New Vista On Excited States

1. New Vista On Excited States

2. Contents • Monte Carlo Hamiltonian: • Effective Hamiltonian in low • energy/temperature window

3. - Spectrum of excited states • - Wave functions • - Thermodynamical functions • - Klein-Gordon model • - Scalar φ^4 theory • - Gauge theory • Summary

4. Critical review of Lagrangian vs Hamiltonian LGT • Lagrangian LGT: • Standard approach- very sucessfull. • Compute vacuum-to-vacuum transition amplitudes • Limitation: Excited states spectrum, • Wave functions

5. Hamiltonian LGT: • Advantage: Allows in principle for computation of excited states spectra and wave functions. • BIG PROBLEM: To find a set of basis states which are physically relevant! • History of Hamilton LGT: - Basis states constructed from mathematical principles (like Hermite, Laguerre, Legendre fct in QM). BAD IDEA IN LGT!

6. Basis constructed via perturbation theory: Examples: Tamm-Dancoff, Discrete Light Cone Field Theory, …. BIASED CHOICE!

7. STOCHASTIC BASIS • 2 Principles: - Randomness: To construct states which sample a HUGH space random sampling is best. - Guidance by physics: Let physics tell us which states are important. Lesson: Use Monte Carlo with importance sampling! Result: Stochastic basis states. Analogy in Lagrangian LGT to eqilibrium configurations of path integrals guided by exp[-S].

8. Construction of Basis

9. Box Functions

10. Monte Carlo Hamiltonian H. Jirari, H. Kröger, X.Q. Luo, K.J.M. Moriarty, Phys. Lett. A258 (1999) 6. C.Q. Huang, H. Kröger,X.Q. Luo, K.J.M. Moriarty, Phys.Lett. A299 (2002) 483. Transition amplitudes between position states. Compute via path integral. Express as ratio of path integrals. Split action: S =S_0 + S_V

11. Diagonalize matrix Spectrum of energies and wave funtions Effective Hamiltonian

12. Many-body systems – Quantum field theory: Essential: Stochastic basis: Draw nodes x_i from probability distribution derived from physics – action. Path integral. Take x_i as position of paths generated by Monte Calo with importance sampling at a fixed time slice.

13. Thermodynamical functions: Definition: Lattice: Monte Carlo Hamiltonian:

14. Klein Gordon Model X.Q.Luo, H. Jirari, H. Kröger, K.J.M. Moriarty, Non-perturbative Methods and Lattice QCD, World Scientific Singapore (2001), p.100.

15. Energy spectrum

16. Free energy beta x F

17. Average energy U

18. Specific heat C/k_B

19. Scalar Model C.Q. Huang, H. Kröger, X.Q. Luo, K.J.M. Moriarty Phys.Lett. A299 (2002) 483.

20. Energy spectrum

21. Free energy F

22. Average energy U

23. Entropy S

24. Specific heat C

25. Lattice gauge theory

26. Principle: Physical states have to be gauge invariant! Construct stochastic basis of gauge invariant states.

27. Abelian U(1) gauge group. Analogy: Q.M. – Gauge theory l = number of links = index of irreducible representation.

28. Fourier Theorem – Peter Weyl Theorem

29. Transition amplitude between Bargman states

30. Transition amplitude between gauge invariant states

31. Result: • Gauss’ law at any vertex i: Plaquette angle:

32. Results From Electric Term…

33. Spectrum 1Plaquette

34. Spectrum 2 Plaquettes

35. Spectrum 4 Plaquettes

36. Spectrum 9 Plaquettes

37. Energy Scaling Window: 1 Plaquette

38. Energy scaling window (fixed basis)

39. Energy scaling window: 4 Plaq

40. 4 Plaquettes: a_s=1

41. Scaling Window: Wave Functions

42. Scaling: Energy vs.Wave Fct

43. Scaling: Energy vs. Wave Fct.

44. Average Energy U

45. Free Energy F

46. Entropy S

47. Specific Heat C

48. Including Magnetic Term…