We write the Dirac equation in curved 4-dimensional Lorentzian spacetime using concepts from the analysis of partial differential equations as opposed to geometric concepts

Vassiliev, D

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September 4, 2017 14:2 ws-procs961x669 MG-14 – Proceedings (Part D) D497 page 37793779A non-geometric representation of the Dirac equationin curved spacetimeDmitri VassilievDepartment of Mathematics, University College London,Gower Street, London WC1E 6BT, UKE-mail: D.Vassiliev@ucl.ac.ukwww.homepages.ucl.ac.uk/∼ucahdva/Supported by EPSRC grant EP/M000079/1We write the Dirac equation in curved 4-dimensional Lorentzian spacetime using conceptsfrom the analysis of partial diﬀerential equations as opposed to geometric concepts.Keywords: Analysis of partial diﬀerential equations; gauge theory; Dirac equation.1. Playing FieldLet M be a connected 4-manifold without boundary. We will work with 2-columnsv : M → C2 of complex-valued half-densities (a half-density is a quantity whichtransforms as the square root of a density under changes of local coordinates).The inner product on such 2-columns is deﬁned as 〈v, w〉 := ∫Mw∗v dx, wherex = (x1, x2, x3, x4) are local coordinates, dx = dx1dx2dx3dx4 and the star standsfor Hermitian conjugation.Let L be a formally self-adjoint ﬁrst order linear diﬀerential operator acting on2-columns of complex-valued half-densities. Our initial objective will be to examinethe geometric content of the operator L. In order to pursue this objective we ﬁrstneed to provide an invariant analytic description of the operator.In local coordinates our operator readsL = Fα(x)∂∂xα+G(x), (1)where Fα(x), α = 1, 2, 3, 4, and G(x) are some 2×2 matrix-functions. The principaland subprincipal symbols of the operator L are deﬁned asLprin(x, p) := iFα(x) pα , (2)Lsub(x) := G(x) +i2(Lprin)xαpα(x) , (3)where p = (p1, p2, p3, p4) is the dual variable (momentum); see Ref. 3. The principaland subprincipal symbols are invariantly deﬁned 2× 2 Hermitian matrix-functionson T ∗M and M respectively which uniquely determine the operator L.Further on in this paper we assume that the principal symbol of our operatorsatisﬁes the following non-degeneracy condition:Lprin(x, p) = 0, ∀(x, p) ∈ T ∗M \ {0}. (4)Condition (4) means that the elements of the 2 × 2 matrix-function Lprin(x, p) donot vanish simultaneously for any x ∈ M and any nonzero momentum p. The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.comby UNIVERSITY COLLEGE LONDON (UCL) on 01/16/19. Re-use and distribution is strictly not permitted, except for Open Access articles.September 4, 2017 14:2 ws-procs961x669 MG-14 – Proceedings (Part D) D497 page 378037802. Lorentzian Metric and Orthonormal FrameObserve that the determinant of the principal symbol is a quadratic form in thedual variable (momentum) p :detLprin(x, p) = −gαβ(x) pαpβ . (5)We interpret the real coeﬃcients gαβ(x) = gβα(x), α, β = 1, 2, 3, 4, appearing informula (5) as components of a (contravariant) metric tensor.The following result was established in Ref. 2.Lemma 2.1. Our metric is Lorentzian, i.e. it has three positive eigenvalues andone negative eigenvalue.Furthermore, the principal symbol of our operator deﬁnes an orthonormal frameejα(x). Here the Latin index j = 1, 2, 3, 4 enumerates the vector ﬁelds, the Greekindex α = 1, 2, 3, 4 enumerates the components of a given vector ej and orthonor-mality is understood in the Lorentzian sense:gαβ ejαekβ =⎧⎪⎨⎪⎪⎩0 if j = k,1 if j = k = 4,−1 if j = k = 4.(6)The orthonormal frame is recovered from the principal symbol as follows. De-composing the principal symbol with respect to the standard basiss1 =(0 11 0), s2 =(0 −ii 0), s3 =(1 00 −1), s4 =(1 00 1)in the real vector space of 2× 2 Hermitian matrices, we get Lprin(x, p) = sjcj(x, p).Each coeﬃcient cj(x, p) is linear in momentum p, so cj(x, p) = ejα(x) pα .The existence of an orthonormal frame implies that our manifold M is paral-lelizable. We see that our analytic non-degeneracy condition (4) has far reachinggeometric consequences.3. Gauge Transformations and Covariant Subprincipal SymbolLet us consider the action (variational functional)∫Mv∗(Lv) dx associated withour operator. Take an arbitrary smooth matrix-functionR : M → SL(2,C) (7)and consider the following transformation of our 2-column of unknowns:v → Rv. (8)We interpret (8) as a gauge transformation because we are looking here at a changeof basis in our vector space of unknowns v : M → C2. The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.comby UNIVERSITY COLLEGE LONDON (UCL) on 01/16/19. Re-use and distribution is strictly not permitted, except for Open Access articles.September 4, 2017 14:2 ws-procs961x669 MG-14 – Proceedings (Part