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Oliver J. Rosten

Publications and source records attributed to Oliver J. Rosten.

At least 19 recordsLinked to original sources
arXiv2022

The Conformal Anomaly and a new Exact RG

For scalar field theory, a new generalization of the Exact RG to curved space is proposed, in which the conformal anomaly is explicitly present. Vacuum terms require regularization beyond that present in the canonical formulation of the Exact RG, which can be accomplished by adding certain free fields, each at a non-critical fixed-point. Taking the Legendre transform, the sole effect of the regulator fields is to remove a divergent vacuum term and they do not explicitly appear in the effective average action. As an illustration, both the integrated conformal anomaly and Polyakov action are recovered for the Gaussian theory in d=2.

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arXiv2019

Wilsonian Ward Identities

For conformal field theories, it is shown how the Ward identity corresponding to dilatation invariance arises in a Wilsonian setting. In so doing, several points which are opaque in textbook treatments are clarified. Exploiting the fact that the Exact Renormalization Group furnishes a representation of the conformal algebra allows dilatation invariance to be stated directly as a property of the action, despite the presence of a regulator. This obviates the need for formal statements that conformal invariance is recovered once the regulator is removed. Furthermore, the proper subset of conformal primary fields for which the Ward identity holds is identified for all dimensionalities.

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arXiv2018

On Functional Representations of the Conformal Algebra

Starting with conformally covariant correlation functions, a sequence of functional representations of the conformal algebra is constructed. A key step is the introduction of representations which involve an auxiliary functional. It is observed that these functionals are not arbitrary but rather must satisfy a pair of consistency equations corresponding to dilatation and special conformal invariance. In a particular representation, the former corresponds to the canonical form of the Exact Renormalization Group equation specialized to a fixed-point whereas the latter is new. This provides a concrete understanding of how conformal invariance is realized as a property of the Wilsonian effective action and the relationship to action-free formulations of conformal field theory. Subsequently, it is argued that the conformal Ward Identities serve to define a particular representation of the energy-momentum tensor. Consistency of this construction implies Polchinski's conditions for improving the energy-momentum tensor of a conformal field theory such that it is traceless. In the Wilsonian approach, the exactly marginal, redundant field which generates lines of physically equivalent fixed-points is identified as the trace of the energy-momentum tensor.

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arXiv2018

A Wilsonian Energy-Momentum Tensor

For local conformal field theories, it is shown how to construct an expression for the energy-momentum tensor in terms of a Wilsonian effective Lagrangian. Tracelessness implies a single, unintegrated equation which enforces both the Exact Renormalization Group equation and its partner encoding invariance under special conformal transformations.

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arXiv2012

Fundamentals of the Exact Renormalization Group

Various aspects of the Exact Renormalization Group (ERG) are explored, starting with a review of the concepts underpinning the framework and the circumstances under which it is expected to be useful. A particular emphasis is placed on the intuitive picture provided for both renormalization in quantum field theory and universality associated with second order phase transitions. A qualitative discussion of triviality, asymptotic freedom and asymptotic safety is presented. Focusing on scalar field theory, the construction of assorted flow equations is considered using a general approach, whereby different ERGs follow from field redefinitions. It is recalled that Polchinski's equation can be cast as a heat equation, which provides intuition and computational techniques for what follows. The analysis of properties of exact solutions to flow equations includes a proof that the spectrum of the anomalous dimension at critical fixed-points is quantized. Two alternative methods for computing the beta-function in lambda phi^4 theory are considered. For one of these it is found that all explicit dependence on the non-universal differences between a family of ERGs cancels out, exactly. The Wilson-Fisher fixed-point is rediscovered in a rather novel way. The discussion of nonperturbative approximation schemes focuses on the derivative expansion, and includes a refinement of the arguments that, at the lowest order in this approximation, a function can be constructed which decreases monotonically along the flow. A new perspective is provided on the relationship between the renormalizability of the Wilsonian effective action and of correlation functions, following which the construction of manifestly gauge invariant ERGs is sketched, and some new insights are given. Drawing these strands together suggests a new approach to quantum field theory.

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arXiv2011

Relationships Between Exact RGs and some Comments on Asymptotic Safety

The standard flow equation for the effective average action can be derived from a Legendre transform of Polchinski's exact renormalization group equation. However, the latter is not well adapted for finding fixed-points with non-zero anomalous dimension. Instead, it is more convenient to use a modified version which ensures that the redundant coupling associated with the normalization of the field never appears in the action. Taking this as the starting point, a Legendre transform is constructed allowing a direct derivation of the corresponding flow equation for the effective average action. This equation is then used to exactly construct some illuminating though essentially trivial) asymptotically safe trajectories emanating from various non-unitary fixed-points. Finally, in the context of asympotically safe quantum gravity, it is pointed out that the standard argument that the anomalous dimension of Newton's constant is necessarily 2-d at a non-trivial fixed-point is incomplete. The implications of this are discussed.

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arXiv2011

Equivalent Fixed-Points in the Effective Average Action Formalism

Starting from a modified version of Polchinski's equation, Morris' fixed-point equation for the effective average action is derived. Since an expression for the line of equivalent fixed-points associated with every critical fixed-point is known in the former case, this link allows us to find, for the first time, the analogous expression in the latter case.

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arXiv2011

Aspects of Manifest Gauge Invariance

The essential elements of manifestly gauge invariant exact renormalization groups are recalled. The pros and cons of the formalism are discussed and it is argued that now is the right time to try to utilize the formalism to provide new insights into the strong coupling domain of QCD.

