Mathematical Principles of Theoretical Physics
3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 147 For the general form of Yang-Mills functional given by (3.4.10) F= ∫ M [ − 1 4 Gab ...
148 CHAPTER 3. MATHEMATICAL FOUNDATIONS It follows that (3.4.19) g ̃ij=−gikgjlg ̃kl. Thus, (3.4.18) is rewritten as (3.4.20) d d ...
3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 149 By the Riemannian Geometry, for each pointx 0 ∈Mthere exists a coordinate system u ...
150 CHAPTER 3. MATHEMATICAL FOUNDATIONS 3.4.4 Variational principle withdivA-free constraint LetMbe a closed manifold. A Riemann ...
3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 151 With (3.4.32), we can define the following derivative operators of the functionalF ...
152 CHAPTER 3. MATHEMATICAL FOUNDATIONS We are now in position to consider the variation with divA-free constraints. We know tha ...
3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 153 We have the following theorems for divA-free constraint variations. Theorem 3.26.L ...
154 CHAPTER 3. MATHEMATICAL FOUNDATIONS 3.4.5 Scalar potential theorem In Theorem3.26, if the vector fieldAinDAis zero, and the ...
3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 155 If the first Betti numberβ 1 (M) 6 =0, then we have the following theorem. Theorem ...
156 CHAPTER 3. MATHEMATICAL FOUNDATIONS It follows that the covector fieldsψj( 1 ≤j≤N)in (3.4.52) satisfy (3.4.55) ∆ψj= 0 for 1≤ ...
3.5.SU(N)REPRESENTATION INVARIANCE 157 For the gauge theory, a vary basic and important problem is that in the gauge transforma- ...
158 CHAPTER 3. MATHEMATICAL FOUNDATIONS andψhas a continuous inverseψ−^1 :Rn→U. The groupSU(N)consists of allN-th order unitary ...
3.5.SU(N)REPRESENTATION INVARIANCE 159 x^1 x^2 N 2 R^2 N 2 SU(N) xk A TASU(N) τA γ(t) Figure 3.2: Tangent spaceTASU(N)atA∈SU(N). ...
160 CHAPTER 3. MATHEMATICAL FOUNDATIONS is derived by (2.2.51) and detΩ=1 forΩas in (3.5.18). In particular, at the unit matrixe ...
3.5.SU(N)REPRESENTATION INVARIANCE 161 Definition 3.30(SU(N)Tensors).Let T be given as T={Tba 11 ······baji| 1 ≤ak,bl≤K=N^2 − 1 ...
162 CHAPTER 3. MATHEMATICAL FOUNDATIONS 2) SU(N)tensor gab. The structure constantsλbca generated by the generatorsωain (3.5.27) ...
3.5.SU(N)REPRESENTATION INVARIANCE 163 3.5.4 Intrinsic Riemannian metric onSU(N) By Theorem3.31,Gab(A)andgabare symmetric. We no ...
164 CHAPTER 3. MATHEMATICAL FOUNDATIONS The structure constants are λabc = 2 fabc, 1 ≤a,b,c≤ 8 , andfabcare anti-symmetric, give ...
3.5.SU(N)REPRESENTATION INVARIANCE 165 whereσk( 1 ≤k≤ 3 )are as in (3.5.36) andλj( 3 ≤j≤ 8 )as in (3.5.38). Corresponding to (3. ...
166 CHAPTER 3. MATHEMATICAL FOUNDATIONS In fact, the following are three terms in (3.5.44), which involve contractions ofSU(N)te ...
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