Structures of spinors fiber bundles with special relativity of Dirac operator using the Clifford algebra

Definition 1

A quadratic form q is an operator from a vector space V _k to its fields k, such that if α ∈ k and δ, ɳ ∈ V _k , then q(αɳ) = α ² q(ɳ) and 2q(δ, ɳ) = q(δ + ɳ) − q(δ) − q(ɳ) is a symmetric bilinear form on V _k [2]. A nondegenerate quadratic form is a quadratic form with the extra condition that q(ɳ) = 0 ⇔ ɳ = 0 [1].

Definition 2

Let V _k be a vector field over a field k and take the tensor algebra T(V _k ) = ∑ i = 0 ∞ ⊕ i V k = k ⊕ V _k ⊕ V _k ⊗ V _k ⊕ … and let q be a quadratic form [1]. A Clifford algebra Cℓ(V _k , q) on V _k is then Cℓ(V _k , q) = T(V _k )/I(V _k , q) with I(V _k , q), the two-sided ideal generated by ɳ ⊗ ɳ + q(ɳ) [1,2]. This is an associative algebra with unit, the exterior algebra can be related with the Clifford algebra with quadratic form 0. Defining the Clifford algebra and deriving its basic properties, we follow [2], but we could have equivalently started with the universal property proven below and worked our way back from there.

Lemma 1

Every vector space with a quadratic form has a q-orthogonal basis {e ₁,…, e _n}, that is, q(e _i , e _j ) = 0 if i ≠ j [1].

Definition 3

(Pin and Spin groups)

The group Pin is the following subgroup of the Clifford group:

i.e., we define P(V _k , q) ⊂ P̌(V _k , q) (where P̌(V _k , q) calls Lipschitz group which is “largest” spinor group) to be the subgroup of Cℓ(V _k , q) generated by the elements ɳ ∈ V _k with q(ɳ) ≠ 0. And that there is a representation

The group spin is the even subgroup of Pin: Spin(V _k , q) = Pin(V _k , q) ∩ Cℓ⁰(V _k , q) [1,2].

Theorem 2

Let k be a spin field. Then, Pin(V _k , q) is the kernel of N : P̌(V _k , q) → k ^×, (where k ^× is nonzero multiples of 1) and the twisted adjoint Ad ˜ |_{Pin(V k,  q)} is a surjection of Pin(V _k , q) on O(V _k , q). Then, the sequence:

with µ ₄(k) the fourth roots of 1 in the field. For example, µ ₄(ℝ) = {−1, 1} ≈ ℤ₂. And µ ₄(ℂ) = {−1, 1, −i, i} ≈ ℤ₄ ^(1,10).

Proof

See [1].□

We can also consider the covering group of SO(V _k , q) = {x ∈ O(V _k , q) | det(x) = 1}. Then, we have the following:

Corollary 1

The group Spin(V _k , q) is a cover of SO(V _k , q), and we have the exact sequence:

with µ ₄(k) as in Theorem 2 [1,3].

Proof

Let x ∈ Pin(V _k , q). Then, Ad ˜ x is equal to the composition of a number of reflections due to Theorem 2. We have that for all v ∈ V _k , det ( Ad ˜ η ) = − 1 . To see this, take a q-orthogonal basis with ɳ ₁ = ɳ and q(ɳ, ɳ _j ) = 0 for all j > 1. Then, Ad ˜ η ( η 1 ) = − η by definition and Ad ˜ η ( η j ) = η j with j > 1 by definition, so det ( Ad ˜ η ) = − 1 . So any element of SO(V _k , q) is generated by an even number of reflection, and so Spin(V _k , q) must be generated by an even number of vectors, so by the properties of the ℤ₂ grading we get that it is also an element of Cℓ ⁰ (V _k , q). The exact sequence follows immediately from Theorem 2 [1].□

Proposition 3

For any x in Spin(n), there exists an even integer p and elements f ₁,…, f _p of norm 1 such that x = f ₁,…, f _p . The reverse statement also holds [4].

Theorem 4

For any n ≥ 2, Spin(n) is connected. For n ≥ 3, Spin(n) is simply connected; hence, Spin(n) is the universal covering Lie group of SO(n).

Proof

Then, the short exact sequence of Lie groups.

Since π ₁SO(n) = ℤ₂ for any n ≥ 3, any connected couple covering of SO(n) is simply connected for n ≥ 3. Hence, it suffices to show that Spin(n) is connected for any n ≥ 2 [3].

