Conformal group

 ( Conformal Geometry ) 

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 conformal orthogonal group is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V
• :$Q(Tx)\; =\; \backslash lambda^2\; Q(x)$

For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations.

• The conformal group of the sphere is generated by the inversions in circles. This group is also known as the Möbius group.
• In Euclidean space E ⁿ, , the conformal group is generated by inversions in .
• In a pseudo-Euclidean space E ^{p, q}, the conformal group is .
  (2025). 9780191085789, Oxford University Press. ISBN 9780191085789

All conformal groups are .

---

Angle analysis In Euclidean geometry one can expect the standard circular angle to be characteristic, but in pseudo-Euclidean space there is also the hyperbolic angle. In the study of special relativity the various frames of reference, for varying velocity with respect to a rest frame, are related by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities. Thus, they are conformal transformations with respect to the hyperbolic angle.

A method to generate an appropriate conformal group is to mimic the steps of the Möbius group as the conformal group of the ordinary complex plane. Pseudo-Euclidean geometry is supported by alternative complex planes where points are split-complex numbers or . Just as the Möbius group requires the Riemann sphere, a compact space, for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping. Nevertheless, the conformal group in each case is given by linear fractional transformations on the appropriate plane. Tsurusaburo Takasu (1941) "Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie", 2, Japan Academy 17(8): 330–8, link from Project Euclid,

---

Mathematical definition Given a (Pseudo-)Riemannian manifold $M$ with conformal class $g$, the conformal group $\backslash text\{Conf\}(M)$ is the group of from $M$ to itself.

More concretely, this is the group of angle-preserving smooth maps from $M$ to itself. However, when the signature of $g$ is not definite, the 'angle' is a hyper-angle which is potentially infinite.

For Pseudo-Euclidean space, the definition is slightly different.

$\backslash text\{Conf\}(p,q)$ is the conformal group of the manifold arising from conformal compactification of the pseudo-Euclidean space $\backslash mathbf\{E\}^\{p,\; q\}$ (sometimes identified with $\backslash mathbb\{R\}^\{p,q\}$ after a choice of orthonormal basis). This conformal compactification can be defined using $S^p\backslash times\; S^q$, considered as a submanifold of null points in $\backslash mathbb\{R\}^\{p+1,\; q+1\}$ by the inclusion $(\backslash mathbf\{x\},\; \backslash mathbf\{t\})\backslash mapsto\; X\; =\; (\backslash mathbf\{x\},\; \backslash mathbf\{t\})$ (where $X$ is considered as a single spacetime vector). The conformal compactification is then $S^p\backslash times\; S^q$ with 'antipodal points' identified. This happens by projective space the space $\backslash mathbb\{R\}^\{p+1,q+1\}$. If $N^\{p,q\}$ is the conformal compactification, then $\backslash text\{Conf\}(p,q)\; :=\; \backslash text\{Conf\}(N^\{p,q\})$. In particular, this group includes inversion of $\backslash mathbb\{R\}^\{p,q\}$, which is not a map from $\backslash mathbb\{R\}^\{p,q\}$ to itself as it maps the origin to infinity, and maps infinity to the origin.

---

Lie algebra of the conformal group For Pseudo-Euclidean space $\backslash mathbb\{R\}^\{p,q\}$, the Lie algebra of the conformal group is given by the basis $\backslash \{M\_\{\backslash mu\backslash nu\},\; P\_\backslash mu,\; K\_\backslash mu,\; D\backslash \}$ with the following commutation relations:

(1997). 9780387947853, Springer. ISBN 9780387947853

and with all other brackets vanishing. Here $\backslash eta\_\{\backslash mu\backslash nu\}$ is the Minkowski metric.

