动量空间方法研究共形场论，为宇宙学关联函数提供了新视角，并更好地揭示了其与散射振幅之间的潜在联系。本论文探讨积分表示与移位算子之间的相互作用。最近，一般标量算子 n 点函数被表示为具有 (n−1)-单纯形拓扑的 Feynman 积分，其中包含动量空间交叉比的任意函数。我们证明，该积分的图多项式都可以用单纯形 Laplacian 矩阵的一阶和二阶子式表示。通过计算相应电网节点间的有效电阻，我们利用 Cayley–Menger 矩阵的行列式和一阶子式找到了逆参数化。这些参数化揭示了可表示为行列式的新型移位算子族，它们将时空维数相差二维的 n 点函数联系起来。此外，它们使共形 Ward 恒等式的有效性能够被直接确立，而无需借助点数递推。随后，我们通过 Gelfand、Kapranov 与 Zelevinsky 提出的一类多变量超几何函数，分析共形积分和 Feynman 积分的表示。除其他优势外，该形式体系能够系统构建被称为“创生”算子的高度非平凡权重移位算子。我们讨论这些算子，强调它们与特殊参数值下出现的谱奇点的密切联系，以及它们与被积函数 Newton 多面体的关系。通过这些方法，我们构建了新的权重移位算子，将不同算子维数和时空维数的接触 Witten 图，以及具有纯非导数顶点的交换图联系起来。
英文摘要：
A momentum-space approach to conformal field theory offers a new perspective on cosmological correlators and better reveals the underlying connections to scattering amplitudes. This thesis explores the interplay between integral representations and shift operators.
A representation for the general $n$-point function of scalar operators was recently proposed in the form of a Feynman integral with the topology of an $(n-1)$-simplex, featuring an arbitrary function of momentum-space cross ratios. We show the graph polynomials for this integral can all be expressed in terms of the first and second minors of the Laplacian matrix for the simplex. Computing the effective resistance between nodes of the corresponding electrical network, an inverse parametrisation is found in terms of the determinant and first minors of the Cayley-Menger matrix. These parametrisations reveal new families of shift operators expressible as determinants that connect $n$-point functions in spacetime dimensions differing by two. Furthermore, they enable the validity of the conformal Ward identities to be established directly without recourse to recursion in the number of points.
We then analyse the representation of conformal and Feynman integrals through a class of multivariable hypergeometric functions proposed by Gelfand, Kapranov \& Zelevinsky. Among other advantages, this formalism enables the systematic construction of highly non-trivial weight-shifting operators known as ``creation'' operators. We discuss these operators emphasising their close connection to the spectral singularities that arise for special parameter values, and their relationship to the Newton polytope of the integrand. Via these methods we construct novel weight-shifting operators connecting contact Witten diagrams of different operator and spacetime dimensions, as well as exchange diagrams with purely non-derivative vertices.
In this thesis we study the momentum space approach to the solution of the CWI's of CFT's in higher dimensions. Our work's goal is to illustrate the essential steps needed to build tensor correlators starting from the scalar solutions, for 3-point functions. In the case of 4-point functions, our attention is centred around scalar correlators for which the CWI's are sufficient to isolate the unique solution if we enhance the symmetry with the addition of a dual conformal symmetry. Dual conformal symmetry in momentum space is obtained once the momentum variables are rewritten in a dual form, as …
I investigate the higher-derivative conformal theory which shows how the Nambu-Goto and Polyakov strings can be told apart. Its energy-momentum tensor is conserved, traceless but does not belong to the conformal family of the unit operator. To implement conformal invariance in this case, I develop the new technique that explicitly accounts for the quantum equation of motion and results in singular products. I show that the conformal transformations generated by such a nonprimary energy-momentum tensor form a Lie algebra with a central extension which in the path-integral formalism gives a loga…
Tractor Calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian ambient space. For dimension d conformally flat manifolds we show that the (d+2)-dimensional Fefferman--Graham ambient space corresponds to the momentum space of a massless scalar field. Hence on the one hand the null cone parameterizes physical momentum excitations, while on the other hand, null rays correspond to points in the underlying conformal manifold. T…
We construct conformal three-point functions in momentum space with a general tensor and conserved currents of spin $1$ and $2$. While conformal correlators in momentum space have been studied especially in the connection with cosmology, correlators involving a tensor of general spin and scaling dimension have not been studied very much yet. Such a direction is unavoidable when we go beyond three-point functions because general tensors always appear as an intermediate state. In this paper, as a first step, we solve the Ward-Takahashi identities for correlators of a general tensor and conserved…
