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Annihilation and creation operators in Lie algebra theory and physics
Jarkovská, Kateřina ; Šmíd, Dalibor (advisor) ; Křižka, Libor (referee)
We show the use of the theory of Lie algebras, especially their oscillator realizations, in the context of quantum mechanics. One can construct oscillator realizations from matrix realizations. In the case of symplectic and special orthogonal algebra, we demonstrate an alternative method of obtaining oscillator realizations from symmetric or exterior power of a vector space of annihilation and creation bosonic or fermionic operators. We find Lie algebra of polynomials of degree at most two in phase space of a mechanical system, which form the semi-direct product of the Heisenberg algebra and symplectic algebra. It is shown that a classical system with Hamiltonian function in this algebra can be quantized by two equivalent representations - Schrödinger or Bargmann-Fock representation. The second mentioned representation generates the same operators of symplectic algebra as we got from their previous formal construction from symmetric power of a vector space of bosonic operators. Quantization is demonstrated on the bosonic harmonic oscillator. We use the similarities between bosonic and fermionic oscillator realizations to define the fermionic harmonic oscillator. Some properties of spinor representations of special orthogonal algebra are illustrated on its state space. Powered by TCPDF (www.tcpdf.org)
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Operads and field theory
Pulmann, Ján ; Jurčo, Branislav (advisor) ; Křižka, Libor (referee)
Operads and their variants, modular and cyclic operads, naturally describe compositions of objects of various types. We provide an accessible introduction to the theory of operads, the formalism for modular operads from [1] and modern application of modular operads to physics, due to Barannikov [2]. Through examples, we introduce Batalin-Vilkovisky formalism as a tool for cohomological integration of path integral in quantum field theories. A master equation, consistency condition for action, follows from this formalism. Solutions to master equation also describe algebras over Feynman transform of a modular operad. We explore the master equation defined in terms of modular operad and review an application to closed string field theory. [1] Martin Doubek, Branislav Jurco, and Korbinian Muenster. Modular operads and the quantum open-closed homotopy algebra. 2013. arXiv: 1308.3223 [math-AT]. [2] Serguei Barannikov. "Modular operads and Batalin-Vilkovisky geometry". In: International Mathematics Research Notices 2007 (2007), rnm075.
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