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


 The purpose of this article is to demonstrate how to use the mathematics of spinor bundles and their category. We have used the methods of principle fiber bundles obey thorough solid harmonic treatment of pseudo-Riemannian manifolds and spinor structures with Clifford algebras, which couple with Dirac operator to study important applications in cohomology theory.


Introduction
The interplay between physics and mathematics has spurred each of the disciplines to great heights. It has led to numerous discoveries, not the least of which is the Dirac operator and the related concept of the spinor. The first groundwork for these concepts was laid down by Clifford in the middle of the nineteenth century as a generalization of the quaternions of Hamilton and the exterior algebra [1] of Grassmann. In 1913 [1], Elie Cartan wrote down the general theory of spinors. Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927 [1], when he introduced his spin matrices. On the physics, Dirac introduced his famed operator in 1928 [1,2] but made no mention of the connection with spinors, and this was only done much later. During the years 1940-1970, the spinors and Clifford algebra of bundle became a fundamental tool of particle physics and came back later, at the forefront of differential geometry and of mathematics in general, with the recognition of the importance of the Dirac operator and theory of spinors in differential geometry.
Connections on fiber bundles and their relation with covariant derivation is discussed. In the final section, spin structures are introduced as nontrivial coverings of SO bundles, followed by a proof of necessary and the sufficient condition for their existence in terms of SO bundle and Dirac operator.

Clifford algebras
A Clifford algebra is a type of associative algebra, which can be considered a generalization of the usual associative fields such as ℝ, ℂ or ℍ. In fact, these fields can be seen as particular examples of Clifford  algebras [1]. It is defined over a vector space V over a field k denoted by V k equipped with a quadratic form q defined as follows: 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].
A Clifford algebra is a generalization of the exterior or Grassmann algebra precisely.
Definition 2. Let V k be a vector field over a field k and take the tensor algebra T( .. 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) ∩ Cℓ 0 (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: Ad with µ 4 (k) the fourth roots of 1 in the field. For example, µ 4 (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: Ad 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 , To see this, take a q-orthogonal basis with ɳ 1 = ɳ and q(ɳ, ɳ j ) = 0 for all j > 1. Then, 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 ℤ 2 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 1 SO(n) = ℤ 2 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].
Second, note that any element y of Spin(n) can be connected with y by the path yγ(t).
Note that here we are using the ungraded tensor product.
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].

Spinors
We will build representations with m-dimensions of spinor on a complex vector, and it will become clear that the complex Clifford algebra has a much simpler structure than the real one, a period of degree 2 instead of degree 8 as in the real case [3]. Now if m is even, m = 2n, we have Welly representation, which restricts to Cℓ 0 (V k , q).
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 To start the induction, we put n = 1, 2. 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 ξ 0 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)) = ℤ 2 . 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: 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)) = ℤ 2 , 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].
Now consider a spin structure ξ: P Spin → P SO . Define α F ∈ π 1 (SO (r,s)), the nontrivial element. The spin structure ξ induces a group homomorphism ξ * : π 1 (P) → π 1 (Q). This subgroup of π 1 (Q) is a subgroup of index 2 because of covering morphism, and P Spin is a double covering of P SO fiber wise.
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)) = ℤ 2 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 × ℤ 2 and π * , the map induced by projection map, maps K isomorphically to π 1 (M).
To prove this theorem, we need a few lemmas. First, we prove the necessity of the conditions, we assume P SO has a spin structure P Spin .
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 h̃into 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, h̃is 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]. □ is trivial, then we can lift α to a path α in P Spin . The loop α lies within a single fiber, so the loop α also does. Because Spin is simply connected, there is a homotopy between α 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 ⊂ i(P Spin ) such that α = αg 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 = ξ(α′δ). We can now conclude that [ ] and that π(α) = π(α). Thus, we have that α = αg for some loop g ⊆ SO; hence, If the fundamental group of SO(r, s) is not ℤ 2 , this result can be generalized, but the double covering space of SO(r, s) will not be a universal covering space.
Instead, one must look at either the universal covering space of SO(r, s), which is then not equal to Spin (r,s) , or one must look at not simply connected Spin (r,s) [1,6].
The condition of Lemma 11 for the existence of a spin structure can be shown to be equivalent with the usual condition that the second Stiefel-Whitney class w 2 of M vanishes [1,6,8]. For the special case of SO 0 (1, 3)-principal fiber bundles over a noncompact 4-manifold M, the case in general relativity, it has been shown that any existing spin structures are trivial, M × Spin(1, 3) and so it has a spin structure, if and only if the SO 0 (1, 3)-bundle is parallelizable, meaning that there is a global section of the SO 0 (1,3) bundle [1].
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).

