2.2 Miscellaneous information

• Conformal group is the largest nontrivial subgroup of diffeomorphisms of $R^d$ (page 6 of 1805.04405)
• In $2d$ ($d>2$) CFT, operator $\mathcal{𝒪}$ with $[K,\mathcal{𝒪}]=0$ is called quasi-primary (primary) (page 7 of 1805.04405)
• CFT spectrum is typically discrete to ensure that thermal partition function is well-defined. (page 8 of 1805.04405) Also, it follows from self-adjointness of Hamiltonian (see page 7 of Commun. math. Phys. 55, 1–28 (1977))
• Examples of nonunitary conformal theories in which the 2pt functions cannot be diagonalized occur in logarithmic CFTs (page 8 of 1805.04405)
• Reflection positivity of correlation functions is called Osterwalder-Schrader reflection positivity (page 11 of 1805.04405)
• Conformal group of Minkowski space is isomorphic to $\mathit{SO}(4,2)∕Z_2\sim \mathit{SU}(2,2)∕Z_4\sim G∕(Z_2\mathit{xZ})$ where $G$ is the universal cover. (page 6 of Commun. math. Phys. 55, 1–28 (1977))
• two and three point functions can be seen as intertwining and Clebsh-Gordon kernels of the conformal group (page 1 of Commun. math. Phys. 40, 15—35 (1975))
• 4+1 and 3+2 de Sitter groups are the only simple Lie groups that can be contracted to the Poincare group (page 1 of J. Math. Phys. 8, 170 (1967))
• Only scalar and spinors can be massless finite-dimensional unitary irrep of conformal group in odd dimensions (when we impose conformal invariance order by order in perturbation theory of free particle). In even dimensions, anything that is (anti)self-dual (chiral) on vector indices are allowed. (Journal of Modern Physics 1989)
• The process of an incoming state becoming excited to a new state after interacting with a (reggeized) graviton is called a tidal excitation (1912.05580)
• Diagonal unitary minimal models are among the few CFT with a Lagrangian description (page 231 of yellow book)
• Conformal Hamiltonian is $H=(P^0+K^0)∕2$ (Commun. math. Phys. 41, 203—234 (1975))
• In a Wightman field theory in Minkowski space, the fields are always assumed to obey the commutation relations $i{\nabla <!--\; FUNCTION\; APPLICATION\; -->}^{\mu }{\varphi }_{\alpha }(x)=[{\varphi }_{\alpha }(x),P^{\mu }]$ with the generators of space time translations $P^{\mu }$. That is, these generators do not act on the indices. This amounts to fixing the choice of basis in index space at different points x relative to each other. This is the natural choice for a theory with Poincare invariance, but is not appropriate for conformal invariance; one should instead, choose conformal hamiltonian to not act on the indices. (page 220 of Commun. math. Phys. 41,203–234 (1975))
• There are physically interesting theories which satisfy all CFT axioms except for the existence of the local stress tensor. Examples include defect and boundary CFTs , and critical points of models with long-range interactions: (see footnote 69 of 1805.04405)
• Central charge typically scales with the number of degrees of freedom. (page 22 of 1805.04405)
• Interacting CFT’s in $d>2$ do not have conserved higher spin currents! (page 23 of 1805.04405)
• In 3d, the three point function of the conserved currents has, generically, three distinct pieces: free boson + free fermion + odd. The odd piece does not appear in free theories and is observed to be present whenever triangular rule is satisfied: $s1<=s2+s3$ (page 11 of 1112.1016)
• 15. 16. In a 3d free fermion theory we do not have a twist one spin zero operator: (page 15 of 1112.1016)
• In 3d, if there is any half integer higher spin current, we should have at least 2 spin-2 conserved current. (page 17 of 1112.1016)
• There are few examples of CFTs in d>2 with exact solutions. They are mostly free theories and MFT’s, but one example is fishnet theories. (page 24 of 1805.04405)
• Free massive fields in AdS are dual to MFT’s in CFT. (page 25 of 1805.04405)
• In AdS, the locality in   AdS length and locality in r«AdS length are conceptually different, and are named as coarse and sharp holographies respectively. The coarse holography is related to color transparency and to energy-radius holography, whereas sharp holography is much more subtle. (0907.0151v3)
• Via holography, a four point CFT correlator can be related to 2-to-2 scattering amplitude; when the operators in the correlators are conformally equivalent to a particular position which corresponds to head-on scattering in AdS, the CFT correlator diverges. A small change in the position of the operators may result in excitation missing in the bulk, hence a rapid drop in the amplitude. This is clear from the bulk locality perspective but the singularity is surprising from pure CFT point of view (and that singularity only emerges in strong-coupling limit) (page 4 of 0907.0151v3)
• In MFT’s, the absence of stress tensor means that the theory is a set of correlators without a notion of causality. From AdS/CFT point of view, this means that one needs to measure the boundary state at every time to reconstruct the bulk state at a single time so there is no holography, consistent with the fact that bulk has no gravity. (see page 10 of 0907.0151v3)
• In AdS calculation, so-called D-function is defined which acts as a building block from which various quartic interaction can be computed. See appendix A of 9903196 for details, and see 0907.0151v3 for similar calculations.
