====== LeviCivitaSimplify ======
----
====Description====
  * ''LeviCivitaSimplify'' simplifies combinations of Levi-Civita tensors. 


  * ''LeviCivitaSimplify.euclidean[eps]'' simplifies combinations of Levi-Civita tensors  (denoted as ''eps'') assuming that space is Euclidean. 


  * ''LeviCivitaSimplify.minkowski[eps]'' simplifies combinations of Levi-Civita tensors  (denoted as ''eps'') assuming that metric has signature {+, -, -, ...}. 


  * ''LeviCivitaSimplify'' works in arbitrary dimensions: if one specified e.g. tensor ''eps_{abcd}'' as Levi-Civita tensor (using e.g. ''LeviCivitaSimplify.euclidean['eps_{abcd}'.t]'' ) then space dimension will be considered equal to 4 (number of Levi-Civita indices) and Kronecker trace will be substituted (i.e. '''d^n_n = 4'.t'' will be applied).


  * When using ''LeviCivitaSimplify'' one should be ensured that symmetries of Levi-Civita tensor are already set up.
  
  * ''%%LeviCivitaSimplify[[Simplifications: tr]]%%'' will apply additional simplifications ''tr'' to each processed product of Levi-Civita tensors and their contractions

  * ''%%LeviCivitaSimplify[[OverallSimplifications: tr]]%%'' will apply additional simplifications ''tr'' to each processed transformed product of tensors
====Examples====

----
Simplify combinations of Levi-Civita tensors in dimension 3 in Euclidean space:
<sxh groovy; gutter: true>
setAntiSymmetric 'e_abc'
println LeviCivitaSimplify.euclidean['e_abc'] >> 'e_abc*e^abd'.t
</sxh>
<sxh plain; gutter: false>
   > 2*d^d_c
</sxh>
<sxh groovy; gutter: true; first-line: 3>
println LeviCivitaSimplify.euclidean['e_abc'] >> 'e_abc*e^abc'.t
</sxh>
<sxh plain; gutter: false>
   > 6
</sxh>
<sxh groovy; gutter: true; first-line: 4>
def t = 'e_abc*e^amd*e_mnk*e^bnk'.t
println LeviCivitaSimplify.euclidean['e_abc'] >> t
</sxh>
<sxh plain; gutter: false>
   > 4*d_{c}^{d}
</sxh>

----
Simplify combinations of Levi-Civita tensors in dimension 4 in Euclidean space:
<sxh groovy; gutter: true>
setAntiSymmetric 'e_abcd'
def t = '4*e^h_d^fb*e_abch*e_e^d_gf'.t
println LeviCivitaSimplify.euclidean['e_abcd'] >> t
</sxh>
<sxh plain; gutter: false>
   > 16*e_{eagc}
</sxh>
Simplify same expression in Minkowski space:
<sxh groovy; gutter: true>
setAntiSymmetric 'e_abcd'
def t = '4*e^h_d^fb*e_abch*e_e^d_gf'.t
println LeviCivitaSimplify.minkowski['e_abcd'] >> t
</sxh>
<sxh plain; gutter: false>
   > -16*e_{eagc}
</sxh>

----
Simplify combinations of Levi-Civita tensors in dimension 5 in Minkowski space:
<sxh groovy; gutter: true>
setAntiSymmetric 'e_abcde'
def t = '''e^{m}_{g}^{kci}*e_{pdj}^{l}_{o}*e_{c}^{n}_{mi}^{p}
           *e_{khnef}*e^{g}_{a}^{efd}*e_{l}^{hj}_{b}^{o}'''.t
println LeviCivitaSimplify.minkowski['e_abcde'] >> t
</sxh>
<sxh plain; gutter: false>
   > 864*g_{ab}
</sxh>
----
Simplify expression where Levi-Civita is contracted with symmetric tensor:
<sxh groovy; gutter: true>
setAntiSymmetric 'e_abcd'
def t = 'e_abcd*(A^a + C^a)*(A^b + C^b)'.t
println LeviCivitaSimplify.minkowski['e_abcd'] >> t
</sxh>
<sxh plain; gutter: false>
   > 0
</sxh>
----
====See also====
  * Related guides: [[documentation:guide:applying_and_manipulating_transformations]], [[documentation:guide:list_of_transformations]]
  * Related transformations: [[documentation:ref:DiracTrace]], [[documentation:ref:UnitaryTrace]]
  * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api//cc/redberry/physics/feyncalc/LeviCivitaSimplifyTransformation.html| LeviCivitaSimplifyTransformation]]
  * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/physics/src/main/java/cc/redberry/physics/feyncalc/LeviCivitaSimplifyTransformation.java|LeviCivitaSimplifyTransformation.java]]