Editing Kinetic term (section)

=== Higher-order derivatives ===

Higher-order derivative kinetic terms are bilinear in fields but have more than two derivatives. Such terms generally break [[perturbation theory|perturbative]] unitarity, giving rise to non-unitary theories. This is because in [[position and momentum spaces|momentum space]], unitarity requires propagators to have an [[asymptotic analysis|asymptotic]] falloff of at most <math>p^{-2}</math> in its [[momentum]], corresponding to kinetic terms with at most two derivatives in position space.<ref name="Schwartz"/>{{rp|470}}

Non-unitary theories with higher-order kinetic terms are useful in a number of areas such as in condensed matter physics where unitarity is not a strict requirement. Here they have been used to study [[elasticity (physics)|elasticity]], [[phase transition]]s, and certain [[polymer]]s.<ref>{{cite journal|last1=Safari|first1=M.|authorlink1=|last2=Vacca|first2=G.P.|authorlink2=|date=2018|title=Multicritical scalar theories with higher-derivative kinetic terms: A perturbative RG approach with the $\ensuremath{\epsilon}$-expansion|url=https://link.aps.org/doi/10.1103/PhysRevD.97.041701|journal=Phys. Rev. D|volume=97|issue=4|pages=041701|doi=10.1103/PhysRevD.97.041701|pmid=|arxiv=1708.09795|s2cid=|access-date=}}</ref> These kinetic terms can also help improve the [[ultraviolet completion|ultraviolet]] behaviour of Feynman diagrams and turn [[renormalization|nonrenormalizable]] theories into renormalizable ones,<ref name="LWM">{{cite journal|last1=Anselmi|first1=D.|authorlink1=|last2=Piva|first2=M.|authorlink2=|date=2017|title=A new formulation of Lee-Wick quantum field theory|url=|journal=JHEP|volume=6|issue=|pages=066|doi=10.1007/JHEP06(2017)066|pmid=|arxiv=|s2cid=|access-date=|hdl=11568/865571|hdl-access=free}}</ref> such as for higher-derivative [[gravity]].<ref>{{cite journal|last1=Stelle|first1=K.S.|authorlink1=|date=1977|title=Renormalization of higher-derivative quantum gravity|url=https://link.aps.org/doi/10.1103/PhysRevD.16.953|journal=Phys. Rev. D|volume=16|issue=4|pages=953–969|doi=10.1103/PhysRevD.16.953|pmid=|arxiv=|s2cid=|access-date=|url-access=subscription}}</ref> A class of higher-derivative theories known as Lee–Wick models,<ref>{{cite journal|last1=Lee|first1=T.D.|authorlink1=Tsung-Dao Lee|last2=Wick|first2=G.C.|authorlink2=Gian Carlo Wick|date=1969|title=Negative metric and the unitarity of the S-matrix|url=https://www.sciencedirect.com/science/article/pii/0550321369900984|journal=Nuclear Physics B|volume=9|issue=2|pages=209–243|doi=10.1016/0550-3213(69)90098-4|pmid=|arxiv=|s2cid=|access-date=|url-access=subscription}}</ref> usually formulate at the [[S-matrix]] level, are claimed to be unitary, with them get around the aforementioned obstruction using cutting equations.<ref name="LWM"/>

When higher-order derivative kinetic terms occur in a Minkowski theory and result in propagators with complex [[zeroes and poles|poles]], the theory is mathematically [[consistency|inconsistent]].<ref>{{cite journal|last1=Aglietti|first1=U.G.|authorlink1=|last2=Anselmi|first2=D.|authorlink2=|date=2017|title=Inconsistency of Minkowski higher-derivative theories|url=|journal=Eur. Phys. J. C|volume=77|issue=2|pages=84|doi=10.1140/epjc/s10052-017-4646-7|pmid=|arxiv=1612.06510|s2cid=|access-date=}}</ref> This is because these kinetic terms give rise to [[nonlocal Lagrangian|non-local]] and non-Hermitian [[ultraviolet divergence]]s that cannot be eliminated using the standard renormalization procedure. These inconsistencies do not affect higher-derivative theories which do not have propagator complex poles or purely [[Euclidean space|Euclidean]] theories.

Free higher-derivative scalar field theories are solvable and do not suffer from instabilities such as [[false vacuum|vacuum decay]].<ref>{{cite journal|last1=Brust|first1=C.|authorlink1=|last2=Hinterbichler|first2=K.|authorlink2=|date=2017|title=Free \ensuremath{\square}$^{k}$ scalar conformal field theory|url=|journal=JHEP|volume=02|issue=|pages=066|doi=10.1007/JHEP02(2017)066|pmid=|arxiv=1607.07439|s2cid=|access-date=}}</ref> They can also be fully conformal. Such non-unitary conformal field theories may be useful for studying the [[dS/CFT correspondence]].