1. Discrete symmetry in QM
2조 김동진

2. Symmetry Continuous symmetry Discrete symmetry Spatial Translation
𝑥 →𝑥+𝐶
Parity
𝑥 →−𝑥
Temporal Translation
𝑥 𝑡 →𝑥(𝑡+𝐶)
Discrete symmetry

3. T-symmetry 𝑥 →𝑥 𝑝 →−𝑝 ( 𝑑𝑥 𝑑𝑡 → 𝑑𝑥 𝑑 −𝑡 ) 𝑖 →−𝑖 𝑇𝑖 𝑇 −1 =−𝑖
T – asymmetry
weak force(CP violation)
second law of thermodynamics
Time Reversal Transformation
𝑇 :𝑡 →−𝑡
𝑥 →𝑥
𝑝 →−𝑝 ( 𝑑𝑥 𝑑𝑡 → 𝑑𝑥 𝑑 −𝑡 )
𝑖 →−𝑖 𝑇𝑖 𝑇 −1 =−𝑖
𝑇 :𝑎𝑛𝑡𝑖−𝑢𝑛𝑖𝑡𝑎𝑟𝑦

4. C-symmetry 𝐶𝜓 𝑞 =𝜓(−𝑞) Charge-conjugation Transformation 𝐶 :𝑞 →−𝑞
Violation C–symmetry
weak interaction
Charge-conjugation Transformation
𝐶 :𝑞 →−𝑞
𝐶𝜓 𝑞 =𝜓(−𝑞)

5. CP-violation 𝑝 →−𝑝 P-symmetry (parity transformation) P:𝑥 →−𝑥
Violation CP–symmetry
decays of neutral kaons
Standard Model
P-symmetry (parity transformation)
P:𝑥 →−𝑥
𝑝 →−𝑝
CP-symmetry (combination C and P)

6. CP-violation CPT symmetry Matter-Antimatter Imbalance CP-violation
Any Lorentz invariant local quantum field theory
with a Hermitian Hamiltonian must have CPT symmetry

7. PT-symmetric QM 𝑃 :𝑥→−𝑥, 𝑝→−𝑝 𝑇 :𝑥→ 𝑥, 𝑝→−𝑝, 𝑖→−𝑖
𝑃 :𝑥→−𝑥, 𝑝→−𝑝
𝑇 :𝑥→ 𝑥, 𝑝→−𝑝, 𝑖→−𝑖
𝑃 :𝑅𝑒 𝑥→−𝑅𝑒 𝑥, 𝐼𝑚 𝑥→−𝐼𝑚 𝑥
𝑅𝑒 𝑝 →−𝑅𝑒 𝑝, 𝐼𝑚 𝑝 →−𝐼𝑚 𝑝
𝑇 :𝑅𝑒 𝑥→𝑅𝑒 𝑥, 𝐼𝑚 𝑥→−𝐼𝑚 𝑥
𝑅𝑒 𝑝 →−𝑅𝑒 𝑝, 𝐼𝑚 𝑝 →𝐼𝑚 𝑝

8. PT-symmetric QM 𝐻= 𝑝 2 + 𝑥 2 (𝑖𝑥) 𝜖 (𝜖: 𝑟𝑒𝑎𝑙) Complex QM Hamiltonian
𝐻= 𝑝 2 + 𝑥 2 (𝑖𝑥) 𝜖 (𝜖: 𝑟𝑒𝑎𝑙)
Complex QM
Hamiltonian
1) 𝜖 ≥0
2) −1<𝜖<0
3) 𝜖 ≤−1
Conditional real energy spectrum

9. Thank you