DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 549–557

A GEOMETRIC DEFINITION OF LIE DERIVATIVE FOR SPINOR FIELDS

LORENZO FATIBENE, MARCO FERRARIS, MAURO FRANCAVIGLIA AND MARCO GODINA

Abstract. Relying on the general theory of Lie derivatives a new geometric definition of Lie derivative for general spinor fields is given, more general than Kosmann’s one. It is shown that for particular infinitesimal lifts, i.e. for Kosmann vector fields, our definition coincides with the definition given by Kosmann more than 20 years ago.

1 Introduction

It is known that starting from some works by physicists (Dirac [1,2], Pauli [3]) the notion of spinor was introduced in Physics, although its discovery is due to E. Cartan in 1913 (cf. [4]) in his researches on the linear representations of some simple Lie groups. A pretty nice and detailed algebraic theory was given later by C. Chevalley [5] and other papers on Clifford–algebras and spinor groups were published, in particular by J. Dieudonné [6], Atiyah, Bott, Shapiro [7], Milnor [8].

Spinors on curved space–time manifolds were introduced and used since 1928 by physicists. In a series of very important papers Yvette Kosmann [9,10,11,12] introduced the notion of Lie derivative for spinor fields on spin manifolds and the study of the problem of transforming a spinor field under one–parameter group of diffeomorphisms of the base manifold.

However, thanks to her beautiful and relevant work, we have the feeling that Kosmann’s theory was not formulated in the clearest possible way and that the geometric meaning of her “ad hoc” definition was still lacking. In fact, it is quite known that the difficulty in defining the Lie derivative of a spinor field on a space (and time) oriented (pseudo) Riemannian manifold (M, g) comes from the fact that there is no natural definition of the image of such a spinor field by a diffeomorphism. This corresponds to the fact that the bundle SO(M, g) together with any spin bundle Spin(M, g) covering SO(M, g) are not natural bundles [17].

1991 Mathematics Subject Classification. 53C25; 19L10, 58G10.

Key words and phrases. G–bundle, connection, spin structure.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Of course, these bundles are particular gauge–natural bundles [17,21] and we will show moreover that it is always possible to lift in a unique way any vetor field ξ ∈ 𝔛(M) of the base manifold (M, g).

The corresponding vector field ξ̃_{K} ∈ 𝔛(Spin(M, g)), projectable over ξ, and ξ_{K} ∈ 𝔛(SO(M, g)) also projectable over ξ, will be called the Kosmann vector fields of ξ ∈ 𝔛(M). We remark that the vector field ξ̃_{K} also projects over ξ_{K}.

The Kosmann lifting is not natural and differs from the Levi–Civita lifting, i.e. the principal lifting induced by the Levi–Civita connection on SO(M, g) (first order derivatives of ξ are involved), although the Ricci rotation coefficients of the Levi– Civita connection appear in the expression which gives the Kosmann lifting of ξ.

In this paper we shall give a new definition of Lie derivative for spinor fields and we will show that for particular infinitesimal lifts, i.e. for Kosmann vector fields, our definition coincides with the definition given by Kosmann more than 20 years ago.

2 Spin structures

Let (P, M, π, G) be a principal fiber bundle over M with structural group a Lie group G. Let ρ: Γ → G be a central homomorphism of a Lie group Γ onto G with kernel K, that is K is discrete and contained in the center of Γ (see Greub and Petry in [13]; see also [22]). We recall that a Γ–structure on P is a principal bundle map η: P̃ → P which is equivariant under the right actions of the structure groups, that is:

η(z̃ · γ) = η(z̃) · ρ(γ) , z̃ ∈ P̃, γ ∈ Γ ,  (2.1)

and where ρ is assumed central by definition. Equivalently the following diagram commutes:

\begin{matrix}P̃ × Γ & ⟶^{η×ρ} & P × G \\ ↓R̃ & & ↓^{R} \\ P̃ & ⟶^{η} & P \\ ↓π̃ & & ↓^{π} \\ M & ⟶^{id_{M}} & M \end{matrix}  (2.2)

where R and R̃ denote right multipication in P and P̃ respectively . This means that for z̃ ∈ P̃, both z̃ and η(z̃) lie over the same point, and that η restricted to a fiber is a “copy” of ρ, i.e. equivalent to ρ.

