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.
In full generality the equation can be defined on
, 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.
doesn't transform as a tensor under a change of coordinates.
If the frame fields are position dependent then greek indices do not necessarily transform tensorially under a change of coordinates.
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
Just as with the Dirac equation on flat spacetime, we make use of the Clifford algebra, a set of four gamma matrices
They can be used to construct a representation of the Lorentz algebra: defining where
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
The abuse of terminology extends to forming this representation at the group level.
We can write a finite Lorentz transformation on
is not well defined: there are sets of generator components
, 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
, 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.
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]
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
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.
We can formulate this theory in terms of an action.
is integrated against the volume form to obtain the Dirac action