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  Quantum mechanical vacuum energy of a system generally obtainable from the metaplectic correction of its geometric quantization?

+ 3 like - 0 dislike

In geometric quantization which assigns a (quantum) Hilbert space to a symplectic manifold ( for example the phase space of a system) the metaplectic correction is needed to  obtain a nonzero Hilbert space when using a real polarization.

In the case of a complex polarization when geometrically quantizing the harmonic oscillator, it allows to reproduce the well-known formula

$$E_n = (1/2 + n)\hbar\omega$$

for the energy levels, with the $1/2$ coming from the metaplectic correction.

Is this just a coincidence or is it generally the case that the vacuum energy of a (geometrically quantizable) system can be obtained by applying an (appropriate) metaplectic correction?

asked Sep 28, 2018 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
edited Sep 29, 2018 by Dilaton

It is the case whenever the manifold is a homogeneous space with respect to some group, and the representation of the group on the Hilbert space is obtainable by restriction from a representation of the symplectic group.

1 Answer

+ 2 like - 0 dislike

I guess that is not a coincidence. Please see the sentences below Eq.(46) of arXiv:1606.06405 by Karabali and Nair. 

answered Oct 2, 2018 by no name [ revision history ]
edited Oct 2, 2018 by Dilaton

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