Reflection positivity constitutes an integral prerequisite in the Osterwalder–Schrader reconstruction theorem which relates quantum field theories defined on Euclidean space to their Lorentzian signature counterparts. In this work, we rigorously prove the violation of reflection positivity in a large class of free scalar fields with a rational propagator. This covers, in particular, higher-derivative theories where the propagator admits a partial fraction decomposition as well as degenerate cases including, e.g., p4-type propagators.
REFERENCES
1.
K. S.
Stelle
, “Renormalization of higher derivative quantum gravity
,” Phys. Rev. D
16
, 953
–969
(1977
).2.
K. S.
Stelle
, “Classical gravity with higher derivatives
,” Gen. Relativ. Gravitation
9
, 353
–371
(1978
).3.
A. S.
Wightman
, “Quantum field theory in terms of vacuum expectation values
,” Phys. Rev.
101
, 860
–866
(1956
).4.
K.
Osterwalder
and R.
Schrader
, “Axioms for Euclidean Green’s functions
,” Commun. Math. Phys.
31
, 83
–112
(1973
).5.
A.
Jaffe
and G.
Ritter
, “Reflection positivity and monotonicity
,” J. Math. Phys.
49
, 052301
(2008
); e-print arXiv:0705.0712 [math-ph].6.
M.
Ostrogradski
, “Mémoires sur les équations différentielles, relatives au problème des isopérimètres
,” Mem. Acad. St. Petersbourg
6
, 385
–517
(1850
), available at http://inspirehep.net/record/1468685?ln=en.7.
J.
Glimm
and A. M.
Jaffe
, “A note on reflection positivity
,” Lett. Math. Phys.
3
, 377
–378
(1979
).8.
A.
Jaffe
and G.
Ritter
, “Quantum field theory on curved backgrounds. I. The Euclidean functional integral
,” Commun. Math. Phys.
270
, 545
–572
(2007
); e-print arXiv:hep-th/0609003 [hep-th].9.
A.
Jaffe
and G.
Ritter
, “Quantum field theory on curved backgrounds. II. Spacetime symmetries
,” e-print arXiv:0704.0052 [hep-th] (2007
).10.
R.
Trinchero
, “Examples of reflection positive field theories
,” Int. J. Geom. Methods Mod. Phys.
15
, 1850022
(2017
); e-print arXiv:1703.07735 [hep-th].11.
L.
Modesto
and L.
Rachwał
, “Nonlocal quantum gravity: A review
,” Int. J. Mod. Phys. D
26
, 1730020
(2017
).12.
A. H.
Chamseddine
, A.
Connes
, and M.
Marcolli
, “Gravity and the standard model with neutrino mixing
,” Adv. Theor. Math. Phys.
11
, 991
–1089
(2007
); e-print arXiv:hep-th/0610241 [hep-th].13.
B.
Iochum
, C.
Levy
, and D.
Vassilevich
, “Spectral action beyond the weak-field approximation
,” Commun. Math. Phys.
316
, 595
–613
(2012
); e-print arXiv:1108.3749 [hep-th].14.
W. D.
van Suijlekom
, “Renormalization of the asymptotically expanded Yang-Mills spectral action
,” Commun. Math. Phys.
312
, 883
–912
(2012
); e-print arXiv:1104.5199 [math-ph].15.
M.
Niedermaier
and M.
Reuter
, “The asymptotic safety scenario in quantum gravity
,” Living Rev. Relativ.
9
, 5
(2006
).16.
M.
Reuter
and F.
Saueressig
, “Quantum Einstein gravity
,” New J. Phys.
14
, 055022
(2012
).17.
R.
Percacci
, An Introduction to Covariant Quantum Gravity and Asymptotic Safety
(World Scientific Publishing Company
, 2017
).18.
D.
Becker
, C.
Ripken
, and F.
Saueressig
, “On avoiding Ostrogradski instabilities within asymptotic safety
,” J. High Energy Phys.
2017
, 121
.19.
C. M.
Bender
and P. D.
Mannheim
, “No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model
,” Phys. Rev. Lett.
100
, 110402
(2008
); e-print arXiv:0706.0207 [hep-th].20.
M.
Niedermaier
, “A quantum cure for the Ostrogradski instability
,” Ann. Phys.
327
, 329
–358
(2012
).21.
M.
Crisostomi
, R.
Klein
, and D.
Roest
, “Higher derivative field theories: Degeneracy conditions and classes
,” J. High Energy Phys.
2017
(06
), 124
; e-print arXiv:1703.01623 [hep-th].22.
23.
24.
R.
Strichartz
, A Guide to Distribution Theory and Fourier Transforms
(CRC Press
, 1994
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.
