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Physics[DiracConjugate] - compute the Dirac conjugate a given mathematical expression

 Calling Sequence DiracConjugate(psi)

Parameters

 psi - any mathematical expression, possibly involving spinors or matrices

Description

 • The DiracConjugate command represents and computes the Dirac conjugate of its argument; the returned result is built as follows:
 - If $\mathrm{\psi }=z$ is a scalar, return its conjugate, $\stackrel{&conjugate0;}{z}$.
 - If $\mathrm{\psi }$ is a spinor, so defined with one spinor index using the Define command, or $\mathrm{\psi }$ is an anticommutative quantum operator (see Setup) with a spinor index, then return unevaluated, as DiracConjugate(psi), displayed as $\overline{\mathrm{\psi }}$, representing $\overline{\mathrm{\psi }}={\mathrm{\psi }}^{†}·{\mathrm{\gamma }}^{0}$, where ${\mathrm{\psi }}^{†}$ is the Hermitian conjugate Dagger(psi) and ${\mathrm{\gamma }}^{0}$ is the contravariant ${0}^{\mathrm{th}}$ Dirac matrix Dgamma[~0].
 - If $\mathrm{\psi }=\overline{A}$ is the Dirac conjugate of - say -  $A$, then return $A$.
 - If $\mathrm{\psi }={\mathrm{\gamma }}^{\mathrm{\mu }}$ is a Dirac matrix (represented by the Dgamma command), then return the matrix ${\mathrm{\gamma }}^{\mathrm{\mu }}$ itself, also when $\mathrm{\mu }=5$.
 - If $\mathrm{\psi }=M$ is a Matrix - say $M$ - return the matrix product ${\mathrm{\gamma }}^{0}·{M}^{†}·{\mathrm{\gamma }}^{0}$, where ${M}^{†}$ is the Dagger(M), the Hermitian conjugate of $M$.
 - If $\mathrm{\psi }$ is a sum of terms, return the sum of the DiracConjugate of each term.
 - If $\mathrm{\psi }=A\mathrm{...}B$ is a product, return $\overline{B}\mathrm{...}\overline{A}$, that is the product of the DiracConjugate of each of the factors with the ordering reversed.
 - If $\mathrm{\psi }$ is one of the d_ or dAlembertian operators, return the operator applied to the DiracConjugate of the first operand of $\mathrm{\psi }$.
 - Otherwise, return the operation unevaluated, DiracConjugate(A).
 • The %DiracConjugate command is the inert form of DiracConjugate; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command.

 • After loading Physics, you indicate the letter to represent a spinor index using the Setup command, for example: Setup(spinorindices = lowercaselatin). A spinor is then any symbol indexed with one spinor index, and defined as a tensor using Define. If the symbol is anticommutative, and has one spinor index, and was defined as a tensor using Define or alternatively was set as a quantum operator using Setup, then it represents a DiracSpinor,
 • You can check whether say ${\mathrm{\psi }}_{j}$ is a spinor using the Library:-PhysicsType:-Spinor, as in type(psi[j], Library:-PhysicsType:-Spinor). Likewise, you can check if the object is a DiracSpinor using the Library:-PhysicsType:-DiracSpinor. Note that the object can have more than one index, but to be a spinor (Dirac or not), only one of them must be a spinor index. The related types Library:-PhysicsType:-SpinorWithoutIndices and Library:-PhysicsType:-DiracSpinorWithoutIndices return true when the argument passed to them, say $\mathrm{\psi }$, has no indices but would be a Spinor if it had a spinor index.
 • Any tensor with more than one spinor index, is not "a spinor" but it is spinorial, for the example the Dirac matrices $\mathrm{\left(γ μ μ\right)j,k}$. You can check whether an object is spinorial using the Library:-IsSpinorial command, say as in Library:-IsSpinorial(Dgamma[mu][j,k]), which will returns true in this case, and false when the object is not spinorial. Likewise, a sum of spinorial objects, or a product of them with at least one free spinor index, is considered spinorial, for which Library:-IsSpinorial returns true.