In general, it is not guaranteed that a manifold admits a spin structure. The existence condition for a Γ–structure on P can be formulated [13,14,22] in terms of Čech cohomology.

A GEOMETRIC DEFINITION OF LIE DERIVATIVE FOR SPINOR FIELDS 551

Let us also recall that for any principal fiber bundle (P, M, π, G) an (principal) automorphism of P is a diffeomorphism ϕ: P → P such that ϕ(u · g) = ϕ(u) · g, for every u ∈ P, g ∈ G. Each ϕ induces a unique diffeomorphism φ: M → M such that π ∘ ϕ = φ ∘ π. Accordingly, we denote by Aut(P) the group of all principal (i.e. equivariant) automorphisms of P. Assume a vector field X ∈ 𝔛(P) on P generates the one–parameter group ϕ_{t}. Then, X is G–invariant if and only if ϕ_{t} is an automorphism of P for every t ∈ IR. Accordingly, we denote by 𝔛_{G}(P) the Lie algebra of G–invariant vector fields of P.

Let IR^{p,q} be the familiar vector space IR^{n} (p+q = n) equipped with the canonical nondegenerate symmetric bilinear form <, > of signature (p, q), i.e. with p “+” signs and with q “-” signs. We will denote by O(p, q) the (pseudo)–orthogonal group with respect to <, >, that is O(p, q) = {L ∈ Gl(IR^{p,q})| < Lu, Lv >=< u, v >}, by SO(p, q) = {L ∈ O(p, q)| det(L) = 1} its special subgroup, and by SO_{0}(p, q) its connected component with identity. In the Euclidean case (p or q = 0), SO(p, q) = SO_{0}(p, q) but in the general case they are not equal. To define the Clifford algebra Cl(p, q) ≡ Cl(IR^{p,q}) we choose any orthonormal basis e_{a} of IR^{p,q} ⊂ Cl(p, q), that is < e_{a}, e_{b} >= η_{ab}. The Clifford algebra Cl(p, q) is the real vector space endowed with an associative product generated (as an algebra) by the unit I and the elements e_{a}, 1 ≤ a ≤ n = p + q, with the relations:

e_{a}e_{b} + e_{b}e_{a} = −2η_{ab}I  (2.3)

where η_{ab} = 0 if a ≠ b, +1 if a = b ≤ p, and −1 if p < a = b. As a real vector space, the Clifford algebra Cl(p, q) has dimension 2^{n}. The set {I, e_{a}, e_{a_{1}}e_{a_{2}}, ..., e_{a_{1}}e_{a_{2}} ⋯ e_{a_{r}}, e_{a_{1}} ⋯ e_{a_{n}} : a_{1} < a_{2} < ⋯ < a_{r}} forms a basis of Cl(p, q). The complexified Clifford algebra I Cl(p, q) is the vector space Cl(p, q) ⊗_{IR} I C, that is the vector space over the complex numbers generated by the previous elements I, e_{a}, ..., e_{a_{1}} ⋯ e_{a_{n}}. In this paper we shall assume that the dimension of IR^{n} is even, that is n = 2d. In this case it can be proved that I Cl(p, q) is isomorphic to the complex algebra M_{2^{d}}(I C) of complex matrices of order 2^{d}. The matrices representing the fundamental elements e_{a} will be denoted by γ_{a}. They automatically carry a representation of various subgroups of Cl(p, q) and in particular of Spin(p, q).

Let us briefly recall how Spin(p, q) is defined. Consider the multiplicative group of units of Cl(p, q), which is defined to be the subset

Cl^{×}(p, q) ≡ {a ∈ Cl(p, q) : ∃a^{−1} such that a^{−1}a = aa^{−1} = 1} .  (2.4)

This group contains all elements v ∈ IR^{p,q} ⊂ Cl(p, q) with < v, v >≠ 0 so it is possible to consider the subgroup generated by its “units length” elements, v ∈ IR^{p,q} ⊂ Cl(p, q) with < v, v >= ±1.

