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

+ 3 like - 0 dislike
129 views

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?

edited Sep 29, 2018

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.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOver$\varnothing$lowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.