Examples

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

The Dirac conjugate of a scalar is the standard conjugate

 > $\mathrm{DiracConjugate}\left(z\right)$
 $\stackrel{{&conjugate0;}}{{z}}$ (1)

In general, the Dirac conjugate of the Dirac conjugate of an object is the object itself

 > $\mathrm{DiracConjugate}\left(\right)$
 ${z}$ (2)

The Dirac conjugate of a Dirac matrix is the Dirac matrix itself

 > $\left(\mathrm{%DiracConjugate}=\mathrm{DiracConjugate}\right)\left(\mathrm{Dgamma}\left[\mathrm{\mu }\right]\right)$
 $\overline{{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}}{=}{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}$ (3)

Note the display is different from the display of conjugate: the above has the bar in black and bold, instead of blue and thin as in (1)

The Dirac conjugate of ${\mathrm{\gamma }}_{5}$ is also equal to itself

 > $\left(\mathrm{%DiracConjugate}=\mathrm{DiracConjugate}\right)\left(\mathrm{Dgamma}\left[5\right]\right)$
 $\overline{{{\mathrm{\gamma }}}_{{5}}}{=}{{\mathrm{\gamma }}}_{{5}}$ (4)

Set coordinates, a quantum operator, an anticommutative prefix and a kind of letter to represent spinor indices

 > $\mathrm{Setup}\left(\mathrm{coordinates}=\mathrm{cartesian},\mathrm{quantumoperator}=A,\mathrm{anticommutativeprefix}=B,\mathrm{spinorindices}=\mathrm{lowercaselatin}\right)$
 $\mathrm{* Partial match of \text{'}}\mathrm{quantumoperator}\mathrm{\text{'} against keyword \text{'}}\mathrm{quantumoperators}\text{'}$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{B}\right\}{,}{\mathrm{coordinatesystems}}{=}\left\{{X}\right\}{,}{\mathrm{quantumoperators}}{=}\left\{{A}\right\}{,}{\mathrm{spinorindices}}{=}{\mathrm{lowercaselatin}}\right]$ (5)

Define then one spinor using the $B$ anticommutative prefix and also a generic noncommutative spinor ${A}_{j}$

 > $\mathrm{Define}\left(A\left[j\right],B\left[j\right]\right)$
 $\mathrm{Defined objects with tensor properties}$
 $\left\{{{A}}_{{j}}{,}{{B}}_{{j}}{,}{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}{,}{{\mathrm{\sigma }}}_{{\mathrm{\mu }}}{,}{{\partial }}_{{\mathrm{\mu }}}{,}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\epsilon }}}_{{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{X}}_{{\mathrm{\mu }}}\right\}$ (6)

Take their product

 > $A\left[j\right]B\left[j\right]$
 ${{A}}_{{j}}{}{{B}}_{{j}}$ (7)

Sum over the repeated indices, then take the Dirac conjugate of the sum

 > $\mathrm{SumOverRepeatedIndices}\left(\right)$
 ${{A}}_{{1}}{}{{B}}_{{1}}{+}{{A}}_{{2}}{}{{B}}_{{2}}{+}{{A}}_{{3}}{}{{B}}_{{3}}{+}{{A}}_{{4}}{}{{B}}_{{4}}$ (8)
 > $\mathrm{DiracConjugate}\left(\right)$
 $\stackrel{{&conjugate0;}}{{{B}}_{{1}}}{}\stackrel{{&conjugate0;}}{{{A}}_{{1}}}{+}\stackrel{{&conjugate0;}}{{{B}}_{{2}}}{}\stackrel{{&conjugate0;}}{{{A}}_{{2}}}{+}\stackrel{{&conjugate0;}}{{{B}}_{{3}}}{}\stackrel{{&conjugate0;}}{{{A}}_{{3}}}{+}\stackrel{{&conjugate0;}}{{{B}}_{{4}}}{}\stackrel{{&conjugate0;}}{{{A}}_{{4}}}$ (9)