First, let us show that 1 and −1 are path connected in Spin(n). Since the dimension is at least 2, there are elements e ₁ and e ₂ in V _k , such that ‖e ₁‖ = ‖e ₂‖ = 1 and ⟨e ₁, e ₂⟩ = 0. For 0 ≤ t ≤ π, we define γ(t) = cos t + e ₁ e ₂ sin t = e ₁(−e ₁ cos t + e ₂ sin t). It is trivial that the norm (−e ₁ cos t + e ₂ sin t) is 1. By Proposition 3, γ(t) is an element of Spin(n) for any t. We have γ(0) = 1 and γ(π) = −1.

Second, note that any element y of Spin(n) can be connected with y by the path yγ(t).

Finally, let x and y be in Spin(n). Since SO(n) is connected, there exists a path from ρ(x) to ρ(y) in SO(n). We may lift this path to a path starting in x and ending in a point y′ of Spin(n). Since the kernel of ρ equals {±1}, it holds that y′ = y or y′ = −y. In the first case, we are done. In the second case, we connect x with −y and then connect −y with y [4].□

Theorem 5

There are isomorphisms

or all n, r, s ≥ 0 [2].

Proof

See [1].□

Remark 1

It is standard notation to write: q _r,s ≡ q, O _r.s ≡ O(V _k , q) and SO _r,s ≡ SO(V _k , q). In accordance, we write Pin _r,s ≡ Pin(V _k , q) and Spin _r,s ≡ Spin(V _k , q). Similarly, it is conventional to write O _n ≡ O _n,0 ≅ O _0,n and SO _n ≡ SO _n,0 ≅ SO _0,n . Thus, we set Pin _n = Pin _n,0 , and Spin _n = Spin _n,0 .

Definition 4

A section of Clifford Cℓ(V _k , q) is called Clifford field [5].

Definition 5

Clifford’s complex algebra Cℓ^ℂ(V _k , Q) is a Clifford algebra that is constructed by starting with a complex vector space V _k ⊗_ℝ ℂ, and Q extends through the complex linearity and then using the definition as real case. By starting with a real vector space V _k of dimension n, then this is denoted by Cℓ^ℂ(n). One can easily see that Cℓ^ℂ(n) = Cℓ(n) ⊗ Cℓ. Build a spin representation as being reversible elements in Cℓ(n) are complexified, producing a structure of Spin(n, ℂ). (The complexification of Spin(n)) is a reversible element in Cℓ^ℂ(n).)

By an inductive argument. The algebras Cℓ ^ℂ (n) constructed to begin

so

and

Theorem 6

Cℓ^ℂ(n + 2) = Cℓ^ℂ(n) ⊗_ℂ Cℓ^ℂ(2) = Cℓ^ℂ(n) ⊗_ℂ M(2, ℂ).

Corollary 2

If n = 2k, Cℓ^ℂ(2k) = M(2, ℂ) ⊗⋯⊗ M(2, ℂ) = M(2 ^K , ℂ), where the product has k factors, and if n = 2k + 1, Cℓ^ℂ(2k + 1) = Cℓ^ℂ(1) ⊗ M(2 ^k , ℂ) = M(2 ^K , ℂ) ⊕ M(2 ^K , ℂ) [3].

Proof

Choose generators g ₁, g ₂ of Cℓ(2), f ₁,…, f _n of Cℓ(n) and ĕ ₁,…, ĕ _n+2 of Cℓ(n + 2). Then, we get the symmetry by the following map:

If n = 2k (even case). In this case, Clifford’s complex algebra is the algebra of 2 ^k by 2 ^k complex matrices.□

Definition 6

A spin structure on a pseudo-Riemannian vector bundle with signature (r, s) E is a principal Spin(r, s) bundle P _Spin(E) together with a two-sheeted covering ξ: P _Spin(E) → P _SO(E), such that ξ(pg) = ξ(p)ξ ₀(g) for all p ∈ P _Spin and g ∈ Spin(r, s) and ξ₀ the covering map Spin(r, s) → SO(r, s) [2,3].

Two spin structures P _Spin(E) and P Spin ' (E) are called equivalent if there is a mapping F, such that F(gh) = F(g)h for g ∈ P _Spin and h ∈ Spin and the following diagram commutes:

This means that if they are equivalent as spin structures, they are equivalent as principal fiber bundles [1,2].

