- Spinors in three dimensions
mathematics, the spinorconcept as specialised to three dimensions can be treated by means of the traditional notions of dot productand cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
This algebra admits a convenient description, due to
William Rowan Hamilton, by means of quaternions. In detail, given a vector x = ("x"1, "x"2, "x"3) of real (or complex) numbers, one can associate the matrix of complex numbers::
Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:
* det "X" = - (length x)2.
* "X"2 = (length x)2"I", where "I" is the identity matrix.
* where "Z" is the matrix associated to the cross product z = x × y.
* If u is a unit vector, then "-UXU" is the matrix associated to the vector obtained from x by reflection in the plane orthogonal to u.
* It is an elementary fact from
linear algebrathat any rotation in 3-space factors as a composition of two reflections. (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if "R" is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector u1 followed by the plane perpendicular to u2, then the matrix "U"2"U"1"XU"1"U"2 represents the rotation of the vector x through "R".
Having effectively encoded all of the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the
column vectors) play. Provisionally, a spinor is a column vector: with complex entries ξ1 and ξ2.
The space of spinors is evidently acted upon by complex 2×2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors. However, there is one important caveat: the factorization of a rotation is not unique. Clearly, if "X" → "RXR"-1 is a representation of a rotation, then replacing "R" by -"R" will yield the same rotation. In fact, one can easily show that this is the only ambiguity which arises. Thus the action of a rotation on a spinor is always "double-valued."
Spinors can be constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that "X" is a matrix representing a vector x in complex 3-space. Suppose further that x is isotropic: i.e.,:Then, from the properties of these matrices, "X"2=0. Any such matrix admits a factorization as an outer product:This factorization yields an
overdetermined systemof equations in the coordinates of the vector x:: (1)subject to the constraint: (2)This system admits the solutions: (3)Either choice of sign solves the system (1). Thus a spinor may be viewed as an isotropic vector, along with a choice of sign. Note that because of the logarithmic branching, it is impossible to choose a sign consistently so that (3) varies continuously along a full rotation among the coordinates x. In spite of this ambiguity of the representation of a rotation on a spinor, the rotations do act unambiguously by a fractional linear transformationon the ratio ξ1:ξ2 since one choice of sign in the solution (3) forces the choice of the second sign. In particular, the space of spinors is a projective representationof the orthogonal group.
As a consequence of this point of view, spinors may be regarded as a kind of "square root" of isotropic vectors. Specifically, introducing the matrix:the system (1) is equivalent to solving "X" = 2 ξ tξ "C" for the undetermined spinor ξ.
"A fortiori", if the rôles of ξ and x are now reversed, the form "Q"(ξ) = x defines, for each spinor ξ, a vector x quadratically in the components of ξ. If this quadratic form is polarized, it determines a bilinear vector-valued form on spinors "Q"(μ,ξ). This bilinear form then transform tensorially under a reflection or a rotation.
The above considerations apply equally well whether the original euclidean space under consideration is real or complex. When the space is real, however, spinors possess some additional structure which in turn facilitates a complete description of the representation of the rotation group. Suppose, for simplicity, that the inner product on 3-space has positive-definite signature: :length(x)2 = "x"12 + "x"22 + "x"32 (4).With this convention, real vectors correspond to hermitian matrices. Furthermore, real rotations preserving the form (4) correspond (in the double-valued sense) to unitary matrices of determinant one. In modern terms, this presents the
special unitary groupSU(2) as a double-cover of SO(3). As a consequence, the spinor hermitian product: (5)is preserved by all rotations, and therefore is canonical.
If, however, the signature of the inner product on 3-space is indefinite (i.e., non-degenerate, but also not positive definite), then the foregoing analysis must be adjusted to reflect this. Suppose then that the length form on 3-space is given by::length("x")2 = "x"12 - "x"22 + "x"32 (4').Then the construction of spinors of the preceding sections proceeds, but with "x"2 replacing "i x"2 in all the formulas. With this new convention, the matrix associated to a real vector ("x"1,"x"2,"x"3) is itself real::.The form (5) is no longer invariant under a real rotation (or reversal), since the group stabilizing (4') is now a
Lorentz groupO(2,1). Instead, the antihermitian form:defines the appropriate notion of inner product for spinors in this metric signature. This form is invariant under transformations in the connected component of the identity of O(2,1).
In either case, the quartic form:is fully invariant under O(3) (or O(2,1), respectively), where "Q" is the vector-valued bilinear form described in the previous section. The fact that this is a quartic invariant, rather than quadratic, has an important consequence. If one confines attention to the group of special orthogonal transformations, then it is possible unambiguously to take the square root of this form and obtain an identification of spinors with their duals. In the language of representation theory, this implies that there is only one irreducible spin representation of SO(3) (or SO(2,1)) up to isomorphism. If, however, reversals (e.g., reflections in a plane) are also allowed, then it is no longer possible to identify spinors with their duals owing to a change of sign on the application of a reflection. Thus there are two irreducible spin representations of O(3) (or O(2,1)), sometimes called the pin representations.