We present a Feynman integral representation for the general momentum-space scalar $n$-point function in any conformal field theory. This representation solves the conformal Ward identities and features an arbitrary function of $n(n-3)/2$ variables which play the role of momentum-space conformal cross-ratios. It involves $(n-1)(n-2)/2$ integrations over momenta, with the momenta running over the edges of an $(n-1)$-simplex. We provide the details in the simplest non-trivial case (4-point functions), and for this case we identify values of the operator and spacetime dimensions for which singula…
We investigate the structure of the constraints on three-point correlation functions emerging when conformal invariance is imposed in momentum space and in arbitrary space-time dimensions, presenting a derivation of their solutions for arbitrary scalar operators. We show that the differential equations generated by the requirement of symmetry under special conformal transformations coincide with those satisfied by generalized hypergeometric functions (Appell's functions). Combined with the position space expression of this correlator, whose Fourier transform is given by a family of generalized…
General principles of quantum field theory imply that there exists an operator product expansion (OPE) for Wightman functions in Minkowski momentum space that converges for arbitrary kinematics. This convergence is guaranteed to hold in the sense of a distribution, meaning that it holds for correlation functions smeared by smooth test functions. The conformal blocks for this OPE are conceptually extremely simple: they are products of 3-point functions. We construct the conformal blocks in 2-dimensional conformal field theory and show that the OPE in fact converges pointwise to an ordinary func…
Energy momentum tensors of higher-derivative free scalar conformal field theories in flat spacetime are discussed. Two algorithms for the computation of energy momentum tensors are described, which accomplish different goals: the first is brute-force and highlights the complexity of the energy momentum tensors, while the second displays some features of their geometric origin as variations of Weyl invariant curved-space actions. New compact expressions for energy momentum tensors are given and specific obstructions to defining them as conformal primary operators in some spacetime dimensions ar…
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli（意大利国家核物理研究所（INFN）那不勒斯分部）；University of Southampton（南安普顿大学）；Indian Institute of Technology Indore（印度理工学院印多尔分校）
The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension $d \geq 3$ (including non-integer $d$). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example,…
All conformal correlation functions of 3 scalar primary operators are constructed in an axiomatic way, relying only on conformal symmetry and causality. The construction makes use of the R-product and its analyticity properties in Minkowski momentum space. The R-product is completely determined by a system of partial differential equations (the Ward identities for conformal transformations) and a boundary condition (permutation symmetry), up to an OPE coefficient. Wightman functions and time-ordered products are then derived from the R-product using operator identities, and the Schwinger funct…
We investigate the mapping of conformal correlators and of their anomalies from configuration to momentum space for general dimensions, focusing on the anomalous correlators $TOO$, $TVV$ - involving the energy-momentum tensor $(T)$ with a vector $(V)$ or a scalar operator ($O$) - and the 3-graviton vertex $TTT$. We compute the $TOO$, $TVV$ and $TTT$ one-loop vertex functions in dimensional regularization for free field theories involving conformal scalar, fermion and vector fields. Since there are only one or two independent tensor structures solving all the conformal Ward identities for the $…
In this work we investigate the matrix elements of the energy-momentum tensor for massless on-shell states in four-dimensional unitary, local, and Poincaré covariant quantum field theories. We demonstrate that these matrix elements can be parametrised in terms of covariant multipoles of the Lorentz generators, and that this gives rise to a form factor decomposition in which the helicity dependence of the states is factorised. Using this decomposition we go on to explore some of the consequences for conformal field theories, deriving the explicit analytic conditions imposed by conformal symmetr…