This result is expressed in terms of the conjugate of the spinor components of $A$ and $B$. Reversing the order of operations results in the same: take first the Dirac conjugate of the product ${A}_{j}{B}_{j}$, then sum over the repeated indices

 > $\mathrm{DiracConjugate}\left(\right)$
 ${\overline{{B}}}_{{j}}{}{\overline{{A}}}_{{j}}$ (10)
 > $\mathrm{SumOverRepeatedIndices}\left(\right)$
 $\stackrel{{&conjugate0;}}{{{B}}_{{1}}}{}\stackrel{{&conjugate0;}}{{{A}}_{{1}}}{+}\stackrel{{&conjugate0;}}{{{B}}_{{2}}}{}\stackrel{{&conjugate0;}}{{{A}}_{{2}}}{+}\stackrel{{&conjugate0;}}{{{B}}_{{3}}}{}\stackrel{{&conjugate0;}}{{{A}}_{{3}}}{+}\stackrel{{&conjugate0;}}{{{B}}_{{4}}}{}\stackrel{{&conjugate0;}}{{{A}}_{{4}}}$ (11)

Unlike conjugate, DiracConjugate allows for constructing true scalars using contracted products of spinors

 > $\mathrm{DiracConjugate}\left(B\right)B$
 $\overline{{B}}{}{B}$ (12)
 > $\mathrm{DiracConjugate}\left(\right)$
 $\overline{{B}}{}{B}$ (13)
 > $\mathrm{DiracConjugate}\left(B\left[j\right]\right)B\left[j\right]$
 ${\overline{{B}}}_{{j}}{}{{B}}_{{j}}$ (14)
 > $\mathrm{DiracConjugate}\left(\right)$
 ${\overline{{B}}}_{{j}}{}{{B}}_{{j}}$ (15)

The Dirac conjugate of a Matrix

 > $M≔\mathrm{Matrix}\left(4,\mathrm{symbol}=m\right)$
 ${M}{≔}\left[\right]$ (16)

The output involves the conjugates of the components of the transpose of $M$ multiplied at both sides by the Dirac matrix ${\mathrm{\gamma }}^{0}$

 > $\mathrm{DiracConjugate}\left(M\right)$
 $\left[\right]$ (17)

If the matrix components are real,

 > $\mathrm{DiracConjugate}\left(M\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{real}$
 $\left[\right]$ (18)
 > $M≔\mathrm{Matrix}\left(4,\left(i,j\right)↦A\left[i\right]\cdot B\left[j\right]\right)$
 ${M}{≔}\left[\right]$ (19)
 > $\mathrm{DiracConjugate}\left(M\right)$
 $\left[\right]$ (20)

The Lagrangian of QED: to load the StandardModel package, clear first the letters used to represent spinor indices

