In
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 ...
:
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 ...
in
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 connections are a class of affine connections that is invariant, although no Weyl connection is individual invariant under Weyl transformations.
Conformal weight
A quantity
has
conformal weight if, under the Weyl transformation, it transforms via
:
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
be the
connection one-form associated to the Levi-Civita connection of
. Introduce a connection that depends also on an initial one-form
via
:
Then
is covariant and has conformal weight
.
Formulas
For the transformation
:
We can derive the following formulas
:
Note that the Weyl tensor is invariant under a Weyl rescaling.
References
*
Conformal geometry
Differential geometry
Scaling symmetries
Symmetry
Theoretical physics
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