Symmetry Operators and Separation of Variables for Diracs Equation on Two-Dimensional Spin Manifolds

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 7 (2011), 057, 13 pages Symmetry Op e rators and Separation of V ariables for Dirac’s Equation on Tw o-Dimensional Spin Manifo lds ⋆ Alb erto CARIGNANO † 1 , L or enzo F A TIBE NE † 2 , R aymond G. McLENAGHAN † 3 and Giovanni RASTELLI † 4 † 1 Dep artment of Engine ering, University of Cambridge, U nite d K ingdom E-mail: ac737@c am.ac.uk † 2 Dip artimento di Matematic a, Universit` a di T orino, Italy E-mail: lor enzo.fa tib ene@ unito.it † 3 Dep artment of Applie d Mathematics, University of Waterlo o, Ontario, Canada E-mail: r gmclena@uwaterlo o.c a † 4 F ormerly at Dip artimento di Matematic a, Universit` a di T orino, Italy E-mail: gi or ast.gior ast@alic e.it Received F ebrua ry 01, 20 11, in f inal form June 02, 201 1; Published online June 15, 201 1 doi:10.38 42/SIGMA.20 11.057 Abstract. A signa ture indep endent for malism is created and utilized to deter mine the g e- neral seco nd-order symmetry op erato r s for Dir a c’s equatio n on tw o-dimensional Lorentzian spin manifolds. The forma lism is used to characterize the o rthonorma l frames and metrics that per mit the solution of Dirac’s equatio n by separ ation of v aria bles in the case where a second-order symmetry oper ator underlies the separation. Separation of v aria bles in com- plex v ariables on tw o-dimensiona l Minko wski space is a lso considered. Key wor ds: Dirac equation; symmetry op era tors; se pa ration o f v ariables 2010 Mathematics Subje ct Classific ation: 70S1 0; 81Q 80 This p ap er is de dic ate d to Pr ofessor Wil lar d M i l ler, Jr. on the o c c asion of his r etir ement fr om the Scho ol of Mathe- matics at the University of Minnesota. 1 In tro duction This pap er is a con tribution to the study of the sep arab ility th eory for Dirac’s equation to which Professor Miller has made imp ortant con tributions [17, 18, 24]. Exact solutions to Dirac’s re- lativistic w a ve equation b y means of the metho d of separation of v ariables ha v e b een studied since the equation w as p ostulated in 192 8. In deed, the sol ution for the h ydrogen atom ma y b e obtained by t his metho d. While there is a we ll deve lop ed theory of separation of v ariables for the Hamilton–Jaco bi equation, and the Sc hr¨ odinger equation based o n the existence of v alence t w o c haracteristic Killing tens ors whic h d ef in e resp ective ly qu adratic f irst in tegrals and second-order symmetry op erators for these equations (see [23 , 16, 1, 12]) an analogous theory f or the Dirac equation is still in its early stage s. T he complications arise from the fact that one is d ealing with a system of f irst-order p artial dif f er ential equations whose deriv ati on from the in v arian t Dirac equation dep ends not only on the choic e of co ordinate system but also on the choice of an orthonormal mo ving frame and represent ation for the Dirac matrices with resp ect to whic h the comp onent s of the unknown spinor are def ined. F urther complications arise if th e bac kground ⋆ This pap er is a con tribution to the Sp ecial Issue “Symmetry , S eparation, Sup er-integra bilit y and Sp ecial F unctions (S 4 )”. The full collection is a v aila ble at http://www.emis .de/journals/SIGMA/S4.h tml 2 A. Carignano, L. F atib ene, R.G. Mc Lenaghan and G. Rastelli space-time is assu med to b e curve d. Muc h of the p rogress in th e theory h as b een stimulate d b y dev elopmen ts in Einstein’s general theory of rela tivit y where one studies f irst quan tized relativis- tic elec trons on curved bac kground space-times of ph ysical interest such as the Sc h w arzsc hild and Ker r blac k h ole space-times. This wo rk required the preliminary d ev elopmen t of a theory of sp inors on general