What is the structure constant in layman's? In the Yang-Mills field strength tensor, there's this symbol f which is the structure constant. What is the definition of this structure constant in layman's terms?
 A: In the Yang-Mills theory describing QCD, the underlying symmetry group is the Lie group SU(3). The Lie algebra of a given Lie group is characterized by how the generators of the group commute. The commutation relations for the generators of a Lie group can be written as linear combinations of the other generators, with coefficients of this linear combination being called "structure constants".
For the group SU(3), the generators are $T^a = \frac{\lambda^a}{2}$, where the $\lambda^a$'s are the 8 Gell-Mann matrices. The commutation relations of these generators are given by $$[T^a, T^b] = i f^{abc}T^c,$$ where $f^{abc}$ is the totally antisymmetric symbol defined here.
Edit: In layman's terms, the structure constants are unique to a specific Lie group (continuous group). They contain valuable information about how the group's generators behave.
A: The structure constants are linear coefficients.
Consider a vector space $V$ with a basis $(\lambda_i)_{i\in I}$. If an additional bilinear operation $V\times V\rightarrow V$ is given on $V$, it is called algebra (see Wikipedia) or Lie algebra (see nlab or Wikipedia) depending on what additional conditions this bilinear operation fulfills. Lie algebras, whose bilinear operation is also called commutator, are very important for physics, especially quantum theory.
To fully describe the commutator, we only need to know the result of it being applied to pairs of basis vectors $(\lambda_i)_{i\in I}$ because of bilinearity. To get the structure constants, we then express the result as a linear combination using the basis vectors again:
$$[\lambda_a,\lambda_b]
=2if_{ab}^c\lambda_c.$$
You can compare this to the angular momentum algebra given by $[L_i,L_j]=\varepsilon_{ijk}L_k$. (A smiliar approach can also be found in other areas of mathematics and physics, the Christoffel symbols $\Gamma_{ij}^k$ are also linear coefficients for example.) In case of Yang-Mills theories, the Lie algebra considered is called the special unitary Lie algebra and defined as:
$$\mathfrak{su}(n)
:=\left\{A\in\mathbb{C}^{n\times n}|A^H=-A,\operatorname{tr}(A)=0\right\}.$$
For $\mathfrak{su}(2)$, a suitable basis are the three ($2^2-1$) Pauli matrices. For $\mathfrak{su}(3)$, a suitable basis are the eight ($3^2-1$) Gell-Mann matrices. This is the Lie algebra used in quantum chromodynamic, every basis vector (also called generator) corresponds to one of the eight quarks.
