×

zbMATH — the first resource for mathematics

Hamiltonian approach to secondary quantization. (English. Russian original) Zbl 1410.81029
Dokl. Math. 98, No. 3, 571-574 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 483, No. 2, 138-142 (2018).
Summary: Structures and objects used in Hamiltonian secondary quantization are discussed. By the secondary quantization of a Hamiltonian system \(\mathcal{H}\), we mean the Schrödinger quantization of another Hamiltonian system \(\mathcal{H}_1\) for which the Hamiltonian equation is the Schrödinger one obtained by the quantization of the original Hamiltonian system \(\mathcal{H}\). The phase space of \(\mathcal{H}_1\) is the realification \(\mathbb{H}_R\) of the complex Hilbert space \(\mathbb{H}\) of the quantum analogue of \(\mathcal{H}\) equipped with the natural symplectic structure. The role of a configuration space is played by the maximal real subspace of \(\mathbb{H}\).

MSC:
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V70 Many-body theory; quantum Hall effect
81T05 Axiomatic quantum field theory; operator algebras
70H05 Hamilton’s equations
81S05 Commutation relations and statistics as related to quantum mechanics (general)
81S10 Geometry and quantization, symplectic methods
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kozlov, V. V.; Smolyanov, O. G., No article title, Dokl. Math., 85, 416-420, (2012) · Zbl 1260.81092
[2] O. G. Smolyanov and E. T. Shavgulidze, Feynman Path Integrals (Lenand, Moscow, 2015) [in Russian].
[3] V. P. Maslov, Complex Markov Chains and Feynman Integral for Nonlinear Systems (Moscow, 1976) [in Russian].
[4] Remizov, I. D., No article title, Dokl. Math., 96, 433-437, (2017) · Zbl 06828239
[5] Dubravnia, V. A., No article title, Izv. Math., 82, 494-511, (2018) · Zbl 1393.81022
[6] F. A. Berezin, Method of Second Quantization (Academic, New York, 1966; Nauka, Moscow, 1965). · Zbl 0131.44805
[7] N. N. Bogolyubov and N. N. Bogolyubov, Jr., Introduction to Quantum Statistical Mechanics (Nauka, Moscow, 1964; World Scientific, Singapore, 1982). · Zbl 0576.60095
[8] J. C. Baez and I. E. Segal, and Z. Zhou, Introduction to Algebraic and Constructive Quantum Field Theory (Princeton Univ. Press, Princeton, 1990). · Zbl 0760.46061
[9] Smolyanov, O. G., Nonlinear pseudodifferential operators in superspaces, 103-106, (1988)
[10] Ratiu, T. S.; Smolyanov, O. G., No article title, Dokl. Math., 87, 289-292, (2013) · Zbl 1312.81143
[11] Smolyanov, O. G.; Weizsäcker, H., No article title, Infinite-Dimensional Anal. Quantum Probab. Related Topics, 2, 51-78, (1999) · Zbl 0936.60052
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.