By Slava Rychkov
This primer develops Conformal box thought (CFT) from scratch, wherein CFT is seen as any conformally-invariant thought that describes a hard and fast element of a renormalization crew circulate in quantum box theory.
The booklet is split into 4 lectures: Lecture 1 addresses the actual foundations of conformal invariance, whereas Lecture 2 examines the limitations imposed through conformal symmetry at the correlation features of neighborhood operators, offered utilizing the so-called projective null cone – a method sometimes called the embedding formalism. In flip, Lecture three makes a speciality of the radial quantization and the operator product growth, whereas Lecture four deals a truly short creation to the conformal bootstrap.
Derived from course-based notes, those lectures are meant as a primary element of access to this subject for grasp and PhD scholars alike.
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Extra info for EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions
The first one is the radial quantization. In this picture the in and out vacuum states sit at 0 and ∞, so that the relation between them is not entirely obvious. The second one is the cylinder. In this picture the in and out states are at the two symmetric ends of the cylinder, and they can be related by the cylinder “time” reflection τ → −τ . 33). We will now present the N-S (North-South pole) quantization picture which is in between the other two. It is almost as efficient as the cylinder, and in addition it takes place in flat space.
Is positive definite, one derives D − 1. e. the constraints we saw above are necessary and sufficient to have unitarity at all levels. 3 Operator Product Expansion (OPE) The idea of OPE is hopefully familiar from the usual QFT: it says that we should be able to replace a product of two local operators, in the limit when they become very close to each other, by a series of operators inserted at the midpoint. This logic still hold in CFT. Moreover, in CFTs the OPE acquires additional and very powerful properties thanks to a connection with the radial quantization.
We will also use the lightcone coordinates X + = X D+2 + X D+1 , X − = X D+2 − X D+1 . 8) In terms of the above, the mostly plus metric η M N in R D+1,1 is D (d X i )2 − d X + d X − . 10) Jμ− = K μ , J+− = D , with μ, ν = 1, . . , D. It is understood that Jμν is antisymmetric under μ ↔ ν. Then one can check that the conformal algebra commutation relations coincide with the S O(D + 1, 1) ones: [JM N , J R S ] = −i(η M R JN S ± permutations) . 11) [Jμ+ , Jν− ] ∝ δμν J+− + δ+− Jμν . 12) One example is Exercise: Check and fix all the constants in the above identification.