D) D497 page 37813781The transformation (8) of the 2-column v induces the following transformationof the action:∫Mv∗(Lv) dx → ∫Mv∗(R∗LRv) dx . This means that our 2 × 2diﬀerential operator L experiences the transformationL → R∗LR . (9)This section is dedicated to the analysis of the transformation (9).Remark 3.1. We chose to restrict our analysis to matrix-functions R(x) of deter-minant one, see formula (7), because we want to preserve our Lorentzian metricdeﬁned in accordance with formula (5).Remark 3.2. In non-relativistic theory one normally looks at the transformationL → R−1LR (10)rather than at (9). The reason we chose to go along with (9) is that we are thinkingin terms of actions and corresponding Euler–Lagrange equations rather than opera-tors as such. We believe that this point of view makes more sense in the relativisticsetting. If one were consistent in promoting such a point of view, then one wouldhave had to deal with actions throughout the paper rather than with operators. Wedid not adopt this ‘consistent’ approach because this would have made the paperdiﬃcult to read. Therefore, throughout the paper we use the concept of an oper-ator, having in mind that we are really interested in the action and correspondingEuler–Lagrange equation.Remark 3.3. The transformations (9) and (10) coincide if the matrix-functionR(x) is special unitary. Applying special unitary transformations is natural inthe non-relativistic 3-dimensional setting when dealing with an elliptic system, seeRef. 1, but in the relativistic 4-dimensional setting when dealing with a hyperbolicsystem special unitary transformations are too restrictive.The transformation (9) of the diﬀerential operator L induces the following trans-formations of its principal (2) and subprincipal (3) symbols:Lprin → R∗LprinR , (11)Lsub → R∗LsubR+ i2(R∗xα(Lprin)pαR− R∗(Lprin)pαRxα) . (12)Comparing formulae (11) and (12) we see that, unlike the principal symbol, thesubprincipal symbol does not transform in a covariant fashion due to the appearanceof terms with the gradient of the matrix-function R(x).It turns out that one can overcome the non-covariance in (12) by introducingthe covariant subprincipal symbol Lcsub(x) in accordance with formulaLcsub := Lsub +i16gαβ{Lprin, adjLprin, Lprin}pαpβ , (13) The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.comby UNIVERSITY COLLEGE LONDON (UCL) on 01/16/19. Re-use and distribution is strictly not permitted, except for Open Access articles.September 4, 2017 14:2 ws-procs961x669 MG-14 – Proceedings (Part D) D497 page 37823782where {F,G,H} := FxαGHpα − FpαGHxα is the generalised Poisson bracket onmatrix-functions and adj is the operator of matrix adjugationF =(a bc d)→(d −b−c a)=: adjF (14)from elementary linear algebra.The following result was established in Ref. 2.Lemma 3.1. The transformation (9) of the diﬀerential operator induces the trans-formation Lcsub → R∗LcsubR of its covariant subprincipal symbol.Comparing formulae (3) and (13) we see that the standard subprincipal sym-bol and covariant subprincipal symbol have the same structure, only the covariantsubprincipal symbol has a second correction term designed to ‘take care’ of speciallinear transformations in the vector space of unknowns v : M → C2. The standardsubprincipal symbol (3) is invariant under changes of local coordinates (its elementsbehave as scalars), whereas the covariant subprincipal symbol (13) retains this fea-ture but gains an extra SL(2,C) covariance property. In other words, the covariantsubprincipal symbol (13) behaves ‘nicely’ under a wider group of transformations.4. Electromagnetic Covector PotentialThe covariant subprincipal symbol can be uniquely represented in the formLcsub(x) = Lprin(x,A(x)), (15)where A = (A1, A2, A3, A4) is some real-valued covector ﬁeld. We interpret thiscovector ﬁeld as the electromagnetic covector potential.Lemma 3.1 and formulae (11) and (15) tell us that the electromagnetic covectorpotential is invariant under gauge transformations (9).5. Adjugate OperatorDeﬁnition 5.1. The adjugate of a formally self-adjoint non-degenerate ﬁrstorder 2× 2 linear diﬀerential operator L is the formally self-adjoint non-degenerateﬁrst order 2 × 2 linear diﬀerential operator AdjL whose principal and covariantsubprincipal symbols are matrix adjugates of those of the operator L.We denote matrix adjugation by adj , see formula (14), and operator adjugationby Adj . Of course, the coeﬃcients of the adjugate operator can be written downexplicitly in local coordinates via the coeﬃcients of the original operator (1), seeRef. 