Analytic Continuation of Conformal Integrals in Momentum Space
Max Planck Institute for Physics, Werner-Heisenberg-Institut（马克斯·普朗克物理研究所）；Max Planck-IAS-NTU Center for Particle Physics, Cosmology and Geometry（马克斯·普朗克粒子物理、宇宙学与几何中心）
In the framework of classical field theory, we first review the Noether theory of symmetries, with simple rederivations of its essential results, with special emphasis given to the Noether identities for gauge theories. Will this baggage on board, we next discuss in detail, for Poincaré invariant theories in flat spacetime, the differences between the Belinfante energy-momentum tensor and a family of Hilbert energy-momentum tensors. All these tensors coincide on shell but they split their duties in the following sense: Belinfante's tensor is the one to use in order to obtain the generators of …
In this paper we consider matter fields in a gravitational background in order to compute the breaking of the conformal current at one-loop order. Standard perturbative calculations of conformal symmetry breaking expressed by the non-zero trace of the energy-momentum tensor have shown that some violating terms are regularization dependent, which may suggest the existence of spurious breaking terms in the anomaly. Therefore, we perform the calculation in a momentum space regularization framework in which regularization dependent terms are judiciously parametrized. We compare our results with th…
We derive and analyze the conformal Ward identities (CWI's) of a tensor 4-point function of a generic CFT in momentum space. The correlator involves the stress-energy tensor $T$ and three scalar operators $O$ ($TOOO$). We extend the reconstruction method for tensor correlators from 3- to 4-point functions, starting from the transverse traceless sector of the $TOOO$. We derive the structure of the corresponding CWI's in two different sets of variables, relevant for the analysis of the 1-to-3 (1 graviton $\to$ 3 scalars) and 2-to-2 (graviton + scalar $\to$ two scalars) scattering processes. The …
We discuss specific hypergeometric solutions of the conformal Ward identities (CWI's) of scalar 4-point functions of primary fields in momentum space, in $d$ spacetime dimensions. We determine such solutions using various dual conformal ansätze (DCA's). We start from a generic dual conformal correlator, and require it to be conformally covariant in coordinate space. The two requirements constrain such solutions to take a unique hypergeometric form. They describe correlators which are at the same time conformal and dual conformal in any dimension. These specific ansätze also show the existence …
Starting from the well-known expression for the trace anomaly we derive the $T\cdot T$ operator product expansion of the energy-momentum tensor in 2D conformal theories defined in the upper halfplane $without$ making use of the additional condition of no energy-momentum flux across the boundary. The OPE turns out to be the same as in the absence of the boundary. For this result it is crucial that the trace anomaly is proportional to the Gauß-Bonnet density. Some relations to the $σ$ - model approach for open strings are discussed.
We find the general solution of the conformal Ward identities for scalar $n$-point functions in momentum space and in general dimension. The solution is given in terms of integrals over $(n-1)$-simplices in momentum space. The $n$ operators are inserted at the $n$ vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators whe…
The energy-momentum tensor for a massless conformally coupled scalar field in de Sitter spacetime in the presence of a couple curved branes is investigated. We assume that the scalar field satisfies the Robin boundary condition on the surfaces. Static de Sitter space is conformally related to the Rindler space, as a result we can obtain vacuum expectation values of energy-momentum tensor for conformally invariant field in static de Sitter space from the corresponding Rindler counterpart by the conformal transformation.
Our previous work [1] constructed, in three-dimensional momentum space, a manifestly crossing symmetric basis for scalar conformal four-point functions, based on the factorization property proposed by Polyakov. This work extends this construction to general dimensional conformal field theory. To facilitate the treatment of symmetric traceless tensors, we exploit techniques of spherical harmonics in general dimensions.
