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Early Results
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A Spinor is a mathematical object which describe's a particle's Spin in a similar way that a Vector describes it's translation. For instance, the rotational modes correspond with Spinors while the vibrational modes correspond with Vectors. It is also possible to understand Spinors as elements of a complex vector space (while vectors are elements of a real vector space), which results in certain strange results, for example, the Spinors only transform under the Orthogonal Group up to a sign, so after a rotation by $ 2\pi $, the components are put into the negative signs, so there needs to be a rotation by $ 4\pi $ to re-obtain the original values.
Spinor IndicesEdit this section
Spinors have 1 Spinor Index, represented in the same way as a Contravariant Index.
For example, if $ \psi^\mu $ is a spinor, then $ \mu $ is the Spinor Index.
Intuitive understandingEdit this section
Spinors can be intuitively understood in the following ways:
- Spinors are Vectors of a space whose transformations are closely related to rotations in a physical space.
- Spinors exist in a complex vector space and therefore to quote Michael Atiyah,
“ | No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of -1 took centuries, the same might be true of spinors. | ” |
ExamplesEdit this section
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Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓ_{p, q}(R). This is an 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, with the product rule for the basis vectors
- $ e_i e_j = \Bigg\{ \begin{matrix} +1 & i=j, \, i \in (1 \ldots p) \\ -1 & i=j, \, i \in (p+1 \ldots n) \\ - e_j e_i & i \not = j. \end{matrix} $
Two dimensions Edit this section
The Clifford algebra Cℓ_{2,0}(R) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ_{1} and σ_{2}, and one unit pseudoscalar i = σ_{1}σ_{2}. From the definitions above, it is evident that (σ_{1})^{2} = (σ_{2})^{2} = 1, and (σ_{1}σ_{2})(σ_{1}σ_{2}) = −σ_{1}σ_{1}σ_{2}σ_{2} = −1.
The even subalgebra Cℓ^{0}_{2,0}(R), spanned by even-graded basis elements of Cℓ_{2,0}(R), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ_{1}σ_{2}. As a real algebra, Cℓ^{0}_{2,0}(R) is isomorphic to field of complex numbers C. As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by
- $ (a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1\, $.
which, by the Clifford relations, can be written
- $ (a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1 = a-b\sigma_1\sigma_2\, $.
The action of an even Clifford element γ ∈ Cℓ^{0}_{2,0}(R) on vectors, regarded as 1-graded elements of Cℓ_{2,0}(R), is determined by mapping a general vector u = a_{1}σ_{1} + a_{2}σ_{2} to the vector
- $ \gamma(u) = \gamma u \gamma^*\, $,
where γ^{∗} is the conjugate of γ, and the product is Clifford multiplication. In this situation, a spinor^{[1]} is an ordinary complex number. The action of γ on a spinor φ is given by ordinary complex multiplication:
- $ \gamma(\phi) = \gamma\phi $.
An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:
- $ \gamma(u) = \gamma u \gamma^* = \gamma^2 u\, $.
On the other hand, comparing with the action on spinors γ(φ) = γφ, γ on ordinary vectors acts as the square of its action on spinors.
Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of θ corresponds to γ^{2} = exp(θ σ_{1}σ_{2}), so that the corresponding action on spinors is via γ = ± exp(θ σ_{1}σ_{2}/2). In general, because of logarithmic branching, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued.
In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by abuse of language, the two are often conflated. One may then talk about "the action of a spinor on a vector." In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to computer graphics) they make sense.
- Examples
- The even-graded element
- $ \gamma = \tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2) \, $
- corresponds to a vector rotation of 90° from σ_{1} around towards σ_{2}, which can be checked by confirming that
- $ \tfrac{1}{2} (1 - \sigma_1 \sigma_2) \, \{a_1\sigma_1+a_2\sigma_2\} \, (1 - \sigma_2 \sigma_1) = a_1\sigma_2 - a_2\sigma_1 \, $
- It corresponds to a spinor rotation of only 45°, however:
- $ \tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2) \, \{a_1+a_2\sigma_1\sigma_2\}= \frac{a_1+a_2}{\sqrt{2}} + \frac{-a_1+a_2}{\sqrt{2}}\sigma_1\sigma_2 $
- Similarly the even-graded element γ = −σ_{1}σ_{2} corresponds to a vector rotation of 180°:
- $ (- \sigma_1 \sigma_2) \, \{a_1\sigma_1 + a_2\sigma_2\} \, (- \sigma_2 \sigma_1) = - a_1\sigma_1 -a_2\sigma_2 \, $
- but a spinor rotation of only 90°:
- $ (- \sigma_1 \sigma_2) \, \{a_1 + a_2\sigma_1\sigma_2\} =a_2 - a_1\sigma_1\sigma_2 $
- Continuing on further, the even-graded element γ = −1 corresponds to a vector rotation of 360°:
- $ (-1) \, \{a_1\sigma_1+a_2\sigma_2\} \, (-1) = a_1\sigma_1+a_2\sigma_2 \, $
- but a spinor rotation of 180°.
