Perturbative renormalization of composite operators via flow equations II: Short distance expansion

Keller, Georg B.; Kopper, Christoph

doi:10.1007/bf02096643

Cited by 37 publications

(73 citation statements)

References 10 publications

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“…Remarkably, for Euclidean perturbative φ 4 -theory, it has been shown that the series on the right side actually converges, at least at arbitrary but fixed order in perturbation theory [51,63]. If, as we expect, this also were found to be true at spacelike related points for theories on Lorentzian curved spacetimes, then this would strongly reinforce the view that the OPE coefficients contain the entire local information about the theory, even on a curved spacetime.…”

Section: Nonperturbative Formulation Of Interacting Qftcssupporting

confidence: 53%

“…We consider a massless quantum Klein-Gordon scalar field in the spacetime (63). This is a (slight) simplification of the more physically relevant problem of starting with a classical solution of the Einstein-scalar-field system with scalar factor of a form approximating (63)-as would occur if the scalar field "slowly rolls" down an extremely flat potential-and then treating the linearized perturbations of this system as quantum fields.…”

Section: D) Cosmological Perturbationsmentioning

confidence: 99%

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Quantum Fields in Curved Spacetime

Hollands¹,

Wald²

2015

General Relativity and Gravitation

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We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states. We review the Unruh and Hawking effects for free fields, as well as the behavior of free fields in deSitter spacetime and FLRW spacetimes with an exponential phase of expansion. We review how nonlinear observables of a free field, such as the stress-energy tensor, are defined, as well as timeordered-products. The "renormalization ambiguities" involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar field, but a brief discussion of gauge fields is included. We conclude with a brief discussion of a possible approach towards a nonperturbative formulation of quantum field theory in curved spacetime and some remarks on the formulation of quantum gravity. Quantum field theory in curved spacetime (QFTCS) is the theory of quantum fields propagating in a background, classical, curved spacetime (M , g). On account of its classical treatment of the metric, QFTCS cannot be a fundamental theory of nature. However, QFTCS is expected to provide an accurate description of quantum phenomena in a regime where the effects of curved spacetime may be significant, but effects of quantum gravity itself may be neglected. In particular, it is expected that QFTCS should be applicable to the description of quantum phenomena occurring in the early universe and near (and inside of) black holesprovided that one does not attempt to describe phenomena occurring so near to singularities that curvatures reach Planckian scales and the quantum nature of the spacetime metric would have to be taken into account.It should be possible to derive QFTCS by taking a suitable limit of a more fundamental theory wherein the spacetime metric is treated in accord with the principles of quantum theory. However, this has not been done-except in formal and/or heuristic ways-simply because no present quantum theory of gravity has been developed to the point where such a well defined limit can be taken. Rather, the framework of QFTCS that we shall describe in this review has been obtained by suitably merging basic principles of classical general relativity with the basic principles of quantum field theory in Minkowski spacetime. As we shall explain further below, the basic principles of classical general relativity are relatively easy to identify and adhere to, but it is far less clear what to identify as the "basic principles" of quantum field theory in Minkowski spacetime. Indeed, many of the concepts normally viewed as fundamental to quantum field theory in Minkowski spacetime, such as Poinca...

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Section: Nonperturbative Formulation Of Interacting Qftcssupporting

confidence: 53%

Section: D) Cosmological Perturbationsmentioning

confidence: 99%

Quantum Fields in Curved Spacetime

Hollands¹,

Wald²

2015

General Relativity and Gravitation

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“…FRG formulations are also suitable for both discussing formal aspects as well as practical applications. The FRG has been introduced with a smooth momentum cut-off for simplifying proofs of perturbative renormalisability and the construction of effective Lagrangians in [6], see also [9,[31][32][33]. More recently, there has been an increasing interest in FRG methods as a computational tool for accessing both perturbative as well as non-perturbative physics, initiated by [10][11][12][13][14].…”

Section: Flowsmentioning

confidence: 99%

Aspects of the functional renormalisation group

2007

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We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being N PI quantities. We discuss optimisation and derive a functional optimisation criterion.Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation.

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“…Using a cut-off function with compact support, Polchinski gave a simple proof of the power-counting renormalization theorem; the proof has been generalized and further simplified in [5,23,24,7]. Analogous results have been obtained using an exponential cut-off in [25]; a proof of renormalizability with a Pauli-Villars cut-off, like that of eq.…”

Section: Exact Renormalization Groupmentioning

confidence: 98%

Hard-soft renormalization and the exact renormalization group

1998

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The Wilsonian exact renormalization group gives a natural framework in which ultraviolet and infrared divergences can be treated separately. In massless QED we introduce, as the only mass parameter, a renormalization scale Λ R > 0. We prove, using the flow equation technique, that infrared convergence is a necessary consequence of any zero-momentum renormalization condition at Λ R compatible with the effective Ward identities and axial symmetry. The same formalism is applied to renormalize gauge-invariant composite operators and to prove their infrared finiteness; in particular we consider the case of the axial current operator and its anomaly. * Work supported in part by M.U.R.S.T.

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Perturbative renormalization of composite operators via flow equations II: Short distance expansion

Cited by 37 publications

References 10 publications

Quantum Fields in Curved Spacetime

Quantum Fields in Curved Spacetime

Aspects of the functional renormalisation group

Hard-soft renormalization and the exact renormalization group

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