Abstract
We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of any dimension invariant and apply these transformations to field equations.
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References
Barut, A.O. and Raczka, R. (1965). “Classification of non-compact real Lie Groups and Groups containing the Lorentz Group”, Proc. Roy. Soc. London Series A, 287, 519–548.
Beckers, J., Harnad, J., Perroud, M. and Winternitz, P. (1978). “Tensor field invariant under subgroups of the conformai group of space-time”, J. Math. Phys., 19, 2126–2153.
Castell, L., Drieschner, M. and Weizsaecker, C.F. ed. (1975-7-9-81-83-85). Quantum Theory and the Structure of Space and Time, Hanser, vol. 1-6.
Cayley, A. (1846). Journal für reine und angewandte Mathematik, 32, 1 (1889). Collected Mathematical Papers, Cambridge, 117.
Earman, J. (1989). World Enough and Space and Time, Relational Theories of Space and Time, Cambridge.
Gel’fan, I.M., Minlos, R.A. and Saphiro, Z. (1963). Representations of the Rotation and Lorentz Groups and their Applications, Pergamon Press.
Grünbaum, A. (1977). “Absolute and Relational Theories of space and Time”, in Minnesota Studies in the Philosophy of Science (J. Earman, C. Glymour, J. Stachel, ed.). University of Minnesota Press, vol. VII.
Helgason, S. (1978). Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, p. 518.
Jammer, M. (1969). Concepts of Space and Time. Cambridge. Harvard U. Press.
Lorente, M. (1974). “Cayley Parametrization of Semisimple Lie Groups and its Application to Physical Laws in a (3+1)-Dimensional Cubic Lattice”, Int. J. Theor. Phys. 11, 213–247.
Lorente, M. (1976). “Basis for a Discrete Special Relativity”, Int. J. Theor. Phys. 12, 927.
Lorente, M. (1986a). “Space-time Groups for the Lattice”, Int. J. Theor. Phys. 25, 55–65.
Lorente, M. (1986b) “A Causal Interpretation of the Structure of Space and Time”. Foundations of Physics, HölderPichler-Tempsky, Viena.
Lorente, M. (1986c). “Physical Models on Discrete Space and Time”, in Symmetries in Science II (B. Gruber, R. Lenczweski, eds.), Plenum Press.
Lorente, M. (1987). “The Method of Finite Differences for Some Operator Field Equations”, Lett. Math. Phys. 13, 229–236.
Lorente, M. (1991). “Lattice Fermions with Axial Anomaly and without Species Doubling”, II Int. Wigner Symposium, Goslar.
Lorente, M. (1992). “A Relativistic Invariant Scheme for the Quantum Klein-Gordon and Dirac Fields on the Lattice”. XIX Int. Colloquium on Group Theoretical Methods in Physics, Salamanca.
Møller, C. (1952). The Theory of Relativity, Oxford p. 42.
Naimark, M.A. (1964). Linear Representation of the Lorentz Group, Pergamon Press, p. 92.
Penrose R. (1971). “Angular momentum: an approach to combinatorial Space-time” in Quantum Theory and Beyond (T. Bastin, ed.), Cambridge.
Schild, A. (1949). “Discrete space-time and integral Lorentz transformations”, Canadian Journal of Mathematics, 1, 29–47.
Wigner, E.P. (1959). Group theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, p. 160.
Yamamoto, H. (1985). Phys. Rev. DS2, 2659.
Yamamoto, H. (1991). “Noether Theorem and Gauge Theory in the Field Theory on Discrete Spacetime”, II Int. Wigner Symposium, Goslar.
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Lorente, M. (1993). Representations of Classical Groups on the Lattice and its Application to the Field Theory on Discrete Space-Time. In: Gruber, B. (eds) Symmetries in Science VI. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1219-0_36
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DOI: https://doi.org/10.1007/978-1-4899-1219-0_36
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