Hans-Peter Eckle
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials
The core of the exposition of the theory of conformal symmetry in statistical mechanics are the concepts of correlation functions of order parameter fields, whose behaviour under conformal ...
More
The core of the exposition of the theory of conformal symmetry in statistical mechanics are the concepts of correlation functions of order parameter fields, whose behaviour under conformal transformations are the defining characteristic of conformal field theories. Chapter 7 discusses the transformation properties of the energy-momentum tensor, the conformal Ward identities, and the operator product expansion lead to the loop or Witt algebra with central extension, the Virasoro algebra, allowing the characterization of the possible universality classes, in particular through the conformal anomaly or central charge. It discusses how the finite-size corrections to thermodynamic quantities, obtained from conformal transformations to finite geometries, can be used to determine critical parameters, especially the central charge.Less
The core of the exposition of the theory of conformal symmetry in statistical mechanics are the concepts of correlation functions of order parameter fields, whose behaviour under conformal transformations are the defining characteristic of conformal field theories. Chapter 7 discusses the transformation properties of the energy-momentum tensor, the conformal Ward identities, and the operator product expansion lead to the loop or Witt algebra with central extension, the Virasoro algebra, allowing the characterization of the possible universality classes, in particular through the conformal anomaly or central charge. It discusses how the finite-size corrections to thermodynamic quantities, obtained from conformal transformations to finite geometries, can be used to determine critical parameters, especially the central charge.
Iosif L. Buchbinder and Ilya L. Shapiro
- Published in print:
- 2021
- Published Online:
- April 2021
- ISBN:
- 9780198838319
- eISBN:
- 9780191874666
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198838319.003.0017
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
This chapter describes in detail basic results concerning the conformal (trace) anomaly and anomaly-induced action in four spacetime dimensions. It is shown how the anomaly appears from the non-local ...
More
This chapter describes in detail basic results concerning the conformal (trace) anomaly and anomaly-induced action in four spacetime dimensions. It is shown how the anomaly appears from the non-local form factors discussed in chapter 16. Starting from the conformal transformations, the necessary invariants and transformation rules are obtained. The simplest derivation of the anomaly in dimensional regularization is explained, followed by the equally simple calculation of the anomaly-induced effective action of gravity. The chapter also briefly discusses applications of the induced effective action in cosmology and black hole physics.Less
This chapter describes in detail basic results concerning the conformal (trace) anomaly and anomaly-induced action in four spacetime dimensions. It is shown how the anomaly appears from the non-local form factors discussed in chapter 16. Starting from the conformal transformations, the necessary invariants and transformation rules are obtained. The simplest derivation of the anomaly in dimensional regularization is explained, followed by the equally simple calculation of the anomaly-induced effective action of gravity. The chapter also briefly discusses applications of the induced effective action in cosmology and black hole physics.
Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question ...
More
This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.Less
This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.
M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko
- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199238743
- eISBN:
- 9780191716461
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199238743.003.0011
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics
This chapter presents a brief review of the Casimir effect in spaces with nontrivial topology (both flat and curved). As an important application of the numerous results obtained in this field, the ...
More
This chapter presents a brief review of the Casimir effect in spaces with nontrivial topology (both flat and curved). As an important application of the numerous results obtained in this field, the vacuum energy-momentum tensor due to the Casimir effect in the closed Friedmann model is considered. A related subject is the role of the Casimir effect in multidimensional Kaluza–Klein theories, where it provides one of the mechanisms for compactification of extra spatial dimensions. This is also reflected in the chapter, which is concluded with a brief discussion of the Casimir effect for topological defects, such as cosmic strings and domain walls. This problem is of interest for cosmology because some grand unification theories predict the formation of such defects in the early Universe.Less
This chapter presents a brief review of the Casimir effect in spaces with nontrivial topology (both flat and curved). As an important application of the numerous results obtained in this field, the vacuum energy-momentum tensor due to the Casimir effect in the closed Friedmann model is considered. A related subject is the role of the Casimir effect in multidimensional Kaluza–Klein theories, where it provides one of the mechanisms for compactification of extra spatial dimensions. This is also reflected in the chapter, which is concluded with a brief discussion of the Casimir effect for topological defects, such as cosmic strings and domain walls. This problem is of interest for cosmology because some grand unification theories predict the formation of such defects in the early Universe.
Iosif L. Buchbinder and Ilya L. Shapiro
- Published in print:
- 2021
- Published Online:
- April 2021
- ISBN:
- 9780198838319
- eISBN:
- 9780191874666
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198838319.003.0015
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
As the main purpose of renormalization is not to remove divergences but to get essential information about the finite part of effective action, this chapter discusses some of the existing methods of ...
More
As the main purpose of renormalization is not to remove divergences but to get essential information about the finite part of effective action, this chapter discusses some of the existing methods of solving this problem; such methods can be denoted the renormalization group. First, the minimal subtraction renormalization group in curved space is formulated. Next, the chapter shows how the overall μ-independence of the effective action enables one to interpret μ-dependence in some situations. As an example, the effective potential is restored from the renormalization group and compared with the expression calculated directly in chapter 13. In addition, the global conformal (scaling) anomaly is derived from the renormalization group.Less
As the main purpose of renormalization is not to remove divergences but to get essential information about the finite part of effective action, this chapter discusses some of the existing methods of solving this problem; such methods can be denoted the renormalization group. First, the minimal subtraction renormalization group in curved space is formulated. Next, the chapter shows how the overall μ-independence of the effective action enables one to interpret μ-dependence in some situations. As an example, the effective potential is restored from the renormalization group and compared with the expression calculated directly in chapter 13. In addition, the global conformal (scaling) anomaly is derived from the renormalization group.
