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Examples of reflection positive field theories

R. Trinchero

Instituto Balseiro y Centro Atmico Bariloche, Bariloche 8400, Argentina

E-mail Address: trincher@cab.cnea.gov.ar

https://doi.org/10.1142/S0219887818500226Cited by:6 (Source: Crossref)

Abstract

The requirement of reflection positivity (RP) for Euclidean field theories is considered. This is done for the cases of a scalar field, a higher derivative scalar field theory and the scalar field theory defined on a non-integer dimensional space (NIDS). It is shown that regarding RP, the analytical structure of the corresponding Schwinger functions plays an important role. For the higher derivative scalar field theory, RP does not hold. However for the scalar field theory on a NIDS, RP holds in a certain range of dimensions where the corresponding Minkowskian field is defined on a Hilbert space with a positive definite scalar product that provides a unitary representation of the Poincaré group. In addition, and motivated by the last example, it is shown that, under certain conditions, one can construct non-local reflection positive Euclidean field theories starting from the corrected two point functions of interacting local field theories.

Keywords:

AMSC: 51P05, 46L60, 81Q35, 81T05

References

1. K. Osterwalder and R. Schrader, Axioms for euclidean green’s functions, Commun. Math. Phys. 31(2) (1973) 83–112. Web of Science, Google Scholar
2. K. G. Wilson and J. B. Kogut, The renormalization group and the epsilon expansion, Phys. Rept. 12 (1974) 75–200. Google Scholar
3. C. G. Bollini and J. J. Giambiagi, Dimensional renorinalization: The number of dimensions as a regularizing parameter, Il Nuovo Cimento B (1971–1996) 12(1) (1972) 20–26. Google Scholar
4. G. Hooft and M. J. G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189–213. Web of Science, Google Scholar
5. A. Connes, Noncommutative Geometry (Elsevier Science, 1995). Google Scholar
6. E. Christensen, C. Ivan and M. L. Lapidus, Dirac operators and spectral triples for some fractal sets built on curves, Adv. Math. 217(1) (2008) 42–78. Web of Science, Google Scholar
7. R. Trinchero, Scalar field on non-integer dimensional spaces, Int. J. Geom. Methods Mod. Phys. 09 (2012) 1250070. Link, Web of Science, Google Scholar
8. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer-Verlag, 1987). Google Scholar
9. F. Guerra, Euclidean field theory, in Encyclopedia of Mathematical Physics (Elsevier Limited, Oxford, 2006), pp. 256–265. Google Scholar
10. C. C. Anderson, Defining Physics at Imaginary Time: Reflection Positivity for Certain Riemannian Manifolds (Harvard University, 2013). Google Scholar
11. A. Uhlmann, Some remarks on reflection positivity, Czech. J. Phys. B 29(1) (1979) 117–126. Google Scholar
12. M. J. G. Veltman, Unitarity and causality in a renormalizable field theory with unstable particles, Physica 29 (1963) 186–207. Web of Science, Google Scholar
13. M. Hogervorst, S. Rychkov and B. C. van Rees, Unitarity violation at the Wilson-Fisher fixed point in 4- $ϵ$ $𝜖$ dimensions, Phys. Rev. D 93(12) (2016) 125025. Web of Science, Google Scholar
14. R. Zwicky, A brief introduction to dispersion relations and analyticity, in Quantum Field Theory at the Limits: From Strong Fields to Heavy Quarks (HQ 2016), Dubna, Russia, July 18–30, 2016, pp. 93–120. Google Scholar