# Matrix operator symmetries of the Dirac equation and separation of variables

@article{Kalnins1986MatrixOS, title={Matrix operator symmetries of the Dirac equation and separation of variables}, author={Ernie G. Kalnins and Willard Miller and G. C. Williams}, journal={Journal of Mathematical Physics}, year={1986}, volume={27}, pages={1893-1900} }

The set of all matrix‐valued first‐order differential operators that commute with the Dirac equation in n‐dimensional complex Euclidean space is computed. In four dimensions it is shown that all matrix‐valued second‐order differential operators that commute with the Dirac operator in four dimensions are obtained as products of first‐order operators that commute with the Dirac operator. Finally some additional coordinate systems for which the Dirac equation in Minkowski space can be solved by…

## 47 Citations

### Symmetry operators and separation of variables for Dirac's equation on two-dimensional spin manifolds with external fields

- Mathematics
- 2014

The second order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin-manifold. It is shown…

### The classification of complete sets of operators commuting with the Dirac operator in Minkowski space-time

- Mathematics
- 1988

Under the action of the Poincare group P(1,3) the three‐, four‐, and five‐dimensional vector spaces of formally self‐adjoint first‐order matrix differential operators commuting among themselves and…

### Symmetry Operators and Separation of Variables for Dirac's Equation on Two-Dimensional Spin Manifolds ?

- Mathematics, Physics
- 2011

A signature independent formalism is created and utilized to determine the ge- neral second-order symmetry operators for Dirac's equation on two-dimensional Lorentzian spin manifolds. The formalism…

### Factorization of Dirac equation in two space dimensions

- Physics
- 2014

We present a systematic approach for the separation of variables for the two-dimensional (2D) Dirac equation in polar coordinates. The three vector potential, which couple to the Dirac spinor via…

### The Dirac equation in external fields: Variable separation in Cartesian coordinates

- Physics, Mathematics
- 1991

The method of separation of variables in the Dirac equation proposed in an earlier work by one of the present authors [J. Math. Phys. 3 0, 2132 (1989)] is developed for the complete set of…

### Dirac equation in external vector fields: Separation of variables

- Mathematics
- 1989

The method of separation of variables in the Dirac equation in the external vector fields is developed through the search for exact solutions. The essence of the method consists of the separation of…

### Symmetry operators and the separability of massive Klein–Gordon and Dirac equations in the general five-dimensional Kerr–(anti-)de Sitter black hole background

- Physics
- 2009

It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr–(anti-)de Sitter black hole with two independent angular momenta. A first-order symmetry operator…

### Separation of variables and exact solution of the Dirac equation in nonstatic Minkowski spacetimes

- Physics
- 1991

Using a second-order formalism, a complete separation of variables in the Dirac equation for a free particle in nonstatic orthogonal curvilinear coordinates of the form t=f(u,v), x=g(u, v), y, z, is…

### Separability of the massive Dirac equation in 5-dimensional Myers-Perry black hole geometry and its relation to a rank-three Killing-Yano tensor

- Physics
- 2008

The Dirac equation for the electron around a five-dimensional rotating black hole with two different angular momenta is separated into purely radial and purely angular equations. The general solution…

### Generalized Killing Tensors and Symmetry of Klein-Gordon-Fock Equations 1

- Mathematics
- 2005

The paper studies non-Lie symmetry of the Klein–Gordon–Fock equation (KGF) in (p + q)-dimensional Minkowsky space. Full set of symmetry operators for the norder KGF equation was explicitly calculated…

## References

SHOWING 1-10 OF 12 REFERENCES

### Separation of variables and symmetry operators for the neutrino and Dirac equations in the space‐times admitting a two‐parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null congruences

- Mathematics, Physics
- 1984

We show that there exist a coordinate system and null tetrad for the space‐times admitting a two‐parameter abelian orthogonally transitive isometry group and a pair of shearfree geodesic null…

### Covariant Perturbed Wave Equations in Arbitrary Type $D$ Backgrounds

- Mathematics, Physics
- 1979

We present an approach to the fundamental tensorial quantities of general relativity which is inherently covariant and based on the irreducible representations of the Lorentz group, O(3,1). Using…

### Quantum numbers for Dirac spinor fields on a curved space-time

- Mathematics
- 1979

The most general first-order differential operator that commutes with the Dirac operator and hence permits the construction of quantum numbers is given. Necessary and sufficient conditions for its…

### The solution of Dirac’s equation in a class of type D vacuum space-times

- PhysicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1977

Chandrasekhar’s treatment of Dirac’s equation for the electron is extended to a class of type D vacuum space-times. It is shown that in all the back-ground geometries which are known as case II in…

### Killing Tensors and Variable Separation for Hamilton-Jacobi and Helmholtz Equations

- Mathematics
- 1980

Every separable coordinate system for the Hamilton-Jacobi equation on a Riemannian manifold $V_n$ corresponds to a family of $n-1$ Killing tensors in involution, but the converse is false. For…

### Generalized total angular momentum operator for the Dirac equation in curved space-time

- Physics
- 1979

It is found that an operator of the form i..gamma../sub 5/..gamma../sup ..mu../(f/sub ..mu..//sup ..nu../del/sub ..nu../ - 1/6..gamma../sup ..nu../..gamma../sup p/f/sub munu(p/) commutes with the…

### Perturbations of a rotating black hole. I. Fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations

- Physics
- 1973

Decoupled, separable equations describing perturbations of a Kerr black hole are derived. These equations can be used to study black-hole processes involving scalar, electromagnetic, neutrino or…

### The Mathematical Theory of Black Holes

- Geology
- 1983

In a course of lectures on the ‘underlying mathematical structures of classical gravitation theory’ given in 1978, Brandon Carter began with the statement ‘If I had been asked five years ago to…

### On separable solutions of Dirac’s equation for the electron

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1982

The properties of coordinate systems that admit separation of the Laplacian and Hamilton–Jacobi operators have been thoroughly explored so that the nature of solutions in separable form of Laplace’s…