Let $K$ be a simply connected compact Lie group and $T^{ast}(K)$ its cotangent bundle. We consider the problem of quantization commutes with reduction for the adjoint action of $K$ on $T^{ast}(K).$ We quantize both $T^{ast}(K)$ and the reduced phase space using geometric quantization with half-forms. We then construct a geometrically natural map from the space of invariant elements in the quantization of $T^{ast}(K)$ to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.
There is a decomposition of a Lie algebra for open matrix chains akin to the triangular decomposition. We use this decomposition to construct unitary irreducible representations. All multiple meson states can be retrieved this way. Moreover, they are the only states with a finite number of non-zero quantum numbers with respect to a certain set of maximally commuting linearly independent quantum observables. Any other state is a tensor product of a multiple meson state and a state coming from a representation of a quotient algebra that extends and generalizes the Virasoro algebra. We expect the representation theory of this quotient algebra to describe physical systems at the thermodynamic limit.
Using the character expansion method, we generalize several well-known integrals over the unitary group to the case where general complex matrices appear in the integrand. These integrals are of interest in the theory of random matrices and may also find applications in lattice gauge theory.
Using the fact that the algebra M := M_N(C) of NxN complex matrices can be considered as a reduced quantum plane, and that it is a module algebra for a finite dimensional Hopf algebra quotient H of U_q(sl(2)) when q is a root of unity, we reduce this algebra M of matrices (assuming N odd) into indecomposable modules for H. We also show how the same finite dimensional quantum group acts on the space of generalized differential forms defined as the reduced Wess Zumino complex associated with the algebra M.
Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion equation is used as an example to illustrate the effectiveness of the Lie group method.
The concept of duality reflects a link between two seemingly different physical objects. An example in quantum mechanics is a situation where the spectra (or their parts) of two Hamiltonians go into each other under a certain transformation. We term this phenomenon as the energy-spectrum reflection symmetry. We develop an approach to this class of problems, based on the global properties of the Riemann surface of the quantum momentum function, a natural quantum-mechanical analogue to the classical momentum. In contrast to the algebraic method, which we also briefly review, our treatment provides an explanation to the long-noticed matching of the perturbative and WKB expansions of dual energy levels. Our technique also reveals the classical origins of duality.
Brian C. Hall
,Benjamin D. Lewis
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(2017)
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"A unitary quantization commutes with reduction map for the adjoint action of a compact Lie group"
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Brian C. Hall
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