From now on, we will only consider r, s such that π ₁(SO(r, s)) = ℤ₂. This makes many proofs much simpler since it makes Spin_(r,s) simply connected. Also, this situation is the one where most examples interesting to physics are, when r ≥ 3 and s = 0 or 1, or vice-versa. The definition of the spin structure gives the following commutative diagram:

Since restriction to fibers gives the covering map ξ ₀, this diagram can be extended [1,2]:

In the above figure, we see that the vertical lines show the inclusions of fibers in the fiber bundle. Now we can consider whether given a two-sheeted covering φ: C ₂ (E) → P _SO(E) of P _SO(E) gives a spin structure. It certainly gives a fiber bundle over E since we can set π′ = π _◦ ξ.

Now we see that this bundle gives a spin structure if the covering is nontrivial on the fibers.

Theorem 7

If π ₁(SO(r, s)) = ℤ₂, then the spin structures are in one-to-one correspondence with two-sheeted coverings of P _SO(E), which are nontrivial on the fibers [1,2].

Lemma 8

Proof

Suppose α _F ∈ ξ _∗ (π ₁(P _Spin)). Then, the inclusion map i: SO → P _SO lifts to a continuous map.

I: SO → P _Spin such that commutes. I (SO) ⊂ P _Spin is contained in one fiber Spin of P _Spin, so I: SO(r, s) → Spin_(r,s) with ξ ₀ ◦ I = IdSO(r, s) = Idℤ ₂ . Then, ξ _0∗ I _∗ = Idπ ₁(SO(r, s)) and π ₁(SO(r, s)) = ℤ₂ and π ₁(Spin(r, s)) = 1, so we get a contradiction, so α _F ∉ ξ _∗ (π ₁(P _Spin)) [6].

This lemma is then used to prove a classification theorem of spin structures, following and using the classification of covering spaces.□

Theorem 9

A SO(r, s)-principal fiber bundle P _SO over a manifold M has a spin structure if and only if there is a short split exact sequence.

meaning that π ₁(P _SO) is isomorphic to K × ℤ₂ and π _∗, the map induced by projection map, maps K isomorphically to π ₁(M).

Lemma 10

The map π′_∗: π ₁(P _Spin) → π ₁(M) is an isomorphism.

Proof

First, we prove surjectivity. Take an element [g] ∈ π ₁(M) represented by a loop g: [0, 1] → M based at m ₀ ∈ M. We can then look at a path g̃: [0, 1] → P _Spin such that πg̃(t) = g(t). This path exists because P _Spin is a fiber bundle over M. This path does not have to be a loop. However, π(g̃(0)) = π(g̃(1)) = m ₀ and the fibers of P _Spin are path connected, so we can find a path δ in spin such that δ(0) = 1 and δ(1)g̃(1) = g(0). The product path δg̃ is now a loop and π(δg̃) = g. This proves surjectivity.

As for injectivity, consider a loop h̃: [0, 1] → P _Spin and assume π(h̃)(t) = h(t) is homotopic to trivial loop, say at the point x ₀ = h(0). There is then a homotopy from the loop h to the point x ₀. This homotopy can be covered by a homotopy of h̃ into a new loop lying in the fiber π ⁻¹(x ₀), or its generalization to fiber bundles over paracompact spaces [7]. Since spin is simply connected, there is then a homotopy to a single point in this fiber; hence, h̃ is homotopic to the trivial loop, so π _∗ is injective.□

Lemma 11

The map π _∗ : π ₁(P _SO) → π ₁(M) restricted to the image of ξ _∗ (π ₁(P _Spin)) is an isomorphism.

Proof

Call the image of ξ _∗ K. We have the following commutative diagram:

The map ξ _∗ is injective, and according to the covering space, morphism is injective. We also know that π′ is an isomorphism, so π _∗ |k is injective. Furthermore, π _∗ |k◦ξ _∗ = π′ is an isomorphism, so π _∗|k must be surjective, hence an isomorphism [1].□

Lemma 12

The map i _∗: π ₁(SO) = ℤ₂ → π ₁(P _SO) is injective, so it is an isomorphism onto its image.

Proof

Suppose [g] ∈ π ₁(SO) and i _∗[g] = [α] ∈ π ₁(P _SO). If [α] is trivial, then we can lift α to a path α ˆ in P _Spin. The loop α lies within a single fiber, so the loop α ˆ also does. Because Spin is simply connected, there is a homotopy between α ˆ and the trivial loop within the fiber. Applying the covering map to of P _SO, SO, so [g] is trivial and i _∗ is injective.□

Lemma 13

The sequence of Lemma 11 is exact, in particular: ker(π _∗) = img(i _∗) [1,2].

