Symmetry Operators for Dirac’s Equation
Symmetry Operators and Separation of Variables for Dirac’s Equation on Two-Dimensional Spin ManifoldsThis paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html
A. Carignano, L. Fatibene, R.G. McLenaghan and G. Rastelli
Alberto CARIGNANO , Lorenzo FATIBENE , Raymond G. McLENAGHAN
and Giovanni RASTELLI
Department of Engineering, University of Cambridge, United Kingdom \EmailDDac737@cam.ac.uk
Dipartimento di Matematica, Università di Torino, Italy \EmailDDlorenzo.firstname.lastname@example.org
Department of Applied Mathematics, University of Waterloo, Ontario, Canada \EmailDDrgmclena@uwaterloo.ca
Formerly at Dipartimento di Matematica, Università di Torino, Italy \EmailDDgiorast.email@example.com
Received February 01, 2011, in final form June 02, 2011; Published online June 15, 2011
A signature independent formalism is created and utilized to determine the general second-order symmetry operators for Dirac’s equation on two-dimensional Lorentzian spin manifolds. The formalism is used to characterize the orthonormal frames and metrics that permit the solution of Dirac’s equation by separation of variables in the case where a second-order symmetry operator underlies the separation. Separation of variables in complex variables on two-dimensional Minkowski space is also considered.
Dirac equation; symmetry operators; separation of variables
This paper is dedicated to Professor Willard Miller, Jr. on the occasion of his retirement from the School of Mathematics at the University of Minnesota.
This paper is a contribution to the study of the separability theory for Dirac’s equation to which Professor Miller has made important contributions [18, 19, 25]. Exact solutions to Dirac’s relativistic wave equation by means of the method of separation of variables have been studied since the equation was postulated in 1928. Indeed, the solution for the hydrogen atom may be obtained by this method. While there is a well developed theory of separation of variables for the Hamilton–Jacobi equation, and the Schrödinger equation based on the existence of valence two characteristic Killing tensors which define respectively quadratic first integrals and second-order symmetry operators for these equations (see [24, 17, 2, 13]) an analogous theory for the Dirac equation is still in its early stages. The complications arise from the fact that one is dealing with a system of first-order partial differential equations whose derivation from the invariant Dirac equation depends not only on the choice of coordinate system but also on the choice of an orthonormal moving frame and representation for the Dirac matrices with respect to which the components of the unknown spinor are defined. Further complications arise if the background space-time is assumed to be curved. Much of the progress in the theory has been stimulated by developments in Einstein’s general theory of relativity where one studies first quantized relativistic electrons on curved background space-times of physical interest such as the Schwarzschild and Kerr black hole space-times. This work required the preliminary development of a theory of spinors on general pseudo-Riemmanian manifolds (see [10, 6, 11, 12]). The solution of the Dirac equation in the Reissner–Nordstrom solution was apparently first obtained by Brill and Wheeler in 1957  who separated the equations for the spinor components in standard orthogonal Schwarzschild coordinates with respect to a moving frame adapted to the coordinate curves. A comparable separable solution in the Kerr solution was found by Chandrasekhar in 1976  by use of an ingenious separation ansatz involving Boyer–Lindquist coordinates and the Kinnersley tetrad. The separability property was characterized invariantly by Carter and McLenaghan  in terms of a first-order differential operator constructed from the valence two Yano–Killing tensor that exists in the Kerr solution, that commutes with the Dirac operator and that admits the separable solutions as eigenfunctions with the separation constant as eigenvalue. Study of this example led Miller  to propose the theory of a factorizable system for first-order systems of Dirac type in the context of which the separability property may be characterized by the existence of a certain system of commuting first-order symmetry operators. While this theory includes the Dirac equation on the Kerr solution and its generalizations  it is apparently not complete since as is shown by Fels and Kamran  there exist systems of the Dirac type whose separability is characterized by second-order symmetry operators. The work begun by these authors has been continued by Smith , Fatibene, McLenaghan and Smith , McLenaghan and Rastelli , and Fatibene, McLenaghan, Rastelli and Smith  who studied the problem in the simplest possible setting namely on two-dimensional Riemmanian spin manifolds. The motivation for working in the lowest permitted dimension is that it is possible to examine in detail the different possible scenarios that arise from the separation ansatz