Spinor Calculus

the mathematical theory that studies the quantities known as spinors. When physical quantities are investigated, they are usually referred to some coordinate system. Depending on the law governing the transformation of the quantities from one coordinate system to another, a distinction is made between quantities of different types, such as tensors and pseudo-tensors. In the study of the phenomenon of the electron’s spin, physical quantities were found that did not belong to the previously known types. For example, the signs of the quantities are ambiguous, since upon rotation of the coordinate system by 2-rr about some axis, all the components of the quantities change sign. Such quantities were considered in 1913 by E. Cartan in his investigations of the representations of groups. B. L. van der Waerden rediscovered the quantities in his studies of quantum mechanics. He named the quantities spinors.

Spinors of rank, or order, one are defined by two complex numbers: (ξ¹, ξ;²). Moreover, in contrast, for example, to tensors, for which different sets of numbers define different tensors, the sets (ξ¹, ξ;²) and (– ξ;¹, – ξ;²) are regarded as defining the same spinor. This situation is accounted for by the law governing the transformation of a spinor from one coordinate system to another. When the coordinate system is rotated through an angle 6 about an axis with the direction cosines cos x₁, cos x₂, and cos X₃ the spinor components are transformed in accordance with the formulas

ξ’ = αξ¹ + βξ²ξ^2’ = γξ¹ + δξ²

Here, α = λ + iμ, β = v + iρ, γ = -β̂, and δ = α̃, here λ = cos θ/2, μ = sin (θ/2) cos X₁, v = sin (θ/2) cos X₂, and ρ = sin (θ/2) cos X₃. In particular, when the coordinate system is rotated through the angle 2π (such a rotation returns the system to its original position), the spinor components change sign. This fact accounts for the identity of the spinors (ξ¹, ξ;²) and (– ξ¹, –ξ²). The wave function of a particle with spin 1/2, such as an electron, is an example of a spinor quantity. The matrix

in this case is a unitary matrix.

Also classified as spinors are quantities whose components ξ̃¹, ξ̃² are complex conjugates of the components of the spinor (ξ¹, ξ²). The transformation matrix of these quantities has the form

Let Oxyz and O’x’y’z’ be two coordinate systems with pairwise parallel axes. Suppose O’x’y’z’ moves relative to Oxyz with the speed v = c tanh 9, where c is the speed of light, in a direction that forms the angles X₁ X₂ and X₃ with the axes. Under the Lor-entz transformations corresponding to the passage from Oxyz to O’x’y’z’, the spinor components are transformed in accordance with the formulas

ξ¹’ = αξ¹ + βξ² ξ^2’λξ¹ + δ¹

Here, α = λ + μ, β = v + iρ, γ = β̂̄ and δ = λ – μ, where λ = cosh θ/2, μ. = sinh (θ/2) cos X₁, v = sinh (θ/2) cos X₂, and ρ = sinh (θ/2) cos X₃. If Lorentz transformations are considered for the case where the coordinate axes are not pairwise parallel, then the transformation matrix cr of the spinor components can be any complex second-order matrix whose determinant is equal to 1—that is, any complex second-order unimodular matrix.

In addition to the contravariant spinor components ξ¹, ξ² introduced above, the covariant components ξ₁, ξ₂ can be introduced by setting ξ_α = ∈_αβξ^β, where

(as usual, summation is performed with respect to repeating indices). In other words, ξ² =ξ¹, and ξ¹ = – ξ₂. The covariant components are transformed by the matrix

Under rotations, this matrix coincides with the matrix σ—that is, under rotations, covariant spinor components are transformed as components of the complex conjugate spinor.

Spinor algebra is constructed in much the same way as conventional tensor algebra. A set of 2^r complex numbers a^λ1λ2 . . . λ_r determined up to sign is called a spinor of rank, or order, r. Under a transformation from one coordinate system to another, this set of complex numbers transforms as the product of r components of spinors of rank one—that is, as ξ^λ1ξ^λ2 . . . ξ^λr. The complex conjugate spinor of rank r, the mixed spinor, the spinor with covariant components, and other spinors are defined analogously. The addition of spinors and the multiplication of a spinor by a scalar are defined componentwise. The product of two spinors is the spinor whose components are paired products of the components of the factors. For example, the spinor of rank five α_λμ b^vρσ can be formed from the spinors of ranks two and three α_λμ and b_vρσ. The Spinor

b^λ3λ4 . . . λ_r = ∈_αβa^αβλ3λ4 . . . λ_r

is called the convolution of the spinor a^λ1λ2 . . . λ_r with respect to the indices λ₁ and λ₂. The following identities are often used in spinor algebra:

ξ_λη^λ = –ξ^λη_λ

b_λc_μd^μ + C^λd_μb^μ + d^λb_μC^μ = 0

An important role is played in quantum mechanics by systems of linear differential equations relating spinor-type quantities, which remain invariant under unimodular transformations. Only such systems of equations are relativistically invariant. The most important applications of spinor analysis are in the theory of the Maxwell and Dirac equations. By writing these equations in spinor form, it is possible to determine immediately their relativistic invariance and to establish the character of the transformation of their constituent quantities. Spinor algebra also finds application in the quantum theory of chemical valence. The theory of spinors in higher-dimensional spaces is associated with representations of the rotation groups of multidimensional spaces. Some problems of non-Euclidean geometry also involve spinor calculus.