Quantum field theory
Feynman diagram
History
Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity Gauge theory Yang–Mills theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Noether charge Topological charge
Tools Anomaly Background field method BRST quantization Correlation function Crossing Effective action Effective field theory Expectation value Feynman diagram Lattice field theory LSZ reduction formula Partition function Path Integral Formulation Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations Schwinger-Dyson equation Renormalization group equation
Standard Model Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism
Incomplete theories String theory Supersymmetry Technicolor Theory of everything Quantum gravity
Scientists Adler Anderson Anselm Bargmann Becchi Belavin Bell Berezin Bethe Bjorken Bleuer Bogoliubov Brodsky Brout Buchholz Cachazo Callan Cardy Coleman Connes Dashen DeWitt Dirac Doplicher Dyson Englert Faddeev Fadin Fayet Fermi Feynman Fierz Fock Frampton Fritzsch Fröhlich Fredenhagen Furry Glashow Gell-Mann Glimm Goldstone Gribov Gross Gupta Guralnik Haag Hagen Han Heisenberg Hepp Higgs 't Hooft Iliopoulos Ivanenko Jackiw Jaffe Jona-Lasinio Jordan Jost Källén Kendall Kinoshita Kim Klebanov Kontsevich Kreimer Kuraev Landau Lee Lee Lehmann Leutwyler Lipatov Łopuszański Low Lüders Maiani Majorana Maldacena Matsubara Migdal Mills Møller Naimark Nambu Neveu Nishijima Oehme Oppenheimer Osborn Osterwalder Parisi Pauli Peccei Peskin Plefka Polchinski Polyakov Pomeranchuk Popov Proca Quinn Rouet Rubakov Ruelle Sakurai Salam Schrader Schwarz Schwinger Segal Seiberg Semenoff Shifman Shirkov Skyrme Sommerfield Stora Stueckelberg Sudarshan Symanzik Takahashi Thirring Tomonaga Tyutin Vainshtein Veltman Veneziano Virasoro Ward Weinberg Weisskopf Wentzel Wess Wetterich Weyl Wick Wightman Wigner Wilczek Wilson Witten Yang Yukawa Zamolodchikov Zamolodchikov Zee Zimmermann Zinn-Justin Zuber Zumino
v t e

Inmathematical physics,the Dirac equation in curved spacetime is a generalization of theDirac equation from flatspacetime (Minkowski space) tocurved spacetime, a generalLorentzian manifold.

In full generality the equation can be defined on${\displaystyle M}$ or${\displaystyle (M,\mathbf{g})}$ apseudo-Riemannian manifold, but for concreteness we restrict to pseudo-Riemannian manifold with signature${\displaystyle (-+++)}$. The metric is referred to as${\displaystyle \mathbf{g}}$, or${\displaystyle g_{ab}}$ inabstract index notation.

We use a set ofvierbein or frame fields${\displaystyle \{e_{\mu }\}=\{e_0,e_1,e_2,e_3\}}$, which are a set of vector fields (which are not necessarily defined globally on${\displaystyle M}$). Their defining equation is

${\displaystyle g_{ab}{e}_{\mu }^a{e}_{\nu }^b={\eta }_{\mu \nu }.}$

The vierbein defines a local restframe, allowing the constantGamma matrices to act at each spacetime point.

In differential-geometric language, the vierbein is equivalent to asection of theframe bundle, and so defines a local trivialization of the frame bundle.

To write down the equation we also need thespin connection, also known as the connection (1-)form. The dual frame fields${\displaystyle \{e^{\mu }\}}$ have defining relation

${\displaystyle e_a^{\mu }{e}_{\nu }^a={\delta }^{\mu }{}_{\nu }.}$

The connection 1-form is then

${\displaystyle {\omega }^{\mu }{}_{\nu a}:=e_b^{\mu }{\mathrm{\nabla }}_a{e}_{\nu }^b}$

where${\displaystyle {\mathrm{\nabla }}_a}$ is acovariant derivative, or equivalently a choice ofconnection on the frame bundle, most often taken to be theLevi-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${\displaystyle {\omega }^{\mu }{}_{\nu a}}$ doesn't transform as a tensor under a change of coordinates.

Mathematically, the frame fields${\displaystyle \{e_{\mu }\}}$ define an isomorphism at each point${\displaystyle p}$ where they are defined from the tangent space${\displaystyle T_{p}M}$ to${\displaystyle {\mathbb{R}}^{1,3}}$. Then abstract indices label the tangent space, while greek indices label${\displaystyle {\mathbb{R}}^{1,3}}$. 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${\displaystyle g_{ab}}$ for latin indices and${\displaystyle {\eta }_{\mu \nu }}$ for greek indices.

The connection form can be viewed as a more abstractconnection on a principal bundle, specifically on theframe bundle, which is defined on any smooth manifold, but which restricts to anorthonormal frame bundle on pseudo-Riemannian manifolds.

The connection form with respect to frame fields${\displaystyle \{e_{\mu }\}}$ defined locally is, in differential-geometric language, the connection with respect to a local trivialization.

Just as with the Dirac equation on flat spacetime, we make use of the Clifford algebra, a set of fourgamma matrices${\displaystyle \{{\gamma }^{\mu }\}}$ satisfying

${\displaystyle \{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }}$

where${\displaystyle \{\cdot ,\cdot \}}$ is theanticommutator.

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

${\displaystyle {\sigma }^{\mu \nu }=-\frac{i}{4}[{\gamma }^{\mu },{\gamma }^{\nu }]=-\frac{i}{2}{\gamma }^{\mu }{\gamma }^{\nu }+\frac{i}{2}{\eta }^{\mu \nu }}$,

where${\displaystyle [\cdot ,\cdot ]}$ is thecommutator.

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

${\displaystyle [{\sigma }^{\mu \nu },{\sigma }^{\rho \sigma }]=(-i)({\sigma }^{\mu \sigma }{\eta }^{\nu \rho }-{\sigma }^{\nu \sigma }{\eta }^{\mu \rho }+{\sigma }^{\nu \rho }{\eta }^{\mu \sigma }-{\sigma }^{\mu \rho }{\eta }^{\nu \sigma })}$

They therefore are the generators of a representation of the Lorentz algebra${\displaystyle \mathfrak{s}\mathfrak{o}(1,3)}$. But they donot generate a representation of the Lorentz group${\displaystyle \text{SO}(1,3)}$, just as the Pauli matrices generate a representation of the rotation algebra${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ but not${\displaystyle \text{SO}(3)}$. They in fact form a representation of${\displaystyle \text{Spin}(1,3).}$ 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${\displaystyle {\mathbb{C}}^4}$ as a vector space. In the classification of Lorentz group representations, the representation is labelled${\displaystyle \left(\frac{1}{2},0\right)\oplus \left(0,\frac{1}{2}\right)}$.

The abuse of terminology extends to forming this representation at the group level. We can write a finite Lorentz transformation on${\displaystyle {\mathbb{R}}^{1,3}}$ as${\displaystyle {\mathrm{\Lambda }}_{\sigma }^{\rho }=\exp \left(\frac{i}{2}{\alpha }_{\mu \nu }{M}^{\mu \nu }\right){}_{\sigma }^{\rho }}$where${\displaystyle M^{\mu \nu }}$ is the standard basis for the Lorentz algebra. These generators have components

${\displaystyle (M^{\mu \nu }{)}_{\sigma }^{\rho }={\eta }^{\mu \rho }{\delta }_{\sigma }^{\nu }-{\eta }^{\nu \rho }{\delta }_{\sigma }^{\mu }}$

or, with both indices up or both indices down, simply matrices which have${\displaystyle +1}$ in the${\displaystyle \mu ,\nu }$ index and${\displaystyle -1}$ in the${\displaystyle \nu ,\mu }$ index, and 0 everywhere else.

If another representation${\displaystyle \rho }$ has generators${\displaystyle T^{\mu \nu }=\rho (M^{\mu \nu }),}$ then we write

${\displaystyle \rho (\mathrm{\Lambda }{)}_j^i=\exp \left(\frac{i}{2}{\alpha }_{\mu \nu }{T}^{\mu \nu }\right){}_j^i}$

where${\displaystyle i,j}$ are indices for the representation space.

In the case${\displaystyle T^{\mu \nu }={\sigma }^{\mu \nu }}$, without being given generator components${\displaystyle {\alpha }_{\mu \nu }}$ for${\displaystyle {\mathrm{\Lambda }}_{\sigma }^{\rho }}$, this${\displaystyle \rho (\mathrm{\Lambda })}$ is not well defined: there are sets of generator components${\displaystyle {\alpha }_{\mu \nu },{\beta }_{\mu \nu }}$ which give the same${\displaystyle {\mathrm{\Lambda }}_{\sigma }^{\rho }}$ but different${\displaystyle \rho (\mathrm{\Lambda }{)}_j^i.}$

Given a coordinate frame${\displaystyle {\mathrm{\partial }}_{\alpha }}$ arising from say coordinates${\displaystyle \{x^{\alpha }\}}$, the partial derivative with respect to a general orthonormal frame${\displaystyle \{e_{\mu }\}}$ is defined

${\displaystyle {\mathrm{\partial }}_{\mu }\psi =e_{\mu }^{\alpha }{\mathrm{\partial }}_{\alpha }\psi ,}$

and connection components with respect to a general orthonormal frame are

${\displaystyle {\omega }^{\mu }{}_{\nu \rho }=e_{\rho }^{\alpha }{\omega }^{\mu }{}_{\nu \alpha }.}$

