Editing Quantum field theory (section)

====Other spacetimes====
The {{math|''ϕ''<sup>4</sup>}} theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional [[Minkowski space]] (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT ''a priori'' imposes no restriction on the number of dimensions nor the geometry of spacetime.

In [[condensed matter physics]], QFT is used to describe [[two-dimensional electron gas|(2+1)-dimensional electron gases]].<ref>{{cite book |last1=Morandi |first1=G. |last2=Sodano |first2=P. |last3=Tagliacozzo |first3=A. |last4=Tognetti |first4=V. |date=2000 |title=Field Theories for Low-Dimensional Condensed Matter Systems |url=https://www.springer.com/us/book/9783540671770 |publisher=Springer |isbn=978-3-662-04273-1 }}</ref> In [[high-energy physics]], [[string theory]] is a type of (1+1)-dimensional QFT,{{r|zee|page1=452}}<ref name="polchinski1" /> while [[Kaluza–Klein theory]] uses gravity in [[extra dimensions]] to produce gauge theories in lower dimensions.{{r|zee|page1=428–429}}

In Minkowski space, the flat [[metric tensor (general relativity)|metric]] {{math|''η<sub>μν</sub>''}} is used to [[raising and lowering indices|raise and lower]] spacetime indices in the Lagrangian, ''e.g.''
:<math>A_\mu A^\mu = \eta_{\mu\nu} A^\mu A^\nu,\quad \partial_\mu\phi \partial^\mu\phi = \eta^{\mu\nu}\partial_\mu\phi \partial_\nu\phi,</math>
where {{math|''η<sup>μν</sup>''}} is the inverse of {{math|''η<sub>μν</sub>''}} satisfying {{math|''η<sup>μρ</sup>η<sub>ρν</sub>'' {{=}} ''δ<sup>μ</sup><sub>ν</sub>''}}. 
For [[quantum field theory in curved spacetime|QFTs in curved spacetime]] on the other hand, a general metric (such as the [[Schwarzschild metric]] describing a [[black hole]]) is used:
:<math>A_\mu A^\mu = g_{\mu\nu} A^\mu A^\nu,\quad \partial_\mu\phi \partial^\mu\phi = g^{\mu\nu}\partial_\mu\phi \partial_\nu\phi,</math>
where {{math|''g<sup>μν</sup>''}} is the inverse of {{math|''g<sub>μν</sub>''}}. 
For a real scalar field, the Lagrangian density in a general spacetime background is
:<math>\mathcal{L} = \sqrt{|g|}\left(\frac 12 g^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi - \frac 12 m^2\phi^2\right),</math>
where {{math|''g'' {{=}} det(''g<sub>μν</sub>'')}}, and {{math|∇<sub>''μ''</sub>}} denotes the [[covariant derivative]].<ref>{{cite book |last1=Parker |first1=Leonard E. |last2=Toms |first2=David J. |date=2009 |title=Quantum Field Theory in Curved Spacetime |url=https://archive.org/details/quantumfieldtheo00park |url-access=limited |publisher=Cambridge University Press |page=[https://archive.org/details/quantumfieldtheo00park/page/n58 43] |isbn=978-0-521-87787-9 }}</ref> The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.