The properties of the Das-Popowicz Moyal momentum algebra that we introduce in hep-th/0207242 are reexamined in details and used to discuss some aspects of integrable models and 2d conformal field theories. Among the results presented, we setup some useful convention notations which lead to extract some non trivial properties of the Moyal momentum algebra. We use the particular sub-algebra sl(n)-{Sigma}_{n}^{(0,n)} to construct the sl(2)-Liouville conformal model and its sl(3)-Toda extension. We show also that the central charge, a la Feigin-Fuchs, associated to the spin-2 conformal current of…
We construct a crossing symmetric basis for conformal four-point functions in momentum space by requiring consistent factorization. Just as scattering amplitudes factorize when the intermediate particle is on-shell, non-analytic parts of conformal correlators enjoy a similar factorization in momentum space. Based on this property, Polyakov, in his pioneering 1974 work, introduced a basis for conformal correlators which manifestly satisfies the crossing symmetry. He then initiated the bootstrap program by requiring its consistency with the operator product expansion. This approach is complement…
We study the conformally invariant quantum field theory in spaces of even dimension D >= 4. The conformal transformations of current j_μand energy-momentum tensor T_{μν} are examined. It is shown that the set of conformal transformations of particular kind corresponds to the canonical (unlike anomalous) dimensions l_j=D-1 and l_T=D of those fields. These transformations cannot be derived by a smooth transiton from anomalous dimensions. The structure of representations of the conformal group, which correspond to these canonical dimensions, is analyzed, and new expressions for the propagators < …
We study logarithmic conformal field theories (LCFTs) through the introduction of nilpotent conformal weights. Using this device, we derive the properties of LCFT's such as the transformation laws, singular vectors and the structure of correlation functions. We discuss the emergence of an extra energy momentum tensor, which is the logarithmic partner of the energy momentum tensor.
We present an explicit momentum space computation of the four-point function of the energy-momentum tensor in 4 spacetime dimensions for the free and conformally invariant theory of a scalar field. The result is obtained by explicit evaluation of the Feynman diagrams by tensor reduction. We work by embedding the scalar field theory in a gravitational background consistently with conformal invariance in order to derive all the terms the correlator consists of and all the Ward identities implied by the requirements of general covariance and anomalous Weyl symmetry. We test all these identities n…
We study the momentum-space 4-point correlation function of identical scalar operators in conformal field theory. Working specifically with null momenta, we show that its imaginary part admits an expansion in conformal blocks. The blocks are polynomials in the cosine of the scattering angle, with degree $\ell$ corresponding to the spin of the intermediate operator. The coefficients of these polynomials are obtained in a closed-form expression for arbitrary spacetime dimension $d > 2$. If the scaling dimension of the intermediate operator is large, the conformal block reduces to a Gegenbauer po…
The implications of conformal invariance, as relevant in quantum field theories at a renormalisation group fixed point, are analysed with particular reference to results for correlation functions involving conserved currents and the energy momentum tensor. Ward identities resulting from conformal invariance are discussed. Explicit expressions for two and three point functions, which are essentially determined by conformal invariance, are obtained. As special cases we consider the three point functions for two vector and an axial current in four dimensions, which realises the usual anomaly simp…
In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell $F_4$ type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their rela…
Given a conserved and traceless energy-momentum tensor and a conformal Killing vector, one obtains a conserved current. We generalise this construction to superconformal theories in three, four, five and six dimensions with various amounts of supersymmetry by working in the appropriate superspaces.
Theories with generalised conformal structure contain a dimensionful parameter, which appears as an overall multiplicative factor in the action. Examples of such theories are gauge theories coupled to massless scalars and fermions with Yukawa interactions and quartic couplings for the scalars in spacetime dimensions other than 4. Many properties of such theories are similar to that of conformal field theories (CFT), and in particular their 2-point functions take the same form as in CFT but with the normalisation constant now replaced by a function of the effective dimensionless coupling $g$ co…
In holographic models of cosmology based on the (A)dS/CFT correspondence, conformal symmetry is implicit in the dual description of the Universe. Generically, however, one cannot expect the (broken) conformal invariance in the cosmic fluctuations as only the scale invariance is manifest in experiments. Also, in order for the prediction of the holographic models to make sense, the conformal symmetry needs to be broken as the scalar mode of the metric fluctuations becomes pure gauge in the conformal limit. We discuss the improvement ambiguity of the energy-momentum tensor in this context and con…
As it was recently shown, the colour singlet BFKL kernel, taken in Moebius representation in the space of impact parameters, can be written in quasi-conformal shape, which is unbelievably simple compared with the conventional form of the BFKL kernel in momentum space. It was also proved that the total kernel is completely defined by its Moebius representation. In this paper we calculated the difference between standard and quasi-conformal BFKL kernels in momentum space and discovered that it is rather simple. Therefore we come to the conclusion that the simplicity of the quasi-conformal kernel…
Using the dipole framework for QCD at small x in the 1/N_c limit, we derive the expression of the 1 -> p dipole multiplicity density in momentum space. This gives an analytical expression for the 1 -> p QCD Pomeron amplitudes in terms of one-loop integration of effective vertices in transverse momentum. Conformal invariance and a Hilbert space construction for dipole correlation functions are the main tools of the derivation. Relations with conformal field theories in the classical limit are discussed.
