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By reading articles I ran several times into two terms, never being defined so I assume they must have well established definitions somewhere.

  1. The first is semi-classical space-time. If I where to guess I would say spin 2 field quantized over a Minkowski space, or qft in curved space-time (but space-time is totally classical in this case).

  2. The second is supersymmetric Minkowski space. Would it be a flat space-time with both commuting and anti-commuting coordinates? (From what I've read the interest in this cosmological solution, is the fact that the Hilbert space describing needs to be infinite dimensional. How is it related to the presence of supersymmetry?)

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1. probably refers to qft in curved spacetimes

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We normally disapprove of just posting links because the linked article can vanish or change. Having said that, I checked the link and the first equation is the one I was going to post as an answer i.e. replace the classical $G_{ab} = 8\pi T_{ab}$ with $G = 8\pi <T>$, where $<T>$ is the expectation value of the stress-energy tensor from quantum mechanics. –  John Rennie Jun 4 '12 at 19:04
    
This answer is wrong. 1 refers to spin 2 gravity on a background. –  Ron Maimon Jul 4 '12 at 15:42
    
@Maimon, Are you sure there is a standard usage? I recall googling this and finding various uses. What did you base this on? –  MadScientist Jul 4 '12 at 16:00

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