2 for details.Applying the analysis from Sections 2–4 to the diﬀerential operator AdjL it iseasy to see that the metric and electromagnetic covector potential encoded withinthe operator AdjL are the same as in the original operator L. Thus, the metricand electromagnetic covector potential are invariant under operator adjugation.It also easy to see that AdjAdjL = L, so operator adjugation is an involution. The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.comby UNIVERSITY COLLEGE LONDON (UCL) on 01/16/19. Re-use and distribution is strictly not permitted, except for Open Access articles.September 4, 2017 14:2 ws-procs961x669 MG-14 – Proceedings (Part D) D497 page 378337836. Main ResultWe deﬁne the Dirac operator as the diﬀerential operatorD :=(L mImI AdjL)(16)acting on 4-columns ψ =(v1 v2 w1 w2)Tof complex-valued half-densities. Herem is the electron mass and I is the 2× 2 identity matrix.The ‘traditional’ Dirac operator Dtrad is written down in Appendix A of Ref. 2and acts on bispinor ﬁelds ψtrad =(ξ1 ξ2 η1˙ η2˙)T. Here we assume, without lossof generality, that the orthonormal frame used in the construction of the operatorDtrad is the one from Section 2.Our main result is the following theorem established in Ref. 2.Theorem 6.1. The two operators, our analytically deﬁned Dirac operator (16) andgeometrically deﬁned Dirac operator Dtrad , are related by the formulaD = | det gκλ|1/4Dtrad | det gμν |−1/4 . (17)Consider now the two Dirac equationsDψ = 0, (18)Dtradψtrad = 0. (19)Formula (17) implies that the solutions of equations (18) and (19) diﬀer only by aprescribed scaling factor: ψ = | det gμν |1/4 ψtrad . This means that for all practicalpurposes equations (18) and (19) are equivalent.7. Spin StructureLet us consider all possible formally self-adjoint non-degenerate ﬁrst order 2 × 2linear diﬀerential operators L corresponding, in the sense of formula (5), to the pre-scribed Lorentzian metric. In this section our aim is to classify all such operators L.Let us ﬁx a reference operator L and let ej be the corresponding orthonormalframe (see Section 2). Let L be another operator and let ej be the correspondingorthonormal frame. We deﬁne the following two real-valued scalar ﬁeldsc(L) := − 14!(e1 ∧ e2 ∧ e3 ∧ e4)κλμν (e1 ∧ e2 ∧ e3 ∧ e4)κλμν , t(L) := − e4α e4α .Observe that these scalar ﬁelds do not vanish; in fact, c(L) can take only two values,+1 or −1. This observation gives us a primary classiﬁcation of operators L intofour classes determined by the signs of c(L) and t(L). The four classes correspondto the four connected components of the Lorentz group.Note thatc(−L) = c(L), t(−L) = −t(L),c(AdjL) = −c(L), t(AdjL) = t(L), The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.comby UNIVERSITY COLLEGE LONDON (UCL) on 01/16/19. Re-use and distribution is strictly not permitted, except for Open Access articles.September 4, 2017 14:2 ws-procs961x669 MG-14 – Proceedings (Part D) D497 page 37843784which means that by applying the transformations L → −L and L → AdjL to agiven operator L one can reach all four classes of our primary classiﬁcation.Further on we work with operators L such that c(L) > 0 and t(L) > 0.We say that the operators L and L˜ are equivalent if there exists a smooth matrix-function (7) such that L˜prin = R∗LprinR. The equivalence classes of operatorsobtained this way are called spin structures.The above 4-dimensional Lorentzian deﬁnition of spin structure is an extension ofthe 3-dimensional Riemannian deﬁnition from Ref. 1. The diﬀerence is that we havenow dropped the condition trLprin(x, p) = 0, replaced the ellipticity condition bythe weaker non-degeneracy condition (4) and extended our group of transformationsfrom special unitary to special linear.One would hope that for a connected Lorentzian 4-manifold admitting a globalorthonormal frame (see (6) for deﬁnition of orthonormality) our analytic deﬁnitionof spin structure would be equivalent to the traditional geometric one. Unfortu-nately, we do not currently have a rigorous proof of equivalence in the 4-dimensionalLorentzian setting.AcknowledgmentsThe author is grateful to Zhirayr Avetisyan and Nikolai Saveliev for helpful advice.References1. Z. Avetisyan, Y.-L. Fang and D. Vassiliev, arXiv:1512.06281 (2015).2. Y.-L. Fang and D. Vassiliev, J. Phys. A 48, 165203 (2015).3. Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partialdiﬀerential operators (American Mathematical Society, Providence, 1997, 1998). The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.comby UNIVERSITY COLLEGE LONDON (UCL) on 01/16/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

A non-geometric representation of the Dirac equation in curved spacetime

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A non-geometric representation of the Dirac equation in curved spacetime

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