32

They're variants, different kinds of quantum field theory, but they're not mutually exclusive. The different adjectives you mention separate quantum field theory to "pieces" in different ways. The different sorts of variants you mention are being used and studied by different people, the classification has different purposes, the degree of usefulness and ...


32

The Weyl transformation and the conformal transformation are completely different things (although they are often discussed in similar contexts). A Weyl transformation isn't a coordinate transformation on the space or spacetime at all. It is a physical change of the metric, $g_{\mu\nu}(x)\to g_{\mu\nu}(x)\cdot \Omega(x)$. It is a transformation that changes ...


31

The operator-state correspondence says that all states in the theory can be created by operators which act locally in a small neighborhood of the origin. That is to say that the entire Hilbert space of a CFT can be thought of as living at a single point. The key here is that for CFTs we have radial quantization, and states evolve radially outwards unitarily ...


26

The answer is not known, but many believe it is: "Yes, every CFT has an AdS dual." However, whether the AdS dual is weakly-coupled and has low curvature -- in other words whether it's easy to do calculations with it -- is a different question entirely. We expect, based on well-understood examples (like $\mathcal N=4$ SYM dual to Type IIB strings on $\...


26

People hope that it may be an example of AdS/CFT correspondence that can be completely understood. AdS/CFT correspondence itself has been an incredibly important idea in the hep-th community over the past almost twenty years. Yet it remains a conjecture. In the typical situation, quantities computed on one side of the duality are hard to check on the ...


23

As Lubos Motl mentions in a comment, for all practical purposes, OP's sought-for eq. (1) is proved via Wick's Theorem. It is interesting to try to generalize Wick's Theorem and to try to minimize the number of assumptions that goes into it. Here we will outline one possible approach. I) Assume that a family $(\hat{A}_i)_{i\in I}$ of operators $\hat{A}_i\in{...


22

Here's a quick answer. There are a few reasons why CFT's are very interesting to study. The first is that at fixed points of RG flows, or at second order phase transitions, a quantum field theory is scale invariant. Scale invariance is a weaker form of conformal invariance, and it turns out in all cases that we know of (or at least the ones I know of) scale ...


19

The S-matrix (scattering matrix) is the unitary operator $S$ that determines the evolution of the initial state at $t=-\infty$ to the final state at $t=+\infty$. $$|\psi(t=+\infty)\rangle = S |\psi(t=-\infty)\rangle$$ This matrix/operator is therefore a collection of complex numbers that are ready to calculate the probabilities of various scattering ...


17

The three classes of QFTs you are referring to are distinguished by different symmetry assumptions (Poincare invariance, conformal invariance, and volume-preserving diffeomorphism invariance) and different background spacetimes (Minkowski, Riemann curve (or families of them), and arbitrary manifolds). Moreover, Wightman axioms only characterize the vacuum ...


16

If the operators $X_i$ can be written as a sum of an annihilation and a creation part$^1$ $$X_i~=~A_i + A^{\dagger}_i, \qquad i~\in ~I, \tag{1}$$ $$ A_i|0\rangle~=~0, \qquad \langle 0 |A^{\dagger}_i~=~0, \qquad i~\in ~I,\tag{2}$$ where $$ [A_i(t),A_j(t^{\prime})] ~=~ 0, \qquad [A^{\dagger}_i(t),A^{\dagger}_j(t^{\prime})] ~=~ 0, \qquad i,j~\in ~I,\...


16

SYK model provides us with the simplest example of holography which is much easier to study than canonical $AdS_5 \times S^5$ case due to much lower dimensionality. It was the initial motivation for Kitaev to study this model. Here is a set of 2 lectures in which he briefly discusses it. Because of its simplicity, it is easy to consider the thermal and ...


15

The Virasoro algebra is a centrally extended algebra. This means that in every representation, its central element must be represented by the unit operator. Thus (for a non-vanishing central charge) it cannot be fully implemented at the quantum level as a symmetry of the vacuum, otherwise one can get a contradiction of the type $1 |0\rangle = 0 |0\rangle $. ...


14

I recommend you Chapter 5 (page 150+) of the AdS Bible, http://arxiv.org/abs/hep-th/9905111 Concerning your individual questions, which are mostly answered at the beginning of that Chapter, the additional Virasoro generators correspond to bulk coordinate reparametrizations that preserve the metric at infinity, but they do map the ground state to excited ...


14

In order to determine the finite SCT from its infinitesimal version, we need to solve for the integral curves of the special conformal killing field $X$ defined by \begin{align} X(x) = 2(b\cdot x) x - x^2 b. \end{align} I explain why this is equivalent to "integrating" the infinitesimal transformation below. This means we need to solve the differential ...


14

Imagine a QFT with some particle content. Some of these fields will be massless and some massive. For simplicity, consider a massles scalar field and a massive scalar field with mass $M$. If we are working at some energy $E\ll M$, we won't see the massive field (as happened with the Higgs before LHC, for example). This is the IR CFT. Why IR? Because we ...