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arXiv2010

An Extension of Pohlmeyer's Theorem

Applying the Exact Renormalization Group to scalar field theory in Euclidean space of general (not necessarily integer) dimension, it is proven that the only fixed-point with vanishing anomalous dimension is the Gaussian one. The proof requires positivity of the two-point connected correlation function together with a technical assumption concerning solutions of the flow equation. The method, in which the representation of the flow equation as a heat equation plays a central role, extends directly to non-gauge theories with arbitrary matter content (though non-linear sigma models are beyond the scope of the current method).

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arXiv2010

On the Renormalization of Theories of a Scalar Chiral Superfield

An exact renormalization group for theories of a scalar chiral superfield is formulated, directly in four dimensional Euclidean space. By constructing a projector which isolates the superpotential from the full Wilsonian effective action, it is shown that the nonperturbative nonrenormalization theorem follows, quite simply, from the flow equation. Next, it is argued that there do not exist any physically acceptable non-trivial fixed points. Finally, the Wess-Zumino model is considered, as a low energy effective theory. Following an evaluation of the one and two loop beta-function coefficients, to illustrate the ease of use of the formalism, it is shown that the beta-function in the massless case does not receive any nonperturbative power corrections.

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arXiv2009

Wilsonian Renormalization of Noncommutative Scalar Field Theory

Drawing on analogies with the commutative case, the Wilsonian picture of renormalization is developed for noncommutative scalar field theory. The dimensionful noncommutativity parameter, theta, induces several new features. Fixed-points are replaced by `floating-points' (actions which are scale independent only up to appearances of theta written in cutoff units). Furthermore, it is found that one must use correctly normalized operators, with respect to a new scalar product, to define the right notion of relevance and irrelevance. In this framework it is straightforward and intuitive to reproduce the classification of operators found by Grosse & Wulkenhaar, around the Gaussian floating-point. The one-loop beta-function of their model is computed directly within the exact renormalization group, reproducing the previous result that it vanishes in the self-dual theory, in the limit of large cutoff. With the link between this methodology and earlier results made, it is discussed how the vanishing of the beta-function to all loops, as found by Disertori et al., should be interpreted in a Wilsonian framework.

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arXiv2009

Triviality from the Exact Renormalization Group

Using the exact renormalization group, it is shown that no physically acceptable non-trivial fixed points, with positive anomalous dimension, exist for (i) O(N) scalar field theory in four or more dimensions, (ii) non-compact, pure Abelian gauge theory in any dimension. It is then shown, for both theories in any dimension, that otherwise physically acceptable non-trivial fixed points with negative anomalous dimension are non-unitary. In addition, a very simple demonstration is given, directly from the exact renormalization group, that should a critical fixed point exist for either theory in any dimension, then the n-point correlation functions exhibit the expected behaviour.

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arXiv2009

Constraints on an Asymptotic Safety Scenario for the Wess-Zumino Model

Using the nonrenormalization theorem and Pohlmeyer's theorem, it is proven that there cannot be an asymptotic safety scenario for the Wess-Zumino model unless there exists a non-trivial fixed point with (i) a negative anomalous dimension (ii) a relevant direction belonging to the Kaehler potential.

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arXiv2008

A Resummable beta-Function for Massless QED

Within the set of schemes defined by generalized, manifestly gauge invariant exact renormalization groups for QED, it is argued that the beta-function in the four dimensional massless theory cannot possess any nonperturbative power corrections. Consequently, the perturbative expression for the beta-function must be resummable. This argument cannot be extended to flows of the other couplings or to the anomalous dimension of the fermions and so perturbation theory does not define a unique trajectory in the critical surface of the Gaussian fixed point. Thus, resummability of the beta-function is not inconsistent with the expectation that a non-trivial fixed point does not exist.

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arXiv2008

Sensitivity of Nonrenormalizable Trajectories to the Bare Scale

Working in scalar field theory, we consider RG trajectories which correspond to nonrenormalizable theories, in the Wilsonian sense. An interesting question to ask of such trajectories is, given some fixed starting point in parameter space, how the effective action at the effective scale, Lambda, changes as the bare scale (and hence the duration of the flow down to Lambda) is changed. When the effective action satisfies Polchinski's version of the Exact Renormalization Group equation, we prove, directly from the path integral, that the dependence of the effective action on the bare scale, keeping the interaction part of the bare action fixed, is given by an equation of the same form as the Polchinski equation but with a kernel of the opposite sign. We then investigate whether similar equations exist for various generalizations of the Polchinski equation. Using nonperturbative, diagrammatic arguments we find that an action can always be constructed which satisfies the Polchinski-like equation under variation of the bare scale. For the family of flow equations in which the field is renormalized, but the blocking functional is the simplest allowed, this action is essentially identified with the effective action at Lambda = 0. This does not seem to hold for more elaborate generalizations.

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arXiv2008

A Comment on the Path Integral Approach to Cosmological Perturbation Theory

It is pointed out that the exact renormalization group approach to cosmological perturbation theory, proposed in Matarrese and Pietroni, JCAP 0706 (2007) 026, arXiv:astro-ph/0703563 and arXiv:astro-ph/0702653, constitutes a misnomer. Rather, having instructively cast this classical problem into path integral form, the evolution equation then derived comes about as a special case of considering how the generating functional responds to variations of the primordial power spectrum.

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arXiv2007

A Manifestly Gauge Invariant and Universal Calculus for SU(N) Yang-Mills

Within the framework of the Exact Renormalization Group, a manifestly gauge invariant calculus is constructed for SU(N) Yang-Mills. The methodology is comprehensively illustrated with a proof, to all orders in perturbation theory, that the beta function has no explicit dependence on either the seed action or details of the covariantization of the cutoff. The cancellation of these non-universal contributions is done in an entirely diagrammatic fashion.

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