Generalized Dirac operators on lorentzian manifolds and propagation of singularities

Generalized Dirac Operators on Lorentzian Manifolds and Propagation of Singularities

PAOLOANTONINI

ABSTRACT- We survey the correct definition of a generalized Dirac operator on a Space-Time and the classical result about propagation of singularities. This says that light travels along light-like geodesics. Finally we show this is also true for generalized Dirac operators.

1. Introduction

The celebrated theorem of Nils Denker about the propagation of sin- gularities of a real principal type system when applied to the Dirac op- erator on a Lorentzian manifolds says the well-known fact that light travels along light-light geodesics with the light speed. In this paperfirst we give an appropiate definition of a generalized Dirac operator on a Lorentzian manifold then we show that the same result is true for gen- eralized Dirac operator. This means that the Denker (partial) connection on the polarization set (along Hamiltonian orbits) of the system is the starting connection lifted to the cotangent bundle. We thank Bernd Am- mann for giving us the correct definition of the generalized Dirac operator.

2. The generalized Dirac operators on Lorentzian manifolds

Let (X;g) be a Lorentzian manifold. With Cl(X) denote the bundle over X whose fiberoverp is the Clifford algebra of T_pX i.e the quotient of the complexified tensoralgebra of T_pX by the bilateral

(*) Indirizzo dell'A.: Dipartimento di Matematica, Sapienza UniversitaÁ di Roma, Piazzale Aldo Moro, 5, 00185 Roma

E-mail: [email protected] [email protected]

ideal generated by the elements of the form vw‡wv‡2g(v;w).

The Lorentzian metric and the connection extend to Cl(X) to give a metric connection which is Leibnitz with respect to the Clifford multiplication.

DEFINITION2.1. A Clifford module of spink=2 over(X;g) is given by a complex vectorbundleSoverXwith a sesquilinearproduct (antilinearin the second)h;iwith connectionr^S togetherwith a smooth sectionQof hom(^kTX;End(S)) satysfiying the following properties

C1 The fibersSp are left modules over the Clifford algebras Cl(TpX) i.e. there's a vector bundle homomorphismTXS !S,ZW7 !ZW such that (ZY‡YZ‡2g(Z;Y))Wˆ0:

C2 The inner product is parallel,dhW;ci ˆ hr^SW;ci ‡ hW;r^Sci:

C3 The Clifford action Cl(X)S !S is parallel, r^S_Z(YW)ˆ (r^g_ZY)W‡Y r^S_ZW:

C4 Clifford multiplication by tangent vectors is symmetrichZW;ci ˆ hW;Zci:

C5 The section Q is parallel with respect to the connection on hom(^kTX;End(S)) induced by the Levi-Civita connectionr^g andr^S.

C6 Q(;. . .;) is simmetric w.r.t.h;i.

C7 If N is a future directed timelike vector then, putting Q_N:ˆQ(N;. . .;N):

1. ZQ_Nˆ Q_NZ;ifg(Z;N)ˆ0 2. NQ_NˆQ_NN:

C8 IfN is a future directed timelike vector then the quadratic form hh;ii_Ndefined byhhW;cii_Nˆ hQNW;ciis positive definite.

From C8 it follows that Q_N is invertible. From the first one of C7 it follows that its spectrum is symmetric w.r.t the origin, which togetherwith C8 implies that the bilinearform h;i has index (1=2 r ank (S); 1=2 r ank (S)):

FromC7 follows that

C7⁰ 1. Forany vectorZwe haveZQY ˆ QYZ;g(Y;Z)ˆ0 2. ZQ_ZˆQ_ZZ;againQ_ZˆQ(Z;. . .;Z).

With such a structure one can Define the generalized Dirac operator D:C¹(X;S) !C¹(X;S) following the composition

C¹(X;S) ^ir!^S C¹(TX;S) !^] C¹(TX;S) ^Cl!C¹(X;S)

where we used the musical isomorphism ]. In a local orthonormal frame e₀;. . .;e_n withe_jˆg(e_j;e_j) one can check the formula

DWˆiXⁿ

jˆ0

e_j r^S_ejW:

It is a first order differential operator whose principal symbol is Clifford multiplication

sD(j)ˆiZ wherej^]ˆZi:e:g(Z;)ˆj;

This can be seen immediately from the formulaD(fW)ˆigradfW‡fDW:

In this sense one says that the Dirac operator is the quantization of the Clifford action.

3. Propagation of singularities

In this sectionM is a manifold without boundary. We are going to ex- plain the theorem about propagation of singularities for systems of real principal type of Denker [1] and apply to the generalized Dirac operator on a Lorentzian manifold. We shall use only classical pseudodifferential op- erators. Now a scalar pseudodifferential operator on X is said of real principal type if its principal symbolsis real and the Hamiltonian vector fieldH_sis non vanishing and does not have the radial direction on the zero set ofs. There is a corresponding notion for systems. An orderm,NN system Aof pseudodifferential operators with principal symbolsm(A) is said of real principal type ath2TMif there exists on a neighborhoodUof hanNNsymbol~sand a scalar symbol of real principal typeqsuch that

~s(j)sm(A)(j)ˆq(j) Id;j2U:

…1†

If this property holds for everyhinTM we sayAis real principal type.

