a vector. This is an essential feature of Dirac's theory, which ties the spinor formalism to the geometry of physical space. A manner of regarding a spinor as acting upon a vector, by an expression such as ψv. In physical terms, this represents an electric current of Maxwell's electromagnetic theory, or more generally a probability current. Summary 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 e±iφ/2 under a rotation by angle φ. In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and quaternionic. The existence of spinors in 3 dimensions follows from the isomorphism of the groups that allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the