 > $\mathrm{Setup}\left(\mathrm{spinorindices}=\mathrm{none}\right):$
 > $\mathrm{with}\left(\mathrm{StandardModel}\right)$
 ${}\mathrm{_______________________________________________________}$
 $\mathrm{Setting}\mathrm{lowercaselatin_is}\mathrm{letters to represent}\mathrm{Dirac spinor}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{lowercaselatin_ah}\mathrm{letters to represent}\mathrm{SU\left(3\right) adjoint representation, \left(1..8\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercaselatin_ah}\mathrm{letters to represent}\mathrm{SU\left(3\right) fundamental representation, \left(1..3\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercaselatin_is}\mathrm{letters to represent}\mathrm{SU\left(2\right) adjoint representation, \left(1..3\right)}\mathrm{indices}$
 $\mathrm{Setting}\mathrm{uppercasegreek}\mathrm{letters to represent}\mathrm{SU\left(2\right) fundamental representation, \left(1..2\right)}\mathrm{indices}$
 ${}\mathrm{_______________________________________________________}$
 $\mathrm{Defined as the electron, muon and tau leptons and corresponding neutrinos:}{{\mathrm{e}}}_{{i}},{{\mathrm{\mu }}}_{{i}},{{\mathrm{\tau }}}_{{i}},{{{\mathrm{\nu }}}^{\left({\mathrm{e}}\right)}}_{{i}},{{{\mathrm{\nu }}}^{\left({\mathrm{\mu }}\right)}}_{{i}},{{{\mathrm{\nu }}}^{\left({\mathrm{\tau }}\right)}}_{{i}}$
 $\mathrm{Defined as the up, charm, top, down, strange and bottom quarks:}{{\mathrm{u}}}_{{A}{,}{i}},{{\mathrm{c}}}_{{A}{,}{i}},{{\mathrm{t}}}_{{A}{,}{i}},{{\mathrm{d}}}_{{A}{,}{i}},{{\mathrm{s}}}_{{A}{,}{i}},{{\mathrm{b}}}_{{A}{,}{i}}$
 $\mathrm{Defined as gauge tensors:}{{\mathrm{B}}}_{{\mathrm{\mu }}},{{\mathrm{𝔹}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}},{{\mathrm{A}}}_{{\mathrm{\mu }}},{{\mathrm{𝔽}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}},{{\mathrm{W}}}_{{\mathrm{\mu }}{,}{J}},{{\mathrm{𝕎}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{J}},{{{\mathrm{W}}}^{{\mathrm{+}}}}_{{\mathrm{\mu }}},{{{\mathrm{𝕎}}}^{{\mathrm{+}}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}},{{{\mathrm{W}}}^{{\mathrm{-}}}}_{{\mathrm{\mu }}},{{{\mathrm{𝕎}}}^{{\mathrm{-}}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}},{{\mathrm{Z}}}_{{\mathrm{\mu }}},{{\mathrm{ℤ}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}},{{\mathrm{G}}}_{{\mathrm{\mu }}{,}{a}},{{\mathrm{𝔾}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{a}}$
 $\mathrm{Defined as Gell-Mann \left(Glambda\right), Pauli \left(Psigma\right) and Dirac \left(Dgamma\right) matrices:}{{\mathrm{\lambda }}}_{{a}},{{\mathrm{\sigma }}}_{{J}},{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}$
 $\mathrm{Defined as the electric, weak and strong coupling constants:}\mathrm{g__e},\mathrm{g__w},\mathrm{g__s}$