pseudo-Riemmanian m an if olds (see [9, 5, 10, 11]). The solution of the Dirac equation in the Reissn er–Nordstrom solution w as apparently f irst obtained by Brill and Wheeler in 1957 [2] who sep arated the equations for the sp inor comp on ents in standard orthogo- nal Sch w arzschild co ordin ates with resp ect to a moving frame adapted to the co ordinate cur v es. A comparable sep arab le solution in the Kerr solution was found b y Chand rasekhar in 1976 [6] by use of an ingenious separation ansatz in v olving Bo y er–Lindquist co ordinates and the Kinnersley tetrad. The separabilit y prop er ty wa s c haracterized inv ariantly b y Carter and McLenaghan [4] in terms of a f irst-order dif ferent ial op erator constructed from the v alence t w o Y ano–Killing tensor that exists in the Kerr so lution, t hat commutes with th e Dirac op erator a nd that admits the s ep arable solutions as eige nfun ctions with the separation constan t as eigen v alue. Stud y of this example led Miller [24] to p rop ose the theory of a factorizable s y s tem for f irst-order sys- tems of Dirac t yp e in the con text of whic h the separabilit y prop ert y ma y b e c haracterized by the existence of a certain sys tem of comm uting f irst-order sym metry op erators. While this theory includes th e Dirac equation on the Kerr solution an d its generalizations [15] it is apparent ly not complete since as is sho wn by F els and Kamr an [15] there exist s ystems of the Dirac typ e w h ose separabilit y is c haracterize d by second-order symm etry op erators. The w ork b egun by these authors h as b een con tin ued b y Smith [25], F atibene, McLenaghan and S mith [13], M cLenaghan and Rastelli [20], and F atib ene, McLenaghan, Rastelli and Smith [1 4] who studied t he problem in the simplest possible setting namely on t w o-dimensional Riemmanian spin man if olds . The mo- tiv ation fo r wo rking in the lo we st perm itted dimension is th at it is p ossible to e xamine in detail the dif ferent p ossible sce narios that arise from the separation ansatz and the imp osition of t he separation paradigm that the sep aration b e c haracterized by a symmetry op erator adm itting the separable solutions as eigenfunctions. Th e insigh t obtained from this app roac h may help suggest an approac h to tak e for the construction of a general separabilit y theory for Dirac typ e equa- tions. Indeed in [20] systems of t w o f irs t-order linear partial equations of Dirac typ e whic h admit m ultiplicativ e separation of v ariables in some arbitrary co ordinate system and whose separation constan ts are asso ciated w ith commuting dif ferentia l op erators are exhaustiv ely c haracterized. The requirement that the original system arises from the Dirac equation on some t w o-dimensional Riemannian spin m an if old allo ws th e lo cal c haracterization o f the orthon orm al fr ames and m et- rics admitting separation of v ariables for the equation and the d etermin ation of the symmetry op erators asso ciated to the s eparation. The pap er [14] tak es this researc h in a dif f er ent but closely r elated d irection. F ollo w in g earlier w ork of McLenaghan and Sp in del [22] and Kamr an and McLenaghan [19] wh er e the f irst-order symmetry op er ators of the Dirac equation wh ere com- puted on four-dimensional Loren tzia n sp in manifolds and McLenaghan, Smith and W alk er [21] where the seco nd-order op erators w ere determined in terms of a t wo-co mp onent spinor formal- ism, the most general se cond-order linear dif ferent ial op erator which commutes with t he Dirac op erator on a general t w o- dimensional Riemannian s pin manifold is obtained. F urther it is s ho wn that the operator is c haracterized in terms of Killing v ect ors and v alence t w o Killing tensors def ined on th e bac kground manifold. The deriv at ion is