In fact, this Lie algebra is isomorphic to the Lie algebra of the Lorentz group with one more space and one more time dimension, that is, $\backslash mathfrak\{conf\}(p,q)\; \backslash cong\; \backslash mathfrak\{so\}(p+1,\; q+1)$. It can be easily checked that the dimensions agree. To exhibit an explicit isomorphism, define It can then be shown that the generators $J\_\{ab\}$ with $a,\; b\; =\; -1,\; 0,\; \backslash cdots,\; n\; =\; p+q$ obey the Lorentz algebra relations with metric $\backslash tilde\; \backslash eta\_\{ab\}\; =\; \backslash operatorname\{diag\}(-1,\; +1,\; -1,\; \backslash cdots,\; -1,\; +1,\; \backslash cdots,\; +1)$.

---

Conformal group in two spacetime dimensions For two-dimensional Euclidean space or one-plus-one dimensional spacetime, the space of conformal symmetries is much larger. In physics it is sometimes said the conformal group is infinite-dimensional, but this is not quite correct as while the Lie algebra of local symmetries is infinite dimensional, these do not necessarily extend to a Lie group of well-defined global symmetries.

For spacetime dimension $n\; >\; 2$, the local conformal symmetries all extend to global symmetries. For $n\; =\; 2$ Euclidean space, after changing to a complex coordinate $z\; =\; x\; +\; iy$ local conformal symmetries are described by the infinite dimensional space of vector fields of the form $$l\_n\; =\; -z^\{n+1\}\backslash partial\_z.$$ Hence the local conformal symmetries of 2d Euclidean space is the infinite-dimensional Witt algebra.

---

Conformal group of spacetime In 1908, Harry Bateman and Ebenezer Cunningham, two young researchers at University of Liverpool, broached the idea of a conformal group of spacetime They argued that the kinematics groups are perforce conformal as they preserve the quadratic form of spacetime and are akin to orthogonal transformations, though with respect to an isotropic quadratic form. The liberties of an electromagnetic field are not confined to kinematic motions, but rather are required only to be locally proportional to a transformation preserving the quadratic form. Harry Bateman's paper in 1910 studied the Jacobian matrix of a transformation that preserves the light cone and showed it had the conformal property (proportional to a form preserver).

Warwick, Andrew (2025). 9780226873756, University of Chicago Press. . ISBN 9780226873756

Bateman and Cunningham showed that this conformal group is "the largest group of transformations leaving Maxwell’s equations structurally invariant."Robert Gilmore (1994) 1974 Lie Groups, Lie Algebras and some of their Applications, page 349, Robert E. Krieger Publishing The conformal group of spacetime has been denoted Boris Kosyakov (2007) Introduction to the Classical Theory of Particles and Fields, page 216, Springer books via Google Books

Isaak Yaglom has contributed to the mathematics of spacetime conformal transformations in split-complex and .Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, Springer, , Since split-complex numbers and dual numbers form rings, not fields, the linear fractional transformations require a projective line over a ring to be bijective mappings.

It has been traditional since the work of Ludwik Silberstein in 1914 to use the ring of to represent the Lorentz group. For the spacetime conformal group, it is sufficient to consider linear fractional transformations on the projective line over that ring. Elements of the spacetime conformal group were called spherical wave transformations by Bateman. The particulars of the spacetime quadratic form study have been absorbed into Lie sphere geometry.

Commenting on the continued interest shown in physical science, A. O. Barut wrote in 1985, "One of the prime reasons for the interest in the conformal group is that it is perhaps the most important of the larger groups containing the Poincaré group."A. O. Barut & H.-D. Doebner (1985) Conformal groups and Related Symmetries: Physical Results and Mathematical Background, Lecture Notes in Physics #261 Springer books, see preface for quotation

---

See also

• Conformal map
• Conformal symmetry

---

Further reading

• S. Kobayashi (1972). 9783540586593, Springer. ISBN 9783540586593
• .
• Peter Scherk (1960) "Some Concepts of Conformal Geometry", American Mathematical Monthly 67(1): 1−30
• Martin Schottenloher, The conformal group, chapter 2 of A mathematical introduction to conformal field theory, 2008 ( pdf)
• Conformal Group in nLab

Post Comment