 $\mathrm{Defined as the charge in units of |}\mathrm{g__e}\mathrm{| for 1\right) the electron, muon and tauon, 2\right) the up, charm and top, and 3\right) the down, strange and bottom:}\mathrm{q__e}={-1},\mathrm{q__u}=\frac{{2}}{{3}},\mathrm{q__d}={-}\frac{{1}}{{3}}$
 $\mathrm{Defined as the weak isospin for 1\right) the electron, muon and tauon, 2\right) the up, charm and top, 3\right) the down, strange and bottom, and 4\right) all the neutrinos:}\mathrm{I__e}={-}\frac{{1}}{{2}},\mathrm{I__u}=\frac{{1}}{{2}},\mathrm{I__d}={-}\frac{{1}}{{2}},\mathrm{I__n}=\frac{{1}}{{2}}$
 $\mathrm{You can use the active form without the % prefix, or the \text{'}value\text{'} command to give the corresponding value to any of the inert representations}\mathrm{q__e},\mathrm{q__u},\mathrm{q__d},\mathrm{I__e},\mathrm{I__u},\mathrm{I__d},\mathrm{I__n}$
 ${}\mathrm{_______________________________________________________}$
 ${}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\mathrm{Minkowski spacetime with signatre}\left(\mathrm{- - - +}\right)$
 ${}\mathrm{_______________________________________________________}$
 $\left[\mathrm{I__d}{,}\mathrm{I__e}{,}\mathrm{I__n}{,}\mathrm{I__u}{,}\mathrm{q__d}{,}\mathrm{q__e}{,}\mathrm{q__u}{,}{\mathrm{BField}}{,}{\mathrm{BFieldStrength}}{,}{\mathrm{Bottom}}{,}{\mathrm{CKM}}{,}{\mathrm{Charm}}{,}{\mathrm{Down}}{,}{\mathrm{ElectromagneticField}}{,}{\mathrm{ElectromagneticFieldStrength}}{,}{\mathrm{Electron}}{,}{\mathrm{ElectronNeutrino}}{,}{\mathrm{FSU3}}{,}{\mathrm{Glambda}}{,}{\mathrm{GluonField}}{,}{\mathrm{GluonFieldStrength}}{,}{\mathrm{HiggsBoson}}{,}{\mathrm{Lagrangian}}{,}{\mathrm{Muon}}{,}{\mathrm{MuonNeutrino}}{,}{\mathrm{Strange}}{,}{\mathrm{Tauon}}{,}{\mathrm{TauonNeutrino}}{,}{\mathrm{Top}}{,}{\mathrm{Up}}{,}{\mathrm{WField}}{,}{\mathrm{WFieldStrength}}{,}{\mathrm{WMinusField}}{,}{\mathrm{WMinusFieldStrength}}{,}{\mathrm{WPlusField}}{,}{\mathrm{WPlusFieldStrength}}{,}{\mathrm{WeinbergAngle}}{,}{\mathrm{ZField}}{,}{\mathrm{ZFieldStrength}}{,}\mathrm{g__e}{,}\mathrm{g__s}{,}\mathrm{g__w}\right]$ (21)
 > $\mathrm{CompactDisplay}\left(\mathrm{Electron}\left(X\right)\right)$
 ${\mathbf{e}}{}\left({x}{,}{y}{,}{z}{,}{t}\right){}{\mathrm{will now be displayed as}}{}{\mathbf{e}}$ (22)
 > $\mathrm{Lagrangian}\left(\mathrm{QED}\right)$
 ${\overline{{\mathbf{e}}}}_{{i}}{}\left({I}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{i}{,}{j}}{}{{▿}}_{{\mathrm{\mu }}}{-}{{m}}_{{\mathbf{e}}}{}{{\mathrm{\delta }}}_{{i}{,}{j}}\right){}{{\mathbf{e}}}_{{j}}{-}\frac{{{\mathbf{𝔽}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{{\mathbf{𝔽}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}{,}{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\mu }}{,}{\mathrm{\nu }}}}{{4}}$ (23)

