Symmetry operators and separation of variables Ray McLenaghan Conference September 2010
Symmetry operators and separation of variables for the Dirac equation on curved space Ray McLenaghan Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada S4 Conference September 2010 Outline Introduction Review Theory Separation in 2-dimensional pseudo-Riemannian spaces Second order symmetry operators First order operators Symmetry and separation of variables Future directions References Hamilton-Jacobi, Helmholtz, and Dirac equations We will be considering the following equations defined on a n-dimensional spin manifold M with contravariant metric tensor g µν and covariant derivative ∇µ : The Hamilton-Jacobi equation for the geodesics: g µν ∂µ W ∂ν W = E . (1) Hψ := g µν ∇µ ∇ν ψ − E ψ = 0. (2) Dψ := iγ a ∇a ψ − mψ = 0. (3) The Helmholtz equation The Dirac equation In the last equation the γ a denote the gamma matrices which satisfy γ a γ b + γ b γ a = 2η ab I, where η ab denotes the (constant) frame metric. Dirac equation In (3) ∇a := eaµ ∇µ , denotes the frame covariant derivative associated to the spin frame eaµ . The covariant derivative of a spinor ψ is defined by ∇µ ψ = ∂µ ψ + 81 Γab µ [γa , γb ]ψ, where the spin connection is given by a α bβ Γab + ∂µ e bα ). µ = eα (Γβµ e In (4) Γα βµ denotes the Levi-Civita connection of the metric gµν = eµa ηab eνb induced by the spin frame. (4) Separation of Variables H-J and H Hamilton-Jacobi equation: sum separability ansatz. The H-J equation is separable if there exists a separable complete integral of the form W (x, c) = n X Wi (x i , c). (5) i=1 Helmholtz equation: product separability ansatz. The H equation is separable if there exists a separable solution (satisfying an appropriate completeness condition) of the form ψ(x, c) = n Y ψi (x i , c), (6) i=1 such that the functions ψi are solutions of a set of ODEs in the variables xi. Separation of variables D Dirac equation: local product separability ansatz. The Dirac equation is said to be separable in a spin frame eaµ with respect to local coordinates x := (x 1 , . . . , x n ) if there exists a separable solution of the form ψ̃ 1 (x) .. ψ(x) = , . (7) m ψ̃ (x) where ψ̃ j (x) = n Y ψij (x i ), (8) i=1 where ψij for each j are solutions of (systems) of ODEs in variables x i . Invariant characterization A valence p symmetric tensor Ki1 ,...,ip that satisfies ∇(µ1 Kµ2 ,...,µp+1 ) = 0 is called a Killing tensor. For p = 1, Kµ is called a Killing vector. Orthogonal separability of the HJ equation is characterized (EKM) by the existence of a set of n − 1 linearly independent Killing 2-tensors which commute among themselves and have a common basis of eigenforms θi . The separable coordinates x i are defined by θi = f i dx i . K is a symmetry operator of H iff [K, H]=0. The first and second order linear symmetry operators are defined by Killing vectors and valence two Killing tensors: K := K µ ∇µ , K = ∇µ K µν ∇ν , where in the second case the following additional condition must be satisfied: ∇µ K ρ[µ R ν] ρ = 0 where R ν ρ is the Ricci tensor. The H equation is orthogonally separable iff the corresponding HJ equation is separable and the Robertson condition K ρ[µ R ν] ρ = 0 is satisfied. The separable solutions are eigenvectors of K with the separation constants as eigenvalues: Kψ = λψ (separation paradigm). Invariant charcterization: Dirac We would like to find an analogous characterization for Dirac of the following