:''See also

Wigner–Weyl transform In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger ...

, for another definition of the Weyl transform.'' In theoretical physics, the Weyl transformation, named after

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

, 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 (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

in conformal field theory. 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 (i.e. affine connection) that preserves th ...

and associated spin connections are not invariant under Weyl transformations. An appropriately invariant notion is the Weyl connection, which is one way of specifying the structure of a conformal connection.

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 bundles; 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

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