On Lie derivation of spinors against arbitrary tangent vector fields

Andras´ LASZL´ O´ [email protected] Wigner RCP, Budapest, Hungary

(joint work with L.Andersson and I.Rácz)

CERS8 Workshop Brno, 17th February 2018

On Lie derivation of spinors against arbitrary tangent vector fields – p. 1 Preliminaries

Ordinary Lie derivation. Take a one-parameter group (φt)t∈R of diffeomorphisms over a manifold M.

Lie derivation against that is defined on the smooth sections χ of the mixed tensor algebra of T (M) and T ∗(M), with the formula:

L φ∗ χ := ∂t φ∗ −t χ
t=0

It has explicit formula:

on T (M) : Lu(χ)=[u,χ] , a on F (M) := M× R : Lu(χ)= u da (χ) , ∗ a a on T (M) : Lu(χ)= u da (χ) + d(u χa), where u is the unique tangent vector field underlying (φt)t∈R.

The map u 7→ Lu is faithful Lie algebra representation.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 2 Lie derivation on a vector bundle. Take a vector bundle V (M) over M. Take a one-parameter group of diffeomorphisms of the total space of V (M), which preserves the vector bundle structure.

The ∂t() against the flows of these are vector bundle Lie derivations. t=0

(I.Kolár,P.Michor,K.Slovák:ˇ Natural operations in differential geometry; Springer 1993)

Explicit expression: take a preferred covariant derivation ∇, then over the sections χ of V (M) one can express these as

∇ A a A A B L χ = u ∇a χ − CB χ (u,C)  

That is: Lie derivations on a vector bundle are uniquely characterized by their horizontal a A part u ∇a and vertical part CB .

They naturally form a Lie algebra (Lie bracket: commutator of differential operators).

On Lie derivation of spinors against arbitrary tangent vector fields – p. 3 Special case: Lie derivations over T (M). One can recover the ordinary (natural) notion of Lie derivation over T (M).

Ordinary (natural) Lie derivations are those general vector bundle Lie derivations, which

on T (M) : theypreserve [ , ] of vector fields, on F (M)= M× R : they preserve constancy of constant scalar fields, on T ∗(M) : they preserve duality pairing form.

M M L∇ Example for T ( ): take on T ( ) a general vector bundle Lie derivation (u,C), with L∇ b a∇ b − b c b parametrization (u,C)(χ )= u aχ Cc χ on some tangent vector field χ .

L∇ If one requires that (u,C) preserves the natural operation [ , ] of vector fields, one gets:

c c a c Cb = ∇bu + u T∇ ab ( ←− torsion tensor) L∇ L and thus (u,C) = u follows.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 4 Special case: Lie derivations over the two-spinor bundle. Lot of things are said in the literature, sometimes contradicting each-other.

A. Lichnerowicz: Spineurs harmoniques; C. R. Acad. Sci. Paris. 257 (1963) 7. Y. Kosmann: Dérivées de Lie des Spineurs; Ann. Mat. Pura Appl. 91 (1972) 317. V. Jhangiani: Geometric significance of the spinor Lie derivative I; Found. Phys. 8 (1978) 445. V. Jhangiani: Geometric significance of the spinor Lie derivative II; Found. Phys. 8 (1978) 593. E. Binz, R. Pferschy: The Dirac operator and the change of the metric; Compt. Rend. Math. Rep. Acad. Sci. Canada 5 (1983) 269. R. Penrose, W. Rindler: Spinors and spacetime 1-2; Cambridge University Press (1984). J. Bourguignon, P. Gauduchon: Spineurs, opérateurs de Dirac et variations de métriques; Commun. Math. Phys. 144 (1992) 581. M. Mauhart, P.W. Michor: Commutators of flows and fields; Archivum Mathematicum 28 (1992) 228. D. J. Hurley, M. A. Vandyck: On the concepts of Lie and covariant derivatives of spinors: Part I; J. Phys. A27 (1994) 4569. D. J. Hurley, M. A. Vandyck: On the concepts of Lie and covariant derivatives of spinors: Part II; J. Phys. A27 (1994) 5941. D. J. Hurley, M. A. Vandyck: On the concepts of Lie and covariant derivatives of spinors: Part III, comparison with the invariant formalism; J. Phys. A28 (1995) 1047. L. Fatibene, M. Ferraris, M. Francaviglia, M. Godina: A geometric definition of Lie derivative for spinor fields; Proceedings of the 6th International Conference on Differential Geometry and Applications (1996). ←− T. Ortín: A note on Lie-Lorentz derivatives; Class. Quant. Grav. 19 (2002) L143. M. Godina, P. Matteucci: Reductive G-structures and Lie derivatives; J. Geom. Phys. 47 (2003) 66. M. Godina, P. Matteucci: The Lie derivative of spinor fields: theory and applications; Int. J. Geom. Methods Mod. Phys. 2 (2005) 159. R. A. Sharipov: A note on Kosmann Lie derivation of Weyl spinors; arXiv:0801.0622. R. Geroch: Quantum field theory – 1971 lecture notes; Minkowski Institute Press (2013). R. Geroch: Spinors; Handbook of Spacetime ed. by Ashtekar and Petkov, Springer (2014). R. F. Leao,ˇ W. A. Rodrigues Jr, S. A. Wainer: Concept of Lie derivative of spinor fields: a geometric motivated approach; arXiv:1411.7845. A. D. Helfer: Spinor Lie derivatives and fermion stress-energies; Proc. Roy. Soc. A472 (2016) 20150757.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 5 The two-spinor bundle