manifestly co v arian t: th e calculations are done in a general orth onormal frame without the c hoice of a particular set of Dirac matrices. The purp ose of the presen t pap er is to extend the resu lts of [20] and [14] describ ed ab o v e to th e case of t w o-dimensional Loren tzian spin manifolds . One of the main ac hiev emen ts of the pap er is the creati on of a formalism that p ermits the simultane ous tr e atment of b oth the Riemannian and Loren tzian cases. F urther, we extend the results of [8], where Hamilton–Jacobi separabilit y separabilit y is studied in complex v ariables on tw o-dimensional Minko wski space, to the Dirac equation. Symmetry Op erators for Dirac’s Equation 3 The pap er is organized as follo ws. I n Section 2 w e summarize the basic p rop erties of tw o - dimensional spin manif olds required for the subsequent calculat ions. S ection 3 is dev oted to the d eriv ation of the form of the general second-order linear dif feren tial op erator whic h com- m utes with the Dirac op erator. W e sho w that this op erator is c haracterize d b y a v alence t w o Killing tensor f ield, tw o Killing v ector f ields and t w o scalar f ields def in ed on the bac kgroun d spin-manifold. In Section 4 w e develo p a formalism based on [20] whic h enables us to study si- m ultaneously s eparation of v ariables f or the Dirac equation on b oth Riemann ian and Lorentz ian spin manifolds. All p ossible cases where separation occurs are d etermined. In Section 5 we e s- tablish a lin k b et w een the se cond-order symmetry op erators obta ined f ormally in Sect ion 3 and the second-order symm etry op erator underlyin g the s ep aration of v ariables scheme considered in Section 4. In Section 6 separation of Dirac’s equation in complex v ariables is considered on t w o- dimensional Mink o wski space. Sectio n 7 cont ains the app endix. The notation and conv entions of this pap er are consisten t with [14]. 2 F ra mew ork Let M b e a connected, p aracompact, t w o -dimensional sp in manifold. Let us consider both the Euclidean η = (2 , 0 ) and the Loren tzian η = (1 , 1) signatures. With a n abuse of nota tion, let η also denote the canonical form induced by the sig nature and the determinan t of suc h quadr atic form, namely one has η = ± 1. W e will k ee p this sign as an undetermined parameter in this pap er, since we wan t to consider b oth c ases at once. W e kno w that a r epresen tation of the Clif ford algebra is indu ced by a set of Dirac m atrices γ a suc h that they satisfy the Dirac condition γ a γ b + γ b γ a = 2 η ab I . The generators o f the ev en Clif ford algebra are I , γ a and γ := γ 1 γ 2 . Th erefore the most general elemen t of the Spin( η ) group is S = aI + bγ with a 2 + η b 2 = 1. F rom the theory , w e kno w that it is p ossible to def ine a co v ering map b etw een Spin( η ) and SO( η ). L et l a b b e a generic elemen t of SO( η ). Th en η ab = l c a η cd l d b and the co v ering map in matrix form is l ( S ) =  a 2 − η b 2 2 η ab − 2 η ab a 2 − η b 2  . Let P → M be a suitable S p in( η ) prin cipal bundle, s u c h that it allo ws global map s Λ : P → L ( M ) of the spin bu ndle in to the general frame bun d le of M . The lo cal expression of such maps is giv en by spin frames e µ a . A spin frame induces a metric of sig nature η and the corresp onding spin connection: g µν = e a µ η ab e b ν , Γ ab µ = e a α  Γ α β µ e bβ + ∂ µ e α c η cb  , where Γ α β µ denotes the Levi-Civita connection of the in duced metric g µν , whic h in turn indu ces the co v arian t d eriv ativ e as ∇ µ = e a µ ∇ a . It follo ws from our s etting that the inner pro d uct η ab will raise an d lo w er Latin indices, while the Greek ones are raise d and low ered by the induced metric g µν . 