theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, the Weyl transformation, named after German mathematician

Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...

, is a local rescaling of the

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...

g_ \rightarrow e^ g_

which produces another metric in the same conformal class. A theory or an expression invariant under this transformation is called conformally invariant, or is said to possess Weyl invariance or Weyl symmetry. The Weyl symmetry is an important

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...

. It is, for example, a symmetry of the Polyakov action. When quantum mechanical effects break the conformal invariance of a theory, it is said to exhibit a conformal anomaly or Weyl anomaly. The ordinary

Levi-Civita connection In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...

and associated

spin connection In differential geometry and mathematical physics, a spin connection is a connection (vector bundle), connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field gene ...

s are not invariant under Weyl transformations.

Weyl connection In differential geometry, a Weyl connection (also called a Weyl structure) is a generalization of the Levi-Civita connection that makes sense on a conformal manifold. They were introduced by Hermann Weyl in an attempt to unify general relativity ...

s are a class of affine connections that is invariant, although no Weyl connection is individual invariant under Weyl transformations.

Conformal weight

A quantity

\varphi

has conformal weight

k

if, under the Weyl transformation, it transforms via :

\varphi \to \varphi e^.

Thus conformally weighted quantities belong to certain

density bundle In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the ...

s; see also conformal dimension. Let

A_\mu

be the connection one-form associated to the Levi-Civita connection of

g

. Introduce a connection that depends also on an initial one-form

\partial_\mu\omega

via :

B_\mu = A_\mu + \partial_\mu \omega.

Then

D_\mu \varphi \equiv \partial_\mu \varphi + k B_\mu \varphi

is covariant and has conformal weight

k - 1

Formulas

For the transformation :

g_ = f(\phi(x)) \bar_

We can derive the following formulas :

\begin
    g^ &= \frac \bar^\\
    \sqrt &= \sqrt f^ \\
    \Gamma^c_ &= \bar^c_ + \frac \left(\delta^c_b \partial_a \phi + \delta^c_a \partial_b \phi - \bar_ \partial^c \phi \right) \equiv \bar^c_ + \gamma^c_ \\
     R_ &= \bar_ + \frac \left((2-D) \partial_a \phi \partial_b \phi - \bar_ \partial^c \phi \partial_c \phi \right) + \frac \left((2-D) \bar_a \partial_b \phi - \bar_ \bar \phi\right) + \frac \frac (D-2) \left(\partial_a \phi \partial_b \phi - \bar_ \partial_c \phi \partial^c \phi \right) \\
     R &= \frac \bar + \frac \left( \frac \partial^c \phi \partial_c \phi + \frac \bar \phi \right) + \frac \frac (D-2) (1-D) \partial_c \phi \partial^c \phi 
\end

Note that the Weyl tensor is invariant under a Weyl rescaling.

References

* Conformal geometry Differential geometry Scaling symmetries Symmetry Theoretical physics {{differential-geometry-stub