This subgroup it is denoted by Pin(p, q). The spin group Spin(p, q) is defined as the subgroup of Pin(p, q) consisting of even elements:

Spin(p, q) = Pin(p, q) ∩ Cl^{0}(p, q) ,  (2.5)

where Cl^{0}(p, q) is a subalgebra of Cl(p, q) called the even part of Cl(p, q).

Now, consider a space (and time) oriented (pseudo)–Riemannian manifold with metric g and dimension n = dim(M). In this case we have G = SO(p, q), Γ = Spin(p, q), P = SO(M, g) ⊂ L(M), P̃ = Spin(M, g) where Spin(p, q) is the usual notation for Spin(IR^{p,q}) and (L(M), Gl(n), M) is the principal bundle of linear frames. We shall say that a spin–structure on a SO(p, q)–bundle is a Spin(p, q)– structure with respect to the covering of Lie groups ρ: Spin(p, q) → SO(p, q) (see Milnor in [8]). For more details in Clifford–algebras and spinor groups see also [7,14,15,16]. When a spin structure is given η: Spin(M, g) → SO(M, g), the principal bundle Spin(M, g) is sometimes called the bundle of spinor frames.

Let us also recall that, in this case, the local isomorphism between Spin(M, g) and SO(M, g) allows the Levi–Civita connection to be lifted to a connection on the principal bundle Spin(M, g), equivariant by Spin(p, q), p + q = n. Let ω: T [SO(M, g)] → 𝔰𝔬(p, q) be the connection one–form determined by the Levi–Civita connection of (M, g). The corresponding connection σ on Spin(M, g)

σ: T [Spin(M, g)] → 𝔰𝔭𝔦𝔫(p, q)

is defined by the composition:

σ = (ρ′)^{−1} ∘ (η^{∗}ω)  (2.6)

where ρ′ = T_{e}ρ: 𝔰𝔭𝔦𝔫(p, q) → 𝔰𝔬(p, q) is the Lie algebras isomorphism induced by ρ.

Spinors are defined via a linear representation of the spin group Spin(p, q) on a complex vector space S, with dim_{C}S = 2^{d}. In fact, let χ: Spin(p, q) × S → S be a spinor representation:

χ: Spin(p, q) × S → S : (h, s) ↦ χ(h, s) ≡ h · s .  (2.7)

We recall that any representation of SO(p, q) yields a representation of its covering group, just by composition with ρ. We shall call a representation of this kind a tensor representation of Spin(p, q). A spinor representation of Spin(p, q) (sometimes called a double representation of SO(p, q)) is by definition a linear representation of Spin(p, q) which cannot be obtained from a representation of SO(p, q).

Finally, consider the vector bundle S(M) ≡ Spin(M, g) × I C^{d}/Spin(p, q) over M associated to the principal fiber bundle Spin(M, g), where the action of Spin(p, q) on I C^{d} is the fundamental one defined by inclusion of Spin(p, q) in the complexified Clifford algebra I Cl(p, q). Then a spinor field ψ is a section x ↦ [ẽ_{x}, Ψ(ẽ_{x})] of S(M), where Ψ: Spin(M, g) → I C^{d}, Ψ(ẽ_{x} · h) = h^{−1} · Ψ(ẽ_{x}), is an equivariant function, i.e. an assignment of components Ψ(ẽ_{x}) in I C^{d} to each “spinor frame” ẽ_{x}.

