By W.-H Steeb
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In the rest frame, p0 m = ±1) 3. Eigenvalue of σ · Sˆ in the rest frame. G. 4. 46) 56 CHAPTER 2. 2). 6. 14) Note that ψ and ψc transform in the same way under a Lorentz transformation. 17) Consequently, charge conjugation takes one from a positive energy solution associated with a charge e to a negative energy solution associated with a charge −e. 17). 4) Note that in Euclidean space, ψc = Cψ ∗T , and if ψ = ψc , then ψ = 0 (only way). e. it’s not always possible to have Majorana condition. 58 CHAPTER 2.
2n+1 ] = Tr[γµ1 . . γµ2n+1 γ5 γ5 ] ; γ5 γ5 = 1 Tr[γ5 γµ1 . . γµ2n+1 γ5 ] (Using property of traces) (−1)2n+1 Tr[γ5 γµ1 . . e. 3) eimt ψ40 Using Boosts: i S = exp − σµν ω µν → Can express in closed form. 4. 6) i ∴ S = exp − ωγ0 γ1 2 ω 0 σ1 = exp − 2 σ1 0 Recall, σ12 0 σ1 σ1 0 =1 → 2 = 1 0 0 1 2 ω 0 σ1 1 ω 1 ω 2 0 σ1 − + σ1 0 2 σ1 0 2! 2 3! 8) But tanh(ω) = −v, and recall the trig identities, sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y) ; ∴ sinh(2θ) = 2 sinh(θ) cosh(θ) cosh(2θ) = cosh2 (θ) + sinh2 (θ) 3 x=y=θ 52 CHAPTER 2.
E. it’s not always possible to have Majorana condition. 58 CHAPTER 2. 2) With time, the situation is slightly different. 7) Here, Φ (t ) = Φ(t) , A (t ) = −A(t). In going from t → −t, we have the same physical process, but using a camera that’s running backwards (sequence of events in reverse order). A Wigner time transformation takes us from ψ(t) to ψ (t ). 9) with 0 = any operator, T is anti-unitary, and T is an ordinary matrix. e. 7. TIME REVERSAL 59 1. ψ (t ) = T ψ ∗ (t) 2. T α∗ T −1 = −α 3.
A handbook of terms used in chaos and quantum chaos by W.-H Steeb