Clearly the existence of ~sis only locally granted and Ucan be assumed conical.

PROPOSITION 3.2. The generalized Diracoperator on a Lorentzian manifold is real principal type.

PROOF. LetUMbe an open set in which a timelike future directed vectorfieldN exists. RememberthatQNis invertible. Define inTU the following simbol,

es(j)ˆ iQ_N¹(aY bN)QN

if j^]ˆZˆaY‡bN with g(Y;N)ˆ0:Then a straightforward computa- tion based onC7 andC1 shows that

e

s(j)s_D(j)ˆ kjk²_gId

wherej7 ! kjk²_gis the Lorentzian norm, in coordinateskjk²_gˆg^ij(x)j_ij_j a well known Hamiltonian who generates the geodesic flow. It is standard the computation that it is of real principal type. The zero set is the light cone and the null bicharacteristics are lightlike geodesic. p For a real principal type system there's an elegant result of Denker about the propagation of singularities [1, 4]. First one defines the C¹- based polarization set of a vector valued distribution. It is a family of vector spaces in C^N (ora vectorbundle S) overthe singularsupport of the distribution conical in the j-variable and serves as an indicator of the direction of the strongest singularity. Here the definition: Let u2 D⁰(M;C^N) a vector valued distribution i.e. a vector of distributions u_i2 D⁰(M). It's wave front set is by definition the union of all the wave front sets of its components

WF(u):ˆ [

iˆ1;::N

WF(ui):

In particular it does not contain any information about the components of distributions that are singular. In order to specify the singular directions in the vectorspaceC^N, one could considervector-valued operators that map the vectorvalued distribution to a smooth scalarfunction, instead of just looking at scalar operators mapping the individual components to smooth functions. This approach leads to the definition of the polarization set.

DEFINITION3.3. The polarization set of a distributionu2 D⁰(M;C^N) is defined as

WF_pol(u):ˆ \

Au2C¹(M)

N_A:

Here, for an operator mapping the vector valued distribution to a smooth function, with (principal) symbola(x;j) define

N_A:ˆ f(x;j;w)2(TMn0)C^N:w2kera(x;j)g:

We stress that the intersection is taken over all 1N systems A2L⁰(M)^N of classycal CDOs. Two important properties proved by Dencker[1]

Let u2 D⁰(M;C^N) andp_1;2:TMC^N !TM be the projection on the cotangent bundle, then

p1;2(WFpol(u)n(TM0))ˆWF(u):

In this way the polarization set is a refinement of the wave front set.

LetAbe anNNsystem of pseudo-differential operators onMwith principal symbola(x;j) andu2 D⁰(M;C^N). Then

a(WF_pol(u))WF_pol(Au);

whereaacts on the fibre:a(x;j;w)ˆ(x;j;a(x;j)w):

IfEis anNNsystem of pseudo-differential operators onMand its principal symbole(y;h)6ˆ0, then

e(WF_pol(u))ˆWF_pol(Eu)

overa conic neighbourhood of (y;h):From this last property we see that the polarization set behaves covariantly under a change of coordinates.

Thus the polarization set can be defined for distributional sections D⁰(M;E) of a vectorbundleEgiving a well defined subset

WFpol(u)pE

of the vectorbundle lifted overthe cotangent bundle.

So return to the setting before and assume we are given an NN systemAof pseudo-differential operators acting onC^N of principal type and (1) valid inU. Define the set

VA:ˆ fj2TX:detsm(A)(j)ˆ0g

locallyVAˆq ¹(0):Furthermore one can show thatAis real principal type aroundhif and only if aroundhthe setV_Ais a hypersurface with non radial Hamiltonian field, the dimension of kers_m(A) is constant inV_A and for every normal vectorr2N(V_A) if

p_C:C^N !C^N=Ims_m(A)(j)ˆcokers_m(A)(j) is the quotient mapping then the map

p_C(@_r)s_m(A)(j):kers_m(A)(j) !cokers_m(A)(j) is an isomorphism.

Now letu2C ¹(X;C^N) a distribution such thatAu2C¹. Let denote NAj_j C^N the kernel ofsm(A) atj. Now if (j;w)2 NA, by definition it follows necessarily that the (vector valued) distribution wave front set is contained inV_A.

