In
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, the Dirac equation in curved spacetime is a generalization of the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
from flat spacetime (
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
) to curved spacetime, a general Lorentzian manifold.
Mathematical formulation
Spacetime
In full generality the equation can be defined on
or
a pseudo-Riemannian manifold, but for concreteness we restrict to pseudo-Riemannian manifold with signature
. The metric is referred to as
, or
in
abstract index notation.
Frame fields
We use a set of
vierbein
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent ...
or frame fields
, which are a set of vector fields (which are not necessarily defined globally on
). Their defining equation is
:
The vierbein defines a local rest
frame
A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent.
Frame and FRAME may also refer to:
Physical objects
In building construction
*Framing (con ...
, allowing the constant
Gamma matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
to act at each spacetime point.
In differential-geometric language, the vierbein is equivalent to a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
, and so defines a local trivialization of the frame bundle.
Spin connection
To write down the equation we also need the
spin connection
In differential geometry and mathematical physics, a spin connection is a 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 generated by local Lorentz tr ...
, also known as the connection (1-)form. The dual frame fields
have defining relation
:
The connection 1-form is then
:
where
is a
covariant derivative, or equivalently a choice of
connection on the frame bundle, most often taken to be the
Levi-Civita connection.
One should be careful not to treat the abstract Latin indices and Greek indices as the same, and further to note that neither of these are coordinate indices: it can be verified that
doesn't transform as a tensor under a change of coordinates.
Mathematically, the frame fields
define an isomorphism at each point
where they are defined from the tangent space
to
. Then abstract indices label the tangent space, while greek indices label
. If the frame fields are position dependent then greek indices do not necessarily transform tensorially under a change of coordinates.
Raising and lowering indices In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Vectors, covectors and the metric
Math ...
is done with
for latin indices and
for greek indices.
The connection form can be viewed as a more abstract
connection on a principal bundle, specifically on the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
, which is defined on any smooth manifold, but which restricts to an ''orthonormal'' frame bundle on pseudo-Riemannian manifolds.
The connection form with respect to frame fields
defined locally is, in differential-geometric language, the connection with respect to a local trivialization.
Clifford algebra
Just as with the Dirac equation on flat spacetime, we make use of the Clifford algebra, a set of four
gamma matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
satisfying
:
where
is the anticommutator.
They can be used to construct a representation of the Lorentz algebra: defining
:
,
where