For local conformal field theories, it is shown how to construct an expression for the energy-momentum tensor in terms of a Wilsonian effective Lagrangian. Tracelessness implies a single, unintegrated equation which enforces both the Exact Renormalization Group equation and its partner encoding invariance under special conformal transformations.
We find the rules of the conformal transformation for the energetic quantities such as the Einstein energy-momentum complex, the Bergmann-Thomson angular momentum complex, the superenergy tensor, and the angular supermomentum tensor of gravitation and matter. We show that the conformal transformation rules for the matter parts of both the Einstein complex and the Bergmann-Thomson complex are fairly simple, while the transformation rules for their gravitational parts are more complicated. We also find that the transformational rules of the superenergy tensor of matter and the superenergy tensor…
Starting with conformally covariant correlation functions, a sequence of functional representations of the conformal algebra is constructed. A key step is the introduction of representations which involve an auxiliary functional. It is observed that these functionals are not arbitrary but rather must satisfy a pair of consistency equations corresponding to dilatation and special conformal invariance. In a particular representation, the former corresponds to the canonical form of the Exact Renormalization Group equation specialized to a fixed-point whereas the latter is new. This provides a con…
The requirements of conformal invariance for the two point function of the energy momentum tensor in the neighbourhood of a plane boundary are investigated, restricting the conformal group to those transformations leaving the boundary invariant. It is shown that the general solution may contain an arbitrary function of a single conformally invariant variable $v$, except in dimension 2. The functional dependence on $v$ is determined for free scalar and fermion fields in arbitrary dimension $d$ and also to leading order in the $\vep$ expansion about $d=4$ for the non Gaussian fixed point in $ϕ^4…
We present a study of the correlation function of three stress-energy tensors in $d$ dimensions using free field theory realizations, and compare them to the exact solutions of their conformal Ward identities (CWI's) obtained by a general approach in momentum space. The identification of the corresponding form factors is performed within a reconstruction method, based on the identification of the transverse traceless components $(A_i)$ of the same correlator. The solutions of the primary CWI' s are found by exploiting the universality of the Fuchsian indices of the conformal operators and a re…
We present a general study of 3-point functions of conformal field theory in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWI' s), introduced in recent works by Bzowski, McFadden and Skenderis (BMS). We investigate and detail the structure of the CWI's, their non-perturbative solutions and the transition to momentum space, comparing them to perturbation theory by taking QED as an example. We then proceed with an analysis of the $TJJ$ correlator, presenting independent and detailed re-derivations of the conform…
The critical behavior of infinite families of shift symmetric interacting theories with higher derivative kinetic terms (non unitary) is considered. Single scalar theories with shift symmetry are classified according to their upper critical dimensions and studied at the leading non trivial order in perturbation theory. For two infinite families, one with quartic and one with cubic interactions, beta functions, criticality conditions and universal anomalous dimensions are computed. At the order considered, the cubic theories enjoy a one loop non renormalization of the vertex, so that the beta f…
Conformal symmetry has important consequences for strong interactions at short distances and provides powerful tools for practical calculations. Even if the Lagrangians of Quantum Chromodynamics (QCD) and Electrodynamics (QED) are invariant under conformal transformations, this symmetry is broken by quantum corrections. The signature of the symmetry breaking is encoded in the presence of massless poles in correlators involving stress-energy tensors. We present a general study of the correlation functions $\langle TJJ\rangle$ and $\langle TTJJ\rangle$ of conformal field theory (CFT) in the flat…
