# Symmetry operators for Dirac's equation on two-dimensional spin manifolds

@article{Fatibene2009SymmetryOF, title={Symmetry operators for Dirac's equation on two-dimensional spin manifolds}, author={Lorenzo Fatibene and Raymond G. McLenaghan and Giovanni Rastelli and Shane N. Smith}, journal={Journal of Mathematical Physics}, year={2009}, volume={50}, pages={053516-053516} }

It is shown that the second order symmetry operators for the Dirac equation on a general two-dimensional spin manifold may be expressed in terms of Killing vectors and valence 2 Killing tensors. The role of these operators in the theory of separation of variables for the Dirac equation is studied.

## 4 Citations

### 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…

### 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…

### A non‐uniqueness problem of the Dirac theory in a curved spacetime

- Mathematics
- 2009

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under…

## References

SHOWING 1-10 OF 18 REFERENCES

### Symmetry operators for neutrino and Dirac fields on curved spacetime

- Physics, Mathematics
- 1984

We characterize tensorially all first-order differential operators whose commutator with the massless Dirac operator is proportional to it on a general curved background in terms of skew-symmetric…

### Symmetry operators for spin-Â½ relativistic wave equations on curved space-time

- PhysicsProceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 2000

The second–order linear covariant differential symmetry operators of the source–free spin–½ relativistic wave equation are given in a canonical form. We find that the operators are constructed from…

### 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…

### Non-factorizable separable systems and higher-order symmetries of the Dirac operator

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1990

It is shown that there exist separable systems for the Dirac operator on four-dimensional lorentzian spin manifolds that are not factorizable in the sense of Miller. The symmetry operators associated…

### Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation

- Mathematics
- 1997

The nonorthogonal separation of variables in the Hamilton–Jacobi equation corresponding to a natural Hamiltonian H=12gijpipj+V, with a metric tensor of any signature, is intrinsically characterized…

### Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds

- Mathematics
- 2008

The problem of solving the Dirac equation on two-dimensional manifolds is approached from the point of separation of variables, with the aim of creating a foundation for analysis in higher…

### A geometrical approach to the problem of integrability of Hamiltonian systems by separation of variables

- Mathematics
- 2001

### Generalized symmetries in mechanics and field theories

- Mathematics
- 2002

Generalized symmetries are introduced in a geometrical and global formalism. Such a framework applies naturally to field theories and specializes to mechanics. Generalized symmetries are…

### Gauge Formalism for General Relativity and Fermionic Matter

- Physics
- 1998

A new formulation for General Relativity is developed; it is a canonical, global and geometrically well posed formalism in which gravity is described using only variables related to spin structures.…

### Natural and gauge natural formalism for classical field theorie[s] : a geometric perspective including spinors and gauge theories

- Mathematics
- 2003

I The Geometric Setting Introduction.- 1. Fiber Bundles.- 2. Jet Bundles.- 3. Principal Bundles and Connections.- 4. Natural Bundles.- 5. Gauge Natural Bundles.- II The Variational Structure of Field…