4 A. Carignano, L. F atib ene, R.G. Mc Lenaghan and G. Rastelli Finally , the co v arian t d er iv ativ e of spinors is d ef ined as ∇ µ ψ := ∂ µ ψ + Γ µ ψ , (1) where Γ µ = 1 8 Γ ab µ [ γ a , γ b ] = 1 4 Γ ab µ ǫ ab γ . The co v ariant deriv ativ e (1) is inv ariant u nder spin transf ormations in view of the follo wing prop erty S ∂ ν S − 1 = 1 4 l b a ∂ ν ¯ l a c ǫ b c · γ . A sp in transformation is an automorph ism of the spin b undle P . If we def ine the f ib er co- ordinates of P as ( x, S ), the local expr essions of the sp in trans formation are x ′ µ = x ′ µ ( x ), S ′ = φ ( x ) S , where φ : U → Spin( η ). As w e kno w, it a cts as a left group on spinors and spin frames ( l b a ≡ l b a ( ϕ )) ψ ′ = φψ , e ′ µ a = J µ ν e ν b ¯ l b a , where l : S pin( η ) → SO( η ) and J µ ν is the Jacobian of the co ordin ate transformation. Therefore the spin connection transforms as Γ ′ ab µ = ¯ J ν µ l a c ( ψ )  Γ cd ν l b d ( ψ ) + ∂ ν ¯ l c d ( ψ ) η db  . Lo oking bac k to the cov ariant deriv ativ e, o ne can pr o v e that the commutato r can b e related t o the curv ature [ ∇ µ , ∇ ν ] ψ = 1 4 γ ψ ǫ ab R ab µν . (2) The adv ant age of working in dimen s ion t wo is that the R iemann te nsor R αβ µν has only one indep en d en t comp onent that can b e written as a function of the Ricci s calar R . Hence the iden tit y (2) ma y b e rewritten as [ ∇ c , ∇ d ] ψ = 1 4 γ ψ ǫ cd R. (3) Similarly the follo wing prop erty holds [ ∇ a , ∇ b ] ∇ c ψ = R 4 γ ǫ ab ∇ c ψ − R 2 ǫ d · c ǫ ab ∇ d ψ . (4) 3 First- and second -order symmetry op erators In our fr amework, the Dirac equ ation has the form D ψ = iγ a ∇ a ψ − mψ = 0 . (5) An operator is a symmetry op erator for the Dirac e quation if [ K , D ] = 0 . (6) The most general op erator of the second-order has the form K = E ab ∇ ab + F a ∇ a + G I , where E ab , F a , G are alg ebraic matrix coef f icien ts to b e determin ed . Symmetry Op erators for Dirac’s Equation 5 W e aim to determine co ef f icien ts so that condition (6) holds tru e. Consid ering the Ricci’s iden tities (2), (3) and (4), we expand th e symmetry equation (6) for the co ef f icien ts and w e obtain E ( ab γ c ) − γ ( c E ab ) = 0 , F ( a γ b ) − γ ( b F a ) = γ c ∇ c E ab , G γ d − γ d G = γ c ∇ c F d − ( E ad γ c + γ c E ad ) γ R 4 ǫ ac + + R 3  1 2 E ba γ c + γ c E ba  ǫ d · a ǫ bc , γ c ∇ c G = R 8 ( F a γ c + γ c F a ) γ ǫ ac +  1 2 γ c E ab + E ab γ c  1 6 γ ǫ bc ∇ b R. (7) T o write the system (7) w e mak e use of a c haracterizatio n of seco nd- and third-order cov ariant deriv ative s whic h is shown in the App endix. Let u s b egin with th e f irst equation in (7) . W e k n o w that the co ef f icien ts of E ab are zero-o rder matrix oper ators (i.e. not dif f er ential) w hic h c an be expanded in the basis of C ( η ) E ab = e ab I + e ab c γ c + ˆ e ab γ , where the co ef f icient s are p oin t functions in M . Here w e are using the fact that I , γ c and γ form a basis, since they are linearly indep endent . Hence, the f irst equation can b e rewritten as − 2 ˆ e ( ab ǫ c ) · d γ d − 2 η e ( ab d ǫ dc ) γ = 0 , whic h ca n b e solv ed to obtain ˆ e ab = 0 , e ab d γ d = 2 α ( a γ b ) , where α a are the co ef f icients a long the fr ame e a := e µ a ∂ µ of an arbitrary v ecto r f ield α . Applying these conditions, the op erator E is of the form E ab = e ab I + 2 α ( a γ b ) . No w w e consider the s econd condition of (7). As w e did for E , we exp and F in the b asis F a = f a I + f a b γ b + ˆ f a γ , and substitute it bac k in the condition (7). W e obtain ∇ c α ( a η b ) c = 0 , f ( a c ǫ b ) c = ∇ c α ( a ǫ b ) c , − 2 ˆ f ( a ǫ b ) · c = ∇ c e ab . (8) The f irs t equation sa ys that α a is a Killing vecto r of g or p ossibly the zero v ecto r. Through an explicit calculatio n, we pro v e that the most general tensor satisfying th e second condition is Aδ a c . Hence f a c = ∇ c α a + Aδ a c . Finally , w e substitute into the third equatio n of (8) obtaining − 2 ˆ f ( a ǫ b ) · c ǫ c · d = ∇ c e ab ǫ c · d . (9) T aking the trac e w e get ˆ f b = 1 3 ∇ c e ab ǫ c · a = 1 3 ∇ c e ab ǫ ca , (10) whic h shows that ˆ f a is uniqu ely determined by e ac . Ho wev er (9) conta ins six equ ations, of w hic h only one has b een u sed. The other f ive equations are exp loited by bac k sub stituting (10) into (9) to o btain an equation for e ac alone, namely ∇ c e ea ǫ ce δ b d + ∇ c e eb ǫ ce δ a d = ∇ c e ab ǫ c · d . 