Clifford algebras and spinor bundles
There are two equivalent ways of defining a Clifford bundle of a pseudo-Riemannian vector bundle E over a manifold X. One is the obvious generalization of Cℓ( n ): where q is a smooth quadratic form on E and q x is the restriction of that form to the fiber over x. This definition emphasizes that the Clifford bundle is a bundle of Clifford algebra's over X. The other definition uses associated bundles and can be used to determine the topology, as follows: An orthogonal transformation with respect to an inner product of signature (r, s) in (r+s) induces an orthogonal transformation in Cℓ r,s , since it preserves the ideal. This induced map preserves the multiplication in Cℓ r,s , so if we take an orthogonal transformation ρ r,s from SO(r, s), we get a map cl(ρ r,s ): SO r,s → Aut(Cℓ r,s ). The bundle associated to this bundle is called the Clifford bundle: where I(E) is the bundle of ideals [2]. It is also clear that all fundamental concepts on Clifford algebras carries over to Clifford bundles. For example, Cℓ(E) = Cℓ 0 (E) ⊕ Cℓ 1 (E) corresponding to the even-odd decomposition of the algebras.
These are the +1 and −1 eigen bundles of the bundle automorphism These two definitions are the same since the fiber at x ∈ E of P SO (E) × cl(ρ r,s ) Cℓ r,s is Cℓ r,s [10]. All notions familiar from Clifford algebras over real vector spaces carry over to Clifford bundles over manifolds. If X is a pseudo-Riemannian manifold, we can construct the Clifford bundle Cℓ(TX) associated with the pseudo-Riemannian form on the tangent bundle TX. We will also call this bundle Cℓ(X), in case there is no confusion possible [10].
Definition 8. A smooth manifold endowed with a spin structure will be called a spin manifold [5].   If the module L(or L ℂ ) is ℤ 2 graded, the corresponding bundle is said to be ℤ 2 graded.
(ii) 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].
A similar remark holds for the complex case. Of course, the bundle Cℓ Spin (X) differs from the Clifford bundle Cℓ(X). They can be compared as follows.
( ( )) → ℓ Ad: Spin Aut C n n 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

Spin Ad
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].
Let us now say a word about the ℤ 2 -graded case. There is irreducible bundle of ℤ 2 -graded modules over Cℓ(X) = Cℓ 0 (X) ⊕ Cℓ 1 (X) and classes irreducible bundle of modules over Cℓ 0 (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: , or the other way around, gives a ℤ 2graded module structure to ( ) S 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]. 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).
(ii) Let S (X) is spinor bundle on Q 2k+i , and let i: Q 2k → Q 2k+i be a smooth hyperplane section. Then ″ S X are the spinor bundles on Q 2k [11]. The embedding S: Q 3 → Gr(l, 3) corresponds to a hyperplane section. If S (X) is the spinor bundle on Q 3 , then S 2 (X) S * (X) = TQ 3 . In fact, TQ 4 /Q 3 ≃ 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 ≃ ℙ 1 , then the spinor bundle can defined to be O ℙ 1 (−1) [11].
Corollary 3. Let l ⊂ Q n (n > 3) be a line and let S (X) on Q n [11]. Then

