Mathematical Principles of Theoretical Physics
2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 67 Consequently, physical laws obeying Principle2.26of Einstein general relativity ...
68 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS 2.Scalar curvature. Again by contraction of the Ricci tensor with the metric ten ...
2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 69 Remark 2.30.The terms in (2.3.38) are Rμ ν− 1 2 gμ νR=δLR, LRas in (2.3.36), Tμ ...
70 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS and the constantλis usually called the cosmological constant. Today, a largenumb ...
2.4 Gauge Invariance However, we note that (2.4.8) ∂μψ ̃=∂μ(eiθψ) =eiθ(∂μ+i∂μθ)ψ, indicating that∂μis not covariant as defined b ...
72 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS whereFμ νis given by (2.4.4). The left-hand side of (2.4.13) is covariant under ...
2.4. GAUGE INVARIANCE 73 gis a coupling constant,τk( 1 ≤k≤N^2 − 1 )are the generators ofSU(N). We recall thatSU(N)is the group c ...
74 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS We remark here that the basis{τk| 1 ≤k≤K}of the tangent space (2.4.24) is usuall ...
2.4. GAUGE INVARIANCE 75 where m= m 1 0 .. . 0 mN (2.4.31) is the the mass matrix, (2.4.32) Dμ=∂μ+igGμaτa, andτa( 1 ...
76 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Usually, (2.4.37) is taken as an infinitesimal transformation as follows Ω=I+iθa ...
2.4. GAUGE INVARIANCE 77 Then, thanks to (2.4.41), we deduce that i g [ D ̃μ,D ̃ν ] Ψ ̃=F ̃μ νaτaΩΨ = i g Ω [ Dμ,Dν ] Ψ (by( 2. ...
78 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS whereλabc are the structure constants ofSU(N). In the classicalSU(N)gauge theory ...
2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 79 Remark 2.35.All current theories about interactions, concluding the Standard Mode ...
80 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS In classical mechanics, a physical motion system can be described by three dynam ...
2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 81 The total energyHof theN-body system is given by (2.5.7) H(x,y) = N ∑ k= 1 1 2 mk ...
82 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS 2.5.3 Classical electrodynamics Electrodynamics consists of two parts: the Maxwe ...
2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 83 It is clear that d dλ ∣ ∣ ∣ λ= 0 L 2 (Aμ+λA ̃μ) = 1 c ∫ QT d dλ (Aμ+λA ̃μ)Jμdxdt= ...
84 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS This is the second pair of the Maxwell equations (2.2.35) and (2.2.36). Since th ...
2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 85 By (2.5.23), we have ∇L= e c ∇~A·v+e∇A 0. Thanks to ∇~A·v=∇(~A·v) (by∂v= 0 ) =(~A ...
86 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS 2.5.4 Lagrangian actions in quantum mechanics PLD is also valid in quantum physi ...
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