It is demonstrated that within the framework of the second quantization, the quantum Hamiltonian operator for a free transverse field reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based on extensions of the quadratic form of the transverse Laplace operator, which are used as a source of spherical basis functions with singularity at the origin. This basis naturally replaces the basis of plane or spherical waves, which is used to separate variables with the help of the Fourier transform or transition to spherical coordinates.
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References
P. A. M. Dirac, “Quantum theory of emission and absorption of radiation,” Proc. Royal Soc. London A, 114, 243 (1927).
C. M. Becchi, “Second quantization,” https://doi.org/10.4249/scholarpedia.7902, http://www.scholarpedia.org/article/Second quantization.
F. A. Berezin and L. D. Faddeev, “A Remark on Schrodinger’s equation with a singular potential,” Sov. Math. Dokl., 2, 372 (1961).
R. Jackiw, “Delta function potentials in two-dimensional and three-dimensional quantum mechanics,” in: R. Jackiw, Diverse Topics in Theoretical and Mathematical Physics (1991), pp. 35–53.
L. D. Faddeev, “Mass in Quantum Yang–Mills Theory (comment on a Clay millenium problem),” Bull. Brazil. Math. Soc., 33, No. 2, 201–212 (2002).
L. D. Faddeev, “Notes on divergences and dimensional transmutation in Yang-Mills theory,” Theor. Math. Phys., 148, 986 (2006).
R. D. Richtmyer, Principles of Advanced Mathematical Physics, Springer-Verlag, New York–Heildelberg–Berlin (1978).
L. D. Faddeev and A. A. Slavnov, “Gauge Fields. Introduction To Quantum Theory,” Front. Phys., 50, No. 1 (1980).
K. Friedrichs, “Spektraltheorie halbbeschr¨ankter Operatoren,” Math. Ann., 109, 465–487 (1934).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-adjointness, Academic Press (1975).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. 1. Functional Analysis, Academic Press, New York–London (1972).
T. A. Bolokhov, “Properties of the l=1 radial part of the Laplace operator in a special scalar product”, J. Math. Sci., 215, 560–573 (2016).
B. F. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press (1982).
T. A. Bolokhov, “Extensions of the quadratic form of the transverse Laplace operator”, J. Math. Sci., 213, 671–693 (2016).
T. A. Bolokhov, “The Scalar products of the regular analytic vectors of the Laplace operator in the solenoidal subspace,” J. Math. Sci., 242, 642–650 (2019).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 487, 2019, pp. 78–99.
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Bolokhov, T.A. Quantum Hamiltonian Eigenstates for a Free Transverse Field. J Math Sci 257, 476–490 (2021). https://doi.org/10.1007/s10958-021-05496-y
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DOI: https://doi.org/10.1007/s10958-021-05496-y