Let M be four dimensional. M has spinor bundle iff ∃ a complex two dimensional vector bundle S(M), such that ∃ T (M) → Re S¯(M) ⊗ S(M) vector bundle isomorphism.

Such an isomorphism can be chosen to be pointwise, and is called: ′ AA Infeld–Van der Waerden symbol, or soldering form, or Pauli map (σa ).

We denote by σ/∼ its equivalence class up to T (M) → T (M) and S(M) → S(M) vector bundle automorphisms, and call it solderability of T (M) to S(M).

A spinor bundle is a triple (M, S(M), σ/∼).

R.Geroch: Spinor structure of space-times in General Relativity I; J.Math.Phys.9(1968)1739. R.Geroch: Spinor structure of space-times in General Relativity II; J.Math.Phys.11(1970)343.

If M is non-compact, then has spinor bundle iff parallelizable, i.e. iff T (M) is trivial. And in that case S(M) is also trivial, and therefore σ/∼ is unique.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 6 The Kosmann Lie derivation

′ AA Take a σa soldering form, an ǫAB spinor maximal form. ′ ′ ′ ′ AA BB AB A B The spacetime metric g(σ, ǫ)ab := σa σb ǫ ǫ¯ , and the unique ∇a spinor Levi-Civita covariant derivation are functions of these. ′ ′ A AC ′ − AC ′ Also, Σ(σ, ǫ)abB := i σa σb BC σb σa BC , called to be the spin tensor.

Let ua be any smooth tangent vector field, and take the differential operator

σ,ǫ A a A i c b A B 1 d A B L χ := u ∇a χ + ∇b(u )Σc B χ − ∇d(u ) δB χ u   4 8 over the sections χA of S(M).

This differential operator is called Kosmann Lie derivation or Kosmann formula.

Y.Kosmann: Derivées de Lie des Spineurs; Ann.Mat.Pura.Appl.91(1972)317.

It is understood to be extended to S¯(M), S∗(M), S¯∗(M) via commutativity with complex ∗ conjugation and duality form. Also to T (M) and T (M) via coincidence with Lu. Also over tensor products: via Leibnitz rule.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 7 Properties. (See: e.g. Penrose-Rindler book.)

Let the vector field ua be conformal Killing.

σ,ǫ Then the followings hold for a Kosmann Lie derivation Lu : (i) It is linear. a (ii) Coincides with the scalar Lie derivation u da on scalars. (iii) Obeys Leibnitz rule for product of scalar and spinor fields. (iv) It is real with respect to complex conjugation. (v) Commutes with duality form (contraction). (vi) Obeys Leibnitz rule over tensor product. ′ σ,ǫ AA σ,ǫ — At this point, Lu σa and Lu ǫAB becomes uniquely defined. — ′ σ,ǫ AA (vii) Lu σa = 0. σ,ǫ σ,ǫ (viii) Lu preserves complex phase of ǫAB . (Lu ǫAB = α ǫAB with real scalar field α.)

Known result: properties (i)–(viii) uniquely determine the Kosmann Lie derivation. Literature: lot of confusion when ua is not conformal Killing. What happens?!

On Lie derivation of spinors against arbitrary tangent vector fields – p. 8 Theorem. (I have not seen similar statement in the literature.)

Let ua and va be arbitrary smooth tangent vector fields. Then:

σ,ǫ σ,ǫ σ,ǫ σ,ǫ A σ,ǫ A i ^ ^ ac bd A B L L −L L χ = L χ + Lugab Lvgcd g Σ B χ  u v v u 
[u,v]
8    

σ,ǫ So, the problem is that u 7→ Lu is Lie algebra homomorphism only for conformal Killings!! This is the origin of all problems. What can we do?