3 Lie derivative of Spinor Fields

Given a spin structure Spin(M, g) → M, a generalized spinorial transformation Φ of Spin(M, g) is an automorphism Φ: Spin(M, g) → Spin(M, g). As discussed above Φ is a diffeomorphism of Spin(M, g) such that Φ(ẽ·h) = Φ(ẽ)·h, for every ẽ ∈ Spin(M, g), h ∈ Spin(p, q). Each Φ induces a unique diffeomorphism φ: M → M such that η_{M} ∘Φ = φ∘η_{M}, with η_{M} = π ∘η. Moreover, in this case, Φ also induces a unique automorphism (i.e. equivariant diffeomorphism) ϕ: SO(M, g) → SO(M, g) such that η ∘ Φ = ϕ ∘ η. Equivalently we have the following diagram:

Spin(M, g) ⟶^{Φ} Spin(M, g)
↓^{η} ↓^{η}
SO(M, g) ⟶^{ϕ} SO(M, g)
↓^{π}  ↓^{π}
M  ⟶^{φ}  M

(3.1)

Furthermore, let S(M) be the fiber bundle of spinors associated to the principal bundle Spin(M, g). For every element Φ ∈ Aut(Spin(M, g)) we obtain an automorphism Φ_{S(M)} of S(M), Φ_{S(M)} : S(M) → S(M), given by Φ_{S(M)}(z) = [Φ(ẽ), s], z = [ẽ, s] ∈ S(M), which is well defined because it does not depend on the representative chosen. In fact, this is so since Φ is equivariant under the right action of the structure group Spin(p, q). A vector field Ξ̃ ∈ 𝔛(Spin(M, g)) generating a one–parameter group of automorphisms of Spin(M, g) defines the generalized Lie derivative of any section ψ: M → S(M), that is of any spinor field, as follows:

L̃Ξ̃ψ = T ψ ∘ ξ − Ξ̃_{S(M)} ∘ ψ ,  (3.2)

where ξ ∈ 𝔛(M) is the only vector field such that T π ∘ Ξ = ξ ∘ π, Ξ is the only vector field such that T η ∘ Ξ̃ = Ξ ∘ η, and for Ξ̃_{S(M)} we have: Ξ̃_{S(M)}(z) = \frac{d}{dt}[(Φ_{t})_{S(M)}(z)]|_{t=0}, for every z ∈ S(M). Once again we have the following diagram:

Spin(M, g) ⟶^{Ξ̃} T [Spin(M, g)]
↓^{η} ↓^{T η}
SO(M, g) ⟶^{Ξ} T [SO(M, g)]
↓^{π}  ↓^{T π}
M  ⟶^{ξ}  T M

(3.3)

Hence the generalized Lie derivative of ψ with respect to Ξ̃, L̃Ξ̃ψ: M → V [S(M)], takes values in the vertical sub-bundle V [S(M)] ⊂ T [S(M)] and being S(M) a vector bundle there is a canonical identification V [S(M)] = S(M) ⊕ S(M). In this case, (3.2) is of the form L̃Ξ̃ψ = (ψ, LΞ̃ψ). The first component being the original section ψ, the second component LΞ̃ψ is also a section of S(M), and it is called the Lie derivative of ψ with respect to Ξ̃. Sometimes, the second component LΞ̃ψ is called the restricted Lie derivative [17]. Using the fact that the second component of L̃Ξ̃ψ is the derivative of (Φ_{−t})_{S(M)} ∘ ψ ∘φ_{t} for t = 0 in the classical sense, one can re-express the restricted Lie derivative in the form

(LΞ̃ψ)(x) = lim_{t→0} \frac{1}{t}((Φ_{−t})_{S(M)} ∘ ψ ∘ φ_{t}(x) − ψ(x)) .  (3.4)

Further details and deeper discussions on the general theory of Lie derivatives may be found in [17,18,19].

We are now seeking a coordinate expression for the restricted Lie derivative given by the equation (3.4). Now, if (x^{λ}, ψ^{i}) denote local fibered coordinates for S(M) then (3.4) reads as:

LΞ̃ψ = [ξ^{a}e_{a}ψ^{i} − Ξ̃_{j}^{i}ψ^{j}]f_{i} ,  (3.5)

where ξ = ξ^{a}(x)e_{a}, e = (e_{a}) = (e_{a}^{μ}(x)∂_{μ}) is a local section of SO(M, g) induced by a local section of Spin(M, g) ẽ: U → Spin(M, g) such that η ∘ ẽ = e, e_{a}(ψ^{i}) is the Pfaff derivative of ψ^{i}, that is e_{a}ψ^{i} = e_{a}^{μ}(x)∂_{μ}ψ^{i} and f_{i} is a basis of [S(M)]_{x} at each x ∈ U.

