ExpandPartialD
ExpandPartialD[exp]
expands noncommutative products of QuantumField
s and partial differentiation operators in exp
and applies the Leibniz rule.
By default the function assumes that there are no expressions outside of exp
on which the derivatives inside exp
could act. If this is not the case, please set the options LeftPartialD
or RIghtPartialD
to True
.
See also
Overview , ExplicitPartialD , LeftPartialD , LeftRightPartialD , PartialDRelations , RightPartialD , LeftRightNablaD , LeftRightNablaD2 , LeftNablaD , RightNablaD .
Examples
RightPartialD[ \ [ Mu]] . QuantumField[ A , LorentzIndex[ \ [ Mu]]] . QuantumField[ A , LorentzIndex[ \ [ Nu]]]
ExpandPartialD[ % ]
∂ ⃗ μ . A μ . A ν \vec{\partial }_{\mu }.A_{\mu }.A_{\nu } ∂ μ . A μ . A ν
A μ . ( ∂ μ A ν ) + ( ∂ μ A μ ) . A ν A_{\mu }.\left(\partial _{\mu }A_{\nu }\right)+\left(\partial _{\mu }A_{\mu }\right).A_{\nu } A μ . ( ∂ μ A ν ) + ( ∂ μ A μ ) . A ν
RightNablaD[ i ] . QuantumField[ A , LorentzIndex[ \ [ Mu]]] . QuantumField[ A , LorentzIndex[ \ [ Nu]]]
ExpandPartialD[ % ]
∇ ⃗ i . A μ . A ν \vec{\nabla }^i.A_{\mu }.A_{\nu } ∇ i . A μ . A ν
− A μ . ( ∂ i A ν ) − ( ∂ i A μ ) . A ν -A_{\mu }.\left(\partial _iA_{\nu }\right)-\left(\partial _iA_{\mu }\right).A_{\nu } − A μ . ( ∂ i A ν ) − ( ∂ i A μ ) . A ν
LeftRightPartialD[ \ [ Mu]] . QuantumField[ A , LorentzIndex[ \ [ Nu]]]
ExpandPartialD[ % ]
∂ ↔ μ . A ν \overleftrightarrow{\partial }_{\mu }.A_{\nu } ∂ μ . A ν
1 2 ( ( ∂ μ A ν ) − ∂ ← μ . A ν ) \frac{1}{2} \left(\left(\partial _{\mu }A_{\nu }\right)-\overleftarrow{\partial }_{\mu }.A_{\nu }\right) 2 1 ( ( ∂ μ A ν ) − ∂ μ . A ν )
LeftRightNablaD[ i ] . QuantumField[ A , LorentzIndex[ \ [ Nu]]]
ExpandPartialD[ % ]
∇ ↔ i . A ν \overleftrightarrow{\nabla }_i.A_{\nu } ∇ i . A ν
1 2 ( ∂ ← i . A ν − ( ∂ i A ν ) ) \frac{1}{2} \left(\overleftarrow{\partial }_i.A_{\nu }-\left(\partial _iA_{\nu }\right)\right) 2 1 ( ∂ i . A ν − ( ∂ i A ν ) )
QuantumField[ A , LorentzIndex[ \ [ Mu]]] . (LeftRightPartialD[ OPEDelta] ^ 2 ) . QuantumField[ A ,
LorentzIndex[ \ [ Rho]]]
ExpandPartialD[ % ]
A μ . ∂ ↔ Δ 2 . A ρ A_{\mu }.\overleftrightarrow{\partial }_{\Delta }^2.A_{\rho } A μ . ∂ Δ 2 . A ρ
1 4 ( A μ . ( ∂ Δ ∂ Δ A ρ ) − 2 ( ∂ Δ A μ ) . ( ∂ Δ A ρ ) + ( ∂ Δ ∂ Δ A μ ) . A ρ ) \frac{1}{4} \left(A_{\mu }.\left(\partial _{\Delta }\partial _{\Delta }A_{\rho }\right)-2 \left(\partial _{\Delta }A_{\mu }\right).\left(\partial _{\Delta }A_{\rho }\right)+\left(\partial _{\Delta }\partial _{\Delta }A_{\mu }\right).A_{\rho }\right) 4 1 ( A μ . ( ∂ Δ ∂ Δ A ρ ) − 2 ( ∂ Δ A μ ) . ( ∂ Δ A ρ ) + ( ∂ Δ ∂ Δ A μ ) . A ρ )
8 LeftRightPartialD[ OPEDelta] ^ 3
8 ∂ ↔ Δ 3 8 \overleftrightarrow{\partial }_{\Delta }^3 8 ∂ Δ 3
( ∂ ⃗ Δ − ∂ ← Δ ) 3 \left(\vec{\partial }_{\Delta }-\overleftarrow{\partial }_{\Delta }\right){}^3 ( ∂ Δ − ∂ Δ ) 3