and the imposition of the separation paradigm that the separation be characterized by a symmetry operator admitting the separable solutions as eigenfunctions. The insight obtained from this approach may help suggest an approach to take for the construction of a general separability theory for Dirac type equations. Indeed in  systems of two first-order linear partial equations of Dirac type which admit multiplicative separation of variables in some arbitrary coordinate system and whose separation constants are associated with commuting differential operators are exhaustively characterized. The requirement that the original system arises from the Dirac equation on some two-dimensional Riemannian spin manifold allows the local characterization of the orthonormal frames and metrics admitting separation of variables for the equation and the determination of the symmetry operators associated to the separation. The paper  takes this research in a different but closely related direction. Following earlier work of McLenaghan and Spindel  and Kamran and McLenaghan  where the first-order symmetry operators of the Dirac equation where computed on four-dimensional Lorentzian spin manifolds and McLenaghan, Smith and Walker  where the second-order operators were determined in terms of a two-component spinor formalism, the most general second-order linear differential operator which commutes with the Dirac operator on a general two-dimensional Riemannian spin manifold is obtained. Further it is shown that the operator is characterized in terms of Killing vectors and valence two Killing tensors defined on the background manifold. The derivation is manifestly covariant: the calculations are done in a general orthonormal frame without the choice of a particular set of Dirac matrices.
The purpose of the present paper is to extend the results of  and  described above to the case of two-dimensional Lorentzian spin manifolds. One of the main achievements of the paper is the creation of a formalism that permits the simultaneous treatment of both the Riemannian and Lorentzian cases. Further, we extend the results of , where Hamilton–Jacobi separability separability is studied in complex variables on two-dimensional Minkowski space, to the Dirac equation.
The paper is organized as follows. In Section 2 we summarize the basic properties of two-dimensional spin manifolds required for the subsequent calculations. Section 3 is devoted to the derivation of the form of the general second-order linear differential operator which commutes with the Dirac operator. We show that this operator is characterized by a valence two Killing tensor field, two Killing vector fields and two scalar fields defined on the background spin-manifold. In Section 4 we develop a formalism based on  which enables us to study simultaneously separation of variables for the Dirac equation on both Riemannian and Lorentzian spin manifolds. All possible cases where separation occurs are determined. In Section 5 we establish a link between the second-order symmetry operators obtained formally in Section 3 and the second-order symmetry operator underlying the separation of variables scheme considered in Section 4. In Section 6 separation of Dirac’s equation in complex variables is considered on two-dimensional Minkowski space. Section 7 contains the appendix. The notation and conventions of this paper are consistent with .
Let be a connected, paracompact, two-dimensional spin manifold. Let us consider both the Euclidean and the Lorentzian signatures. With an abuse of notation, let also denote the canonical form induced by the signature and the determinant of such quadratic form, namely one has . We will keep this sign as an undetermined parameter in this paper, since we want to consider both cases at once.
We know that a representation of the Clifford algebra is induced by a set of Dirac matrices such that they satisfy the Dirac condition
The generators of the even Clifford algebra are , and . Therefore the most general element of the ) group is
From the theory, we know that it is possible to define a covering map between and . Let be a generic element of . Then
and the covering map in matrix form is
Let be a suitable ) principal bundle, such that it allows global maps of the spin bundle into the general frame bundle of . The local expression of such maps is given by spin frames .
A spin frame induces a metric of signature and the corresponding spin connection:
where denotes the Levi-Civita connection of the induced metric , which in turn induces the covariant derivative as .
It follows from our setting that the inner product will raise and lower Latin indices, while the Greek ones are raised and lowered by the induced metric .
Finally, the covariant derivative of spinors is defined as
The covariant derivative (1) is invariant under spin transformations in view of the following property
A spin transformation is an automorphism of the spin bundle . If we define the fiber coordinates of as , the local expressions of the spin transformation are , , where .
