New Vista On Excited States
Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window
- Spectrum of excited states - Wave functions - Thermodynamical functions - Klein-Gordon model - Scalar φ^4 theory - Gauge theory Summary
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
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!
-Basis constructed via perturbation theory: Examples: Tamm-Dancoff, Discrete Light Cone Field Theory, …. BIASED CHOICE!
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].
Construction of Basis
Box Functions
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
Diagonalize matrix Spectrum of energies and wave funtions Effective Hamiltonian
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.
Thermodynamical functions: Definition: Lattice: Monte Carlo Hamiltonian:
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.
Energy spectrum
Free energy beta x F
Average energy U
Specific heat C/k_B
Scalar Model C.Q. Huang, H. Kröger, X.Q. Luo, K.J.M. Moriarty Phys.Lett. A299 (2002) 483.
Energy spectrum
Free energy F
Average energy U
Entropy S
Specific heat C
L attice gauge theory
Principle: Physical states have to be gauge invariant! Construct stochastic basis of gauge invariant states.
Abelian U(1) gauge group. Analogy: Q.M. – Gauge theory l = number of links = index of irreducible representation.
Fourier Theorem – Peter Weyl Theorem
Transition amplitude between Bargmann states
Transition amplitude between gauge invariant states
Result: Gauss’ law at any vertex i: Plaquette angle:
Electric Hamiltonian… Lattice results versus analytical results
Energy Spectrum 2x2
Energy Spectrum 3x3
Energy Spectrum 4x4
Energy Spectrum 10x10 4x4
Scaling of energy levels 2x2
Scaling of wave functions 2x2
Scaling of excited states: energy - wave fct. 2x2
Scaling of exited states: energy - wave fct. 2x2
Energy scaling: 3x3, a_s=1
Energy scaling: 3x3, a_s=0.05
Energy scaling: 6x6
Wave fct scaling: 6x6
Wave fct scaling: ground state + 1st excited state: 6x6
Wave fct scaling. 2 nd excited state: 6x6
Wave fct scaling: 3 rd excited state: 6x6
Spectrum: 8x8
Spectrum -Degeneracy: 8x8
Spectrum - Error estimate: 8x8
Energy scaling: 8x8
Thermodynamics: Average energy U: 2x2
Free energy F
Entropy S
Specific heat C
Including Magnetic Term…
Comparison of electric and...
... full Hamiltonian: 2x2, a_s=a_t=1
a_s=1, a_t=0.05
Application of Monte Carlo Hamiltonian - Spectrum of excited states -Wave functions -Hadronic structure functions (x_B, Q^2) in QCD - S-matrix, scattering and decay amplitudes. IV. