Dirac equation in curved spacetime

Short description: Generalization of the Dirac equation

For broader coverage of this topic, see Quantum field theory in curved spacetime.

Quantum field theory

Feynman diagram

History

Background

* Field theory
* Electromagnetism
* Weak force
* Strong force
* Quantum mechanics
* Special relativity
* General relativity
* Gauge theory

Symmetries

* Symmetry in quantum mechanics
* C-symmetry
* P-symmetry
* T-symmetry
* Space translation symmetry
* Time translation symmetry
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* Poincaré symmetry
* Gauge symmetry
* Explicit symmetry breaking
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* Noether charge
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Tools

* Anomaly
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* Expectation value
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* Faddeev–Popov ghosts
* Feynman diagram
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* LSZ reduction formula
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* Vacuum state
* Wick's theorem
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* Feynman parametrization

Equations

* Dirac equation
* Klein–Gordon equation
* Proca equations
* Wheeler–DeWitt equation
* Bargmann–Wigner equations
* Joos–Weinberg equation
* Weyl equation

Standard Model

* Quantum electrodynamics
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Incomplete theories

* Topological quantum field theory
* String theory
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* M-Theory
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* Supergravity
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* Quantum gravity

Scientists

* C. D. Anderson
* P. W. Anderson
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* v
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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.

## Contents

* 1 Mathematical formulation
* 1.1 Spacetime
* 1.2 Frame fields
* 1.3 Spin connection
* 1.4 Clifford algebra
* 1.5 Covariant derivative for fields in a representation of the Lorentz group
* 2 Dirac equation on curved spacetime
* 2.1 Recovering the Klein–Gordon equation from the Dirac equation
* 3 Action formulation
* 4 See also
* 5 References

## Mathematical formulation

### Spacetime

In full generality the equation can be defined on M or (M,𝐠) a pseudo-Riemannian manifold, but for concreteness we restrict to pseudo-Riemannian manifold with signature (−+++). The metric is referred to as 𝐠, or gab in abstract index notation.

### Frame fields

We use a set of vierbein or frame fields {eμ}={e0,e1,e2,e3}, which are a set of vector fields (which are not necessarily defined globally on M). Their defining equation is

gabeμaeνb=ημν.

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

To write down the equation we also need the spin connection, also known as the connection (1-)form. The dual frame fields {eμ} have defining relation

eaμeνa=δμν.

The connection 1-form is then

ωμνa:=ebμ∇aeνb

where ∇a 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 ωμνa doesn't transform as a tensor under a change of coordinates.

Mathematically, the frame fields {eμ} define an isomorphism at each point p where they are defined from the tangent space TpM to ℝ1,3. Then abstract indices label the tangent space, while greek indices label ℝ1,3. 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 gab for latin indices and ημν 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 {eμ} defined locally is, in differential-geometric language, the connection with respect to a local trivialization.

### Clifford algebra

See also: Dirac algebra

See also: Physics:Representation theory of the Lorentz group

Just as with the Dirac equation on flat spacetime, we make use of the Clifford algebra, a set of four gamma matrices {γμ} satisfying

{γμ,γν}=2ημν

where {⋅,⋅} is the anticommutator.

They can be used to construct a representation of the Lorentz algebra: defining

σμν=−i4[γμ,γν]=−i2γμγν+i2ημν,

where [⋅,⋅] is the commutator.

It can be shown they satisfy the commutation relations of the Lorentz algebra:

[σμν,σρσ]=(−i)(σμσηνρ−σνσημρ+σνρημσ−σμρηνσ)

They therefore are the generators of a representation of the Lorentz algebra 𝔰𝔬(1,3). But they do not generate a representation of the Lorentz group SO(1,3), just as the Pauli matrices generate a representation of the rotation algebra 𝔰𝔬(3) but not SO(3). They in fact form a representation of Spin(1,3). 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 ℂ4 as a vector space. In the classification of Lorentz group representations, the representation is labelled (12,0)⊕(0,12).

The abuse of terminology extends to forming this representation at the group level. We can write a finite Lorentz transformation on ℝ1,3 as Λσρ=exp⁡(i2αμνMμν)σρ where Mμν is the standard basis for the Lorentz algebra. These generators have components

(Mμν)σρ=ημρδσν−ηνρδσμ

or, with both indices up or both indices down, simply matrices which have +1 in the μ,ν index and −1 in the ν,μ index, and 0 everywhere else.

If another representation ρ has generators Tμν=ρ(Mμν), then we write

ρ(Λ)ji=exp⁡(i2αμνTμν)ji

where i,j are indices for the representation space.

In the case Tμν=σμν, without being given generator components αμν for Λσρ, this ρ(Λ) is not well defined: there are sets of generator components αμν,βμν which give the same Λσρ but different ρ(Λ)ji.

### Covariant derivative for fields in a representation of the Lorentz group

Given a coordinate frame ∂α arising from say coordinates {xα}, the partial derivative with respect to a general orthonormal frame {eμ} is defined

∂μψ=eμα∂αψ,

and connection components with respect to a general orthonormal frame are

ωμνρ=eραωμνα.

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

ψ↦ρ(Λ)ψ,

if we define the covariant derivative

Dμψ=∂μψ+12(ωνρ)μσνρψ,

then Dμψ transforms as

Dμψ↦ρ(Λ)Dμψ

This generalises to any representation R for the Lorentz group: if v is a vector field for the associated representation,

Dμv=∂μv+12(ωνρ)μR(Mνρ)v=∂μv+12(ωνρ)μTνρv.

When R is the fundamental representation for SO(1,3), 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 Dαψ in a coordinate basis is a vector-valued 1-form, which at each point p is an element of Ep⊗Tp*M. The covariant derivative Dμψ in an orthonormal basis uses the orthonormal frame {eμ} to identify the vector-valued 1-form with a vector-valued dual vector which at each point p is an element of Ep⊗ℝ1,3, using that ℝ1,3*≅ℝ1,3 canonically. We can then contract this with a gamma matrix 4-vector γμ which takes values at p in End(Ep)⊗ℝ1,3

## Dirac equation on curved spacetime

Recalling the Dirac equation on flat spacetime,

(iγμ∂μ−m)ψ=0,

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]

Dirac equation on curved spacetime

(iγμDμ−m)Ψ=0.

where Ψ is a spinor field on spacetime. Mathematically, this is a section of a vector bundle associated to the spin-frame bundle by the representation (1/2,0)⊕(0,1/2).

### Recovering the Klein–Gordon equation from the Dirac equation

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

(1−det⁡g𝒟μ(−det⁡ggμν𝒟ν)−14R+ie2Fμνsμν−m2)Ψ=0.

where R is the Ricci scalar, and Fμν is the field strength of Aμ. 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

We can formulate this theory in terms of an action. If in addition the spacetime (M,𝐠) is orientable, there is a preferred orientation known as the volume form ϵ. One can integrate functions against the volume form:

∫Mϵf=∫Md4x−gf

The function Ψ¯(iγμDμ−m)Ψ is integrated against the volume form to obtain the Dirac action

Dirac action on curved spacetime

IDirac=∫Md4x−gΨ¯(iγμDμ−m)Ψ.

## See also

* Dirac equation in the algebra of physical space
* Dirac spinor
* Maxwell's equations in curved spacetime
* Two-body Dirac equations

## References

1. ↑ Lawrie, Ian D.. A Unified Grand Tour of Theoretical Physics.
2. ↑ Pollock, M.D. (2010), On the Dirac equation in curved space-time, https://www.actaphys.uj.edu.pl/R/41/8/1827/pdf

* M. Arminjon, F. Reifler (2013). "Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations". Brazilian Journal of Physics 43 (1–2): 64–77. doi:10.1007/s13538-012-0111-0. Bibcode: 2013BrJPh..43...64A.
* M.D. Pollock (2010). "on the dirac equation in curved space-time". Acta Physica Polonica B 41 (8): 1827. http://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=41&page=1827.
* J.V. Dongen (2010). Einstein's Unification. Cambridge University Press. p. 117. ISBN 978-0-521-883-467. https://books.google.com/books?id=T8Fde0u_XyIC&dq=Dirac+equation+in+curved+spacetime&pg=PA117.
* L. Parker, D. Toms (2009). Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press. p. 227. ISBN 978-0-521-877-879. https://books.google.com/books?id=5nNuGMBBTjMC&q=L.+Parker,+D.+Toms+(2009).+Quantum+Field+Theory+in+Curved+Spacetime:+Quantized+Fields+and+Gravity.
* S.A. Fulling (1989). Aspects of Quantum Field Theory in Curved Spacetime. Cambridge University Press. ISBN 0-521-377-684. https://books.google.com/books?id=Zo5_3cmtFEUC&dq=Dirac+equation+in+curved+spacetime&pg=PA223.

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