Dirac equation in curved spacetime
Generalization of the Dirac equation
In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved spacetime, a general Lorentzian manifold.
Mathematical formulation[edit]
Spacetime[edit]
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[edit]
We use a set of vierbein 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, allowing the constant Gamma matrices to act at each spacetime point.
In differential-geometric language, the vierbein is equivalent to a section of the frame bundle, and so defines a local trivialization of the frame bundle.
Spin connection[edit]
To write down the equation we also need the spin connection, 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 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, 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[edit]
Just as with the Dirac equation on flat spacetime, we make use of the Clifford algebra, a set of four gamma matrices
satisfying
where
is the anticommutator.
They can be used to construct a representation of the Lorentz algebra: defining
- ,
where
is the commutator.
It can be shown they satisfy the commutation relations of the Lorentz algebra:
They therefore are the generators of a representation of the Lorentz algebra
. But they do not generate a representation of the Lorentz group
, just as the Pauli matrices generate a representation of the rotation algebra
but not
. They in fact form a representation of
However, it is a standard abuse of terminology to any representations of the Lorentz algebra as representations of the Lorentz group, even if they do not arise as representations of the Lorentz group.
The representation space is isomorphic to
as a vector space. In the classification of Lorentz group representations, the representation is labelled
.
The abuse of terminology extends to forming this representation at the group level. We can write a finite Lorentz transformation on
as
where
is the standard basis for the Lorentz algebra. These generators have components
or, with both indices up or both indices down, simply matrices which have
in the
index and
in the
index, and 0 everywhere else.
If another representation
has generators
then we write
where
are indices for the representation space.
In the case
, without being given generator components
for
, this
is not well defined: there are sets of generator components
which give the same
but different
Covariant derivative for fields in a representation of the Lorentz group[edit]
Given a coordinate frame
arising from say coordinates
, the partial derivative with respect to a general orthonormal frame
is defined
and connection components with respect to a general orthonormal frame are
These components do not transform tensorially under a change of frame, but do when combined. Also, these are definitions rather than saying that these objects can arise as partial derivatives in some coordinate chart. In general there are non-coordinate orthonormal frames, for which the commutator of vector fields is non-vanishing.
It can be checked that under the transformation
if we define the covariant derivative
- ,
then
transforms as
This generalises to any representation
for the Lorentz group: if
is a vector field for the associated representation,
When
is the fundamental representation for
, this recovers the familiar covariant derivative for (tangent-)vector fields, of which the Levi-Civita connection is an example.
There are some subtleties in what kind of mathematical object the different types of covariant derivative are. The covariant derivative
in a coordinate basis is a vector-valued 1-form, which at each point
is an element of
. The covariant derivative
in an orthonormal basis uses the orthonormal frame
to identify the vector-valued 1-form with a vector-valued dual vector which at each point
is an element of
using that
canonically. We can then contract this with a gamma matrix 4-vector
which takes values at
in
Dirac equation on curved spacetime[edit]
Recalling the Dirac equation on flat spacetime,
the Dirac equation on curved spacetime can be written down by promoting the partial derivative to a covariant one.
In this way, Dirac’s equation takes the following form in curved spacetime:[1].
where
is a spinor field on spacetime. Mathematically, this is a section of a vector bundle associated to the spin-frame bundle by the representation
Recovering the Klein–Gordon equation from the Dirac equation[edit]
The modified Klein–Gordon equation obtained by squaring the operator in the Dirac equation, first found by Erwin Schrödinger as cited by Pollock
[2] is given by
where
is the Ricci scalar, and
is the field strength of
. An alternative version of the Dirac equation whose Dirac operator remains the square root of the Laplacian is given by the Dirac–Kähler equation; the price to pay is the loss of Lorentz invariance in curved spacetime.
Note that here Latin indices denote the “Lorentzian” vierbein labels while Greek indices denote manifold coordinate indices.
Action formulation[edit]
We can formulate this theory in terms of an action. If in addition the spacetime
is orientable, there is a preferred orientation known as the volume form
.
One can integrate functions against the volume form:
The function
is integrated against the volume form to obtain the Dirac action
See also[edit]
References[edit]
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