Recalling that Ξ̃ projects on Ξ and taking into account the Lie algebras isomorphism ρ′ = T_{e}ρ: 𝔰𝔭𝔦𝔫(p, q) → 𝔰𝔬(p, q), we find the following relations:

Ξ̃_{j}^{i} = −\frac{1}{4}Ξ^{a}_{b}(γ_{a}γ^{b})_{j}^{i} = −\frac{1}{4}Ξ_{ab}(γ^{a}γ^{b})_{j}^{i}  (3.6a)
Ξ̃^{a} = Ξ^{a} = ξ^{a}  (3.6b)

where indices in (3.6a) are lowered and raised with respect to the (pseudo) Riemannian metric g, i.e. η_{ab} = g(e_{a}, e_{b}). Recall that for Ξ, being a SO(p, q)-invariant vector field of SO(M, g), in the orthonormal basis (e_{a}^{μ}(x)∂_{μ}), we have Ξ_{ab} = −Ξ_{ba}. We also note that the sign in (3.6a) depends on the choice of sign for the Clifford algebra, and the choice between L^{a}_{b}γ_{a} and L_{b}^{a}γ_{a}, where L^{a}_{b} and L_{b}^{a} are any matrices. Finally, for the spin connection coefficients we have:

σ_{jc}^{i} = −\frac{1}{4}ω^{a}_{bc}(γ_{a}γ^{b})_{j}^{i} = −\frac{1}{4}ω_{abc}(γ^{a}γ^{b})_{j}^{i} .  (3.7)

So we can re–write the (3.5) as follows:

LΞ̃ψ = \{ξ^{a}∇_{a}ψ^{i} − [−\frac{1}{4}(ω_{[ab]c}ξ^{c} + Ξ_{ab})(γ^{a}γ^{b})_{j}^{i}ψ^{j}] \}f_{i} ,  (3.8)

where in the orthonormal frame (e_{a}^{μ}(x)∂_{μ}) we have ω_{abc} = −ω_{bac} and square brackets on indices mean complete antisymmetrization, although in this case ω_{[ab]c} = ω_{abc}. In fact, the coefficients ω_{abc} are the Ricci rotation coefficients defined by:

ω_{abc} = η_{ad}ω^{d}_{bc} .  (3.9)

Moreover, for ∇_{a}ψ^{i} we have the local expression:

∇_{a}ψ^{i} = e_{a}ψ^{i} + σ_{ja}^{i}ψ^{j} .  (3.10)

In this notation we have, in the same orthonormal basis (e_{a}^{μ}(x)∂_{μ}), the following: ∇_{e_{a}}e_{b} = ω^{c}_{ba}e_{c} for the connection coefficients and ω^{a}_{b} = ω^{a}_{bc}(x)θ^{c} for the connection one-forms, where θ^{c} is the dual basis of e_{a}. Hence, in our notation we get for the covariant derivative ∇_{a}ξ_{b} the following local expression:

∇_{a}ξ_{b} = e_{a}ξ_{b} − ω^{c}_{ba}ξ_{c} = e_{a}ξ_{b} − ω_{cba}ξ^{c} ,  (3.11)

where ξ_{c} = η_{cb}ξ^{b}.

Given a space (and time) oriented (pseudo) Riemannian manifold (M, g) and a vector field ξ it is possible to give its lift to SO(M, g), denoted by ξ_{K}. The vector field ξ_{K} has the following coordinated expression:

ξ_{K} = (ξ_{K})^{a}e_{a} + (ξ_{K})_{ab}A^{ab} ,  (3.12)

where the coefficients (ξ_{K})^{a} and (ξ_{K})_{ab} are given by:

(ξ_{K})^{a} = ξ^{a}  (ξ_{K})_{ab} = −{∇_{[a}ξ_{b]} + ω_{[ab]c}ξ^{c}} ,  (3.13)

and where ∇_{[a}ξ_{b]} = e_{[a}ξ_{b]} − ω_{c}_{[ba]}ξ^{c}. The vector fields A^{ab} are local right SO(p, q)– invariant vector fields on SO(M, g) and in a suitable chart (x^{μ}, u_{a}^{b}) are defined as follows:

