Unitarity in “ quantization commutes with reduction ”
نویسندگان
چکیده
Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M/G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515–538] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M/G and the G-invariant subspace of the quantum Hilbert space over M. We prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck’s constant. We then modify the quantization procedure by the “metaplectic correction” and show that in this setting there is still a natural invertible map between the Hilbert space over M/G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin–Sternberg map is asymptotically unitary to leading order in Planck’s constant.
منابع مشابه
Unitarity in “ quantization commutes with reduction ” Brian
Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M/G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515–538] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M/G and the G-invariant subspa...
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Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In [Invent. Math. 67 (1982), no. 3, 515–538], Guillemin and Sternberg showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M//G. This map, though, is not in general unitary, even to lea...
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Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In [Invent. Math. 67 (1982), no. 3, 515–538], Guillemin and Sternberg showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M/G. This map, though, is not in general unitary, even to lead...
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