form: Geometric conditions on M imply there exists a spin transformation which induces the following transformations on spinors and spin frames: ψ 0 = Sψ, ea0µ = Jνµ ebν `ba , where ` is the image of S in SO(η) and J is the Jacobian of the transformation, such that the transformed Dirac equation D0 ψ 0 = SDψ = 0 is separable. Kerr solution Separated by Chandrasekhar WRT Boyer-Lidquist coordinates in the Kinnersley (null) spin frame ` = (r 2 +a2 )∆−1 ∂t +∂r +a∆−1 ∂φ , n = (2Σ)−1 ((r 2 +a2 )∂t −∆∂r +a∂φ ), √ m = ( 2(r + ia cos(θ))−1 (ia sin(θ)∂t + ∂θ + i csc(θ)∂φ ), where Σ + r 2 + a2 cos(θ), ∆ = r 2 − 2ma + a2 Special case of Schwarzschild (a = 0) separated earlier by Brill and Wheeler. Chandra procedure analysed by Carter & McLenaghan who discovered the underlying symmetry operators which characterize the separation K = i(k µ − (1/4)γ µ γ ν ∇ν kµ ), L = iγ5 γ µ (fµ ν ∇ν − (1/6)γ ν γ ρ ∇ρ fµν ), where ∇(µ kν) = 0 and ∇(µ fν)ρ = 0. Type D vacuum solution I Most general first-order commuting symmetry operator (McLenaghan & Spindel) is a sum of the two operators given previously and M = y µνρ γνρ ∇µ − (3/4)∇µ y µ γ5 where ∇(µ yν)ρσ = 0 (Killing-Yano equation) I Chandra procedure extended to the class of Petrov type D vacuum solutions and its characterization by first order symmetry operators (Kamran & McLenaghan). I Most general first order R-commuting symmetry operators for the massless Dirac equation constructed (Kamran & McLenaghan). Factorizable systems First general theory of Dirac separability is that of factorizable systems proposed by Miller  as part of his general theory of mechanisms for variable separation in PDEs. Miller studies systems of the form Dψ := H i ∂i ψ + V ψ = λ1 ψ A factorizable system for the above is set of n equations ∂i ψ = (Cij λj − Ci )ψ A factorizable system is separable if ∂k Cij = ∂k Ci = 0 for k 6= i. The integrability conditions for a factorizable are satisfied iff there exist a a system of n − 1 first order differential operators that commute amongst themselves and with D and have the solutions of the fs as eigenspinors. Non-factorizable systems Fels & Kamran show  I that factorizable systems contain all the the separable systems for the Dirac equation on Petrov type D vacuum spacetimes. I that for n = 4 there exist non-factorizable systems by giving an example of a metric with a 1-parameter isometry group and for which the Dirac equation admits a non-factorizable separable system characterized by second order symmetry operators. They also provide an example for n = 2. It thus seems that a complete theory requires the study of second order symmetry operators. The first steps in this direction were taken by McLenaghan, Smith & Walker  and Smith  who in the case n = 4 computed the general second-order symmetry operator using a two-component spinor formalism. The remaining slides describe joint work with Carignano, Fatibene, Smith & Rastelli for the case n = 2. Separability in 2-dimensions: HJ and H equations If M admits a non-trivial valence two Killing tensor K, there exists a coordinate system (u, v ) such that ds 2 = (A(u) + B(v )) du 2 + dv 2 K = (A(u) + B(v )) −1 (Liouville metric) (9) (B(v )∂u ⊗ ∂u − A(v )∂v ⊗ ∂v ) where A and B are arbitrary smooth functions. The Helmhotz equation reads 2 2 Hψ = (A + B)−1 ∂uu ψ + ∂vv ψ − Eψ = 0 Since Rµν = 1/2Rgµν , H admits a second order symmetry operator K: 2 2 Kψ = (A + B)−1 B ∂uu ψ − A ∂vv ψ With the product ansatz ψ(u, v ) = a(u)b(v ), the Helmholtz equation implies ba00 + ab 00 − E (A + B)ab = 0, which separates yielding a00 /a − EA = −b 00 /b + EB = λ, where λ is a separation constant. It follows that separable solution satisfies K(ab) = λab Dirac equation separability in 2-dimensions [5,7] A choice of gamma matrices valid for both signatures is 1 0 0 −k 1 2 γ = , γ = , 0 −1 k 0 √ where k = −η, η = det(ηab ), and ηab = diag(1, ±1) is the frame metric. The Dirac equation may be written as e 1 ψ + B∂ e 2ψ + C e ψ − λψ = 0 Dψ := A∂ where e= A A1 −A2 A2 −A1 e= B B1 −B2 B2 −B1 e= C C1 C2 −C2 −C1 with A1 = ie11 A2 = −ike21 B1 = ie12 B2 = −ike22 (10) C1 = −(i/2)ke2µ Γ12 µ C2 = −(i/2)e1µ Γ12 µ Separation With product separability assumption ψi = ai (x)bi (y ) the D equation reads A1 ȧ1 b1 + A2 ȧ2 b2 + B1 a1 ḃ1 + B2 a2 ḃ2 + C1 a1 b1 − C2 a2 b2 − λa1 b1 = 0 A2 ȧ1 b1 + A1 ȧ2 b2 + B2 a1 ḃ1 + B1 a2 ḃ2 − C2 a1 b1 + C1 a2 b2 + λa2 b2 = 0 Definition The Dirac equation is separable in the coordinates (x, y ) if there exist nonzero functions Ri (x, y ) such the equations can be written as R1 ai bj (E1x + E1y ) = 0 R2 ak b` (E2x + E2y ) = 0 for a suitable choice of indices i, j, k, `, where Eix (x, aj , ȧj ) and Eiy (y , bj , ḃj ). Moreover, the equations Eix (x, aj , ȧj ) = µi = −Eiy (y , bj , ḃj ) define the separation constants µi . Additional assumptions 1. We construct eigenvalue-type operators Lψ = µψ with eigenvalues µ(µi ) making use only of the terms Eix and Eiy . 2. We require that the operators L are independent of λ. 3. We require that [L, D]ψ = 0 for all ψ. 4. We assume λ 6= 0. TYPES OF SEPARATION I. a2 6= a2 and b1 6= b2 II. a1 = a2 = a and b1 6= b2 (or vice-versa) III. a1 = a2 = a and b1 = cb2 = b (c constant) Results Proposition The only type I separation is associated with the nonsingular Dirac operator and associated symmetry operator of the forms 2 0 1 1 0 ∂x 0 D := ∂x + (i/R1 (y )) ∂y L= −1 0 0 −1 0 ∂x2 The separable spin frame is given by e11 = e22 = 0 e12 = 1 e21 = R1 (y ) The corresponding coordinates separate the geodesic Hamilton-Jacobi equation. If the Riemannian manifold is the Euclidean plane, the coordinates, up to a rescaling, coincide with polar or Cartesian coordinates. Separability is also possible for equations of Type II which give rise to first order operators. Separability is not possible for Type III. Conclusions from the analysis I Dirac separability implies Hamilton-Jacobi and Helmholtz separability. I Dirac separability implies M admits a 1-parameter isometry group. Second order symmetry operators [6,7] A second order symmetry operator for the Dirac equation is an operator of the form K = Eab ∇ab + Fa ∇a + GI, which satisfies the defining relation [K, D] = 0. ab a The coefficients E , F , G are matrix zero-order operators and ∇ab = 12 (∇a ∇b + ∇b ∇a ). The condition (11) is equivalent to E(ab γ c) − γ (c Eab) = 0 (a b) (b a) c ab F γ − γ F = γ ∇ c E Gγ a − γ a G = γ c ∇c Fa − R4 Eab γ c + γ c Eab bc γ+ R Ebd γ c + 2γ c Ebd a d bc 6 γ a ∇ G = R Fa γ b + γ b Fa γ + 1 2Eab γ c + γ c Eab γ ∇ R a ab ac b 8 12 where R denotes the Ricci scalar of gµν . (11) Solution of defining equations Expand Eab in the basis (I, γ1 , γ2 , γ) of the Clifford algebra C(2): Eab = e ab I + ecab γ c + ê ab γ, where the coefficients e ab , ecab , ê ab are functions in M. The solution of the first equation of (12) is given by Eab = e ab I + 2α(a γ b) , where the e ab and the αa are the frame components of arbitary valence