Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions

Abstract

Meta-conformal transformations are constructed as sets of time-space transformations which are not angle-preserving but contain time-and space translations, time-space dilatations with dynamical exponent z = 1 and whose Lie algebras contain conformal Lie algebras as sub-algebras. They act as dynamical symmetries of the linear transport equation in d spatial dimensions. For d = 1 spatial dimensions, meta-conformal transformations constitute new representations of the conformal Lie algebras, while for d = 1 their algebraic structure is different. Infinite-dimensional Lie algebras of meta-conformal transformations are explicitly constructed for d = 1 and d = 2 and they are shown to be isomorphic to the direct sum of either two or three centre-less Virasoro algebras, respectively. The form of co-variant two-point correlators is derived. An application to the directed Glauber-Ising chain with spatially long-ranged initial conditions is described.1 In memoriam Vladimir Rittenberg "Whenever you have to do with a structure-endowed entity, try to determine its group of automorphisms. You can expect to gain a deep insight." H. Weyl, Symmetry, Princeton University Press (1952) 1 See [81] and refs. therein for the considerable recent interest into the case r ∈ R d with d > 2.

Meta-Schrödinger Transformations

“…such that for p, q ∈ Z + 1 2 we have the commutator [Y y p ,Y y q ] = (p − q)M p+q . Next, following [28], define a new family of generators…”

“…) . (28) and the normalisation constant G 0 . This form combines aspects of Schrödingerinvariance in the transverse coordinate y and of meta-conformal invariance in the parallel coordinate x.…”

“…[21,24,16,17,32,1] (including its infinite-dimensional extension), which also arises independently in gravitational physics [7,2,3]. On the other hand, the dynamical symmetries of simple ballistic transport equations has recently been identified as the new meta-conformal algebra meta(1, d) [28], whose infinite-dimensional extensions are isomorphic to the direct sum of two Virasoro algebras in d = 1 spatial dimensions and of three Virasoro algebras for d = 2. Again, the non-relativistic limit of these are the conformal Galilean algebras.…”

“…Schrödinger-invariance is realised in quenches to the ordered phase, see [26] for a detailed review. Meta-conformal invariance may be realised if the underlying microscopic dynamics has a directional bias [20,28].…”

Boundedness of meta-conformal two-point functions in one and two spatial dimensions

“…Their solutions are holomorphic in the variables z p (or anti-holomorphic in the variables zp ) [46]. For example, the ortho-conformal two-point function has the well-known form (C 0 is a normalisation constant) C [2] ortho (z 1 , z1 , z 2 , z2 In this work, we are interested in new types of (meta-)conformal invariance [35,44,57] where the implicit hypothesis of holomorphy in the physical coordinates z p , zp (or equivalently t p , r p , with p = 1, . .…”