Proof

Take a [g] ∈ π ₁(SO). Then, i _∗[g] has a representative lying within a single fiber. Then, π _∗ i _∗[g] = [e] and hence, im(i _∗) ⊆ ker(π _∗). Take a loop α ⊆ P _SO. We will construct a loop α ˆ ⊂ i(P _Spin) such that α = α ˆ g with g a loop in SO. First, let α′ = π_◦ α be a loop in M. Then, we lift this loop one in P _Spin by lifting and multiplying with a path δ with δ(0) = 1 and α′(1)δ(1) = α(0). Now define α ˆ = ξ(α′δ). We can now conclude that [ α ˆ ] = ξ ∗ π ' ∗ − 1 π ∗ ( [ α ] ) and that π(α) = π( α ˆ ). Thus, we have that α = α ˆ g for some loop g ⊆ SO; hence, [α] = [ α ˆ g] = [ α ˆ ]i _∗[g], so if [α] ∈ ker(π_∗) we have that because π_∗ is an isomorphism on the image of ξ_∗, that [α] = [ α ˆ ]i _∗[g] = [e]i _∗[g], so ker(π _∗) ⊆ im(i _∗) [1].□

Definition 7

We call global sections of P _SO(1,3)(M) Lorentz frames and global sections of P _Spin(1,3)(M) spin frames [5].

Remark 2

When we will use the concept of the spin structure in physics, the space–time is fourdimensional manifold with a metric of signature (3, 1) or (1, 3), we will naturally use SL(2, ℂ) as group instead of Spin(p, q).

Definition 8

A smooth manifold endowed with a spin structure will be called a spin manifold [5].

Definition 9

(Spinor bundles). Let X be a smooth manifold with a spin structure ξ: P _Spin(X) → P _SO(X). A real or complex spinor bundle of X is a bundle of the form [1–3]

where L is a left module of Cℓ _r,s and µ: Spin(r, s) → End (L) is left-multiplication of elements in Spin(r, s), and where L _ℂ is a complex left module for Cℓ(ℝ ⁿ ) ⊗ ℂ and µ: Spin(r, s) → End (L _ℂ) is the left-multiplication of elements in Spin(r, s). When (r, s) = (1, 3) defined as certain equivalence classes in appropriate sets, and a first definition for field of these object on living Minkowski space time was given [5].

Remark 3

The complexified left spin Clifford bundle denoted by

and the complexified right Clifford bundle by

Remark 4

Taking, e.g., L _ℂ = ℂ⁴ and µ _ℂ the D ^(1\2,0)   ⊕ D ^(O,1\2) of Spin_1,3 ≌ SL(2, ℂ) in End(ℂ⁴) and L is covariant spinor bundle [5] (where D is Dirac operator see the last sections).

If the module L(or L _ℂ ) is ℤ₂ graded, the corresponding bundle is said to be ℤ₂ graded.

Example 1

Consider Cℓ(ℝ ⁿ ) as a module over itself by left multiplication l. The corresponding real spinor bundle Cℓ_Spin(X) = P _Spin(X) × _l Cℓ(ℝ ⁿ ), then:

1. Cℓ_Spin(X) is a “principal Cℓ(ℝ ⁿ )-bundle,” i.e., it admits a free action of Cℓ(ℝ ⁿ ) on the right.

2. There is a natural embedding P _Spin(X) ⊂ Cℓ_Spin(X), which comes from the embedding Spin _n ⊂ Cℓ(ℝ ⁿ ). Hence, every real spinor bundle for X can be captured from this one [2].

Of course, the bundle Cℓ_Spin(X) differs from the Clifford bundle Cℓ(X). They can be compared as follows.

given by Ad _g (ϕ) = gφg ^−l for g ∈ Spin ⁿ ⊂ Cℓ(ℝ ⁿ ). Clearly Ad₋₁ = identity, and this representation come to acting Ad′ of SO _n . One easily checks that Ad′ is just the representation Cℓ( ρ n ) given by

Two spinor bundles of X are equivalent if they are equivalent as bundles of Cℓ(X)-modules.

Real bundle, complex bundle, graded, and ungraded bundle of Cℓ(X) modules is called irreducible if at each x fiber is irreducible as a (real or complex, graded or ungraded) module over Cℓ(X _x ) [2].