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

${\displaystyle \psi \mapsto \rho (\mathrm{\Lambda })\psi ,}$

if we define the covariant derivative

${\displaystyle D_{\mu }\psi ={\mathrm{\partial }}_{\mu }\psi +\frac{1}{2}({\omega }_{\nu \rho }{)}_{\mu }{\sigma }^{\nu \rho }\psi }$,

then${\displaystyle D_{\mu }\psi }$ transforms as

${\displaystyle D_{\mu }\psi \mapsto \rho (\mathrm{\Lambda })D_{\mu }\psi }$

This generalises to any representation${\displaystyle R}$ for the Lorentz group: if${\displaystyle v}$ is a vector field for the associated representation,

${\displaystyle D_{\mu }v={\mathrm{\partial }}_{\mu }v+\frac{1}{2}({\omega }_{\nu \rho }{)}_{\mu }R(M^{\nu \rho })v={\mathrm{\partial }}_{\mu }v+\frac{1}{2}({\omega }_{\nu \rho }{)}_{\mu }{T}^{\nu \rho }v.}$

When${\displaystyle R}$ is the fundamental representation for${\displaystyle \text{SO}(1,3)}$, 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${\displaystyle D_{\alpha }\psi }$ in a coordinate basis is a vector-valued 1-form, which at each point${\displaystyle p}$ is an element of${\displaystyle E_p\otimes T_p^{\ast }M}$. The covariant derivative${\displaystyle D_{\mu }\psi }$ in an orthonormal basis uses the orthonormal frame${\displaystyle \{e_{\mu }\}}$ to identify the vector-valued 1-form with a vector-valued dual vector which at each point${\displaystyle p}$ is an element of${\displaystyle E_p\otimes {\mathbb{R}}^{1,3},}$ using that${\displaystyle {{\mathbb{R}}^{1,3}}^{\ast }\cong {\mathbb{R}}^{1,3}}$ canonically. We can then contract this with a gamma matrix 4-vector${\displaystyle {\gamma }^{\mu }}$ which takes values at${\displaystyle p}$ in${\displaystyle \text{End}(E_p)\otimes {\mathbb{R}}^{1,3}}$

Recalling the Dirac equation on flat spacetime,

${\displaystyle (i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }-m)\psi =0,}$

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]

where${\displaystyle \mathrm{\Psi }}$ is a spinor field on spacetime. Mathematically, this is a section of a vector bundle associated to the spin-frame bundle by the representation${\displaystyle (1/2,0)\oplus (0,1/2).}$

The modifiedKlein–Gordon equation obtained by squaring the operator in the Dirac equation, first found byErwin Schrödinger as cited by Pollock^[2] is given by

${\displaystyle \left(\frac{1}{\sqrt{-detg}}{\mathcal{D}}_{\mu }\left(\sqrt{-detg}{g}^{\mu \nu }{\mathcal{D}}_{\nu }\right)-\frac{1}{4}R+\frac{ie}{2}{F}_{\mu \nu }{s}^{\mu \nu }-m^2\right)\mathrm{\Psi }=0.}$

where${\displaystyle R}$ is the Ricci scalar, and${\displaystyle F_{\mu \nu }}$ is the field strength of${\displaystyle A_{\mu }}$. An alternative version of the Dirac equation whoseDirac operator remains the square root of theLaplacian is given by theDirac–Kähler equation; the price to pay is the loss ofLorentz invariance in curved spacetime.

Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denotemanifold coordinate indices.

We can formulate this theory in terms of an action. If in addition the spacetime${\displaystyle (M,\mathbf{g})}$ isorientable, there is a preferred orientation known as thevolume form${\displaystyle ϵ}$.One can integrate functions against the volume form:

${\displaystyle {\int }_Mϵf={\int }_M{d}^{4}x\sqrt{-g}f}$

The function${\displaystyle \overline{\mathrm{\Psi }}(i{\gamma }^{\mu }{D}_{\mu }-m)\mathrm{\Psi }}$is integrated against the volume form to obtain the Dirac action

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

• 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.arXiv:1103.3201.Bibcode:2013BrJPh..43...64A.doi:10.1007/s13538-012-0111-0.S2CID 38235437.
• M.D. Pollock (2010)."on the dirac equation in curved space-time".Acta Physica Polonica B.41 (8): 1827.
• J.V. Dongen (2010).Einstein's Unification. Cambridge University Press. p. 117.ISBN 978-0-521-883-467.
• 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.
• S.A. Fulling (1989).Aspects of Quantum Field Theory in Curved Spacetime. Cambridge University Press.ISBN 0-521-377-684.

• Quantum field theory
• Spinors
• Partial differential equations
• Fermions
• Dirac equation

• Articles with short description
• Short description is different from Wikidata