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formal adjoint differential operator

Definition

Over a smooth manifold $\Sigma$ of dimension $p+1$, let $E \overset{fb}{\to} \Sigma$ be a smooth vector bundle and $\tilde E^\ast \coloneqq E^\ast \otimes_{\Sigma} \wedge_\Sigma^{p+1} T^\ast \Sigma$ 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

$$P, P^\ast \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(\tilde E^\ast)$$

are called formally adjoint if there exists a bilinear differential operator

(1)$$K \;\colon\; \Gamma_\Sigma(E) \otimes \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(\wedge^{p} T^\ast \Sigma)$$

such that for all $\Phi_1, \Phi_2 \in \Gamma_\Sigma(E)$ we have

$$P(\Phi_1) \cdot \Phi_2 - \Phi_1 \cdot P^\ast(\Phi_2) \;=\; d K(\Phi_1, \Phi_2)$$

This implies by Stokes' theorem, in the case of compact support, that under an integral $P$ and $P^\ast$ 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 $\Sigma = \mathbb{R}^{p,1}$ be Minkowski spacetime with Minkowski metric $\eta$ and let $E \coloneqq \Sigma \times \mathbb{R}$ be the trivial line bundle. The canonical volume form $dvol_\Sigma$ induces an isomorphism $\tilde E^\ast \simeq E$.

Consider then the Klein-Gordon operator

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

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

$$K(\Phi_1, \Phi_2) \;\coloneqq\; \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \,.$$

Proof

$$\begin{aligned} d K(\Phi_1, \Phi_2) & = d \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \\ &= \left( \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 + \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu} \right) - \left( \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu} + \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) \right) dvol_\Sigma \\ & = \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 - \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) dvol_\Sigma \\ & = \Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2) \end{aligned}$$

Example

(Dirac operator on Dirac spinors is formally self-adjoint differential operator)

The Dirac operator on Dirac spinors is a formally anti-self adjoint (def. ):

$$D^\ast = - D \,.$$

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 $\overline{(-)}$:

$$\array{ \Gamma_\Sigma(\Sigma \times S) &\overset{D}{\longrightarrow}& \Gamma_\Sigma(\Sigma \times S^\ast) \\ \Psi &\mapsto& \overline{(-)} \gamma^\mu \partial_\mu \Psi }$$

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

$$\overline{\Psi_2}\gamma^\mu (\partial_\mu \Psi_1) - \overline{\Psi_1}\gamma^\mu (-\partial_\mu \Psi_2) \;=\; d K(\psi_1, \psi_2) \,.$$

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

$$\overline{\partial_\mu \Psi_1}\gamma^\mu \Psi_2 + \overline{\Psi_1}\gamma^\mu (\partial_\mu \Psi_2) \;=\; d K(\psi_1, \psi_2) \,.$$

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

$$K(\Psi_1, \Psi_2) \;\coloneqq\; \left( \overline{\Psi_1} \gamma^\mu \Psi_2\right) \, \iota_{\partial_\mu} dvol \,.$$

Related concepts

• generalized solution of a differential equation

• Green hyperbolic differential operator

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)