• Similar to the decomposition of a general Hamiltonian into the free Hamiltonian plus an interaction Hamiltonian, i.e. $H=H_0+H_I$, the dilatation operator D of a CFT can be decomposed into two parts; in fact, when studying CFT operators and states with dimension small compared to the central charge $N^2$, we can separate the dilatation operator into $D=D_0+D_I∕N$. (page 3 of 1112.4845 )
• The Symanzik star formula enables passing from integration in coordinate space to integration in mellin space (page 3 of 1107.1504)
• Via AdS/CFT, we know that sum over k-trace operators in the conformal block decomposition reduces to a phase space integral over k-particle states. One might also wonder whether all operators that can be exchanged in the conformal block decomposition are really k-trace operators, and what role is played by the operators dual to unstable particles. The S-Matrix connects in and out states composed of exactly stable particles, and so scattering amplitudes between unstable particles are not well-defined and do not appear in the unitarity relation. The qualitative difference between stable and unstable particles emerges only in the flat space limit, when the original primary operators dual to unstable particles get lost on the sea of multi-trace operators with which they mixes. (page 6 of 1112.4845)
• One can construct double-twist operators explicitly, by taking linear combinations of n $P^2$ and l $P^{\mu }$ acting on $O_1$ and $O_2$. One then fixes the coefficients of the terms by demanding that the resultant combination is annihilated by K and is an eigenfunction of D. One can actually find a recursive relation for such a construction, see page 8 and appendix C of 1112.4845.
• The conformal invariance tells us that we can get $<O_1(x_1)O_2(x_2)>$ from $<O(e)O(-e)>$ for the unit vector e. The stabilizer group SO(1,1)xSpin(d) then restrict that $O_1$ and $O_2$ have same scaling dimension. However, there is the exception: $<O_1(x_1){Õ}_2(x_2)>={\delta }^d(x_1-x_2)$. In other words, two point function of an operator with its shadow is a dirac-delta function, which is consistent with the fact that $<O(e)O(-e)>$ is zero for kronecker delta function, so we do not get the constraint ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2$.
