Lamia
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Hi.
General question: Is there a fixed way to find all invariant tensor for a generic representation?
Example problem: Suppose you search for all indipendent quartic interactions of a scalar octet field ## \phi^{a} ## in the adjoint representation of SU(3). They will be terms like
## L_{int}=\phi^{\dagger a}\phi^{b}\phi^{\dagger c}\phi^{d} T^{abcd}##,
where ##T^{abcd}## is an invariant tensor of ##8 \otimes8 \otimes8 \otimes8##.
So, using ##8 \otimes 8=1\oplus8\oplus27\oplus8\oplus10\oplus\bar{10}##, plus the fact that the adjoint is real, plus the fact that ##N \otimes \bar{N}=1\oplus...##, plus the fact that an invariant tersor exists for each siglets in irreps reduction, one can conclude that there are seven ##T^{abcd}## invariant tensor of ##8 \otimes8 \otimes8 \otimes8##. But what are they?
I'm trying to combine SU(3) main tensors, such as ##\delta^{ab}##, generators of algebra, ##d^{abc}## completely simmetric tensor, etc.., but i would like to know if is there a more systematic approach.
General question: Is there a fixed way to find all invariant tensor for a generic representation?
Example problem: Suppose you search for all indipendent quartic interactions of a scalar octet field ## \phi^{a} ## in the adjoint representation of SU(3). They will be terms like
## L_{int}=\phi^{\dagger a}\phi^{b}\phi^{\dagger c}\phi^{d} T^{abcd}##,
where ##T^{abcd}## is an invariant tensor of ##8 \otimes8 \otimes8 \otimes8##.
So, using ##8 \otimes 8=1\oplus8\oplus27\oplus8\oplus10\oplus\bar{10}##, plus the fact that the adjoint is real, plus the fact that ##N \otimes \bar{N}=1\oplus...##, plus the fact that an invariant tersor exists for each siglets in irreps reduction, one can conclude that there are seven ##T^{abcd}## invariant tensor of ##8 \otimes8 \otimes8 \otimes8##. But what are they?
I'm trying to combine SU(3) main tensors, such as ##\delta^{ab}##, generators of algebra, ##d^{abc}## completely simmetric tensor, etc.., but i would like to know if is there a more systematic approach.