− ∂ ← Δ . ∂ ← Δ . ∂ ← Δ + 3 ∂ ← Δ . ∂ ← Δ . ∂ ⃗ Δ − 3 ∂ ← Δ . ∂ ⃗ Δ . ∂ ⃗ Δ + ∂ ⃗ Δ . ∂ ⃗ Δ . ∂ ⃗ Δ -\overleftarrow{\partial }_{\Delta }.\overleftarrow{\partial }_{\Delta }.\overleftarrow{\partial }_{\Delta }+3 \overleftarrow{\partial }_{\Delta }.\overleftarrow{\partial }_{\Delta }.\vec{\partial }_{\Delta }-3 \overleftarrow{\partial }_{\Delta }.\vec{\partial }_{\Delta }.\vec{\partial }_{\Delta }+\vec{\partial }_{\Delta }.\vec{\partial }_{\Delta }.\vec{\partial }_{\Delta } − ∂ Δ . ∂ Δ . ∂ Δ + 3 ∂ Δ . ∂ Δ . ∂ Δ − 3 ∂ Δ . ∂ Δ . ∂ Δ + ∂ Δ . ∂ Δ . ∂ Δ
LC[ \ [ Mu], \ [ Nu], \ [ Rho], \ [ Tau]] RightPartialD[ \ [ Alpha], \ [ Mu], \ [ Beta ], \ [ Nu]]
ExpandPartialD[ % ]
ϵ ˉ μ ν ρ τ ∂ ⃗ α . ∂ ⃗ μ . ∂ ⃗ β . ∂ ⃗ ν \bar{\epsilon }^{\mu \nu \rho \tau } \vec{\partial }_{\alpha }.\vec{\partial }_{\mu }.\vec{\partial }_{\beta }.\vec{\partial }_{\nu } ϵ ˉ μν ρ τ ∂ α . ∂ μ . ∂ β . ∂ ν
0 0 0
CLC[ i , j , k ] RightNablaD[ i , j , k ]
ExpandPartialD[ % ]
ϵ ˉ i j k ∇ ⃗ i . ∇ ⃗ j . ∇ ⃗ k \bar{\epsilon }^{ijk} \vec{\nabla }^i.\vec{\nabla }^j.\vec{\nabla }^k ϵ ˉ ijk ∇ i . ∇ j . ∇ k
0 0 0
RightPartialD[ CartesianIndex[ i ]] . QuantumField[ S , x ]
% // ExpandPartialD
∂ ⃗ i . S x \vec{\partial }_i.S^x ∂ i . S x
( ∂ i S x ) \left(\partial _iS^x\right) ( ∂ i S x )
RightPartialD[{ CartesianIndex[ i ], x }] . QuantumField[ S , x ]
% // ExpandPartialD
∂ ⃗ { i , x } . S x \vec{\partial }_{\{i,x\}}.S^x ∂ { i , x } . S x
( ∂ { i , x } S x ) \left(\partial _{\{i,x\}}S^x\right) ( ∂ { i , x } S x )
By default the derivative won’t act on anything outside of the input expression. But it can be made to by setting the option RightPartialD
to True
ExpandPartialD[ RightPartialD[ \ [ Mu]] . QuantumField[ A , LorentzIndex[ \ [ Mu]]] . QuantumField[ A , LorentzIndex[ \ [ Nu]]]]
A μ . ( ∂ μ A ν ) + ( ∂ μ A μ ) . A ν A_{\mu }.\left(\partial _{\mu }A_{\nu }\right)+\left(\partial _{\mu }A_{\mu }\right).A_{\nu } A μ . ( ∂ μ A ν ) + ( ∂ μ A μ ) . A ν
ExpandPartialD[ RightPartialD[ \ [ Mu]] . QuantumField[ A , LorentzIndex[ \ [ Mu]]] . QuantumField[ A , LorentzIndex[ \ [ Nu]]], RightPartialD -> True ]
A μ . ( ∂ μ A ν ) + ( ∂ μ A μ ) . A ν + A μ . A ν . ∂ ⃗ μ A_{\mu }.\left(\partial _{\mu }A_{\nu }\right)+\left(\partial _{\mu }A_{\mu }\right).A_{\nu }+A_{\mu }.A_{\nu }.\vec{\partial }_{\mu } A μ . ( ∂ μ A ν ) + ( ∂ μ A μ ) . A ν + A μ . A ν . ∂ μ
The same applies also to LeftPartialD
ExpandPartialD[ QuantumField[ A , LorentzIndex[ \ [ Nu]]] . LeftNablaD[ i ]]
− ( ∂ i A ν ) -\left(\partial _iA_{\nu }\right) − ( ∂ i A ν )
ExpandPartialD[ QuantumField[ A , LorentzIndex[ \ [ Nu]]] . LeftNablaD[ i ], LeftPartialD -> True ]
− ( ∂ i A ν ) − ∂ ← i . A ν -\left(\partial _iA_{\nu }\right)-\overleftarrow{\partial }_i.A_{\nu } − ( ∂ i A ν ) − ∂ i . A ν