compute the Hermitian conjugate or Adjoint of a given mathematical object - Maple Programming Help

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Physics[Dagger] - compute the Hermitian conjugate or Adjoint of a given mathematical object

 Calling Sequence Dagger(A)

Parameters

 A - any mathematical expression

Description

 • 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).

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)
 > $\mathrm{Bra}\left(A,n\right)$
 $⟨\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{{A}}_{{n}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|$ (2)
 > $\mathrm{Dagger}\left(\right)$
 $\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{{A}}_{{n}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩$ (3)

You can also use the inert form of Dagger by prefixing the command's name with %.

 > $\mathrm{%Dagger}\left(\right)$
 ${\left|\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{{A}}_{{n}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right⟩}^{{†}}$ (4)
 > $\mathrm{value}\left(\right)$
 $⟨\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{{A}}_{{n}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|$ (5)
 > $\mathrm{%Dagger}\left(\mathrm{Bracket}\left(A,B\right)\right)$
 ${⟨\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{B}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}⟩}^{{†}}$ (6)
 > $\mathrm{value}\left(\right)$
 $⟨\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{B}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{A}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}⟩$ (7)

For Annihilation and Creation operators, Dagger return the dual, respectively.

 > $\mathrm{am}≔\mathrm{Annihilation}\left(A,1\right)$
 ${\mathrm{am}}{≔}{{a}}^{{-}}$ (8)
 > $\mathrm{ap}≔\mathrm{Creation}\left(A,1\right)$
 ${\mathrm{ap}}{≔}{{a}}^{{+}}$ (9)
 > $\mathrm{Dagger}\left(\mathrm{am}\right)$
 ${{a}}^{{+}}$ (10)
 > $\mathrm{Dagger}\left(\mathrm{ap}\right)$
 ${{a}}^{{-}}$ (11)

As a shortcut to Dagger(ap) you can also use ap^*

 > ${\mathrm{ap}}^{\mathrm{%H}}$
 ${{a}}^{{-}}$ (12)

The Dagger of Dirac matrices

 > $\mathrm{%Dagger}\left({\mathrm{Dgamma}}_{1}\right)$
 ${{{\mathrm{\gamma }}}_{{1}}}^{{†}}$ (13)
 > $\mathrm{value}\left(\right)$
 ${-}{{\mathrm{\gamma }}}_{{1}}$ (14)

As it happens with tensors of the Physics package, when the index has the value 0, it is automatically mapped into the spacetime dimension. So the ${\mathrm{gamma}}_{0}$ Dirac matrix is automatically represented by ${\mathrm{gamma}}_{4}$,

 > $\mathrm{%Dagger}\left({\mathrm{Dgamma}}_{0}\right)$
 ${{{\mathrm{\gamma }}}_{{4}}}^{{†}}$ (15)
 > $\mathrm{value}\left(\right)$
 ${{\mathrm{\gamma }}}_{{4}}$ (16)

For sums and products, Dagger maps itself over the operands, reversing the order of the arguments in the case of noncommutative products or scalar products. First set a prefix to identify noncommutative symbols.

 > $\mathrm{Setup}\left(\mathrm{noncommutativeprefix}=Z\right)$
 $\left[{\mathrm{noncommutativeprefix}}{=}\left\{{Z}\right\}\right]$ (17)
 > $\mathrm{Commutator}\left({Z}_{1},{Z}_{2}\right)$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{-}}$ (18)
 > $=\mathrm{expand}\left(\right)$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{-}}{=}{{Z}}_{{1}}{}{{Z}}_{{2}}{-}{{Z}}_{{2}}{}{{Z}}_{{1}}$ (19)
 > $\mathrm{Dagger}\left(\right)$
 ${\left[{{{Z}}_{{2}}}^{{†}}{,}{{{Z}}_{{1}}}^{{†}}\right]}_{{-}}{=}{{{Z}}_{{2}}}^{{†}}{}{{{Z}}_{{1}}}^{{†}}{-}{{{Z}}_{{1}}}^{{†}}{}{{{Z}}_{{2}}}^{{†}}$ (20)

Thus, the Dagger of an AntiCommutator of Hermitian operators is equal to itself (however, the product of two Hermitian operators is Hermitian only if they commute).

 > $\mathrm{AntiCommutator}\left({Z}_{1},{Z}_{2}\right)$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{+}}$ (21)
 > $=\mathrm{expand}\left(\right)$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{+}}{=}{{Z}}_{{1}}{}{{Z}}_{{2}}{+}{{Z}}_{{2}}{}{{Z}}_{{1}}$ (22)
 > $\mathrm{Dagger}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{Hermitian}$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{+}}{=}{{Z}}_{{1}}{}{{Z}}_{{2}}{+}{{Z}}_{{2}}{}{{Z}}_{{1}}$ (23)

In the generic, non-Hermitian case:

 > $\mathrm{Dagger}\left(\right)$
 ${\left[{{{Z}}_{{1}}}^{{†}}{,}{{{Z}}_{{2}}}^{{†}}\right]}_{{+}}{=}{{{Z}}_{{1}}}^{{†}}{}{{{Z}}_{{2}}}^{{†}}{+}{{{Z}}_{{2}}}^{{†}}{}{{{Z}}_{{1}}}^{{†}}$ (24)

For linear operators, differential and others, Dagger is applied to the first operand.

 > $\mathrm{Setup}\left(\mathrm{diff}=X\right)$
 ${\mathrm{* Partial match of \text{'}Physics:-diff\text{'} against keyword \text{'}differentiationvariables\text{'}}}$
 ${\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}}\left\{{X}{=}\left({\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}\right)\right\}$
 ${\mathrm{Systems of spacetime Coordinates are:}}\left\{{X}{=}\left({\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}\right)\right\}$
 ${\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}}\left\{{X}{=}\left({\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}\right)\right\}$
 $\left[{\mathrm{differentiationvariables}}{=}\left[{X}\right]\right]$ (25)
 > $\mathrm{d_}[\mathrm{μ}]\left(Z[1]\left(X\right)\right)\mathrm{d_}[\mathrm{ν}]\left(Z[2]\left(X\right)\right)+{\mathrm{g_}}_{\mathrm{μ},\mathrm{ν}}\mathrm{dAlembertian}\left(F\left(X\right)\right)$
 $\left({{\partial }}_{{\mathrm{\mu }}}{}\left({{Z}}_{{1}}{}\left({X}\right)\right)\right){}\left({{\partial }}_{{\mathrm{\nu }}}{}\left({{Z}}_{{2}}{}\left({X}\right)\right)\right){+}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{\mathrm{\square }}{}\left({F}{}\left({X}\right)\right)$ (26)
 > $\mathrm{Dagger}\left(\right)$
 $\left({{\partial }}_{{\mathrm{\nu }}}{}\left({{{Z}}_{{2}}{}\left({X}\right)}^{{†}}\right)\right){}\left({{\partial }}_{{\mathrm{\mu }}}{}\left({{{Z}}_{{1}}{}\left({X}\right)}^{{†}}\right)\right){+}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{\mathrm{\square }}{}\left(\stackrel{{&conjugate0;}}{{F}{}\left({X}\right)}\right)$ (27)
 > 
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