representation of conformal group

2. In P. Di Francesco, P. Mathieu, D. Snchal they fix the generators of the conformal group acting on a scalar field by somewhat arbitrarily defining. Φ′(x) = Φ(x) − iωaGaΦ(x) Φ ′ (x) = Φ (x) − i ω a G a Φ (x) and by arbitrary I mean the sign. The "full" transformation would then be given by the exponentiation.

In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.. Several specific conformal groups are particularly important: The conformal orthogonal group.If V is a vector space with a quadratic form Q, then the ...

1 A linear representation of the conformal group We wish to find a linear representation of the conformal group. On flat spacetime, in Cartesian coordinates, we can represent the action of the conformal group as the set of transformations x˜a = La bx b x˜a = xa +ba x˜a = lxa x˜a = xa +x2ca 1+2c axa +c2x2 where h cdL c a L d b =h ab so ...

The Conformal Group Definition 2.1. The conformal group Conf(Rp,q) is the connected component con-taining the identity in the group of conformal diffeomorphisms of the conformal compactification of Rp,q. In this definition, the group of conformal diffeomorphisms is considered as a topo-logical group with the topology of compact convergence ...

eigenvalue on an irreducible representation. We work in Euclidean signature, but generalizations are easy to nd. The conventions are the same as Simmons-Du ns notes. 1 Conformal Algebra The conformal group in ddimensions is composed of 1 dilation operator D, dtranslations P i, dspecial conformal transformations K i and d(d 1) 2 rotations J

group and by doing so will discover a surprise, namely that the addition of scale transformations allows for a fourth kind of transformations to be added to the group structure. 3.1 Requirements for conformal invariance By de nition, conformal transformations only preserve angles ( gure 2) (un-like Lorentz transformations which also keep lengths).

replace origin with conformal representation of the point . Unification L7 S13 In conformal geometric algebra we can use rotors to perform translations and dilations, as well as rotations ... generate the conformal group Reflect the conformal vector in e The is the result of inverting space in the origin. Can translate to invert about any point ...

Conformalization adds two new degrees of freedom to the representation, thus adding an internal symmetry \(\SO(1,1)\) (the dilation), together with one translation and one conformal translation (the two sets of null rotations) for each existing degree of freedom. Subsection 8.5.2 \(E_7\) as the Conformal Group of \(E_6\)

representation {j 1,j 2) oϊ 5L(2, C). The dimension d of the field (see (58), d= — / in the notation of Ref. [1]) is such that n = d-\-j 1 +/ 2 is an integer for the discrete series. Projective representations of the conformal group or unitary representations of the infinitesimal conformal group are

The Conformal group contains the Poincare group. Typically, if you take a representation of a group and then look at it as a representation of a subgroup, the representation will be reducible.

The Conformal Group Revisited. J.-F. Pommaret. Since 100 years or so, it has been usually accepted that the " conformal group " could be defined in an arbitrary dimension n as the group of transformations preserving a non degenerate flat metric up to a nonzero invertible point depending factor called " conformal factor ". However, when n > 2 ...

1. Different non-linear representations of the conformal group! Weyl and DBI! 2. Complicated field redefinition maps Weyl Galileons into DBI Galileons! 3. S-matrix equivalent theories, but superluminality…! 4. Inequivalent when the whole series of operators matters! 5. The symmetry breaking pattern is not enough. How general?! 6.

I work with Francesco's Conformal Field Theory. He stats in equation 4.30 that. eixρPρKμe − ixρPρ = Kμ + 2xμD − 2xνLμν + 2xμ(xνPν) − x2Pμ. Exactly here I have my problem. He derives that using the Hausdorff Formula and the commutation relations of the conformal generators. But I would argue that.

Hence, in particular, it follows that the group of automorphisms of $ K $( which is the same as the group of conformal transformations of any Riemannian metric subordinate to $ K $) is a Lie group of dimension $ \leq ( n + 1 ) ( n + 2 ) / 2 $, while the isotropy representation of its stationary subgroup in the tangent space of the second order ...

Conformal Field Theory (CFT) is a branch of physics with origins in solvable lattice models and string theory. But the mathematics that it has inspired has applications in pure mathematics in modular forms, representation theories of various infinite-dimensional Lie algebras and vertex algebras, Monstrous Moonshine, geometric Langlands theory ...

Conformalization adds two new degrees of freedom to the representation, thus adding an internal symmetry \(\SO(1,1)\) (the dilation), together with one translation and one conformal translation (the two sets of null rotations) for each existing degree of freedom.

The Lie algebra of the conformal group has the following representation: [2] ... The largest possible global symmetry group of a non-supersymmetric interacting field theory is a direct product of the conformal group with an internal group. [4] Such theories are known as conformal field theories. This section needs expansion.

The layout of the paper is as follows. We rewrite the conformal algebra in terms of the orthonormal basis of SO∗(d + 2), the complexification of the conformal group in d dimensions, in section 2. In section 3 we construct the characters of any positive energy unitary irreducible representation of the conformal group.

Field representations of the conformal group with continuous mass spectrum. W. Rühl. Physics, Mathematics. 1973. The discrete series of the conformal groupSU (2, 2) is realized on a Hilbert space of holomorphic functions over a bounded domain or the field theoretic tube domain. The boundary values of these….

While the theory of representations of finite dimensional groups such as the ones Weyl studied in 1925-6 was a well-developed part of mathematics by the 1960s, little was known about the representations of infinite dimensional groups such as the group of conformal transformations in two dimensions. Without some restrictive condition on the ...

Minimal representations of reductive groups G are the 'smallest' infinite dimensional irreducible unitary representations.. The Weil (metaplectic, oscillator, the Segal-Shale-Weil, harmonic) representation, known by a prominent role in number theory, consists of two minimal representations of the metaplectic group \(Mp(n, \mathbb{R})\).The minimal representation of a conformal group SO ...

The first equality for translations is true for any field in general. Since translations form an Abelian group, their irreducible representations are one-dimensional. Thus, under a translation, it is generally true for an irreducible field (for any spin) $\Phi'(x+a) = \Phi(x)$. Of course, the situation changes when the Lorentz group is considered.

The conformal group is defined for any spacetime you want. The conformal group of d-dimensional Euclidean space, which has isometry group SO(d), is SO(d+1,1). The conformal group of d+1 dimensional Minkowski space, whose isometry group is SO(d,1), is SO(d+1,2). The defining property of the conformal group is that its the set of transformations ...