Compliant killing vector field

A conformal killing vector field is a vector field on a semi-Riemannian manifold whose flow is angle-preserving.

The concept of the conformal killing vector field is an extension of the concept of the killing vector field . Conforming killing vector fields scale the metric by a smooth function, while killing vector fields do not scale the metric. The conformal killing vectors are the infinitesimal generators of conformal transformations , which include isometries , but also dilation and special conformal transformations .

definition

A vector field is a conforming Killing vector field if the Lie derivative of the metric is proportional to the metric with respect to ${\displaystyle X}$ ${\displaystyle g}$ ${\displaystyle X}$

${\displaystyle {\mathcal {L}}_{X}g=\Omega \,g.\,.}$

There is a smooth function on the manifold and is called a conformal killing factor . In the expression relating to the Levi-Civita context , this means ${\displaystyle \Omega }$

${\displaystyle g(\nabla _{Y}X,Z)+g(Y,\nabla _{Z}X)=\Omega \,g(Y,Z)}$

for all vectors and . In local coordinates this leads to what is known as the conformal Killing equation ${\displaystyle Y}$ ${\displaystyle Z}$

${\displaystyle \nabla _{i}X_{j}+\nabla _{j}X_{i}=\Omega \,g_{ij}.}$

One can see from all these equations that a conforming killing vector field is a killing vector field if and only if the conforming killing factor is zero.

meaning

The conformal group generated by the conformal Killing vector fields is a frequently used symmetry group , especially in solid state physics . It is assumed that the physical system looks the same across many size scales. The quantum field theoretical description of such systems is carried out using conformant quantum field theories . Conformal quantum field theories in two spacetime dimensions also play a major role in string theory .