Closest result in literature:

L.Fatibene,M.Ferraris,M.Francaviglia,M.Godina: A geometric definition of Lie derivative for spinor fields; Proceedings of the 6th International

Conference on Differential Geometry and Applications (1996). σ,ǫ Realizes that the mapping u 7→ Lu fails to be Lie algebra homomorphism! (But no explicit formula for the residual term.)

On Lie derivation of spinors against arbitrary tangent vector fields – p. 9 What can we do? On S(M) there is still the notion of general vector bundle Lie L∇ a A derivations (u,C) for arbitrary tangent vector field u and spinortensor CB .

Explicit expression:

∇ A a A A B L χ = u ∇a χ − CB χ (u,C)  
7→ L∇ The mapping (u, C) (u,C) is faithful Lie algebra homomorphism, by construction.

Preserves vector bundle structure of S(M), and solderability σ/∼ of S(M) to T (M). Thus, it preserves exactly the spinor bundle (M, S(M), σ/∼).

But does not preserve: ′ AA the actual instance of soldering form σa .

On Lie derivation of spinors against arbitrary tangent vector fields – p. 10 ′ L∇ M AA What are those vector bundle Lie derivations (u,C) of S( ), which preserves also σa ?

These are of the form:

L∇ A Lσ,ǫ A A (u,C) χ  = u χ  + i ϕχ

σ,ǫ where u is conformal Killing (i.e., Lu is Kosmann), and ϕ is real valued scalar field.

′ AA ⇔ The Lie derivations which preserve σa are Kosmann + U(1) gauge transformations.

If, in addition we require preservation of the phase of ǫAB ⇔ only Kosmann.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 11 Summary

Lie derivations against arbitrary vector fields is meaningful on spinor bundle. L∇ M But for this, we need to allow all the vector bundle Lie derivations (u,C) on S( ). They exactly preserve the spinor bundle (M, S(M), σ/∼). ′ AA They do not all preserve a concrete instance of a soldering form σa . ′ AA The sub-Lie algebra which preserves σa is: Kosmann Lie derivations against conformal Killing fields + U(1) gauge transformations.

If phase of spinor maximal form ǫAB is also to be preserved: only Kosmann Lie derivations against conformal Killing fields respect all these.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 12 Backup

On Lie derivation of spinors against arbitrary tangent vector fields – p. 13 Generic vector bundle Lie derivations do naturally form a Lie algebra:

L∇ L∇ −L∇ L∇  (u,C) (v,D) (v,D) (u,C) =

∇ L a b a a ([u,v], [C,D]−u v F∇ ab−u ∇a(D)+v ∇a(C)) where F∇ ab is the curvature tensor of the reference covariant derivation ∇a.

So, commutator of two vector bundle Lie derivations is a vector bundle Lie derivation.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 14 L∇ It is convenient to express a generic spinor Lie derivation (u,C) against Kosmann formula:

L∇ A Lσ,ǫ A − A B Lσ,ǫ A (u,C) χ  = u χ  δCB χ =: (u,δC) χ 

A A − − i ∇ c b A 1 ∇ d A where δCB := CB 4 b(u )ΣcB + 8 d(u ) δB is corresponding σ,ǫ a
spinortensor and Lu is Kosmann formula with arbitrary u .

Then, again 7→ Lσ,ǫ is faithful Lie algebra homomorphism, no problem there. (u,δC) (u,δC)

Lσ,ǫ Lσ,ǫ −Lσ,ǫ Lσ,ǫ  (u,δC) (v,δD) (v,δD) (u,δC) =

Lσ,ǫ i ^ ^ ac bd σ,ǫ σ,ǫ ([u,v], − 8 (Lugab)(Lv gcd) g Σ +Lu (δD)−Lv (δC)−[δC,δD])

σ,ǫ σ,ǫ The commutator of the two Kosmann expression Lu and Lv gives a contribution − i L^ L^ ac bd 8 ( ugab)( vgcd) g Σ to the vertical part whenever u and v not conformal Killing. No philosophical problem there, but vertical part needs to be allowed then.

On Lie derivation of spinors against arbitrary tangent vector fields – p. 15 In Dirac bispinor (Clifford) formalism, one has

i Σab = (γa γb − γb γa) 2

′ ′ b A AC b cb AC d In two-spinor formalism ′ − ′ . Σ(σ)a B := i σa σBC g(σ, ǫ) σc g(σ, ǫ)da σBC 

On Lie derivation of spinors against arbitrary tangent vector fields – p. 16