A^{ab} = \frac{1}{2}(η^{ac}δ_{d}^{b} − η^{bc}δ_{d}^{a})u_{e}^{d} \frac{∂}{∂u_{e}^{c}} .  (3.14)

In order to define local coordinates (x^{μ}, u_{a}^{b}) which appear in (3.14) we remind that, as above, the local section e: U → SO(M, g) is induced by a local section of Spin(M, g) ẽ: U → Spin(M, g), so that we have η ∘ ẽ = e. Moreover, if (x^{μ}) are local coordinates of a chart of the base manifold M with the same domain U, we can define local coordinates u_{a}^{μ} by (u_{a}^{μ}(u)∂_{μ}) where u = (u_{a}) is any frame in the open π^{−1}(U) of L(M) such that g(u_{a}, u_{b}) = η_{ab}. Finally, with the local section e: U → SO(M, g) we define local coordinates u_{a}^{b} by u_{a} = u_{a}^{b}e_{b}.

Accordingly, ξ_{K} transforms as a SO(p, q)–invariant vector field on SO(M, g) and projects over ξ. An alternative geometric definition of ξ_{K} is the following one. Given any vector field ξ of the base manifold M,it admits a unique natural lift ξ̂ to the linear frame bundle L(M). Then, ξ_{K} is by definition the skew of ξ̂ with respect to the (pseudo) Riemannian metric g.

Furthermore, when a spin structure is given η: Spin(M, g) → SO(M, g), any vector field ξ lifts to a vector field on the spin bundle Spin(M, g). In fact, its lift ξ̃_{K} is defined by:

ξ̃_{K} = (ξ̃_{K})^{a}e_{a} + (ξ̃_{K})_{j}^{i}E_{i}^{j} ,  (3.15)

and the coefficients (ξ̃_{K})^{a} , (ξ̃_{K})_{j}^{i} are given by:

(ξ̃_{K})^{a} = ξ^{a}  (ξ̃_{K})_{j}^{i} = −\frac{1}{4}(ξ_{K})_{ab}(γ^{a}γ^{b})_{j}^{i} ,  (3.16)

so that ξ̃_{K} projects over ξ_{K} and hence over ξ. Accordingly, the following diagram commutes:

Spin(M, g) ⟶^{ξ̃_{K}} T [Spin(M, g)]
↓^{η} ↓^{T η}
SO(M, g) ⟶^{ξ_{K}} T [SO(M, g)]
↓^{π}  ↓^{T π}
M  ⟶^{ξ}  T M

(3.17)

The vector field ξ̃_{K} is well defined because the one–parameter group of automorphisms {ϕ_{t}} of ξ_{K} can be lifted (see below) to an one–parameter group of automorphisms \{ϕ̃_{t}\} so that ξ̃_{K}(ẽ) = \frac{d}{dt}[ϕ̃_{t}(ẽ)]|_{t=0}, for every ẽ ∈ Spin(M, g).

The vector fields ξ̃_{K}, ξ_{K}, invariant under the actions of Spin(p, q) and SO(p, q), are called the Kosmann vector fields of the vector field ξ. Let us also remark that the Kosmann lifting ξ ⟼ ξ_{K} is not a Lie–algebra homomorphism although the following remarkable relation holds:

[ξ, ζ]_{K} = [ξ_{K}, ζ_{K}] − \frac{1}{2}L_{ξ}(g_{λμ})g^{μν}L_{ζ}(g_{νσ})A^{λσ}  (3.18a)

where

A^{λσ} = e_{a}^{λ}e_{b}^{σ}A^{ab} .  (3.18b)

We are now in a good position to state the following result.