Conformal group of transformations in the momentum space, consisting of translations $p'_μ=p_μ+h_μ$, rotations $p'_μ=Λ^ν_μp_ν$, dilatation $p'_μ=λp_μ$ and inversion $p'_μ= -M^2p_μ/p^2$ of the four-momentum $p_μ$, is used for the five dimensional generalization of the equations of motion for the interacting massive particles. It is shown, that the ${\cal S}$-matrix of the charged and the neutral particles scattering is invariant under translations in a four-dimensional momentum space $p'_μ=p_μ+h_μ$. In the suggested system of equations of motion, the one-dimensional equations over the fifth coo…
Some results from arguments of research dealt with R. Raczka are exposed and extended. In particular new arguments are brought in favor of the conjecture, formulated with him, that both space-time and momentum may be conformally compactified, building up a compact phase space of automorphism for the conformal group, where conformal reflections determine a convolution between space-time and momentum space which may have consequences of interest for both classical and quantum physics.
I present a Mathematica package designed for manipulations and evaluations of triple-K integrals and conformal correlation functions in momentum space. Additionally, the program provides tools for evaluation of a large class of 2- and 3-point massless multi-loop Feynman integrals with generalized propagators. The package is accompanied by five Mathematica notebooks containing detailed calculations of numerous conformal 3-point functions in momentum space.
In four dimensional unitary scale invariant theories, arguments based on the proof of the a-theorem suggest that the trace of the energy-momentum tensor T vanishes when the momentum is light-like, p^2=0. We show that there exists a local operator O such that the trace is given as T=\partial^2 O, which establishes the equivalence of scale and conformal invariance. We define the operator as O=\partial^{-2} T, and explain why this is a well-defined local operator. Our argument is based on the assumptions that: (1) A kind of crossing symmetry for vanishing matrix elements holds regardless of the e…
We study a series of the Wess-Zumino actions obtained by repeatedly integrating conformal anomalies with respect to the conformal-factor field that appear at higher loops. We show that they arise as physical quantities required to make nonlocal loop correction terms diffeomorphism invariant. Specifically, in a conformally flat spacetime $ds^2=e^{2ϕ}(-dη^2 + d{\bf x}^2)$, we find that effective actions are described in terms of momentum squared expressed as a physical $Q^2 = q^2/e^{2ϕ}$ for $q^2$ measured by the flat metric, which recalls the relationship between physical momentum and comoving …
We present a general study of 3-point functions of conformal field theory (CFT) in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWIs), introduced in recent works. We investigate and detail the structure of the CWIs and their non-perturbative solutions, and compare them to perturbation theory, taking QED and QCD as examples. Exact solutions of CFT's in the flat background limit in momentum space are matched by the perturbative realizations in free field theories, showing that the origin the conformal anomaly is…
The general solution of the conformal Ward identities (CWI's) in momentum space, and their matching to perturbation theory, allows to uncover some specific characteristics of the breaking of conformal symmetry, induced by the anomaly. It allows to compare perturbative features of the 1-particle irreducible (1PI, nonlocal) anomaly action with the prediction of a similar (but exact) nonlocal action identified by the CWI's. The two predictions can be exactly matched at the level of 3-point functions. The analysis of the $TJJ$ and $TTT$ shows that both approaches - based either on 1PI or on the ex…
In Minkowski spacetime it is well-known that the canonical energy-momentum current is involved in the construction of the globally conserved currents of energy-momentum and total angular momentum. For the construction of conserved currents corresponding to (approximate) scale and proper conformal symmetries, however, an improved energy-momentum current is needed. By extending the Minkowskian framework to a genuine metric-affine spacetime, we find that the affine Noether identities and the conformal Killing equations enforce this improvement in a rather natural way. So far, no gravitational dyn…