6 A. Carignano, L. F atib ene, R.G. Mc Lenaghan and G. Rastelli This is an in tegrabilit y cond ition for e ab whic h ma y b e written as ∇ ( a e bc ) = 0 . This equatio n sho ws that e ab is a Killing tensor of g . W e shall n o w co nsider the third condition of (7) G γ d − γ d G = γ c ∇ c F d − ( E ad γ c + γ c E ad ) γ R 4 ǫ ac + R 3  1 2 E ba γ c + γ c E ba  ǫ d · a ǫ bc . Considering the us u al expansion G = g I + g a γ a + ˆ gγ , w e ob tain the follo wing system of equations 0 = ∇ a A − R 2 α a I + R 2 α a I , 2 g b ǫ ba ·· =  ∇ b A − R 2 α b  ǫ ba ·· − R 2 α b ǫ ba ·· − R 2 α b ǫ ab ·· , − 2 η ˆ g ǫ ab ·· γ b = ∇ b f a + η 3 ǫ dc ǫ e b · ∇ e ∇ d e ac + R 2 e ab + R 2 e cd ǫ a · d ǫ c b · . The f irs t equation means that A ∈ C , while the second imp lies g b = − R 4 α b . By sp litting the third in to its symmetric a nd an tisymmetric parts, it results t hat ˆ g = 1 4 ǫ ba ∇ b f a for the antisymmetric part, while the symm etric part can b e expand ed to obtain ∇ ( a ( f b ) − ∇ c e b ) c ) = 0 . It f ollo w s that th ere exists a Killing v ecto r or a zero-v ector ζ suc h th at ζ a = f a − ∇ c e ac from whic h w e obtain f a = ζ a + ∇ c e ac . W e summarize a ll the r esults o btained so far E ab = e ab I + 2 α ( a γ b ) , F a = ( ζ a + ∇ c e ac ) I + ( γ c ∇ c α a + Aγ a ) + 1 3 ǫ bc ∇ b e ac γ , G = g I − R 4 α b γ b + 1 4 ǫ ba ∇ b ζ a γ , where A ∈ C , ∇ ( d e ab ) = 0 , ∇ ( a α b ) = 0 , ∇ ( d ζ b ) = 0 . It remains to consider the fourth equation in (7) γ c ∇ c G = R 8 ( F a γ c + γ c F a ) γ ǫ ac +  1 2 γ c E ab + E ab γ c  1 6 γ ǫ bc ∇ b R, whic h implies the only additional condition: ∇ a g = − 1 4 ∇ b ( Re ab ) . (11) This equation lo cally determines g if and only if the right hand side is a closed 1-form. W e call (11) the inte gr ability c ondition . W e no w f o cus o n f indin g condition for f irst-order op erators. W e can easily obtain the conditions by setting to zero e ab and α a . In particular (11) is trivially satisf ied and g ∈ C . W e th us obtain that the most general f irst-order symm etry op erator ma y b e written as E ab = 0 , F a = ζ a I + Aγ a , G = g I + 1 4 ǫ ba ∇ b ζ a γ , where A, g ∈ C and ∇ ( d ζ b ) = 0. Symmetry Op erators for Dirac’s Equation 7 4 Separation of v ariables Let us start with the Dirac condition γ a γ b + γ b γ a = 2 η ab I . (12) W e f ix th e metric conv entio n to b e η ab = d iag(1 , ± 1). A c h oice of gamma matrices v alid for b oth sig natures is γ 1 =  1 0 0 − 1  , γ 2 =  0 − k k 0  , where k = i for Eu clidean and k = 1 for Lorentzia n signature. Using the gamma matrices and (1), w e can rewrite Dirac ’s equations (5) as ˜ A∂ 1 ψ + ˜ B ∂ 2 ψ + ˜ C ψ − λψ = 0 , where ˜ A =  A 1 A 2 − A 2 − A 1  , ˜ B =  B 1 B 2 − B 2 − B 1  , ˜ C =  C 1 − C 2 C 2 − C 1  , and A 1 = ie 1 1 , A 2 = − ik e 1 2 , B 1 = ie 2 1 , B 2 = − ik e 2 2 , C 1 = − i 2 k e µ 2 Γ 12 µ , C 2 = − i 2 e µ 1 Γ 12 µ . Let ( x, y ) b e a co ordinate system on the t wo-dimensional manifold an d ψ =  ψ 1 ( x, y ) ψ 2 ( x, y )  . W e no w m ake the imp ortant assum p tion that the spinor ψ is m ultiplica tiv ely separable ψ i = a i ( x ) b i ( y ) . The Dirac equation then reads: A 1 ˙ a 1 b 1 + A 2 ˙ a 2 b 2 + B 1 a 1 ˙ b 1 + B 2 a 2 ˙ b 2 + C 1 a 1 b 1 − C 2 a 2 b 2 − λa 1 b 1 = 0 , − A 2 ˙ a 1 b 1 − A 1 ˙ a 2 b 2 − B 2 a 1 ˙ b 1 − B 1 a 2 ˙ b 2 + C 2 a 1 b 1 − C 1 a 2 b 2 − λa 2 b 2 = 0 . (13) W e no w app ly the general results o f [20] to the Loren tzian case. Definition 1 . The Dirac equation (13) and the op erator D are separated in ( x, y ) if ther e exists nonzero functions R i ( x, y ) suc h that (13) can b e rewritten as R 1 a r b s ( E x 1 + E y 1 ) = 0 , R 2 a t b u ( E x 2 + E y 2 ) = 0 (14) for suitable ind ices r , s , t , u where E x i ( x, a j , ˙ a j ), E y i ( y , b j , ˙ b j ). Moreo ver, the equ ations E x i = µ i = − E y i (15) def ine the separation constan ts µ i . 