formal adjoint differential operator



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



Over a smooth manifold Σ\Sigma of dimension p+1p+1, let EfbΣE \overset{fb}{\to} \Sigma be a smooth vector bundle and E˜ *E * Σ Σ p+1T *Σ\tilde E^\ast \coloneqq E^\ast \otimes_{\Sigma} \wedge_\Sigma^{p+1} T^\ast \Sigma the tensor product of vector bundles of the dual vector bundle with the differential (p+1)-form bundle.


(formally adjoint differential operators)

Two differential operators

P,P *:Γ Σ(E)Γ Σ(E˜ *) P, P^\ast \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(\tilde E^\ast)

are called formally adjoint if there exists a bilinear differential operator

(1)K:Γ Σ(E)Γ Σ(E)Γ Σ( pT *Σ) K \;\colon\; \Gamma_\Sigma(E) \otimes \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(\wedge^{p} T^\ast \Sigma)

such that for all Φ 1,Φ 2Γ Σ(E)\Phi_1, \Phi_2 \in \Gamma_\Sigma(E) we have

P(Φ 1)Φ 2Φ 1P *(Φ 2)=dK(Φ 1,Φ 2) P(\Phi_1) \cdot \Phi_2 - \Phi_1 \cdot P^\ast(\Phi_2) \;=\; d K(\Phi_1, \Phi_2)

This implies by Stokes' theorem, in the case of compact support, that under an integral PP and P *P^\ast are related via integration by parts.

(Khavkine 14, def. 2.4)

See also (Vinogradov-Krasilshchik 99, chapter 5, §2.3)



(Klein-Gordon operator is formally self-adjoint differential operator)

Let Σ= p,1\Sigma = \mathbb{R}^{p,1} be Minkowski spacetime with Minkowski metric η\eta and let EΣ×E \coloneqq \Sigma \times \mathbb{R} be the trivial line bundle. The canonical volume form dvol Σdvol_\Sigma induces an isomorphism E˜ *E\tilde E^\ast \simeq E.

Consider then the Klein-Gordon operator

(m 2):Γ Σ(Σ×)Γ Σ(Σ×)dvol Σ. (\Box - m^2) \;\colon\; \Gamma_\Sigma(\Sigma \times \mathbb{R}) \longrightarrow \Gamma_\Sigma(\Sigma \times \mathbb{R}) \otimes \langle dvol_\Sigma\rangle \,.

This is its own formal adjoint (def. ) witnessed by the bilinear differential operator (1) given by

K(Φ 1,Φ 2)(Φ 1x μΦ 2Φ 1Φ 2x μ)η μνι νdvol Σ. K(\Phi_1, \Phi_2) \;\coloneqq\; \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \,.
dK(Φ 1,Φ 2) =d(Φ 1x μΦ 2Φ 1Φ 2x μ)η μνι νdvol Σ =((η μν 2Φ 1x μx νΦ 2+η μνΦ 1x μΦ 2x ν)(η μνΦ 1x νΦ 2x μ+Φ 1η μν 2Φ 2x νx μ))dvol Σ =(η μν 2Φ 1x μx νΦ 2Φ 1η μν 2Φ 2x νx μ)dvol Σ =(Φ 1)Φ 2Φ 1(Φ 2) \begin{aligned} d K(\Phi_1, \Phi_2) & = d \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \\ &= \left( \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 + \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu} \right) - \left( \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu} + \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) \right) dvol_\Sigma \\ & = \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 - \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) dvol_\Sigma \\ & = \Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2) \end{aligned}

We spell out the proof over Minkowski spacetime. Then general case is a straightforward generalization

Regard the Dirac operator as taking values in the dual spin bundle by using the Dirac conjugate ()¯\overline{(-)}:

Γ Σ(Σ×S) D Γ Σ(Σ×S *) Ψ ()¯γ μ μΨ \array{ \Gamma_\Sigma(\Sigma \times S) &\overset{D}{\longrightarrow}& \Gamma_\Sigma(\Sigma \times S^\ast) \\ \Psi &\mapsto& \overline{(-)} \gamma^\mu \partial_\mu \Psi }

Then we need to show that there is K(,)K(-,-) such that for all pairs of spinor sections Ψ 1,Ψ 2\Psi_1, \Psi_2 we have

Ψ 2¯γ μ( μΨ 1)Ψ 1¯γ μ( μΨ 2)=dK(ψ 1,ψ 2). \overline{\Psi_2}\gamma^\mu (\partial_\mu \Psi_1) - \overline{\Psi_1}\gamma^\mu (-\partial_\mu \Psi_2) \;=\; d K(\psi_1, \psi_2) \,.

But the spinor-to-vector pairing is symmetric (see at spin representation), hence this is equivalent to

μΨ 1¯γ μΨ 2+Ψ 1¯γ μ( μΨ 2)=dK(ψ 1,ψ 2). \overline{\partial_\mu \Psi_1}\gamma^\mu \Psi_2 + \overline{\Psi_1}\gamma^\mu (\partial_\mu \Psi_2) \;=\; d K(\psi_1, \psi_2) \,.

By the product law of differentiation, this is solved, for all Ψ 1,Ψ 2\Psi_1, \Psi_2, by

K(Ψ 1,Ψ 2)(Ψ 1¯γ μΨ 2)ι μdvol. K(\Psi_1, \Psi_2) \;\coloneqq\; \left( \overline{\Psi_1} \gamma^\mu \Psi_2\right) \, \iota_{\partial_\mu} dvol \,.


  • Peter Olver, chapter 5.3, around p. 328-330 of Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.

  • Alexandre Vinogradov, I. S. Krasilshchik (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, vol. 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. (pdf)

  • Igor Khavkine, Covariant phase space, constraints, gauge and the Peierls formula, Int. J. Mod. Phys. A, 29, 1430009 (2014) (arXiv:1402.1282)

Last revised on November 30, 2017 at 07:16:17. See the history of this page for a list of all contributions to it.