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| + | ====== UnitaryTrace ====== | ||
| + | ---- | ||
| + | |||
| + | ====Description==== | ||
| + | * ''%% UnitaryTrace%%'' calculates the trace of SU(N) matrices in expressions. | ||
| + | |||
| + | |||
| + | * By default ''UnitaryTrace'' uses notation ''T_A'' for SU(N) matrices, ''f_ABC'' for SU(N) structural constants, ''d_ABC'' for $d$-constatnts and ''N'' for dimension. | ||
| + | |||
| + | |||
| + | |||
| + | * ''%%UnitaryTrace[[Matrix: T, f: f, d: d, N:n]]%%'' specifies the notation for SU(N) matrices, structural constants, $d$-constants and dimension . | ||
| + | ====Examples==== | ||
| + | ---- | ||
| + | Calculate different traces: | ||
| + | <sxh groovy; gutter: true> | ||
| + | //set up matrices | ||
| + | defineMatrices 'T_A', Matrix2.matrix | ||
| + | //structure constants are antisymmetric | ||
| + | setAntiSymmetric 'f_ABC' | ||
| + | //d-constants are symmetric | ||
| + | setSymmetric 'd_ABC' | ||
| + | println UnitaryTrace >> 'Tr[T_A*T_B]'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > (1/2)*g_AB | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true; first-line: 8> | ||
| + | UnitaryTrace >> 'Tr[T_A*T_B*T_C]'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > (1/4*I)*f_{CAB}+(1/4)*d_{CAB}' | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true; first-line: 9> | ||
| + | UnitaryTrace >> 'Tr[T_A*T^A + 1]'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > N-1/2+(1/2)*N**2 | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true; first-line: 10> | ||
| + | UnitaryTrace >> 'Tr[T_A*T_B*T_C*T^A]'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > (-(1/4)*N**(-1)+(1/4)*N)*g_{BC} | ||
| + | </sxh> | ||
| + | ---- | ||
| + | Use another notation for SU(N) objects: | ||
| + | <sxh groovy; gutter: true> | ||
| + | //set up matrices | ||
| + | defineMatrices 'm_a', Matrix4.matrix | ||
| + | //structure constants are antisymmetric | ||
| + | setAntiSymmetric 'a_abc' | ||
| + | //d-constants are symmetric | ||
| + | setSymmetric 's_abc' | ||
| + | def uTrace = UnitaryTrace[[Matrix: 'm_a', f: 'a_abc', d: 's_abc', N: 'n']] | ||
| + | println uTrace >> 'Tr[m^a*m_b*m^c*(p^b*m_a + p_a*m^b)*m_n]'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > (-1/4*I)*n**(-1)*a_{n}^{ac}*p_{a}-(1/4)*n**(-1)*p_{a}*s_{n}^{ac} | ||
| + | </sxh> | ||
| + | ====See also==== | ||
| + | * Related guides: [[documentation:guide:applying_and_manipulating_transformations]], [[documentation:guide:Setting up matrix objects]], [[documentation:guide:list_of_transformations]] | ||
| + | * Related tutorials: [[documentation:tutorials:Compton scattering in QCD]] | ||
| + | * Related transformations: [[documentation:ref:unitarysimplify]], [[documentation:ref:diractrace]] | ||
| + | * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api//cc/redberry/physics/feyncalc/UnitaryTraceTransformation.html| UnitaryTraceTransformation]] | ||
| + | * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/physics/src/main/java/cc/redberry/physics/feyncalc/UnitaryTraceTransformation.java|UnitaryTraceTransformation.java]] | ||