Proposition. The Lie derivative L_{ξ}ψ of a spinor field ψ: M → S(M) with respect to a vector field ξ ∈ 𝔛(M), as defined by Kosmann in [9], is the second component of the generalized Lie derivative of ψ with respect to the Kosmann lift ξ̃_{K} of ξ. That is if L̃ξ̃_{K} ψ is the generalized Lie derivative of ψ and Lξ̃_{K} ψ its second component then we get L_{ξ}ψ = Lξ̃_{K} ψ.

Proof: It is an immediate consequence of the above considerations and from the fact that the local expression of the Lie derivative given by Kosmann is

L_{ξ}ψ = [ξ^{a}∇_{a}ψ^{i} − \frac{1}{4}∇_{[a}ξ_{b]}(γ^{a}γ^{b})_{j}^{i}ψ^{j}]f_{i} .  (3.19)

(Q.E.D)

We remark that the above proposition gives us a geometric meaning to the Kosmann’s formula (3.19) which is taken by Kosmann as a definition of the Lie derivative of ψ with respect to ξ. We shall use the following notation: L̃_{ξ}ψ := L̃ξ̃_{K} ψ. We finally remark that there is an easier interpretation of the above facts in the case in which ξ is an infinitesimal isometry, i.e. a Killing vector field on M.

In fact, given a spin structure η: Spin(M, g) → SO(M, g), let us consider a Killing vector field ξ, its flow denoted by {φ_{t}} and the principal bundle map ϕ_{t}: SO(M, g) → SO(M, g) which is defined by restriction of L_{φ_{t}}: L(M) → L(M) to SO(M, g). The automorphisms ϕ_{t} are well defined since the diffeomorphisms φ_{t} are isometries. Moreover being η: Spin(M, g) → SO(M, g) a covering space it is possible to lift ϕ_{t} to a bundle map ϕ̃_{t}: Spin(M, g) → Spin(M, g) in the following way. For any spinor frame ẽ ∈ Spin(M, g) over η(ẽ) = e, from the theory of covering spaces it follows that for the curve γ_{e}: IR → SO(M, g), based at e, that is γ_{e}(0) = e, and defined by γ_{e}(t) := ϕ_{t}(e) there exists a unique curve γ̃ẽ: IR → Spin(M, g), based at ẽ, such that η ∘ γ̃ẽ = γ_{e}. It is possible to define a principal bundle map ϕ̃_{t}: Spin(M, g) → Spin(M, g), covering ϕ_{t}, by letting ϕ̃_{t}(ẽ) := γ̃ẽ(t). The one–parameter group of automorphisms of Spin(M, g) {ϕ̃_{t}} defines a vector field Ξ̃(ẽ) = \frac{d}{dt}[ϕ̃_{t}(ẽ)]|_{t=0}, for all ẽ ∈ Spin(M, g). It is then easy to verify that the vector field Ξ̃ coincides with the Kosmann vector field ξ̃_{K} of the vector field ξ. In fact, in this case, the lift ξ̂is already skewsymmetric. Hence the Lie derivative of ψ with respect to ξ is still given by the Kosmann’s formula (3.19) (cf. [23,24]) or, better, by the following coordinate–free expression

(L_{ξ}ψ)(x) = lim_{t→0} \frac{1}{t}((ϕ̃_{−t})_{S(M)} ∘ ψ ∘ φ_{t}(x) − ψ(x)) .  (3.20)

Moreover, even for an arbitrary vector field ξ the formula (3.20) is still valid but where now {ϕ_{t}} is the flow of the vector field ξ_{K}, and {ϕ̃_{t}} is the flow of ξ̃_{K}. Now, ξ is not an infinitesimal isometry but ξ_{K} generates a one–parameter group of principal automorphisms of SO(M, g) which lift to Spin(M, g) so that even in this case (3.20) gives us a further interpretation of Kosmann’s formula, i.e. a direct classical definition of the Lie derivative for spinor fields.

4 Acknowledgments

We are deeply grateful to A. Borowiec and G. Magnano for their useful remarks. One of us (M.G) is greatly indebted to G. Chaix from Paris for his editorial assistance during the preparation of this work. This work is sponsored by GNFM, MURST (40% Project “Metodi Geometrici e Probabilistici in Fisica Matematica”).

References

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