WF(u)V_A:

One can show as in the scalarDuistermaat HoÈrmander result [2] the wave front set is union of null bicharacteristics ofq inV_A. Well here q is not unique but it can be shown to be unique modulo the multiplication by a nevervanishing function onTM. This does not affect the bicharacteristics since the wave front is conical in the j-variable. We shall call them bi- characteristics again. It remains to study the polarization vectors over these bicharacteristics. The idea of Denker is to introduce a partial con- nection and show that they follow the parallel transport along these null bicharacteristics and these vector fields form completely the polarization set. The Denker connection depends also from the subprincipal symbol ([2]

Sect 5.2) of the operator (¹) The principal symbol ofAhas an asymptotic homogeneous expansion

s(A)(j)ˆs_m(A)(j)‡p_{m 1}(j)‡p_{m 2}(j)‡;. . .: The subprincipal symbol is defined as

s(A)^s(j)ˆp_{m 1}(j) 1 2i

@²s_m(A)(j)

@x^j@j_j ;

form the ``poisson bracket'' of matrices (this is not anti symmetric !) f~s;s_m(A)g:ˆ@~s

@jj

@s_m(A)

@x^j

@~s

@x^j

@s_m(A)

@j_j :

letH_qdenote the Hamiltonian vectorfield ofqdefined bydq(Y)ˆv(H_q;Y) forevery Y2TM, then fora smooth section w of the trivial bundle (TXno)C^Ndefine the Denkerconnection

r^AwˆH_qw‡1=2f~s;s_m(A)gw‡i~ss(A)^sw:

…2†

One can show thatr^A resctricts toN_A i.e. ifwis a vectorfield along a bicharacteristicginVAthen

r^A_{g_}w2kers_m(A)(g(t))ˆ N_Ajg(t) iff w(g(t))2kers_m(A):

(¹) This is a genuine feature of the system nature, in the scalar case only the principal symbol contributes.

This means that the equation r^Awˆ0 can be solved in NA which is a necessary condition to be in WFpol(u):

DEFINITION3.4 (Denker[1]). A Hamiltonian orbit of the systemAis a line bundleL NAjgwhere,

gis an integral curve of the Hamiltonian field ofV_A. Lis spanned by aC¹ sectionwsuch thatr^Awˆ0:

Finally we can state the theorem about the propagation of the polar- ization set.

THEOREM3.4 (Denker[1]). Supposeh2V_Ais a point where A is real principal type. If u is a vector valued distribution such thath2=WF(Au) (vector valued wave front) then over a neighborhood ofhinV_A,WF_polis a union of Hamiltonian orbits of A.

If Au2C¹ we are saying that that the non trivial part of the polar- ization set of u is parallel inN_Aalong the flow ofV_A:

We are going to apply this result to the generalized Dirac operator. We know from proposition 3.2 thatVDˆ fj:kjk²_ggis the light cone. Then the null bicharacteristics are the null geodesics. Let's compute explicitly the Denkerconnection.

THEOREM 3.5. The Denker connection for the generalized Dirac operator is r^S lifted to the cotangent bundle (along null geodesics), p:TX !X:

In particular the polarization set of the generalized Dirac operator is union of Hamiltonian orbits i.e. curves t7 !g(t)ˆ(x(t);j(t);s(t))2pS such that

x(t)is a null geodesic,

j(t)^]is the velocity of the geodesic

s(t)is a section of S alonggparallel with respect to the connectionpr^S. PROOF. On a point (p;j) with j^]ˆZˆaY‡bN on the light cone choose coordinates such that r^g_@xkZˆ0 and r^g_@xk(aY bN)ˆ0. This is possible sinceZis lightlike meaning thatY6ˆ0. FromC8 since we know the signature of h;i on S we can choose a trivialization s1;. . .;s_N=2, s_N=2‡1;. . .;s_Nsuch that (r^S_eie^j)(p)ˆ0:Now the principal symbols1(D) over

TUcorresponds to a matrix

f_ij(x;j)ˆ h(s(D)(j)s_i;s_j)e_ijˆ h iZs_i;s_ji wheree_ijˆ hs_i;s_ji. Then in the point (p;j) we have

@x_kfijˆ(pr^S)@_xk( iZsi;sj)eijˆ ihr^S_@xk(Zsi);sjieijˆ ih(r^g_@xkZ)si;sjiˆ0 stating no more than the compatibility with the Clifford action, the Levi Civita connection and the innerproduct onS and a corresponding state- ment forthe symbol ~s. In particular in the point j on the lightcone the second and third terms in (2) vanish and remains only the Lie derivative alongH_kk²

g. The same forpr^S. p

REFERENCES

[1] N. DENCKER,On the propagation of singularities of systems of real principal type, J. Funct. Anal.,46, no. 3 (1982), pp. 351-372.

[2] J. J. DUISTERMAAT - L. HOÈRMANDER, Fourier integral operators II, Acta Math.,128(1972), pp. 183-269.

[3] L. HOÈRMANDER,The analysis of linear partial differential operators, vol. 1-4 (Springer-Verlag, 1983).

[4] K. KRATZERT,Singularity structure of the two point function of the free Dirac field on a globally hyperbolicspacetime. Annalen Phys.,9(2000), pp. 475-498.

Manoscritto pervenuto in redazione l'1 aprile 2011.