compute the Hermitian conjugate or Adjoint of a given mathematical object

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

	Calling Sequence
	Dagger(A)

Parameters

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 .

•	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 is Hermitian, then return .

- If is already the Dagger of , then return .

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

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

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

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

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

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

- If is linear operator such as diff, d_, or dAlembertian, then return the linear operator applied to the Dagger of the first operand of .

- If 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

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

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

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Set the representation for Dirac matrices to be the standard one.

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Matrix(%id = 18446790986281764614)

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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.

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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).

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In the generic, non-Hermitian case:

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

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (x1, x2, x3, x4)}$

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	See Also
	Annihilation, AntiCommutator, Bra, Bracket, Commutator, Creation, d_, dAlembertian, g_, Ket, Physics, Physics conventions, Physics examples, Setup, value