13

A conformal field theory is a quantum field theory which is invariant under conformal transformations. Due to this invariance, correlation functions must obey linear equations called conformal Ward identities. Conformal blocks are not just solutions of the conformal Ward identities, but actually elements of a particular basis of solutions. Let us focus on ...


13

I) Here we discuss the problem of defining a connection on a conformal manifold $M$. We start with a conformal class $[g_{\mu\nu}]$ of globally$^{1}$ defined metrics $$\tag{1} g^{\prime}_{\mu\nu}~=~\Omega^2 g_{\mu\nu}$$ given by Weyl transformations/rescalings. Under mild assumption about the manifold $M$ (para-compactness), we may assume that there ...


13

The Virasoro algebra is a true symmetry of the theory, in the sense that the action of a conformal field theory is conformally invariant if it exists, and in the sense that the algebra elements map solutions to the equations of motion (quantumly: eigenstates of the Hamiltonian) to solutions of the equations of motion. However, the generators indeed do not ...


12

A conformal transformation is a space-time transformation which leaves the metric invariant up to scale and thus preserves angles. A Weyl transformation actively scales the metric. More formally: Let $M, N$ be two manifolds with inner products $g, h$ and coordinates $x=(x^i), y=(y^j)$ respectively. A map $f:M\rightarrow N$ is called conformal if there ...


12

From the point of view of non-linear dynamics where self-similarity plays an important role if the attractor is a fractal I would say that the difference is one between continuous and discrete transformations. A self-similar transformation like the one producing the Cantor set or the Sierpinski triangle proceeds by discrete stages. The fractal which is the ...


12

Let $\sigma^1$ and $\sigma^2$ be real coordinates on $\mathbb R^2$. Using the results on page 33, we find that \begin{align} \partial_zv^z &= \frac{1}{2}(\partial_1 -i\partial_2)(v^1 + iv^2) = \frac{1}{2}(\partial_1v^1 + i\partial_1v^2 - i\partial_2v^1 + \partial_2v^2) \\ \partial_{\bar z}v^{\bar z} &= \frac{1}{2}(\partial_1 +i\...


12

The precise statement should be that massless fields in conformal field theories in 3+1 dimensions are necessarily free. This result was first proved by Buchholz and Fredenhagen. There are two modern proofs of this fact, one by Steven Weinberg (please see arXiv: hep-th/1210.3864v1) and the other by Yoh Tanimoto in the framework of algebraic quantum field ...


12

Let $$\begin{align} \overline{\mathbb{R}^{p,q}}~~:=&~~\left\{y\in \mathbb{R}^{p+1,q+1}\backslash\{0\}\mid \eta^{p+1,q+1}(y,y)=0\right\}/\mathbb{R}^{\times} \cr ~~\subseteq &~~ \mathbb{P}_{p+q+1}(\mathbb{R})~~\equiv~~(\mathbb{R}^{p+1,q+1}\backslash\{0\})/\mathbb{R}^{\times}, \qquad \mathbb{R}^{\times}~~\equiv~~\mathbb{R}\backslash\{0\}, \end{align}\...


12

I am no expert, but here is what I understand about classifying CFTs in two dimensions (so I will only try to answer your first question, I do not know enough to tackle the others): The defining property of a CFT is that the primary fields are invariant under the transformations generated by the Virasoro algebra $\mathrm{Vir}_c$ (or rather the sum of the ...


11

Whether the conformal symmetry is local or global depends on the theory! More precisely, the symmetry that may be local is not really conformal symmetry but ${\rm diff}\times {\rm Weyl}$. For example, in all the CFTs we use in the AdS/CFT correspondence, for example the famous ${\mathcal N}=4$ gauge theory in $d=4$, the conformal symmetry is global – and, ...


10

Conformal field theories do not have a mass-gap, which is one of the assumptions [for the strong conclusions of non-mixing of Poincare spacetime symmetries vs internal symmetries] of the Coleman-Mandula no-go theorem. Similarly, for its superversion: the Haag-Lopuszanski-Sohnius no-go theorem. [In the supercase, the Poincare algebra is replaced with the ...


10

The proper definition of an (unbroken) conformal field theory is that all scalar n-point functions (or equivalently, their generating functional, a kind of partition function) remain unchanged when each field (including the metric, if it is a field) transforms according to a representation of the conformal group and the volume element (if the metric is not a ...


9

Not so simple. The equation $Z_{QG}=Z_{CFT}$ has to be interpreted correctly. AdS has a (conformal) boundary at space-like infinity and in order to define Quantum Gravity in AdS one has to supply boundary conditions on this conformal boundary. $Z_{QG}=Z_{CFT}$ is really a dictionary telling us which boundary conditions to choose at space-like infinity in ...


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