two tensor and vector fields. Setting Fa = f a I + fba γ b + fˆa γ, we find that the most general solution of the first and second equations of (12) may be written as ( Eab = K ab I + 2α(a γ b) Fa = f a I + (γ c ∇c αa + Aγ a ) + 13 bc ∇b K ac γ Solution of defining equations where ( ∇(a K bc) = 0 ∇(a αb) = 0 Observe that K ab and is necessarily a valence two Killing tensor and αa a Killing vector. Writing G = g I + ga γ a + ĝ γ we find that the most general solution of the first three equations of (12) is given by ab ab (a b) E = K I + 2α γ Fa = (ζ a + ∇c K ac )I + (γ c ∇c αa + Aγ a ) + 13 bc ∇b K ac γ G = g I − R4 αb γ b + 14 ba ∇b ζ a γ where A ∈ C ∇(d K ab) = 0 (a b) ∇ α = 0, ∇(d ζ b) = 0 Solution of defining equations Finally the fourth equation of (12) yields ∇a g = − 14 ∇b RKa b ⇒ ∂µ g = − 14 ∇ν (RKµν ) (12) This equation locally detemines g and hence the most general second order symmetry operator of the Dirac equation if and only if the right hand side is a closed 1-form. If the space is flat, R = 0 and the solution of is g = constant. The integrability condition for (12) is ∇[µ ∇ρ RKν]ρ = 0 (13) CONCLUSION: The most general second order symmetry operator is defined by a Killing tensor Kµν , two Killing vectors αµ , ζµ , a scalar function g , and a constant A, provided these objects exist on M. Integrability condition solution For the Liouville metric (9) the integrability condition (13) reads (A + B)2 (A0 B 000 + A000 B 0 ) + 6A0 B 0 (A0 )2 + (B 0 )2 − 6A0 B 0 (A + B)(A00 + B 00 ) = 0 (14) If one of A or B is constant, (14) is trivially satisfied in which case the space admits a Killing vector. In the case A0 B 0 6= 0, the integrability condition (14) implies that ( (A0 )2 = kA4 + a3 A3 + a2 A2 + a1 A + a0 (B 0 )2 = −kB 4 + a3 B 3 − a2 B 2 + a1 B − a0 where ai , i = 0, 1, 2, 3, and k are arbirary constants. If A and −B are taken as coordinates, the metric may be written as dA2 dB 2 ds 2 = (A − B) − , (15) p4 (A) p4 (B) where p4 (A) := kA4 + a3 A3 + a2 A2 + a1 A + a0 . The Ricci scalar is given by R = (A + B)k + 12 a3 . If k = 0, (15) is the metric of a space of constant curvature. First order operators and reducibility A first order operator is said to be trivial if it has the form αD + βI. Thus the most general first order operator has the form ab ( E = 0 A, g ∈ C where Fa = ζ a I + Aγ a ∇(d ζ b) = 0 G = g I + 41 ba ∇b ζ a γ There is a one-to-one correspondence between non-trivial first order operators and Killing vectors ζ on M. A second order operator is said to be trivial if it has the form D ◦ K1 + K01 , where K1 and K01 denote any first order operators. Thus the most general non-trivial second order operator has the form ab ab (d ab) =0 E = K I ∇ K 1 a ac b ac ab where K 6= λη ab F = ∇c K I + 3 bc ∇ K γ G = gI ∇[µ ∇λ RKν]λ = 0 and where g a solution of equation (13). There is a one-to-one correspondence between non-trivial second order operators andKilling tensors K ab on M such that K ab 6= λη ab and ∇[µ ∇λ RKν]λ = 0. Separation of variables It is known that the Dirac equation separates only in coordinate systems separating the Helmholtz equation for the corresponding metric tensor, and only if at least one of these coordinates is associated to a Killing vector. This is the case for the coordinate system (u, v ) for which the metric is in Liouville form, and u is an ignorable coordinate. The appropriate spin-frame in this case is given by √ 1 0 A(u)+B(v ) , (eaµ ) = √ 1 − 0 A(u)+B(v ) where A = 0 and B(v ) = β(v )−2 . Assuming that vectors α and ζ are zero, the Dirac operator D and second order symmetry operator K may be written in matrix form as 0 −1 i 0 i 0 D=β ∂u + ∂v − 12 β0, 1 0 0 −i 0 −i and 1 0 2 K= ∂ . 