8 A. Carignano, L. F atib ene, R.G. Mc Lenaghan and G. Rastelli The ab o v e d ef in ition r efers to so-called “naive ” separation of v ariables that is not alwa ys t he most general. In order to f ind symmetry op erators, we adopt f or our analysis some assum ptions. First of all, we assume a giv en co ord inate system ( x, y ) and imp ose a separation of (13) according to the previous def inition. This can b e done in three d if feren t w ays T yp e I: a 1 6 = a 2 and b 1 6 = b 2 . T yp e I I : a 1 = a 2 and b 1 6 = b 2 (or vicev ersa). T yp e I I I: a 1 = ca 2 = a and b 1 = db 2 = b ( c, d ∈ C ). F ollo wing the pro cedure laid out in [20], w e bu ild eigen v alue-t yp e op erators L ψ = µψ with eigen v alues µ ( µ i ) making use of the terms E x i and E y i in (15) only . W e requir e that the op era- tors L are indep endent of λ . F ur thermore, λ 6 = 0. Finally , we r equire the symmetry condition, that is [ L , D ] ψ = ( LD − DL ) ψ = 0 f or all ψ . A op erator L whic h satisf ies the condition, is called a symmetry op er ator since it maps solutions in to so lutions. The symmetry op erators are directly generated b y the separate d equ ations and having them enables one to immediately write do wn th e same separated equations. In ad d ition, the sep aration constan ts are asso ciated with eig en v alues of symmetry operators. The only r elev an t case is T yp e I sep aration, since it is the on ly one asso ciated with non- trivial second-order op erators. W e shall no w mak e use of the naiv e separation assumption and of def inition (14). W e w ould like to d etermine the indices r , s , t , u and E x i ( x, a j , ˙ a j ), E y i ( y , b j , ˙ b j ). Th us, w e fo cus on what w e can fac torize from (13). By insp ection, we n otice that the on ly allo wed factoriza tions are: • F actorize a 1 b 2 in the f irst equation a nd a 2 b 1 in the second equation. • F actorize a 2 b 1 in the f irst equation a nd a 1 b 2 in the second equation. W e consider the f irst factoriza tion, whic h implies A 1 = B 2 = 0. W e d ivide in to t wo parts b oth the equations: one part which is a function of x and the other a function of y , in order to ha v e E x 1 , E y 1 and E x 2 , E y 2 . Hence E x 1 = A 2 R 1 ˙ a 2 a 1 − C 2 R 1 a 2 a 1 = µ 1 , E y 1 = B 1 R 1 ˙ b 1 b 2 + C 1 R 1 b 1 b 2 − λR 1 b 1 b 2 = − µ 1 , and E x 2 = A 2 R 2 ˙ a 1 a 2 − C 2 R 2 a 1 a 2 = µ 2 , E y 2 = B 1 R 2 ˙ b 2 b 1 + C 1 R 2 b 2 b 1 + λR 2 b 2 b 1 = − µ 2 , where µ 1 and µ 2 are the separation constan ts. The seco nd-order op erator L is therefore def ined b y the follo wing equations A 2 R 1 ˙ a 2 − C 2 R 1 a 2 = µ 1 a 1 , A 2 R 1 ˙ a 1 − C 2 R 1 a 1 = µ 2 a 2 . If w e set µ 1 µ 2 = µ , the op erator is gi v en b y L ψ :=  A 2 2 ( x, y ) R 2 1 ( y ) 0 0 A 2 2 ( x, y ) R 2 1 ( y )  ∂ 2 x ψ +  A 2 R 2 1 ∂ x A 2 − C 2 A 2 R 2 1 − C 2 R 2 1 A 2 0 0 A 2 R 2 1 ∂ x A 2 − C 2 A 2 R 2 1 − C 2 R 2 1 A 2  ∂ x ψ Symmetry Op erators for Dirac’s Equation 9 +     C 2 2 ( x, y ) R 2 1 ( y ) − A 2 ( x, y ) R 2 1 ( y ) ∂ x C 2 ( x, y ) 0 0 C 2 2 ( x, y ) R 2 1 ( y ) − A 2 ( x, y ) R 2 1 ( y ) ∂ x C 2 ( x, y )     ψ = µ ψ . W e notice that the functions A 2 R 1 and C 2 R 1 are functions only of x . Finally , w e lo ok for the conditions th at hav e to b e applied in order to ha v e the op erator comm uting with the Dirac op erator D : it r