This Lagrangian is a scalar, constructed as a sum of products, where each term and each product involves noncommutative objects; the first term includes the contracted spinor product of the DiracConjugate of the electron field ${{\mathbf{e}}}_{j}$. Computing the Dirac conjugate of this Lagrangian is thus expected to result in several intermediate computations such that, at the end, the result is the same Lagrangian

 > $\mathrm{DiracConjugate}\left(\right)$
 $\overline{{{\mathbf{e}}}_{{j}}}{}\left({-}{I}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{j}{,}{i}}{}\stackrel{{&conjugate0;}}{{{▿}}_{{\mathrm{\mu }}}}{-}\overline{{{\mathrm{\delta }}}_{{i}{,}{j}}}{}{{m}}_{{\mathbf{e}}}\right){}{{\mathbf{e}}}_{{i}}{-}\frac{{{\mathbf{𝔽}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}{,}{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{{\mathbf{𝔽}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}}{{4}}$ (24)

For Annihilation and Creation operators, DiracConjugate returns the same as the Dagger command, that is the dual, respectively.

 > $\mathrm{am}≔\mathrm{Annihilation}\left(A,1\right)$
 ${\mathrm{am}}{≔}{{a}}^{{-}}$ (25)
 > $\mathrm{ap}≔\mathrm{Creation}\left(A,1\right)$
 ${\mathrm{ap}}{≔}{{a}}^{{+}}$ (26)
 > $\mathrm{DiracConjugate}\left(\mathrm{am}\right)$
 ${{a}}^{{+}}$ (27)
 > $\mathrm{DiracConjugate}\left(\mathrm{ap}\right)$
 ${{a}}^{{-}}$ (28)

DiracConjugate understands Commutator and AntiCommutator

 > $\mathrm{Setup}\left(\mathrm{noncommutativeprefix}=Z\right)$
 $\left[{\mathrm{noncommutativeprefix}}{=}\left\{{Z}\right\}\right]$ (29)
 > $\mathrm{Commutator}\left(Z\left[1\right],Z\left[2\right]\right)$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{-}}$ (30)
 > $=\mathrm{expand}\left(\right)$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{-}}{=}{{Z}}_{{1}}{}{{Z}}_{{2}}{-}{{Z}}_{{2}}{}{{Z}}_{{1}}$ (31)
 > $\mathrm{DiracConjugate}\left(\right)$
 ${\left[{{{Z}}_{{2}}}^{{†}}{,}{{{Z}}_{{1}}}^{{†}}\right]}_{{-}}{=}{{{Z}}_{{2}}}^{{†}}{}{{{Z}}_{{1}}}^{{†}}{-}{{{Z}}_{{1}}}^{{†}}{}{{{Z}}_{{2}}}^{{†}}$ (32)

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

 > $\mathrm{AntiCommutator}\left(Z\left[1\right],Z\left[2\right]\right)$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{+}}$ (33)
 > $=\mathrm{expand}\left(\right)$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{+}}{=}{{Z}}_{{1}}{}{{Z}}_{{2}}{+}{{Z}}_{{2}}{}{{Z}}_{{1}}$ (34)
 > $\mathrm{DiracConjugate}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Hermitian}$
 ${\left[{{Z}}_{{1}}{,}{{Z}}_{{2}}\right]}_{{+}}{=}{{Z}}_{{1}}{}{{Z}}_{{2}}{+}{{Z}}_{{2}}{}{{Z}}_{{1}}$ (35)

In the generic, non-Hermitian case:

 > $\mathrm{DiracConjugate}\left(\right)$
 ${\left[{{{Z}}_{{1}}}^{{†}}{,}{{{Z}}_{{2}}}^{{†}}\right]}_{{+}}{=}{{{Z}}_{{1}}}^{{†}}{}{{{Z}}_{{2}}}^{{†}}{+}{{{Z}}_{{2}}}^{{†}}{}{{{Z}}_{{1}}}^{{†}}$ (36)

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

 > $\mathrm{Setup}\left(\mathrm{diff}=X\right)$
 ${}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{differentiationvariables}}{=}\left[{X}\right]\right]$ (37)
 > $\mathrm{d_}\left[\mathrm{\mu }\right]\left(Z\left[1\right]\left(X\right)\right)\mathrm{d_}\left[\mathrm{\nu }\right]\left(Z\left[2\right]\left(X\right)\right)+\mathrm{g_}\left[\mathrm{\mu },\mathrm{\nu }\right]\mathrm{dAlembertian}\left(F\left(X\right)\right)$
 ${{\partial }}_{{\mathrm{\mu }}}{}\left({{Z}}_{{1}}{}\left({X}\right)\right){}{{\partial }}_{{\mathrm{\nu }}}{}\left({{Z}}_{{2}}{}\left({X}\right)\right){+}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{\mathrm{\square }}{}\left({F}{}\left({X}\right)\right)$ (38)
 > $\mathrm{DiracConjugate}\left(\right)$
 ${{\partial }}_{{\mathrm{\nu }}}{}\left({{{Z}}_{{2}}{}\left({X}\right)}^{{†}}\right){}{{\partial }}_{{\mathrm{\mu }}}{}\left({{{Z}}_{{1}}{}\left({X}\right)}^{{†}}\right){+}{\mathrm{\square }}{}\left(\stackrel{{&conjugate0;}}{{F}{}\left({X}\right)}\right){}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}$ (39)

Compatibility

 • The Physics[DiracConjugate] command was introduced in Maple 2024.
 • For more information on Maple 2024 changes, see Updates in Maple 2024.