0 1 uu Separation of variables The expressions for D and K coincide with the separation scheme earlier. It is an instance of non-factorizable separation of Fels & Kamran. By computing Dψ − mψ = 0 and Kψ − µψ = 0 (16) with the separation ansatz ψ= a1 (u)b1 (v ) a2 (u)b2 (v ) we obtain the separated equations ( a0 b0 − a12 = −i b12 + a10 a2 b20 = i b1 − iβ 0 b1 m b1 2β b2 + β b2 iβ b2 m b2 2β b1 + β b1 0 and, from (16) we have ( a100 = −µa1 a200 = −µa2 (17) Separation of variables By introducing separation constants µ1 and µ2 such that µ1 µ2 = µ, the first system gives for the ai ( a20 = −µ1 a1 (18) a10 = µ2 a2 and for the bi ( 0 β b1 + −ib10 + i 2β ib20 − i β0 2β b2 + m β b1 = µ1 b2 m b β 2 = µ2 b1 (19) It is evident that K provides, via (17), a decoupling relation for the equations (18). On the other hand, (17) can be obtained by applying twice equations (18). The symmetry operators associated with separation are generated by this means in . The solutions for the ai are ( √ √ a1 = c1 sin( µu) + c2 cos( µu) q √ √ a2 = µµ12 [−c2 sin( µu) + c1 cos( µu)] where c1 , c2 ∈ C. Separation of variables The general solution of (19) can be easily computed in Cartesian coordinates, β(v ) = 1: ( b1 = d1 sin(Mv ) + d2 cos(Mv ) b2 = µ11 [(d1 + iMd2 ) sin(Mv ) + (d2 − iMd1 ) cos(Mv )] p where d1 , d2 ∈ C and M = m2 − µ. It is remarkable that, even if the ai and the bi depend on the µj , the products ψi = ai bi depend on µ only. By setting β equal to 1, e v , sinh(v ), cosh(v ), k − cos(v ), with k > 1, respectively, the corresponding Riemannian manifolds are: I the Euclidean plane in Cartesian and polar coordinates I the sphere I the pseudo-sphere I the torus For polar coordinates in the Euclidean plane and in the cases of sphere and pseudo-sphere, equations (19) can be integrated, to obtain solutions respectively in terms of Bessel functions and, on the sphere and pseudosphere, in terms of hypergeometric functions. Future directions I Give a complete invariant characterization for Dirac equation separability comparable to that for the Hamilton-Jacobi and Helmholtz equations. I Determine the procedure for the transformation to separable form. I Extend the analysis to four dimensions. References 1. W. Miller, Jr., Mechanism for variable separation in partial differential equations and their relationship to group theory. In Symmetries and Nonlinear Phenomena, World Scientific, Singapore, 1988, 188-221. 2. M.Fels and N.Kamran, Proc. Roy. Soc. London A 428, 229-249, 1990. 3. R.G. McLenaghan and S.N. Smith, and D.M. Walker, Proc. Roy. Soc. London A 456, 2629-2643, 2000. 4. S. Smith: Symmetry operators and separation of variables for the Dirac equation on curved space-times. PhD thesis, University of Waterloo, 2002. 5. R.G. McLenaghan and G. Rastelli, Separation of variables for systems of first-order partial differential equations and the Dirac equation in two-dimensional manifolds. IMA volumes in mathematics and its applications. M.Eastwood and W. Miller Jr. eds. Vol. 144, 471-496, Springer 2008 6. L. Fatibene, R.G. McLenaghan, G. Rastelli, and S. Smith, J. Math. Phys. 50, 053516, 2009. 7. A. Carignano, Separation of variables for the Dirac equation, Tesi di Laurea, Universitá di Torino, 2010.