esu lts already th at [ L , D ] = 0, so n o other cond itions are needed. It follo ws from a detailed analysis of this case (D5 s eparation sc heme in [20]) that w e obtain a Liouville metric with one ignorable co ordinate. This case will b e discussed in d etail in the next s ection. Another case is also p ossible whic h corresp onds to the D7 separation sc heme in [20]. Ho wev er, it may b e sho wn that it is equiv ale n t to previous one mo dulo the sign of the Loren tz met ric. Separabilit y for equ ations of T yp e I I and of T yp e II I giv es rise to f irst-order oper ators. 5 Liouville metric In Section 2 we concluded our analysis of sym metry op erators with the cond ition (11) on the second-order operator: ∇ a g = − 1 4 ∇ b ( Re ab ) . Its a nalysis requires kno wledge regardin g whic h spin manifolds M admit n on trivial v ale nce t w o Killing tensors. T o further proceed with our analysis, w e recall the foll o wing imp ortan t result. A t wo-dimensional Riemannian or Loren tzian space admits a n on-trivial v alence t wo Killing tensor if and only if it is a Liouville surface in whic h case there exists a system of coord inates ( u, v ) with resp ect to whic h the metric g an d Killing tensor K ha v e the f ollo wing forms g = ( A ( u ) + B ( v ))  du 2 + η dv 2  , K = B ( v ) A ( u ) + B ( v ) ∂ u ⊗ ∂ u − η A ( u ) A ( u ) + B ( v ) ∂ v ⊗ ∂ v = K ab e a ⊗ e b , where A and B are arbitrary sm o oth fun ctions. F urthermore, the frame comp onent of K are giv en b y  K ab  = d iag  B ( v ) , − η A ( u )  . The spin frame corresp ondin g to D5 separation discussed in the previous sec tion is g iv en b y ( e µ a ) =   0 1 √ A ( u )+ B ( v ) − 1 √ A ( u )+ B ( v ) 0   , where A = 0 and B ( v ) = 1 /R 1 ( v ) 2 . It follo ws fr om the previous section th at the only Typ e I separation, other than th e equiv alent one discus s ed at the end of the previous sectio n, is asso ciated with the nonsingular Dirac op erator and associated symmetry operator of t he form D := R 1 ( y )  0 k − k 0  ∂ x + i  1 0 0 − 1  ∂ y  + i 2 R ′ 1 ( y )  1 0 0 − 1  , K =  ∂ 2 x 0 0 ∂ 2 x  . 10 A. Carignano, L. F atib ene, R.G. Mc Lenaghan and G. Rastelli The op erator K written ab ov e agrees with the second-order op erator of Sectio n 3 computed for the Liouville metric under the assum ption α = ζ = 0. W e observ e that it is in fact the square of the f irst-order op erator L =  1 0 0 1  ∂ x , corresp ondin g to the Killing v ector asso ciated to the ignorable co ord inate x of the Liouville metric where, as w e said b efore, k = i for Euclidean and k = 1 for Loren tzian signature. The corresp ondin g coord inates separate the geod esic Hamilton–Jacobi equation. If the mani- fold is the Euclidean plane, the coord inates, up to a rescaling, coincide with polar or Cartesian co ordinates. In the Mink o wski p lane the co ordinates corresp ond to p s eudo-Euclidean or pseudo- p olar coordinates. 6 Separation in complex v ariables On real p seudo-Riemannian m anifolds the Hamilton–Jacobi equ ation c an b e separated not only in stand ard separable co ordinates but also in complex v ariables [8]. As in classical separation of v ariables theory , complex separable v ariables are determined by eigen v alues and eigen v ecto rs of second-order Killing tensors. I f the manifold is p seudo-Riemannian, pairs of complex-conjugate eigen v ectors and eigen v alues of second-order real Killing tensors can exist in some part of the manifold, together w ith real on es. Where complex eig en v ecto rs app ear, it is imp ossible to determine real orthogonal separable coordinates, and th e int ro du ction of complex v ariables is necessary . The complex v ariables b eha v e in all r esp ects as complex co ordinates, but they are not indep endent b ecause of the conju gation relation. In the follo wing th eir lac k of in dep end