An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 15.1

Non-Abelian Gauge Invariance — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · non-abelian-gauge-invariance

Problem

**Brute-force computations in SU(3)SU(3).** The standard basis for the fundamental representation of SU(3)SU(3) is t1=12(010100000),t2=12(0−i0i00000),t3=12(1000−10000),t^1 = \frac{1}{2}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad t^2 = \frac{1}{2}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad t^3 = \frac{1}{2}\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, t4=12(001000100),t5=12(00−i000i00),t^4 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \quad t^5 = \frac{1}{2}\begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}, t6=12(000001010),t7=12(00000−i0i0),t8=123(10001000−2).t^6 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad t^7 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}, \quad t^8 = \frac{1}{2\sqrt{3}}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}. (a) Explain why there are exactly eight matrices in the basis. (b) Evaluate all the commutators of these matrices, to determine the structure constants of SU(3)SU(3). Show that, with the normalizations used here, fabcf^{abc} is totally antisymmetric. (This exercise is tedious; you may wish to check only a representative sample of the commutators.) (c) Check the orthogonality condition (15.78), and evaluate the constant C(r)C(r) for this representation. (d) Compute the quadratic Casimir operator C2(r)C_2(r) directly from its definition (15.92), and verify the relation (15.94) between C2(r)C_2(r) and C(r)C(r).

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