number dimensions geometry spacetime condensed matter physics qft used describe dimensional electron gases high energy physics string theory type dimensional qft ref name polchinski kaluza klein theory uses gravity extra dimensions produce gauge theories lower dimensions minkowski space flat metric used raise lower spacetime indices lagrangian ''eg'' math mu mu eta mu nu mu nu quad partial mu phi partial mu phi eta mu nu partial mu phi partial nu phi math inverse satisfying '' sup sup sub sub '' qfts curved spacetime hand general metric
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