The Dagger command returns the Hermitian conjugate, also called adjoint, of its argument, so, for example, if A is a square matrix, then Dagger(A) computes the complex conjugate of the transpose of $A$. As a shortcut to Dagger(A) you can also use A^*.

The %Dagger command is the inert form of Dagger; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command.

The result returned by Dagger is built as follows:

- If $A$ is Hermitian, then return $A$.

- If $A$ is already the Dagger of $B$, then return $B$.

- If $A$ is a (commutative) constant or a Bracket, then return the conjugate of $A$

- If $A$ is a Bra or a Ket, then return the dual; that is, the corresponding Ket or Bra, respectively.

- If $A$ is an Annihilation or a Creation operator, then return the corresponding Creation or Annihilation, respectively.

- If $A$ is a Matrix, then return the conjugate of the transpose of $A$.

- If $A$ is a sum of terms, then return the sum of the Dagger of each term.

- If $A$ is a (noncommutative) product, then return the product of the Dagger of each factor, after reversing their order in the product.

- If $A$ is d_, or dAlembertian, then return the operator applied to the Dagger of the first operand of $A$.

- If $A$ is an object of the Physics package, such as Inverse, Trace, Dgamma, or KroneckerDelta, then return according to the properties of this object.

- Otherwise, return the operation unevaluated, Dagger(A).

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Bra}\left(A\,n\right)$

$\left.⟨A_n\right|$

$\mathrm{Dagger}\left(\right)$

$\left|A_n\right.⟩$

$\mathrm{\%Dagger}\left(\right)$

${\left|A_n\right.⟩}^{†}$

$\mathrm{value}\left(\right)$

$\left.⟨A_n\right|$

$\mathrm{\%Dagger}\left(\mathrm{Bracket}\left(A\,B\right)\right)$

${⟨A|B⟩}^{†}$

$\mathrm{value}\left(\right)$

$⟨B|A⟩$

$\mathrm{am}≔\mathrm{Annihilation}\left(A\,1\right)$

$\mathrm{am}≔a^{-}$

$\mathrm{ap}≔\mathrm{Creation}\left(A\,1\right)$

$\mathrm{ap}≔a^{+}$

$\mathrm{Dagger}\left(\mathrm{am}\right)$

$a^{+}$

$\mathrm{Dagger}\left(\mathrm{ap}\right)$

$a^{-}$

${\mathrm{ap}}^{\mathrm{\%H}}$

$a^{-}$

$\mathrm{\%Dagger}\left(\mathrm{Dgamma}\left[1\right]\right)$

${{\mathrm{\gamma }}_1}^{†}$

$\mathrm{value}\left(\right)$

$-{\mathrm{\gamma }}_1$

$\mathrm{\%Dagger}\left(\mathrm{Dgamma}\left[0\right]\right)$

${{\mathrm{\gamma }}_4}^{†}$

$\mathrm{value}\left(\right)$

${\mathrm{\gamma }}_4$

$\mathrm{Setup}\left(\mathrm{noncommutativeprefix}=Z\right)$

$\left[\mathrm{noncommutativeprefix}=\left\{Z\right\}\right]$

$\mathrm{Commutator}\left(Z\left[1\right]\,Z\left[2\right]\right)$

${\left[Z_1,Z_2\right]}_{-}$

$=\mathrm{expand}\left(\right)$

${\left[Z_1,Z_2\right]}_{-}=Z_1{Z}_2-Z_2{Z}_1$

$\mathrm{Dagger}\left(\right)$

${\left[{Z_2}^{†},{Z_1}^{†}\right]}_{-}={Z_2}^{†}{Z_1}^{†}-{Z_1}^{†}{Z_2}^{†}$

$\mathrm{AntiCommutator}\left(Z\left[1\right]\,Z\left[2\right]\right)$

${\left[Z_1,Z_2\right]}_{+}$

$=\mathrm{expand}\left(\right)$

${\left[Z_1,Z_2\right]}_{+}=Z_1{Z}_2+Z_2{Z}_1$

$\mathrm{Dagger}\left(\right)\mathbf{assuming}\mathrm{Hermitian}$

${\left[Z_1,Z_2\right]}_{+}=Z_1{Z}_2+Z_2{Z}_1$

$\mathrm{Dagger}\left(\right)$

${\left[{Z_1}^{†},{Z_2}^{†}\right]}_{+}={Z_1}^{†}{Z_2}^{†}+{Z_2}^{†}{Z_1}^{†}$

$\mathrm{Setup}\left(\mathrm{diff}=X\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{differentiationvariables}=\left[X\right]\right]$

$\mathrm{d\_}\left[\mathrm{\mu }\right]\left(Z\left[1\right]\left(X\right)\right)\mathrm{d\_}\left[\mathrm{\nu }\right]\left(Z\left[2\right]\left(X\right)\right)+\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]\mathrm{dAlembertian}\left(F\left(X\right)\right)$

${\partial }_{\mathrm{\mu }}\left(Z_1\left(X\right)\right){\partial }_{\mathrm{\nu }}\left(Z_2\left(X\right)\right)+g_{\mathrm{\mu },\mathrm{\nu }}\mathrm{\square }\left(F\left(X\right)\right)$

$\mathrm{Dagger}\left(\right)$

${\partial }_{\mathrm{\nu }}\left({Z_2\left(X\right)}^{†}\right){\partial }_{\mathrm{\mu }}\left({Z_1\left(X\right)}^{†}\right)+g_{\mathrm{\mu },\mathrm{\nu }}\mathrm{\square }\left(\stackrel{\&conjugate0;}{F\left(X\right)}\right)$