As we know, it acts as a left group on spinors and spin frames ()
where and is the Jacobian of the coordinate transformation. Therefore the spin connection transforms as
Looking back to the covariant derivative, one can prove that the commutator can be related to the curvature
The advantage of working in dimension two is that the Riemann tensor has only one independent component that can be written as a function of the Ricci scalar . Hence the identity (2) may be rewritten as
Similarly the following property holds
3 First- and second-order symmetry operators
In our framework, the Dirac equation has the form
An operator is a symmetry operator for the Dirac equation if
The most general operator of the second-order has the form
where , , are algebraic matrix coefficients to be determined.
To write the system (7) we make use of a characterization of second- and third-order covariant derivatives which is shown in the Appendix.
Let us begin with the first equation in (7). We know that the coefficients of are zero-order matrix operators (i.e. not differential) which can be expanded in the basis of
where the coefficients are point functions in . Here we are using the fact that , and form a basis, since they are linearly independent. Hence, the first equation can be rewritten as
which can be solved to obtain
where are the coefficients along the frame of an arbitrary vector field .
Applying these conditions, the operator is of the form
Now we consider the second condition of (7). As we did for , we expand in the basis
and substitute it back in the condition (7). We obtain
The first equation says that is a Killing vector of or possibly the zero vector. Through an explicit calculation, we prove that the most general tensor satisfying the second condition is . Hence
Finally, we substitute into the third equation of (8) obtaining
Taking the trace we get
which shows that is uniquely determined by . However (9) contains six equations, of which only one has been used. The other five equations are exploited by back substituting (10) into (9) to obtain an equation for alone, namely
This is an integrability condition for which may be written as
This equation shows that is a Killing tensor of .
We shall now consider the third condition of (7)
Considering the usual expansion , we obtain the following system of equations
The first equation means that , while the second implies . By splitting the third into its symmetric and antisymmetric parts, it results that
for the antisymmetric part, while the symmetric part can be expanded to obtain
It follows that there exists a Killing vector or a zero-vector such that from which we obtain
We summarize all the results obtained so far
It remains to consider the fourth equation in (7)
which implies the only additional condition:
This equation locally determines if and only if the right hand side is a closed 1-form. We call (11) the integrability condition.
We now focus on finding condition for first-order operators.
We can easily obtain the conditions by setting to zero and . In particular (11) is trivially satisfied and . We thus obtain that the most general first-order symmetry operator may be written as
where and .
4 Separation of variables
Let us start with the Dirac condition
We fix the metric convention to be .
A choice of gamma matrices valid for both signatures is
where for Euclidean and for Lorentzian signature.
Let be a coordinate system on the two-dimensional manifold and
We now make the important assumption that the spinor is multiplicatively separable
The Dirac equation then reads:
We now apply the general results of  to the Lorentzian case.
for suitable indices , , , where , . Moreover, the equations
define the separation constants .
The above definition refers to so-called “naive” separation of variables that is not always the most general. In order to find symmetry operators, we adopt for our analysis some assumptions.
First of all, we assume a given coordinate system and impose a separation of (13) according to the previous definition. This can be done in three different ways
- Type I:
- Type II:
and (or viceversa).
- Type III:
Following the procedure laid out in , we build eigenvalue-type operators with eigenvalues making use of the terms and in (15) only. We require that the operators are independent of . Furthermore, .
Finally, we require the symmetry condition, that is for all . A operator which satisfies the condition, is called a symmetry operator since it maps solutions into solutions.
The symmetry operators are directly generated by the separated equations and having them enables one to immediately write down the same separated equations. In addition, the separation constants are associated with eigenvalues of symmetry operators.
The only relevant case is Type I separation, since it is the only one associated with non-trivial second-order operators. We shall now make use of the naive separation assumption and of definition (14). We would like to determine the indices , , , and , . Thus, we focus on what we can factorize from (13).
By inspection, we notice that the only allowed factorizations are:
Factorize in the first equation and in the second equation.
Factorize in the first equation and in the second equation.
We consider the first factorization, which implies . We divide into two parts both the equations: one part which is a function of and the other a function of , in order to have , and , .
where and are the separation constants.