Three dimensions Edit this section
- Main articles Spinors in three dimensions, Quaternions and spatial rotation
The Clifford algebra Cℓ_{3,0}(R) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ_{1}, σ_{2} and σ_{3}, the three unit bivectors σ_{1}σ_{2}, σ_{2}σ_{3}, σ_{3}σ_{1} and the pseudoscalar i = σ_{1}σ_{2}σ_{3}. It is straightforward to show that (σ_{1})^{2} = (σ_{2})^{2} = (σ_{3})^{2} = 1, and (σ_{1}σ_{2})^{2} = (σ_{2}σ_{3})^{2} = (σ_{3}σ_{1})^{2} = (σ_{1}σ_{2}σ_{3})^{2} = −1.
The sub-algebra of even-graded elements is made up of scalar dilations,
- $ u^{\prime} = \rho^{(1/2)} u \rho^{(1/2)} = \rho u, $
and vector rotations
- $ u^{\prime} = \gamma \, u \, \gamma^*, $
where
- $ \left.\begin{matrix} \gamma & = & \cos(\theta/2) - \{a_1 \sigma_2\sigma_3 + a_2 \sigma_3\sigma_1 + a_3 \sigma_1\sigma_2\} \sin(\theta/2) \\ & = & \cos(\theta/2) - i \{a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3\} \sin(\theta/2) \\ & = & \cos(\theta/2) - i v \sin(\theta/2) \end{matrix}\right\} $ (1)
corresponds to a vector rotation through an angle θ about an axis defined by a unit vector v = a_{1}σ_{1} + a_{2}σ_{2} + a_{3}σ_{3}.
As a special case, it is easy to see that, if v = σ_{3}, this reproduces the σ_{1}σ_{2} rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ_{3} direction invariant, since
- $ (\cos(\theta/2) - i \sigma_3 \sin(\theta/2)) \, \sigma_3 \, (\cos(\theta/2) + i \sigma_3 \sin(\theta/2)) = (\cos^2(\theta/2) + \sin^2(\theta/2)) \, \sigma_3 = \sigma_3. $
The bivectors σ_{2}σ_{3}, σ_{3}σ_{1} and σ_{1}σ_{2} are in fact Hamilton's quaternions i, j and k, discovered in 1843:
- $ \begin{matrix}\mathbf{i} = -\sigma_2 \sigma_3 = -i \sigma_1 \\ \mathbf{j} = -\sigma_3 \sigma_1 = -i \sigma_2 \\ \mathbf{k} = -\sigma_1 \sigma_2 = -i \sigma_3. \end{matrix} $
With the identification of the even-graded elements with the algebra H of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.^{[2]} Thus the (real^{[3]}) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.
Note that the expression (1) for a vector rotation through an angle θ, the angle appearing in γ was halved. Thus the spinor rotation γ(ψ) = γψ (ordinary quaternionic multiplication) will rotate the spinor ψ through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with (180° + θ/2) in place of θ/2 will produce the same vector rotation, but the negative of the spinor rotation.
The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.
Explicit constructions Edit this section
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A space of spinors can be constructed explicitly with concrete and abstract constructions. The equivalence of these constructions are a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see spinors in three dimensions.
Component spinorsEdit this section
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ(V, g) can be defined as follows. Choose an orthonormal basis e^{1} … e^{n} for V i.e. g(e^{μ}e^{ν}) = η^{μν} where η^{μμ} = ±1 and η^{μν} = 0 for μ ≠ ν. Let k = ⌊ n/2 ⌋. Fix a set of 2^{k} × 2^{k} matrices γ^{1} … γ^{n} such that γ^{μ}γ^{ν} + γ^{ν}γ^{μ} = η^{μν}1 (i.e. fix a convention for the gamma matrices). Then the assignment e^{μ} → γ^{μ} extends uniquely to an algebra homomorphism Cℓ(V, g) → Mat(2^{k}, C) by sending the monomial e^{μ1} … e^{μk} in the Clifford algebra to the product γ^{μ1} … γ^{μk} of matrices and extending linearly. The space Δ = C^{2k} on which the gamma matrices act is a now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the Pauli sigma matrices gives rise to the familiar two component spinors used in non relativistic quantum mechanics. Likewise using the 4 × 4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic quantum field theory. In general, in order to define gamma matrices of the required kind, one can use the Weyl–Brauer matrices.
