Symmetry operators and separation of variables Ray McLenaghan Conference September 2010

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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
Separation in 2-dimensional pseudo-Riemannian spaces
Second order symmetry operators
First order operators
Symmetry and separation of variables
Future directions
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 .
Hψ := g µν ∇µ ∇ν ψ − E ψ = 0.
Dψ := iγ a ∇a ψ − mψ = 0.
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
α bβ
+ ∂µ e bα ).
µ = eα (Γβµ e
In (4)
denotes the Levi-Civita connection of the metric
gµν = eµa ηab eνb
induced by the spin frame.
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) =
Wi (x i , c).
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) =
ψi (x i , c),
such that the functions ψi are solutions of a set of ODEs in the variables
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) = 
ψ̃ (x)
ψ̃ j (x) =
ψij (x i ),
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
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 ),
Σ + r 2 + a2 cos(θ),
∆ = r 2 − 2ma + a2
Special case of Schwarzschild (a = 0) separated earlier by Brill and
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
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)
Chandra procedure extended to the class of Petrov type D vacuum
solutions and its characterization by first order symmetry operators
(Kamran & McLenaghan).
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 [1] 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 [2]
that factorizable systems contain all the the separable systems for
the Dirac equation on Petrov type D vacuum spacetimes.
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 [3] and Smith [4] 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 ))
(Liouville metric)
(B(v )∂u ⊗ ∂u − A(v )∂v ⊗ ∂v )
where A and B are arbitrary smooth functions. The Helmhotz equation
Hψ = (A + B)−1 ∂uu
ψ + ∂vv
ψ − Eψ = 0
Since Rµν = 1/2Rgµν , H admits a second order symmetry operator K:
Kψ = (A + B)−1 B ∂uu
ψ − A ∂vv
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
γ =
γ =
0 −1
where k = −η, η = det(ηab ), and ηab = diag(1, ±1) is the frame
The Dirac equation may be written as
e 1 ψ + B∂
e 2ψ + C
e ψ − λψ = 0
Dψ := A∂
A1 = ie11
A2 = −ike21
B1 = ie12
B2 = −ike22
C1 =
−(i/2)ke2µ Γ12
C2 =
−(i/2)e1µ Γ12
With product separability assumption ψi = ai (x)bi (y ) the D equation
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
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.
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)
The only type I separation is associated with the nonsingular Dirac
operator and associated symmetry operator of the forms
0 1
1 0
∂x 0
D :=
∂x + (i/R1 (y ))
−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
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
Dirac separability implies Hamilton-Jacobi and Helmholtz
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.
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)
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+
Ebd γ c + 2γ c Ebd a d bc
γ a ∇ G = R Fa γ b + γ b Fa γ + 1 2Eab γ c + γ c Eab γ ∇ R
ac b
where R denotes the Ricci scalar of gµν .
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
∇(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
(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
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 )
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
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
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
dB 2
ds 2 = (A − B)
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
E = 0
A, g ∈ C
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
(d ab)
E = K I
∇ K
b ac
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
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
A(u)+B(v ) 
(eaµ ) =  √ 1
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
∂u +
∂v − 12
1 0
0 −i
0 −i
1 0 2
∂ .
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
Dψ − mψ = 0
Kψ − µψ = 0
with the separation ansatz
a1 (u)b1 (v )
a2 (u)b2 (v )
we obtain the separated equations
( a0
− a12 = −i b12 +
= i b1 −
iβ 0 b1
m b1
2β b2 + β b2
iβ b2
m b2
2β b1 + β b1
and, from (16) we have
a100 = −µa1
a200 = −µa2
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
a10 = µ2 a2
and for the bi
b1 +
−ib10 + i 2β
ib20 − i
2β b2
β b1 = µ1 b2
β 2 = µ2 b1
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 [5]. The solutions for the ai are
a1 = c1 sin( µu) + c2 cos( µu)
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 )]
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
Give a complete invariant characterization for Dirac equation
separability comparable to that for the Hamilton-Jacobi and
Helmholtz equations.
Determine the procedure for the transformation to separable form.
Extend the analysis to four dimensions.
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,
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.
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