Abstract
Considering a complex Lagrange space ([24]), in this paper the complex electromagnetic tensor fields are defined as the sum between the differential of the complex Liouville 1-form and the symplectic 2-form of the space relative to the adapted frames of the Chern-Lagrange complex nonlinear connection. In particular, an electrodynamics theory on a complex Finsler space is obtained.
We show that our definition of the complex electrodynamics tensors has physical meaning and these tensors generate an adequate field theory which offers the opportunity of coupling with the gravitation. The generalized complex Maxwell equations are written.
A gauge field theory of electrodynamics on the holomorphic tangent bundle is put over T’M and the gauge invariance to phase transformations is studied. An extension of the Dirac Lagrangian on T’M coupled with the electrodynamics Lagrangian is studied and it offers the framework for a unified gauge theory of fields.
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References
M. Abate and G. Patrizio, Finsler Metrics — A Global Approach, Lecture Notes in Math., Vol. 1591 (Springer-Verlag, 1994).
T. Aikou, A partial connection on complex Finsler bundle and its applications, Illinois J. Math. 42 (1998) 481–492.
T. Aikou, Finsler geometry on complex vector bundles, Riemann Finsler Geometry, MSRI Publications 50 (2004) 85–107.
N. Aldea, Complex Finsler spaces of constant holomorphic curvature, Diff. Geom. and Its Appl. in Proc. Conf. Prague 2004, Charles Univ. Prague, Czech Republic (2005), pp. 179–190.
N. Aldea and G. Munteanu, On complex Finsler spaces with Randers metric, J. Korean Math. Soc. 46(5) (2009) 949–966.
G. S. Asanov, Finsler Geometry, Relativity and Gauge Theories (D. Reidel Publ. Co., Dordrecht, 1985).
G. S. Asanov, Finsleroid-relativistic time-asymmetric space and quantized fields, Reports Math. Physics 57 (2006) 199–231.
M. F. Atiyah, Geometry of Yang–Mills Fields (Pisa, 1979).
E. Barletta, S. Dragomir and H. Urakawa, Yang–Mills fields on CR Manifolds, arXiv:math.DG/0605388v1, May 2006.
A. Bejancu, Finsler Geometry and Appl. (Ellis Harwood, 1990).
D. Bleeker, Gauge Theory and Variational Principles (Addison-Wesley Publ. Co. Inc., 1984).
M. Born and L. Infeld, Fundations of the new field theory, Proc. Royal Soc. London A 144 (1934) 425–451.
M. Calixto, V. Aldaya, F. Lopez-Ruiz and E. Sanchez-Sastre, Coupling nonlinear Sigma-Matter to Yang–Mills fields: Symmetry breaking patterns, J. Nonlinear Math. Physics 15 suppl. 3 (2008) 91–101.
M. Chaichian and N. F. Nelipa, Introduction to Gauge Field Theories (Springer-Verlag, 1984).
S. Donev, Complex structures in electrodynamics, arXiv:math-ph/0106008v3, Nov. 2001.
G. Esposito, Complex Geometry of nature and General Relativity, Kluwer Acad. Publ., FTPH 69 (1995) arXiv:gr-qc/991105v1, Nov. 1999.
R. Friedman and J. Morgan, Gauge Theory and Topology of Four-Manifolds, ed. IAS/ PARK CITY, Math. Series, Vol. 4 (AMS, 1998).
M. Gondran and A. Kenoufi, Complex Faraday’s tensor for the Born-Infeld theory, arXiv:math-ph/0708.0547v1, Aug. 2007.
M. Green, J. Schwarz and E. Witten, Superstring Theory, Vols. 1 and 2 (Cambridge Univ. Press, 1987).
C. Hong-Mo, J. Faridani and T. S. Tsun, A nonabelian Yang–Mills analogue of classical electromagnetic duality, arXiv:hep-th/9503106v4, Sep. 1995.
Y. I. Manin, Gauge Field Theory and Complex Geometry (Springer-Verlag, 1997).
R. Miron, The geometry of Ingarden spaces, Rep. Math. Physics 54 (2004) 131–147.
R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces. Theory and Appl., Fundamental Theories of Physics, Vol. 59 (Kluwer Acad. Publ., 1994).
G. Munteanu, Complex Spaces in Finsler, Lagrange and Hamilton Geometries, Fundamental Theories of Physics, Vol. 141 (Kluwer Acad. Publ., 2004).
G. Munteanu, Gauge field theory in terms of Hamilton geometry, Balkan J. Geom. Appl. 12(1) (2007) 107–121.
G. Munteanu, The Lagrangian–Hamiltonian Formalism in Gauge Complex Field Theories, Hypercomplex number in geomery and physics 2(6), Vol. 3 (2007) 123–133.
L. Nicolaescu, Notes in Seiberg–Witten Theory, Graduate Stud. in Math., Vol. 28 (AMS, 2000).
R. Palais, The Geometrization of Physics, Lecture Notes in Math. (Hsinchu, Taiwan, 1981).
M. Peskin, Duality in supersymmetric Yang–Mills theory, arXiv:hep-th/9702094v1, Feb. 1997.
N. Seiberg and E. Witten, Electric-Magnetic duality, monopole condensation, and condinement in N = 2 supersymmetric Yang–Mills theory, Nucl. Physics B 431 (1994) 19–52.
L. Silberstein, Nachtrag zur Abhandlung ber Electromagnetische Grundgleichungen in bivek-torieller Behandlung, Ann. Phys. Lpz. 24 (1907) 783.
I. Suhendro, A new Finslerian unified field theory of physical interactions, Progress in Physics 4 (2009) 81–90.
P.-Mann Wong, A survey of complex Finsler geometry, Advanced Studied Pure Math., Math. Soc. Japan 48 (2007) 375–433,
C. N. Yang, Collection of Papers in Chern Symposium, Vol. 247 (Springer-Verlag, 1979).
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Munteanu, G. A Yang—Mills Electrodynamics Theory on the Holomorphic Tangent Bundle. J Nonlinear Math Phys 17, 227–242 (2010). https://doi.org/10.1142/S1402925110000738
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DOI: https://doi.org/10.1142/S1402925110000738