In this construction the representation of the Clifford algebra Cℓ(V, g), the Lie algebra so(V, g), and the Spin group Spin(V, g), all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2^{k} complex numbers and is denoted with spinor indices (usually α, β, γ). In the physics literature, abstract spinor indices are often used to denote spinors even when an abstract spinor construction is used.
Abstract spinorsEdit this section
There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of Cℓ(V, g) on itself. These are subspaces of the Clifford algebra of the form Cℓ(V, g)ω, admitting the evident action of Cℓ(V, g) by left-multiplication: c : xω → cxω. There are two variations on this theme: one can either find a primitive element ω that is a nilpotent element of the Clifford algebra, or one that is an idempotent. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it.^{[4]} In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of V, and then specify the action of the Clifford algebra externally to that vector space.
In either approach, the fundamental notion is that of an isotropic subspace W. Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of V is given.
As above, we let (V, g) be an n-dimensional complex vector space equipped with a nondegenerate bilinear form. If V is a real vector space, then we replace V by its complexification V ⊗_{R} C and let g denote the induced bilinear form on V ⊗_{R} C. Let W be a maximal isotropic subspace, i.e. a maximal subspace of V such that g|_{W} = 0. If n = 2k is even, then let W′ be an isotropic subspace complementary to W. If n = 2k + 1 is odd let W′ be a maximal isotropic subspace with W ∩ W′ = 0, and let U be the orthogonal complement of W ⊕ W′. In both the even and odd dimensional cases W and W′ have dimension k. In the odd dimensional case, U is one dimensional, spanned by a unit vector u.
Minimal ideals Edit this section
Since W′ is isotropic, multiplication of elements of W′ inside Cℓ(V, g) is skew. Hence vectors in W′ anti-commute, and Cℓ(W′, g|_{W′}) = Cℓ(W′, 0) is just the exterior algebra Λ^{∗}W′. Consequently, the k-fold product of W′ with itself, W′^{k}, is one-dimensional. Let ω be a generator of W′^{k}. In terms of a basis w′_{1},..., w′_{k} of in W′, one possibility is to set
- $ \omega=w'_1w'_2\cdots w'_k. $
Note that ω^{2} = 0 (i.e., ω is nilpotent of order 2), and moreover, w′ω = 0 for all w′ ∈ W′. The following facts can be proven easily:
- If n = 2k, then the left ideal Δ = Cℓ(V, g)ω is a minimal left ideal. Furthermore, this splits into the two spin spaces Δ_{+} = Cℓ^{even}ω and Δ_{−} = Cℓ^{odd}ω on restriction to the action of the even Clifford algebra.
- If n = 2k + 1, then the action of the unit vector u on the left ideal Cℓ(V, g)ω decomposes the space into a pair of isomorphic irreducible eigenspaces (both denoted by Δ), corresponding to the respective eigenvalues +1 and −1.
In detail, suppose for instance that n is even. Suppose that I is a non-zero left ideal contained in Cℓ(V, g)ω. We shall show that I must be equal to Cℓ(V, g)ω by proving that it contains a nonzero scalar multiple of ω.
Fix a basis w_{i} of W and a complementary basis w_{i}′ of W′ so that
- w_{i}w_{j}′ +w_{j}′ w_{i} = δ_{ij}, and
- (w_{i})^{2} = 0, (w_{i}′)^{2} = 0.
Note that any element of I must have the form αω, by virtue of our assumption that I ⊂ Cℓ(V, g) ω. Let αω ∈ I be any such element. Using the chosen basis, we may write
- $ \alpha = \sum_{i_1<i_2<\cdots<i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p} + \sum_j B_jw'_j $
where the a_{i1…ip} are scalars, and the B_{j} are auxiliary elements of the Clifford algebra. Observe now that the product
- $ \alpha\omega = \sum_{i_1<i_2<\cdots<i_p} a_{i_1\dots i_p}w_{i_1}\cdots w_{i_p} \omega. $
Pick any nonzero monomial a in the expansion of α with maximal homogeneous degree in the elements w_{i}:
- $ a = a_{i_1\dots i_{max}}w_{i_1}\dots w_{i_{max}} $ (no summation implied),
then
- $ w_{i_{max}}\cdots w_{i_1}\alpha\omega = a_{i_1\dots i_{max}}\omega $
is a nonzero scalar multiple of ω, as required.