Relations between spinor bundles and spinor structures
For every x ∈ M, the spinor representation x defines the real line a(τ x ) and the circle c(τ x ) [12]. The set x M x has the structure of a real line bundle over M. The set x M x has the structure of a bundle of circles over M: It is a principal U(1)-bundle. If this bundle is trivial, i.e., if it has a (global) section C: M → c(τ), then the real line bundle over M defined as follows: , H o m , .
x M x Proposition 17. Let (M, g) be an smooth Riemannian manifold with (V, h) as the local model. (i) 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]. (ii) The Clifford c structure can be reduced to a spin structure iff the line bundle above is trivial.
(iii) The Clifford c structure can be reduced to a Clifford structure iff the bundle of circles above is trivial. (iv) 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 is a complex line bundle: each fiber contains a Cℓ(T p M )-linear isomorphism τ p : S p → S̄p and by irreducibility of S p and S̄p and Schur's Lemma any Cℓ(T p M )-linear map ′ τ p : S̄p → S p satisfies ′ τ p • τ p = λ/S p for some λ ∈ . In case of a spin structure τ defines a non-vanishing section 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 ∈ ∞ (M, ) . But λ(p) τ̃p(v) = τ̃p • τ̃p • τ̃p(v) = ( ( ) ) ( ) ( ) = τ λ p v λ p τ ṽ˜, i.e., λ ∈ ∞ (M) is real valued, and replacing τ̃by 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 p v at p as the subspace of T P p , which is tangent to the fiber of the projection P →

Note that
[ ] = ⊕ TP T P T P 12 .

V H
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 = ⊕ TP T P T P V H is Euclidean) on its orthonormal frame bundle Reversely, any orthogonal connection on the frame bundle P ( ) → SO 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 V G coming from a linear representation G → GL(V) [4]. 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 where v w , is the curvature transformation of E x .
Dirac operator D acting on sections of spinor bundle (S (X) ) Σ → M is globally defined as follows.
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: , and for all Γ C , Γ The operator D is elliptic if the linear map σ ξ (D): E x → E x is an isomorphism for all ξ ≠ 0.
Structures of spinors fiber bundles using the Clifford algebra  421 Lemma 22. If D is the Dirac operator of the bundle S defined above. Then, for any 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]. Lemma 24. Let D, S and X be as above. Then for any ( ) , we have that . Having discussed Dirac bundles in general terms, it is now time to look hard at some important examples. We begin with the basic ones. 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 = Δ, where Δ = −∑∂ 2 /∂x 2 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 ℓ = = ≅ V C 1 2 . Then we have [3,8] the generator of a basic semi-group of unitary operators on L 2 . Let n = 2, so that ℓ = = ≅ ⊗ V C 2 . The construction of into ⊗ is natural and corresponds to the ℤ 2 -grading ℓ ⊗ ℓ C C Let n = 3, so that ℓ = ⊗ C 3 and V = ℍ. Cℓ 3 has two representations on ℍ given as Identify ℝ 3 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 Cℓ 4 = ℍ(2) and = = ⊗ V 2 . To describe the full Dirac operator, we consider first the following ℍ under the basis (1, i, j, k): 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 Cℓ 3 in complex form [2,3]. 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].
Notation. For spin manifolds X of even dimension, we shall denote the (unique) irreducible complex spinor bundle by $ℂ; and when dim(X) ≢ 3(mod 4), we denote the irreducible real spinor bundle by $. In both cases, the Atiyah-Singer operator will be written as .
These basic examples each generate large families of new examples by the following construction. Let S and E be a given Dirac bundle with connection ∇ s and ∇ E over a Riemannian manifold X. Then, the tensor product S ⊗ E is again a bundle of left modules over Cℓ(X), where for φ ∈ Cℓ(X), ∈ σ S, e ∈ E, ( ) ( ) ⊗ = ⊗ φ σ e φ σ e .
Furthermore, we can equip S ⊗ E with the canonical tensor product connection, ∇ = ∇ ⊗ ∇ E s , which is defined on sections of the form ⊗ σ e by the formula:

Conclusion
The researches have explained how Riemannian geometry, with the theory of spinor fiber bundle, fits into the general meaning of principal fiber bundles. We see that the spinor bundles defined as vector bundles whose fibers carry spinor representations of the Clifford algebras Cℓ(g), spinor fields are sections of spinor bundles.
With this constructive classification tool, we have investigated the usefulness of spinor structures of Clifford algebra. We have also given an explicit description of an important physical applications of spinor bundle and used a spinor connection and the constant Dirac matrices of special relativity to define the Dirac operator using the Clifford algebra of space-time and Dirac operation.