Editing Quantum field theory (section)

====Topological quantum field theory====
{{Main|Topological quantum field theory}}

The correlation functions and physical predictions of a QFT depend on the spacetime metric {{math|''g<sub>μν</sub>''}}. For a special class of QFTs called [[topological quantum field theories]] (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric.<ref>{{cite arXiv |last1=Ivancevic |first1=Vladimir G. |last2=Ivancevic |first2=Tijana T. |eprint=0810.0344v5 |title=Undergraduate Lecture Notes in Topological Quantum Field Theory |class=math-th |date=2008-12-11 }}</ref>{{rp|36}} QFTs in curved spacetime generally change according to the ''geometry'' (local structure) of the spacetime background, while TQFTs are invariant under spacetime [[diffeomorphism]]s but are sensitive to the ''[[topology]]'' (global structure) of spacetime. This means that all calculational results of TQFTs are [[topological invariant]]s of the underlying spacetime. [[Chern–Simons theory]] is an example of TQFT and has been used to construct models of quantum gravity.<ref>{{cite book |last=Carlip |first=Steven |author-link=Steve Carlip |date=1998 |title=Quantum Gravity in 2+1 Dimensions |url=https://www.cambridge.org/core/books/quantum-gravity-in-21-dimensions/D2F727B6822014270F423D82501E674A |publisher=Cambridge University Press |pages=27–29 |isbn=9780511564192 |doi=10.1017/CBO9780511564192 |arxiv=2312.12596 }}</ref> Applications of TQFT include the [[fractional quantum Hall effect]] and [[topological quantum computer]]s.<ref>{{cite journal |last1=Carqueville |first1=Nils |last2=Runkel |first2=Ingo |arxiv=1705.05734 |title=Introductory lectures on topological quantum field theory |journal=Banach Center Publications |year=2018 |volume=114 |pages=9–47 |doi=10.4064/bc114-1 |s2cid=119166976 }}</ref>{{rp|1–5}} The world line trajectory of fractionalized particles (known as [[anyons]]) can form a link configuration in the spacetime,<ref>{{Cite journal |author-link=Edward Witten |first=Edward |last=Witten |title=Quantum Field Theory and the Jones Polynomial |journal=[[Communications in Mathematical Physics]] |volume=121 |issue=3 |pages=351–399 |year=1989 |mr=0990772 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 |s2cid=14951363 |url=http://projecteuclid.org/euclid.cmp/1104178138 }}</ref> which relates the braiding statistics of anyons in physics to the
link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.<ref>{{Cite journal |first1=Pavel|last1=Putrov |first2=Juven |last2=Wang | first3=Shing-Tung | last3=Yau    |title=Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions |journal=[[Annals of Physics]] |volume=384 |issue=C |pages=254–287 |year=2017|doi =10.1016/j.aop.2017.06.019|arxiv=1612.09298 |bibcode=2017AnPhy.384..254P |s2cid=119578849 }}</ref>