Note that for n even, this computation also shows that
- $ \Delta = \mathrm{C}\ell(W)\omega = (\Lambda^* W)\omega $.
as a vector space. In the last equality we again used that W is isotropic. In physics terms, this shows that Δ is built up like a Fock space by creating spinors using anti-commuting creation operators in W acting on a vacuum ω.
Exterior algebra construction Edit this section
The computations with the minimal ideal construction suggest that a spinor representation can also be defined directly using the exterior algebra Λ^{∗} W = ⊕_{j} Λ^{j} W of the isotropic subspace W. Let Δ = Λ^{∗} W denote the exterior algebra of W considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors.^{[5]}
The action of the Clifford algebra on Δ is defined first by giving the action of an element of V on Δ, and then showing that this action respects the Clifford relation and so extends to a homomorphism of the full Clifford algebra into the endomorphism ring End(Δ) by the universal property of Clifford algebras. The details differ slightly according to whether the dimension of V is even or odd.
When dim(V) is even, V = W ⊕ W′ where W′ is the chosen isotropic complement. Hence any v ∈ V decomposes uniquely as v = w + w′ with w ∈ W and w′ ∈ W′. The action of v on a spinor is given by
- $ c(v) w_1 \wedge\cdots\wedge w_n = (\epsilon(w) + i(w'))\left(w_1 \wedge\cdots\wedge w_n\right) $
where i(w′) is interior product with w′ using the non degenerate quadratic form to identify V with V^{∗}, and ε(w) denotes the exterior product. It may be verified that
- c(u)c(v) + c(v)c(u) = 2 g(u,v),
and so c respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ).
The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group^{[6]} (the half-spin representations, or Weyl spinors) via
- $ \Delta_+ = \Lambda^{even} W,\, \Delta_- = \Lambda^{odd} W $.
When dim(V) is odd, V = W ⊕ U ⊕ W′, where U is spanned by a unit vector u orthogonal to W. The Clifford action c is defined as before on W ⊕ W′, while the Clifford action of (multiples of) u is defined by
- $ c(u) \alpha = \left\{\begin{matrix} \alpha&\hbox{if } \alpha\in \Lambda^{even} W\\ -\alpha&\hbox{if } \alpha\in \Lambda^{odd} W \end{matrix}\right. $
As before, one verifies that c respects the Clifford relations, and so induces a homomorphism.
Hermitian vector spaces and spinors Edit this section
If the vector space V has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural.
The main example is the case that the real vector space V is a hermitian vector space (V, h), i.e., V is equipped with a complex structure J that is an orthogonal transformation with respect to the inner product g on V. Then V ⊗_{R} C splits in the ±i eigenspaces of J. These eigenspaces are isotropic for the complexification of g and can be identified with the complex vector space (V, J) and its complex conjugate (V, −J). Therefore for a hermitian vector space (V, h) the vector space ΛTemplate:SuV (as well as its complex conjugate ΛTemplate:SuV) is a spinor space for the underlying real euclidean vector space.
With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an almost Hermitian manifold and is the reason why every almost complex manifold (in particular every symplectic manifold) has a Spin^{c} structure. Likewise, every complex vector bundle on a manifold carries a Spin^{c} structure.^{[7]}
NotesEdit this section
- ↑ These are the right-handed Weyl spinors in two dimensions. For the left-handed Weyl spinors, the representation is via γ(ϕ) = γϕ. The Majorana spinors are the common underlying real representation for the Weyl representations.
- ↑ Since, for a skew field, the kernel of the representation must be trivial. So inequivalent representations can only arise via an automorphism of the skew-field. In this case, there are a pair of equivalent representations: γ(ϕ) = γϕ, and its quaternionic conjugate γ(ϕ) = ϕγ.
- ↑ The complex spinors are obtained as the representations of the tensor product H ⊗_{R} C = Mat_{2}(C). These are considered in more detail in spinors in three dimensions.
- ↑ This construction is due to Cartan. The treatment here is based on Script error.
- ↑ One source for this subsection is Script error.
- ↑ Via the even-graded Clifford algebra.
- ↑ Script error.