Then, when X is a spin manifold;

1. the components of Cℓ (X) = P _Spin(X) ×_Ad Cℓ(ℝ ⁿ ) are homomorphism classes [(β, α)] of pairs (β, α), where β ∈ P _Spin(X), α ∈ Cℓ(ℝ ⁿ ) and (β, α) ∼ (β′, α′) ⇔ β′ = βu ⁻¹, α′ = uαu ⁻¹, for some u ∈ Spin.

2. the components of C ℓ l _Spin(X) are homomorphism classes of pairs (β, α) where β ∈ P _Spin(X), α ∈ C ℓ ( ℝ n ) and ( β , α ) ∼ ( β ′ , α ′ ) ⇔ β ′ = β u − 1 , α ′ = α u − 1 , for some u ∈ Spin .

3. the components of C ℓ r _Spin(X) are homomorphism classes of pairs (β, α), where β ∈ P _Spin(X), α ∈ C ℓ ( ℝ n ) and ( β , α ) ∼ ( β ' , α ' ) ⇔ β ' = β u − 1 , α ' = α u − 1 , for some u ∈ Spin .

Proposition 14

There is a natural pairing secCℓ ^l _Spin(X) × secCℓ ^r _Spin(X) = secCℓ(X).

Proof

Given δ ∈ secCℓ ^l _Spin(X), ρ ∈ secCℓ ^r _Spin(X) select representatives (β, α) for δ(x) and (β, b) for ρ(x), (with β ∈ π ⁻¹(x)) and define (δρ)(x) ≔ [(β, αb)] ∈ Cℓ (X). If alternative representatives (βu ⁻¹ , uα) and (βu ⁻¹ , bu ⁻¹ ) are chosen for δ(x) and ρ(x), we have (βu ⁻¹ , uαbu ⁻¹ ) ∼ (β, αb) and thus, (δρ)(x) is well define component of Cℓ(X) [5].

Let us now say a word about the ℤ₂-graded case. There is irreducible bundle of ℤ₂-graded modules over Cℓ(X) = Cℓ⁰(X) ⊕ Cℓ¹(X) and classes irreducible bundle of modules over Cℓ⁰(X). Given a bundle S(X) = S ⁰(X) ⊕ S ¹(X) of the first kind, S ⁰(X) is of the second. Given an S ⁰(X) of the second kind, the bundle

Suppose now that n = 2 m and S _ℂ(X) is the irreducible complex spinor bundle of X. We will show clearly how to split S _ℂ(X) into a direct sum:

of Cℓ⁰(X) modules. Interpreting S ℂ + ( X ) as S ℂ 0 ( X ) and S ℂ − ( X ) as S ℂ 1 ( X ) , or the other way around, gives a ℤ₂-graded module structure to S ℂ ( X ) .

There is a similar construction in the real case.

Recall that every module for Cℓ(ℝ ⁿ ) is a direct sum of irreducible ones, and there are at most two homomorphism classes of irreducible modules [2].□

Proposition 15

If S(X) is a real spinor bundle of X, then S(X) is a bundle of modules over the bundle of algebras Cℓ(X). In particular, the sections of the spinor bundle are a module over the sections of the Clifford bundle [5].

Proof

See [2].□

Remark 5

The corresponding fact holds in the complex and ℤ₂-graded cases and Sections of S(X) are called spinors.

Definition 10

The dual spinor bundle S ^∗ (X) is a real or a complex spinor bundle

where L ^∗ is a right module of Cℓ _r,s and µ: Spin(r, s) → End(L) is the representation given by right-multiplication of (inverse) elements in Spin(r, s), and where L ^∗ _ℂ is a complex right module for Cℓ(ℝ ⁿ ) ⊗ ℂ and µ: Spin(r, s) → End(L _ℂ) is the representation given by right-multiplication of inverse elements in Spin(r, s) [5].

Definition 11

If U is a normal bundle of the Grassmannian. Then, if s ^∗ U = S _(X) the spinor bundle on Q _2k+1. Its rank is 2 ^k . We call s ′ ^∗ U = S ′ ( X ) and s ″ ^∗ U ≃ S ″ ( X ) the two spinor bundles on Q _2k . Their rank is 2 ^k−1.

