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formal adjoint differential operator

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

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.

Definition

(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)

Examples

Example

(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 \,.
Proof
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}
Proof

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 \,.

References

  • 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.