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 — documentation:ref:unitarytrace [2015/11/21 12:33] (current) Line 1: Line 1: + ====== 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: + + //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 + ​ + + > (1/2)*g_AB + ​ + + UnitaryTrace >> '​Tr[T_A*T_B*T_C]'​.t + ​ + + > (1/​4*I)*f_{CAB}+(1/​4)*d_{CAB}'​ + ​ + + UnitaryTrace >> '​Tr[T_A*T^A + 1]'.t + ​ + + > N-1/​2+(1/​2)*N**2 + ​ + + UnitaryTrace >> '​Tr[T_A*T_B*T_C*T^A]'​.t + ​ + + > (-(1/​4)*N**(-1)+(1/​4)*N)*g_{BC} + ​ + ---- + Use  another notation for SU(N) objects: + + //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 + ​ + + > (-1/​4*I)*n**(-1)*a_{n}^{ac}*p_{a}-(1/​4)*n**(-1)*p_{a}*s_{n}^{ac} + ​ + ====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]]