If f is an automorphism of Q _2k that exchanges the two families of k-planes, we have

Clear that S _(X) (spinor bundles) on all quadrics Q are homogeneous, i.e., f ^∗ S _(X) ≃ S _(X) for all f ∈ Aut(Q)₀, where Aut(Q)₀ is the connected component of the identity in Aut(Q).

Theorem 16

1. Let S ′ ( X ) , S″_(X) are spinor bundles on Q _2k , and let i: Q _2k-1 → Q _2k be a smooth hyper plane section. Then i ^∗ S ′ ( X ) ≃ i ^∗ S ″ ( X ) ≃ S _(X), where S _(X) is the spinor bundle on Q _2k−1.

2. Let S _(X) is spinor bundle on Q _2k+i , and let i: Q _2k → Q _2k+i be a smooth hyperplane section.

Then, i ^∗ S _(X) ≃ S ′ ( X ) ⊕ S ″ ( X ) , where S ′ ( X ) and S ″ ( X ) are the spinor bundles on Q _2k [11].

Example 2

Two embeddings s′: Q₄ → Gr(l, 3) and S″: Q ₄ → Gr(l, 3) are isomorphisms from the definition of spinor bundles. So S _(X) on Q ₄ are the normal bundle and the dual of the quotient bundle.

The embedding S: Q ₃ → Gr(l, 3) corresponds to a hyperplane section. If S _(X) is the spinor bundle on Q₃, then S ² _(X) S ^∗ _(X) = TQ₃. In fact, TQ₄/Q₃ ≃ S ^∗ _(X) ⊕ S ^∗ _(X) ≃ S ² _(X) S ^∗ _(X) ⊕ (1) and the exact sequence splits

On Q ₂ the two of S _(X) are the duals of the two line bundles corresponding to two skew-lines on Q ₂. On Q ₁ ≃ ℙ¹, then the spinor bundle can defined to be _ℙ¹ (−1) [11].

Corollary 3

Let l ⊂ Q _n (n > 3) be a line and let S _(X) on Q _n [11]. Then

Proposition 17

Let (M, g) be an smooth Riemannian manifold with (V, h) as the local model.

1. There corresponds a Clifford ^c structure to every spinor bundle such that the associated spinor bundle is isomorphic to τ: Cℓ(g) → End Σ [3].

2. The Clifford ^c structure can be reduced to a spin structure iff the line bundle above is trivial.

3. The Clifford ^c structure can be reduced to a Clifford structure iff the bundle of circles above is trivial.

4. The resulting Clifford structure can be reduced to a spin structure iff the real line bundle above is trivial [13,14].

Proof

We already know that Hom C ℓ M ℂ ( S , S ¯ ) is a complex line bundle: each fiber contains a Cℓ(T _p M ℂ )-linear isomorphism τ _p : S _p → S̄ _p and by irreducibility of S _p and S̄ _p and Schur’s Lemma any Cℓ(T _p M ℂ )-linear map τ p ′ : S̄ _p → S _p satisfies τ p ′ ◦ τ _p = λ/S _p for some λ ∈ ℂ . In case of a spin structure τ defines a non-vanishing section in Hom C ℓ M ℂ ( S , S ¯ ) , hence

Hom C ℓ M ℂ ( S , S ¯ ) is trivial. Conversely, if this bundle is trivial and if τ̃′ is a nonvanishing section, then τ̃′² = λ/ _S for some nonvanishing map λ ∈ ℂ ∞ (M, ℂ ) . But λ(p) τ̃ _p (v) = τ̃ _p ◦ τ̃ _p ◦ τ̃ _p (v) = τ ˜ ( λ ( p ) v ) = λ ( p ) ¯ τ ˜ ( v ) , i.e., λ ∈ ℂ ∞ (M) is real valued, and replacing τ̃ by τ = |λ|^−1/2 τ̃, we obtain a structural map.

Now given an irreducible complex spinor bundle S and a structural map τ, we can choose a Riemannian structure compatible with Clifford multiplication and such that τ is an isometry.□

Definition 12

Suppose E is a smooth Riemannian vector bundle over a manifold X and that ξ: P _Spin(E) → P _SO(E) is a spin structure on E. Then, of course, any connection on P _SO(E) can be lifted via ξ to a connection on P _Spin(E), and this, in turn, defines a connection on the associated spinor bundles [2].

We give two equivalent definitions of a connection on a principal G-bundle with projection P → π X.