The second-order operator is therefore defined by the following equations
If we set , the operator is given by
We notice that the functions and are functions only of .
Finally, we look for the conditions that have to be applied in order to have the operator commuting with the Dirac operator : it results already that , so no other conditions are needed. It follows from a detailed analysis of this case (D5 separation scheme in ) that we obtain a Liouville metric with one ignorable coordinate. This case will be discussed in detail in the next section. Another case is also possible which corresponds to the D7 separation scheme in . However, it may be shown that it is equivalent to previous one modulo the sign of the Lorentz metric.
Separability for equations of Type II and of Type III gives rise to first-order operators.
5 Liouville metric
Its analysis requires knowledge regarding which spin manifolds admit nontrivial valence two Killing tensors.
To further proceed with our analysis, we recall the following important result.
A two-dimensional Riemannian or Lorentzian space admits a non-trivial valence two Killing tensor if and only if it is a Liouville surface in which case there exists a system of coordinates with respect to which the metric and Killing tensor have the following forms
where and are arbitrary smooth functions. Furthermore, the frame component of are given by
The spin frame corresponding to D5 separation discussed in the previous section is given by
where and .
It follows from the previous section that the only Type I separation, other than the equivalent one discussed at the end of the previous section, is associated with the nonsingular Dirac operator and associated symmetry operator of the form
The operator written above agrees with the second-order operator of Section 3 computed for the Liouville metric under the assumption . We observe that it is in fact the square of the first-order operator
corresponding to the Killing vector associated to the ignorable coordinate of the Liouville metric where, as we said before, for Euclidean and for Lorentzian signature.
The corresponding coordinates separate the geodesic Hamilton–Jacobi equation. If the manifold is the Euclidean plane, the coordinates, up to a rescaling, coincide with polar or Cartesian coordinates. In the Minkowski plane the coordinates correspond to pseudo-Euclidean or pseudo-polar coordinates.
6 Separation in complex variables
On real pseudo-Riemannian manifolds the Hamilton–Jacobi equation can be separated not only in standard separable coordinates but also in complex variables . As in classical separation of variables theory, complex separable variables are determined by eigenvalues and eigenvectors of second-order Killing tensors. If the manifold is pseudo-Riemannian, pairs of complex-conjugate eigenvectors and eigenvalues of second-order real Killing tensors can exist in some part of the manifold, together with real ones. Where complex eigenvectors appear, it is impossible to determine real orthogonal separable coordinates, and the introduction of complex variables is necessary. The complex variables behave in all respects as complex coordinates, but they are not independent because of the conjugation relation. In the following their lack of independence will be irrelevant. Let us consider the 2-dimensional Minkowski manifold with pseudo-Cartesian coordinates . The geodesic Hamiltonian is given by
The space of valence two Killing tensors is 6-dimensional and there are ten different types of separable webs real in some part of the space . Another separable web, everywhere complex, is defined by 
and is determined by the eigenvectors of the Killing tensor whose non-null components in are
and the associated polynomial first integral is
By defining the canonical momenta as
and a real complete separated integral of the Hamilton–Jacobi equation can be determined . Because are both ignorable variables, they should also separate the Dirac equation (indeed, in Minkowski space they are the only complex separable web with at least one ignorable variable). With respect to the Dirac operator can be written as
where are the components of the spin frame. Because the can be assumed to be constant, and since
In order to write in the form
corresponding to Type I separation in the Minkowski space, the components of in must be
Therefore, by using and for raising and lowering indices,
It is remarkable that the spin frame base allowing the separation of variables in is essentially coincident with .
Both and are ignorable variables, therefore, both
can be used as differential operators associated with separation of variables. The integration of the is in all respects the same as for the separation in real coordinates.
For second-order covariant derivatives the following equation hold
This last equation can be rewritten as
Similar equations for third-order covariant derivative require more calculations and we will show only the main passages
Now by using (16) and expanding:
Now the first part of the right hand side is just . The second part can be rewritten to obtain
The authors wish to thank their reciprocal institutions, the Dipartimento di Matematica, Università di Torino and the Department of Applied Mathematics, University of Waterloo for hospitality during which parts of this paper were written. The research was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
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