• One can use conglomeration and Mellin amplitudes together to extract information about conformal blocks (page 15 of 1112.4845)
• In AdS, special conformal generators in the flat-space limit become the momentum operators (page 22 1112.4845)
• The usual Dyson series for the S-Matrix can be constructed in AdS using the dilatation operator, and that at first order it gives rise to the prescription for the S-Matrix in terms of the anomalous dimension matrix. (page 30 of 1112.4845)
• The most significant principle behind the simplicity of the Mellin amplitude is unitarity, in the guise of the Operator Product Expansion. Using the convergent OPE, it is possible to express correlation functions involving many operators as a sum over products of correlators involving fewer operators. The Mellin amplitude displays these OPE decompositions of correlation functions as a sum over poles, with residues given by lower-point Mellin amplitudes. These poles are analogous to the multi-particle factorization channels of scattering amplitudes, but in fact they are even more constrained: in unitary CFTs the Mellin amplitude can only have simple poles in a certain region of the real axis. (page 1 of 1111.6972)
• In mellin space, the witten amplitude of a diagram and the same diagram where one of the propagator is removed (made a contact interaction) are related by a mere difference equation! In position space, the fact that these witten diagrams are related follows from bulk Klein-Gordon differential equation where the relation turns into an algebraic one in Mellin space. (page 6 of 1111.6972)
• In Mellin space, tree level AdS amplitudes can be computed diagrammatically. (page 7 of 1111.6972)
• 33. 34. Scanning over ratios of OPE coefficients, in addition to scanning over scaling dimensions, decreases the sizes of islands in numerical bootstrap. (page 4 of 1912.03324)
• For various reasons, the most practical experimental representative of the O(2) universality class is the superfluid transition in Helium along the so called $\lambda -$line. One of the reasons for that is its relative robustness against gravitational effects which allow it to reach very low reduced temperatures, unlike similar experiments. (page 6 of 1912.03324)
• Parity transformation, e.g. $(x_0,x_1,\dots ,x_d)->(x_0,-x_1,\dots ,-x_d)$, is part of the connected group for even d, so is not the appropriate transformation considered in odd d, say in standard d=3 case. Instead, one should consider reflection, e.g. $(x_0,x_1,x_2,\dots ,x_d)->(x_0,-x_1,x_2,\dots ,-x_d)$, which has determinant -1 in all d. For example, one should consider CRT instead of CPT in even d. See page 5 of 1508.04715 for this explanation, and see 1905.01311 for some usage.
• $O(2)\cong <!--\; FUNCTION\; APPLICATION\; -->U(1)⋉Z_2$, hence its irreducible representations compose of irreps of SO(2) upto overall parity. In particular, q=0 irrep of SO(2) decomposes into two irreps under Z. For details ,see page 10 of 1912.03324
• Pure Maxwell theory in $d\ne 4$ dimensions is a unitary scale invariant theory which is not conformally invariant. It is because the lowest dimensional gauge invariant operator, $F_{\mathit{\mu \nu }}$, has a two-point function which is not consistent with two-point function of a primary operator. As it cannot be a descendant either, it is neither a primary nor a descendant hence the theory is not conformally invariant (page 6 of 1101.5385).
• An argument why scale invariance generically imply conformal invariance is as follows: For a scale invariant theory, the trace of stress tensor is a total divergence, i.e. $T={\partial }_{\mu }{k}^{\mu }$ for the virial current k. This equation implies there has to be an operator k which does not receive anomalous dimension, and unless there is an accidental coincidence, only the conserved operators receive no anomalous dimension. But if k is conserved, than the trace of T is zero, hence we actually have conformal invariance. Hence, scale invariant theories are conformally invariant unless there exist a non-conserved vector which receives no anomalous dimension. However, no such theory has been found yet, so the only counter-examples are theories without stress tensor or non-unitary theories (page 1,2 of 1101.5385).
• Neutron Interferometry Experiment to study 2pi rotations and spinors in general.
• One usually writes D(R) for the induced rotations in Hilbert space, such as $|a{>}_R=D(R)|a>$ . Here, D stands for Drehung, rotation for German.
• Gram matrix: Gram matrix $G_{\mathit{ij}}$ of a set of vectors $v_i$ is the inner product of them: $G_{\mathit{ij}}=<v_i,v_j>$
• Classical Reality vs Quantum Reality (from a presentation of Roger Penrose): In CR: One can ask of a system “what is your state?”, and it can correctly answer “My state is X”. In QR: One can only ask to a system “Is your state X?”, and if you are correct, the system replies with certainty “Yes my state is indeed X”.
• A. Einstein: If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding lo this physical quantity. (1935 EPR paper)
• Fixed angle scattering can show time advance → causality controls Regge limit: 0908.0004
• In-in formalism = Schwinger-Keldysh formalism, see 1610.01940, and also Introduction to the Keldysh Formalism by Alex Kamenev
• Gravity and gauge groups do not combine as direct product, but rather as semi-direct product: if a fixed choice of gauge is to be maintained, every coordinate transformation must be accompanied by an electromagnetic gauge transformation. see Dewitt "Quantum Theory of Gravity. I. The Canonical Theory*"