First, we define the vertical tangent space T p v P at p as the subspace of T p P , which is tangent to the fiber of the projection P → π X. A connection on P → π X is a smoothly varying family of linear subspaces ( T p H ) _p∈P of the tangent bundle T P, which is everywhere complementary to the vertical distribution ( T p v V ) _p∈P and which is invariant under the action of the group G. The distribution ( T p H ) _p∈P is called the horizontal distribution. Note that

Second, to give a connection on P → π X is to give a connection 1-form ω on P with values in g satisfying two conditions.

It transforms by the adjoint action, i.e., for any ɡ in G, p in P and any γ in T _p P, we have

For any a in g, the associated vector field Va on P defined by the tangent vector of the curve pe ^ta at any p ∈ P. It holds that ω(V _a ) = a.

Note that the horizontal distribution can be recaptured by taking the kernel of the connection 1-form assigned to it.

Lemma 18

Given a Euclidean connection on a real vector bundle E, there is a canonical orthogonal connection (i.e., the decomposition T P = T V P ⊕ T H P is Euclidean) on its orthonormal frame bundle P → SO ( n ) X . Reversely, any orthogonal connection on the frame bundle P → S O ( n ) X induces a Euclidean connection on E. Furthermore, these operations are inverse to each other [4].

Lemma 19

Given a Lie group G and a connection on a principal G-bundle P → π X, there is an induced connection on any vector bundle P × G V coming from a linear representation G → GL(V) [4].

The curvature g-valued 2-form Ω is

Example 3

(Orthogonal connections). Let P = P _SO(E), where E is a smooth, oriented Riemannian vector bundle. The Lie algebra SO _n of real, skew-symmetric n × n-matrices. Hence, a connection 1-form ω can be considered as an n × n-matrix of 1-forms ω = (ω _ij ), where ω i j = − ω j i ″ .

The corresponding curvature is a matrix of 2-forms Ω = Ω _ij, where

Suppose that μ = (e ₁,…, e _n ) is just a section of P _SO(E) over U ⊆ X , and it can be lifted to a section μ ˜ of P _Spin(E) over U. There are two possible such liftings. They satisfy the relation:

The connection 1-form on P _Spin(E) is just the lift ξ ^∗ ω (the pull down) of the connection 1-form ω on P _SO(E) [2].

Theorem 20

Let ω be the connection i-form on P _SO(E) and let S(E) be any spinor bundle associated to E. Then, the covariant derivative ∇ ^s on S(E) is given locally by the formula [2]:

where μ = (e ₁,…, e _n ) is a local section of P _SO(E), ω = μ ^∗(ω), and where E = (σ ₁,…, σ _n ) is a local section of P _SO(S(E)) determined by ω.

Theorem 21

Let Ω be the curvature 2-form on P _SO(E) and let S(E) be any spinor bundle associated to E. Then, the curvature ℛ by the formula S of S(E) is locally given by

where μ = (e ₁,…, e_n ) is a section of P _SO(E), Ω ˜ = μ ∗ ( Ω ) and σ is any section of S(E) [2].

Then, any two tangent vectors V and W at x ∈ X , the curvature transformation ℛ V , W s : S(E _x ) → S(E _x ) is

where ℛ v , w is the curvature transformation of E _x .

Definition 13

(Dirac operator). Let U _ι be an open subset of M and let e = (e _µ ) _µ=1,…,m be a field of (not necessarily orthonormal) frames on U _ι . For every p ∈ U _ι , the components of the metric tensor g with respect to e at p are g _µν (p) = g(e _µ (p), e _ν (p)) and there is the inverse g ^µν (p) of g _µν (p). The restriction of the Dirac operator to U _ι is [2] expressed as follows:

The Dirac operator on M well defined by its restrictions to the sets (U) providing an open cover of M.

∇ ^TM is connection on the spinor bundle to be metric, but may have torsion.

Definition 14

A Dirac bundle is a bundle S over a Riemannian manifold X of left modules over Cℓ(X) together with a Riemannian metric and connection on S with properties:

e 1 σ 1 , e 2 σ 2 = σ 1 , σ 2 and ∇ ( φ σ ) = ( ∇ φ ) σ + φ ( ∇ σ ) for all φ ∈ Γ ( C ℓ ( X )   ) , σ ∈ Γ ( S ) .

The operator D is elliptic if the linear map σ _ξ (D): E _x → E _x is an isomorphism for all ξ ≠ 0.

