Quantum Simulation of Conformal Field Theory
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Conformal field theory, describing systems with scaling symmetry, plays a crucial role throughout physics. We describe a quantum algorithm to simulate the dynamics of conformal field theories, including the action of local conformal transformations. A full analysis of the approximation errors suggests near-term applicability of our algorithm: promising results for conformal field theories with central charge c=1/2 are obtained already with 128 logical qubits.Keywords:
Conformal anomaly
Central charge
Weyl transformation
Boundary conformal field theory (BCFT) is the study of conformal field theory (CFT) on manifolds with a boundary. We can use conformal symmetry to constrain correlation functions of conformal invariant fields. We compute two-point and three-point functions of conformal invariant fields which live in semi-infinite space. For a situation with a boundary condition in surface $z=\bar{z}$ ($t=0$), the results agree with gravity dual results. We also explore representations of conformal group in two dimensions.
Conformal anomaly
Extremal length
Weyl transformation
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Conformal geometry
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Conformal anomaly
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We review conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to BPZ equations and conformal blocks. Examples include Liouville theory, (generalized) minimal models, free bosonic theories, the $H_3^+$ model, and the $SU_2$ and $\widetilde{SL}_2(\mathbb{R})$ WZW models. We also discuss relations between some of these models, and limits of these models when the central charge and/or conformal dimensions tend to particular values.
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Conformal symmetry is a powerful tool used to study field theories at their critical points. The enlargement of the space-time symmetry allows to classify local operators according to their conformal dimension. We propose a similar classification of nonlocal objects, like defects, vortices or Wilson loops, which may exist in conformal field theories. The definition relies on representations of subgroups of the conformal group and this new notion analogous to the conformal dimension may have wide applications in a variety of conformal field theories. We demonstrate the uses of this method to the study of Wilson loops in a conformal gauge theory.
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The conformal bootstrap method is a non-perturbative method that uses the symmetry in a conformal field theory to constrain and solve for the observables in the theory. We consider a conformal fiel ...
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In a previous paper conformal gravity was derived by means of a precise action principle on the hypercone in the conformal space. Here it is shown that the same technique used to construct conformal spin two theory as represented by linear conformal gravity may also be used for the construction of conformal higher spin theories. The basic ingredients in these constructions are gauge invariant field strengths (curvatures). In fact, their very existence as manifestly conformal fields requires the present conformal theory. The general form of the actions for the free theories on the hypercone are given. In spacetime these actions are shown to be expressed in terms of squares of generalized Weyl tensors. Conformal spin three and four are calculated explicitly. The theories are proposed to be equivalent to the free theories given by Fradkin and Linetsky. Since the equations are of order 2s a consistent quantization is difficult to achieve but might exist. Interactions are expected to exist and is the motivation behind this approach.
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Conformal theories in a d dimensional spacetime may be expressed as manifestly conformal theories in a d+2 dimensional conformal space as first proposed by Dirac. The reduction to d dimensions goes via the d+1 dimensional hypercone in the conformal space. Here we give a rather extensive expose of such theories. We review and extend the theory of spinning conformal particles. We give a precise and geometrical formulation of manifestly conformal fields for which we give a consistent action principle. The requirement of invariance under special gauge transformations off the hypercone plays a fundamental role here. Maxwell's theory and linear conformal gravity are derived in the conformal space and are treated in detail. Finally, we propose a consistent coordinate invariant action principle in the conformal space and give an action that should correspond to conformal gravity.
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Abstract 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.
Conformal anomaly
Operator product expansion
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Weyl transformation
Virasoro algebra
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We consider the construction of gauge theories of gravity that are invariant under local conformal transformations. We first clarify the geometric nature of global conformal transformations, in both their infinitesimal and finite forms, and the consequences of global conformal invariance for field theories, before reconsidering existing approaches for gauging the conformal group, namely auxiliary conformal gauge theory and biconformal gauge theory, neither of which is generally accepted as a complete solution. We then demonstrate that, provided any matter fields belong to an irreducible representation of the Lorentz group, the recently proposed extended Weyl gauge theory (eWGT) may be considered as an alternative method for gauging the conformal group, since eWGT is invariant under the full set of local conformal transformations, including inversions, as well as possessing conservation laws that provide a natural local generalization of those satisfied by field theories with global conformal invariance, and also having an ``ungauged'' limit that corresponds to global conformal transformations. By contrast, although standard Weyl gauge theory also enjoys the first of these properties, it does not share the other two, and so cannot be considered a valid gauge theory of the conformal group.
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