The differences between these two signatures can be codified by the notion of a "
reality structure" on the space of spinors. Informally, this is a prescription for taking a complex conjugate of a spinor, but in such a way that this may not correspond to the usual conjugate per the components of a spinor. Specifically, a reality structure is specified by a hermitian 2 × 2 matrix "K" whose product with itself is the identity matrix: "K"2 = "Id". The conjugate of a spinor with respect to a reality structure "K" is defined by:
The particular form of the inner product on vectors (e.g., (4) or (4')) determines a reality structure (up to a factor of -1) by requiring:, whenever "X" is a matrix associated to a real vector.Thus "K" = "i C" is the reality structure in euclidean signature (4), and "K" = "Id" is that for signature (4'). With a reality structure in hand, one has the following results:
* "X" is the matrix associated to a real vector if, and only if, .
* If μ and ξ is a spinor, then the inner product :::determines a hermitian form which is invariant under proper orthogonal transformations.
Examples in physics
pinors of the Pauli spin matrices
Often, the first example of spinors that a student of physics encounters are the 2x1 spinors used in Pauli's theory of electron spin.The
Pauli matricesare a vector of three 2x2 matricesthat are used as spin operators.
unit vectorin 3 dimensions, for example (a,b,c), one takes a dot productwith the Pauli spin matrices to obtain a spin matrix forspin in the direction of the unit vector.
eigenvectors of that spin matrix are the spinors forspin-1/2 oriented in the direction given by the vector.
Example: u = (0.8, -0.6, 0) is a unit vector. Dotting this with the Paulispin matrices gives the matrix:
The eigenvectors may be found by the usual methods of
linear algebra, but a convenient trickis to note that the Pauli spin matrices are square roots of unity, that is, the squareof the above matrix is the identity matrix.Thus a (matrix) solution to the eigenvector problem with eigenvalues of is simply. That is,
One can then choose either of the columns of the eigenvectormatrix as the vector solution, provided that the column chosenis not zero. Taking the first column of the above,eigenvector solutions for the two eigenvalues are:
The trick used to find the eigenvectors is related to the concept of
ideals, that is, the matrix eigenvectors are
projection operatorsor idempotents and therefore each generates an
ideal in the Pauli algebra. The same trickworks in any
Clifford algebra, in particularthe Dirac algebra that are discussed below. These projectionoperators are also seen in density matrixtheorywhere they are examples of pure density matrices.
More generally, the projection operator for spin in the (a,b,c) directionis given by
and any non zero column can be taken as the projection operator. While thetwo columns appear different, one can use to show that they are multiples (possibly zero) of the same spinor.
In atomic physics and quantum mechanics, the property of "spin" plays a major role. In addition to their other properties all particles possess a non-classical property, i.e., which has no correspondence at all in conventional physics, namely the spin, which is a kind of "intrinsic angular momentum". In the position representation, instead of a wavefunction without spin, , one has with spin: , where belongs to the following discrete set of values: . The operator of the "total angular momentum", , of a particle corresponds to the "sum" of the "orbital angular momentum" (i.e., there only integers are allowed) and the "intrinsic part", the "spin". One distinguishes
bosons (S = 0 or ±1 or ±2 or ...) and fermions (S = ±1/2 or ±3/2 or ±5/2 or ...)
*cite book | author=Cartan, Élie|title=The Theory of Spinors | publisher=Dover |year=1981 |id=ISBN 0-486-64070-1
Wikimedia Foundation. 2010.
Look at other dictionaries:
Spinor — In mathematics and physics, in particular in the theory of the orthogonal groups (such as the rotation or the Lorentz groups), spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the… … Wikipedia
List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia
Supergravity — In theoretical physics, supergravity (supergravity theory) is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry (in contrast to… … Wikipedia
Weyl-Brauer matrices — In mathematics, particularly in the theory of spinors, the Weyl Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra. They generalize to n dimensions the Pauli matrices. They are named for Richard Brauer and… … Wikipedia
Parity (physics) — Flavour in particle physics Flavour quantum numbers: Isospin: I or I3 Charm: C Strangeness: S Topness: T Bottomness: B′ Related quantum numbers: Baryon number: B Lepton number: L Weak isospin: T or T3 Electric charge: Q … Wikipedia
Neutrino theory of light — The neutrino theory of light is the proposal that the photon is a composite particle formed of a neutrino antineutrino pair. It is based on the idea that emission and absorption of a photon corresponds to the creation and annihilation of a… … Wikipedia
Cross product — This article is about the cross product of two vectors in three dimensional Euclidean space. For other uses, see Cross product (disambiguation). In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on… … Wikipedia
Rotation (mathematics) — Rotation of an object in two dimensions around a point O. In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a… … Wikipedia
Supersymmetry nonrenormalization theorems — In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory. Renormalization theorems are… … Wikipedia
Pythagorean theorem — See also: Pythagorean trigonometric identity The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c) … Wikipedia