Lattice calculations on the spectrum of Dirac and Dirac-Kähler operators. (English) Zbl 1192.81121
Summary: We use a lattice formulation to study the spectra of the Dirac and the Dirac-Kähler operators on the 2-sphere. This lattice formulation uses differentiation matrices which yield exact values for the derivatives of polynomials, preserving the Leibniz rule in subspaces of polynomials of low degree and therefore, this formulation can be used to study the fermion-boson symmetry on the lattice. In this context, we find that the free Dirac and Dirac-Kähler operators on the 2-sphere exhibit fermionic as well as bosonic-like eigensolutions for which the corresponding eigenvalues and the number of states are computed. In the Dirac case these solutions appear in doublets, except for the bosonic mode with zero eigenvalue, indicating the possible existence of a supersymmetry of the square of the Dirac operator.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 42A05 | Trigonometric polynomials, inequalities, extremal problems |
| 26D05 | Inequalities for trigonometric functions and polynomials |
| 39A70 | Difference operators |
Keywords:
Dirac operator; Dirac-Kähler operator; spectral problem; differentiation matrices; trigonometric polynomialsReferences:
| [1] | Trautman A., Acta Phys. Polon. B 26 pp 1283– |
| [2] | DOI: 10.1016/0393-0440(95)00042-9 · Zbl 0865.53044 · doi:10.1016/0393-0440(95)00042-9 |
| [3] | DOI: 10.1142/S0217751X02010261 · doi:10.1142/S0217751X02010261 |
| [4] | DOI: 10.2969/jmsj/04810069 · Zbl 0848.58046 · doi:10.2969/jmsj/04810069 |
| [5] | DOI: 10.1007/BF00844970 · Zbl 0796.53076 · doi:10.1007/BF00844970 |
| [6] | DOI: 10.1023/A:1015280310677 · Zbl 1026.81021 · doi:10.1023/A:1015280310677 |
| [7] | DOI: 10.1007/BF01214659 · Zbl 0527.58023 · doi:10.1007/BF01214659 |
| [8] | Graf W., Ann. Inst. Henri Poincaré pp 85– |
| [9] | DOI: 10.1007/BF01614426 · doi:10.1007/BF01614426 |
| [10] | DOI: 10.1017/CBO9780511470783 · doi:10.1017/CBO9780511470783 |
| [11] | DOI: 10.1142/S0217751X04017628 · Zbl 1080.81535 · doi:10.1142/S0217751X04017628 |
| [12] | DOI: 10.1016/S0375-9601(02)00383-3 · Zbl 0994.81083 · doi:10.1016/S0375-9601(02)00383-3 |
| [13] | DOI: 10.1016/j.physleta.2006.09.019 · Zbl 1170.81353 · doi:10.1016/j.physleta.2006.09.019 |
| [14] | Campos R. G., Bol. Soc. Mat. Mexicana 12 pp 121– |
| [15] | Campos R. G., Il Nuovo Cimento B 116 pp 31– |
| [16] | DOI: 10.1103/PhysRevD.14.487 · doi:10.1103/PhysRevD.14.487 |
| [17] | DOI: 10.1016/j.aop.2004.09.002 · Zbl 1074.81017 · doi:10.1016/j.aop.2004.09.002 |
| [18] | DOI: 10.1007/BF01205500 · Zbl 0528.58034 · doi:10.1007/BF01205500 |
| [19] | DOI: 10.1063/1.530630 · Zbl 0799.58077 · doi:10.1063/1.530630 |
| [20] | Friedrich T., Coll. Math. 47 pp 57– |
| [21] | DOI: 10.1016/j.aop.2004.08.006 · Zbl 1086.81048 · doi:10.1016/j.aop.2004.08.006 |
| [22] | Campos R. G., Nuovo Cimento B 100 pp 485– |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
