Time-ordered ovolutions on the Feynman-Dyson Hilbert opace
Resumen
In this work, we consider hyperbolic and parabolic evaluation problems on the Feynman-Dyson Hilbert space, $\mathcal{FD}_\otimes^2$. We use some important properties given in $\mathcal{FD}_\otimes^2$ to find solutions for both homogeneous and non-homogeneous cases. Therefore, we first focus the structure of the Feynman-Dyson Hilbert space from a mathematical perspective in terms of the construction of this space and the lifting of operator theory to this time-ordered setting. We then observe that $\mathcal{FD}_\otimes^2$ allows operators acting at different times to commute, while maintain their relative position on paper. We also deal with a time-ordered version of the Hille-Yosida theorem for semigroups of operators. This approach has the added advantage of requiring the weakest known domain and continuity conditions. We show these advantages for the generic classes of time-dependent homogeneous hyperbolic and parabolic problems. We also see that the theory has advantages for operators with no time dependence.
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R. P. Feynman, The theory of positrons, Phys. Rev. 76, 749-759, (1949).
R. P. Feynman, Space-time approach to quantum electrodynamics, Phys. Rev. 76, 769-789, (1949).
T. L. Gill and W. W. Zachary, Foundations for Relativistic Quantum Theory I: Feynman’s Operator Calculus and the Dyson Conjectures, J. Math. Phys. 43, 69-93, (2002).
T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus, Springer, New York, (2016).
F. J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman, Physical Review 75, 486-502, (1949).
I. Fujiwara, Operator calculus of quantized operator, Prog. Theor. Phys. 7, 433-448, (1952).
W. L. Miranker and B. Weiss, The Feynman operator calculus, SIAM Rev. 8, 224-232, (1966).
E. Nelson, Operants: a functional calculus for non-commuting operators, in “Functional Analysis and Related Fields”, Springer, Berlin, (1970).
H. Araki, Expansional in Banach algebras, Ann. scient. Ec. Norm. Sup. Math. 4, 67-84, (1973).
V. P. Maslov, Operational Methods, English translation 1976 (revised from the Russian edition), Moscow, (1973).
G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus, Oxford U. Press, New York, (2000).
T. L. Gill and G. A. De Parga, Foundations for QED, Feynman operator calculus, Dyson conjectures, and Einstein’s dual theory, J. Phys.: Conf. Ser. 2482(1), 012015, (2023).
A. Constantin, The Construction of an Evolution System in the Hyperbolic Case and Applications, Math. Nachr. 224, 49-73, (2001).
G. Da Prato and M. Iannelli, On a method for studying abstract evolution equations in the Hyperbolic case, Comm. in Partial Diff. Eqs. 1, 585-608, (1976).
T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan 5, 208-234, (1953).
K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan 31(4), 647-654, (1979).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, (Appl. Math. Sci., vol. 44), Springer-Verlag, New York, (1983).
L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations, Discrete and Continuous Dynamical Systems - B 14(4), 1601-1620, (2010).
S. Pankavich, P. Radu, Nonlinear instability of solutions in parabolic and hyperbolic diffusion, Evolution Equations and Control Theory 2(2), 403-422, (2013).
X. Dai, Hyperbolic and parabolic equations with several opposite-sign source at critical initial energy level in heat and vibrating systems, Nonlinear Analysis 195, 111752, (2020).
A. Agresti and M. Veraar, Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence, Nonlinearity 35(8), 4100, (2022).
T. L. Gill and W. W. Zachary, Feynman operator calculus: the constructive theory, Expo. Math. 29, 165–203, (2011).
H. O. Fattorini, The Cauchy Problem, Addison-Wesley, London, (1983).
Derechos de autor 2025 Boletim da Sociedade Paranaense de Matemática
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Funding data
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Yildiz Teknik Üniversitesi
Grant numbers FBA-2023-5765
