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
withPhysics:
Setupmathematicalnotation=true
mathematicalnotation=true
BraA,n
An
Dagger
You can also use the inert form of Dagger by prefixing the command's name with %.
%Dagger
An†
value
%DaggerBracketA,B
A|B†
B|A
For Annihilation and Creation operators, Dagger return the dual, respectively.
am≔AnnihilationA,1
am≔a−
ap≔CreationA,1
ap≔a+
Daggeram
a+
Daggerap
a−
As a shortcut to Dagger(ap) you can also use ap^*
ap%H
The Dagger of Dirac matrices
%DaggerDgamma1
γ1†
−γ1
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 γ0 Dirac matrix is automatically represented by γ4,
%DaggerDgamma0
γ4†
γ4
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.
Setupnoncommutativeprefix=Z
noncommutativeprefix=Z
CommutatorZ1,Z2
Z1,Z2−
=expand
Z1,Z2−=Z1Z2−Z2Z1
Z2†,Z1†−=Z2†Z1†−Z1†Z2†
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).
AntiCommutatorZ1,Z2
Z1,Z2+
Z1,Z2+=Z1Z2+Z2Z1
DaggerassumingHermitian
In the generic, non-Hermitian case:
Z1†,Z2†+=Z1†Z2†+Z2†Z1†
For linear operators, differential and others, Dagger is applied to the first operand.
Setupdiff=X
Systems of spacetime coordinates are:X=x1,x2,x3,x4
Default differentiation variables for d_, D_ and dAlembertian are:X=x1,x2,x3,x4
_______________________________________________________
differentiationvariables=X
d_μZ1Xd_νZ2X+g_μ,νdAlembertianFX
∂μZ1X∂νZ2X+gμ,ν□FX
∂νZ2X†∂μZ1X†+gμ,ν□FX&conjugate0;
See Also
Annihilation, AntiCommutator, Bra, Bracket, Commutator, Creation, d_, dAlembertian, DiracConjugate, g_, Ket, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup, value
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