Conformal invariance

Group theory says a system is conformally invariant if it has the same form in the new variables here, that is... 

II. Conformally Invariant Ansatzes for an Arbitrary Vector Field... 

To the best of our knowledge, the first paper devoted to symmetry reduction of the 57/(2) Yang-Mills equations in Minkowski space has been published by Fushchych and Shtelen [27] (see also Ref. 21). They use two conformally invariant ansatzes in order to perform reduction of Eqs. (1) to systems of ordinary differential equations. Integrating the latter yields several exact solutions of Yang-Mills equations (1). 

CONFORMALLY INVARIANT ANSATZES FOR AN ARBITRARY VECTOR FIELD... 

In this section we describe the general approach to constructing conformally invariant ansatzes applicable to any (linear or nonlinear) system of partial differential equations, on whose solution set a linear covariant representation of the conformal group 0(1,3) is realized. Since the majority of the equations of the relativistic physics, including the Klein-Gordon-Fock, Maxwell, massless Dirac, and Yang-Mills equations, respect this requirement, they can be handled within the framework of this approach. 

Now we turn to the problem of constructing conformally invariant ansatzes that reduce systems of partial differential equations invariant under the group C(1,3) to systems of ordinary differential equations. 

Classification of inequivalent subalgebras of the algebras p(l,3), p(1.3), c(l,3) within actions of different automorphism groups [including the groups P(l, 3), P(l, 3) and 0(1,3)] is already available [30]. Since we will concentrate on conformally invariant systems, it is natural to restrict our disscussion to the classification of subalgebras of c(l, 3) that are inequivalent within the action of the conformal group 0(1, 3). 

Summarizing we conclude that the problem of constructing conformally invariant ansatzes reduces to finding the fundamental solution of the system of linear partial differential equations (33) and particular solutions of first-order systems of nonlinear partial differential equations (39). 

To obtain the full description of conformally invariant ansatzes it suffices to consider the subalgebras Cj, (j = 1,2,..., 14) listed in Assertion 3. 

Conformally invariant ansatzes for the Yang-Mills field, that reduce equations... 

Note that in contrast to the case of the nonlinear Dirac equation, it is not possible to construct the general solutions of the reduced systems (59)-(61). For this reason, we give whenever possible their particular solutions, obtained by reduction of systems of equations in question by the number of components of the dependent function. Let us emphasize that the miraculous efficiency of the t Hooft ansatz [5] for the Yang-Mills equations is a consequence of the fact that it reduces the system of 12 differential equations to a single conformally invariant wave equation. 

Thus, to get the full description of conformally invariant solutions of the Maxwell equations, it suffices to consider the following subalgebras of the conformal algebra c(l,3) (note, that we have also made use of the discrete symmetry group in order to simplify their basis elements) ... 

After some algebra, we obtain the following form of the conformally invariant ansatz for the Maxwell fields ... 

Furthermore, the general method presented in this chapter applies directly to solving the full Maxwell equations with currents. It can also be used to construct exact classical solutions of Yang-Mills equations with Higgs fields and their generalizations. Generically, the method developed in this chapter can be efficiently applied to any conformally invariant wave equation, on the solution set of which a covariant representation of the conformal algebra in Eq. (15) is realized. 

The main idea is as follows. Let us consider the plane in which our chain is placed as a complex one, z = x + iy. (z = z(x, >)) and let us find the conformal transformation, z = z( ), of the plane z with the obstacle to the Riemann surface, = + b], which does not contain an obstacle (such a transformation means the transfer to the covering space). Due to the conformal invariance of Brownian motion1, in the covering space a random process will be obtained corresponding to the initial one on the plane z but without any topological constraints. 

Conformal invariance of random walk means that after the conformal transformation this process will be random again. 

Conformational invariance means that molecular descriptor values are independent of the conformational changes in molecules. Conformations of molecules are the different atom dispositions in the 3D space, i.e. configurations that flexible molecules can assume without any change to their connectivity. Usually interest in different conformations of a molecule is related to those conformations where the total energies are relatively close to the minimum energy, i.e. within a cut-off energy value of some kcal/mol. 

Molecular descriptors can be devided into four classes according to their conformational invariance degree as suggested by Charton [Charton, 1983] ... 

It should be noted that some invariance properties such as invariance to atom numbering and roto-translations are mandatory for molecular descriptors used in QSAR/ QSPR modelling in several cases, chemical invariance is required, particularly when dealing with a series of compounds with different substituents moreover, conformational invariance is closely dependent on the considered problem. 

Conformational-Independent Chirahty Code chirality descriptors ( Chirality Codes) conformational invariance molecular descriptors (0 invariance properties of molecular... 

As a final point of this section, we return to tricritical phenomena in d = 2 dimensions. The tricritical exponents are known exactly from conformal invariance (Cardy, 1987). For the Ising case, the results are (Pearson, 1980 Nienhuis, 1982)... 

Also the tricritical 3-state Potts exponents (for a phase diagram, see fig. 28c) can be obtained from conformal invariance (Cardy, 1987). But in this case the standard Potts critical exponents are related to an exactly solved hard core model, namely the hard hexagon model (Baxter, 1980), and not the tricritical ones. The latter have the values crt = 5/6, = 1/18, Yt = 19/18, <5t = 20, ut = 7/12, rjt = 4/21,

tricritical exponents coincide (den Nijs, 1979). 

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