ence will b e irrelev ant . Let us consider th e 2-dimensional Minko w s ki manifold w ith ps eudo-Cartesian co ordinates ( x, y ). Th e geo desic Hamiltonian is giv en by H = 1 2  p 2 x − p 2 y  . The sp ace of v alence t w o Killing tensors is 6-dimensional and there are ten dif ferent types of separable w ebs real in some part o f the sp ace [7]. Another separable w eb, ev erywhere complex, is def ined b y [8 ] z = x + iy , ¯ z = x − iy , and is determined by the eigen v ect ors of the Killing tensor whose non-null comp onen ts in ( x, y ) are K 12 = K 21 = 1 , and the asso ciated p olynomial f irs t in tegral is L = p x p y . By def ining the canonical momen ta as P = 1 2 ( p x − ip y ) , ¯ P = 1 2 ( p x + ip y ) , w e ha ve H = P 2 + ¯ P 2 , L = i  P 2 − ¯ P 2  , Symmetry Op erators for Dirac’s Equation 11 and a rea l complete separated integral o f the Hamilton–Jac obi equation can b e dete rmined [8]. Because ( z , ¯ z ) are b oth ignorable v ariables, they should also separate the Dirac equation (indeed, in Minko wski space they are the only complex separable w eb with at least one ignorable v ariable). With resp ect to ( x, y ) the Dirac o p erator can b e writte n as D = i " e 1 1 − e 1 2 e 1 2 − e 1 1 ! ∂ x + e 2 1 − e 2 2 e 2 2 − e 2 1 ! ∂ y # , where ( e µ a ) are the comp onen ts of the spin frame. Because th e ( e µ a ) can b e assumed to b e constan t, and since ∂ z = 1 2 ( ∂ x − i∂ y ) , ∂ ¯ z = 1 2 ( ∂ x + i∂ y ) , w e ha ve D = i " e 1 1 + ie 2 1 − e 1 2 − ie 2 2 e 1 2 + ie 2 2 − e 1 1 − ie 2 1 ! ∂ z + e 1 1 − ie 2 1 − e 1 2 + ie 2 2 e 1 2 − ie 2 2 − e 1 1 + ie 2 1 ! ∂ ¯ z # . In order to write D in the form D =  0 − 1 1 0  ∂ z + i  1 0 0 − 1  ∂ ¯ z  , corresp ondin g to T yp e I separation in th e Minko ws k i sp ace, the components of ( e µ a ) in ( x, y ) m ust be e 1 1 = i 2 , e 2 1 = − 1 2 , e 1 2 = 1 2 , e 2 2 = − i 2 . Therefore, b y u s ing ( e µ a ) and ( e a µ ) = ( e µ a ) − 1 for raising and lo w ering indices, ( e a ) = 1 2 ( ∂ x + i∂ y , ∂ x − i∂ y ) . It is remark able that the spin frame base allo wing the s ep aration of v aria bles in ( z , ¯ z ) is essen tially coinciden t with ( ∂ ¯ z , ∂ z ). Both z and ¯ z are ig norable v ariables, therefore, b oth  1 0 0 1  ∂ 2 z and  1 0 0 1  ∂ 2 ¯ z , can b e used as dif fer ential op erators asso ciated with separation o f v ariables. Th e in tegration of the ψ i = a i b i is i n all respects the sa me a s for th e separation in r eal c o ordinates. 7 App endix System (7) is obtained by using equations that c haracterize second- and third-order co v arian t deriv ative s. Su c h equ ations mak e use of Ricci’s identi ties (3) and (4). F or second-order co v arian t d eriv ativ es the follo wing equation hold ∇ a ∇ c ψ = 1 2 [ ∇ a , ∇ c ] ψ + ∇ ac ψ = ∇ ac ψ + R 8 ǫ ac γ ψ . This last equation can b e rewritten a s ∇ c ∇ a ψ = ∇ a ∇ c ψ + R 4 ǫ ca γ ψ . (16) 12 A. Carignano, L. F atib ene, R.G. Mc Lenaghan and G. Rastelli Similar equ ations for third-order co v arian t der iv ativ e require more calculati ons and we will sh o w only the main passages ∇ ab ∇ c ψ = 1 6 (3 ∇ a ∇ b ∇ c + 3 ∇ b ∇ a ∇ c ) ψ = 1 6 ( ∇ a ∇ b ∇ c + ∇ b ∇ a ∇ c + ∇ a ∇ b ∇ c + 2 ∇ b ∇ a ∇ c ) ψ No w b y using (16) and expandin g: ∇ ab ∇ c ψ = 1 6 ( ∇ a ∇ b ∇ c + ∇ b ∇ a ∇ c + ∇ a ∇ c ∇ b + ∇ b ∇ c ∇ a + ∇ a ∇ c ∇ b + ∇ b ∇ c ∇ a ) ψ + 1 12 ∇ a Rǫ bc γ ψ + 1 12 ∇ b Rǫ ac γ ψ + 1 12 Rǫ bc γ ∇ a ψ + R 12 ǫ ac γ ∇ b ψ . No w the f irst part of the righ t h and side is just ∇ abc . The second part can b e rewritten to obtain ∇ ab ∇ c ψ = ∇ abc ψ + 1 12 ( ∇ a Rǫ bc + ∇ b Rǫ ac ) γ ψ + R 8 ǫ ac γ ∇ b ψ + R 8 ǫ bc γ ∇ a + R 12 ( − η ac ∇ b − η bc ∇ a + 2 η ab ∇ c ) . (17) Equations (17) and (16) together with Ricci’s identit ies a re all the to ols needed to expand equation (6) in system ( 7). 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