Lemma 22

If D is the Dirac operator of the bundle S defined above. Then, for any ξ ∈ T ∗ ( X ) ≅ T ( X ) , we have that

where the symbol on the right denotes Clifford multiplication by the vector ξ and the scalar ∥ ξ 2 ∥ . In particular, both D and D ² are elliptic operators.

Proof

See [2].□

Theorem 23

Let X be a complete Riemannian manifold and if D is the Dirac operator of Dirac bundle S over X. Then, the closure of D in L ²(S) is a self-adjoint operator. Furthermore, ker(D) = ker(D ²) on L ²(S), where L is a canonical bundle map L: Cℓ(X) → Cℓ(X) defined by L ( φ ) = − ∑ e j φ e j [2].

Lemma 24

Let D, S and X be as above. Then for any f ∈ C ∞ ( X ) and any φ ∈ Γ ( S ) ⊂ C ∞ ( X ) , we have that

Proof

D(fφ) = ∑ e j ⋅ ∇ e j ( f φ )   = ∑ e j { ( e j f ) φ   +   f ∇ e j φ } = ∑ ( ( e j f )   e j ⋅ φ )   +   f D φ )   = (grad f) ⋅ φ + fDφ [2].

Having discussed Dirac bundles in general terms, it is now time to look hard at some important examples. We begin with the basic ones.□

Example 4

(An historical case). Let X = R ⁿ , Euclidean n-space, and let S = R ⁿ × V, where V some module for Cℓ _n . Here, D is a constant on V-valued functions [2]

and each γ k is a linear map γ k :V → V and γ j γ k + γ k γ j = − 2 δ j k .

This particular operator has historical roots in physics. In the 1920s [2], the physicist P.A.M. Dirac was searching for a Lorentz-invariant first-order differential operator whose square would be the Klein-Gordon operator. Thus, he was essentially led to search for a first-order operator D of the form above, which satisfied the equation D ² = ∆, where ∆ = −∑∂²/∂x² is the positive Laplacian in ℝ ⁿ . Realizing that the γ _k s must be matrices, he was led immediately by this equation to the aforementioned relations, which we recognize now as the generating relations of a representation of Cℓ _n .

Let n = 1, so that C ℓ 1 = V = ℂ ≅ ℝ 2 . Then we have [3,8]

the generator of a basic semi-group of unitary operators on L ².

Let n = 2, so that C ℓ 2 = V = ℍ ≅ ℂ ⊗ ℂ . The construction of ℍ into ℂ ⊗ ℂ is natural and corresponds to the ℤ₂-grading C ℓ 2 0 ⊗ C ℓ 2 1 [3]

has the form D = 0 − ∂ ∂ z ∂ ∂ z ¯ 0 , where ∂ / ∂ z ¯ = ∂ / ∂ x 1 + i ∂ / ∂ x 2 .

Let n = 3, so that C ℓ 3 = ℍ ⊗ ℍ and V = ℍ. Cℓ₃ has two representations on ℍ given as Identify ℝ³ with Im(ℍ), by letting i, j, and k act on either the right or the left in ℍ. On the left, we get Dirac operator on ℍ:

Let n = 4, so that Cℓ₄ = ℍ(2) and V = ℍ 2 = ℍ ⊗ ℍ . To describe the full Dirac operator, we consider first the following ℍ under the basis (1, i, j, k):

then D = 0 − ∂ ∂ q ∂ ∂ q ¯ 0 .

Thus, left multiplication be represented by complex 2 × 2-matrices σ ₀, σ ₂ and σ ₃ respectively, then the operator ∂ / ∂ q ¯ becomes

The matrices σ _k can be chosen to be the classical Pauli matrices:

Note that these matrices generate the fundamental representation of Cℓ₃ in complex form [2,3].

Example 5

(The Clifford bundle). Let S = Cℓ(X) with its canonical Riemannian connection, and view Cℓ(X) as a bundle of left modules over itself by left Clifford multiplication. The Dirac operator D in this case is a square root of the classical Hodge Laplacian [2].

Example 6

(The spinor bundles). Suppose X is a spin manifold with a spin structure on its tangent bundle. Let S _(X) be any spinor bundle associated with T(X). Then, S _(X) is a bundle of modules over Cℓ(X), and S _(X) carries a canonical Riemannian connection, which has property of Definition 14. The Dirac operator in this case was first written down by Atiyah and Singer in their work on the Index theorem. Finding this operator was a major accomplishment, and for this reason, we shall call it the Atiyah-Singer operator [2,15].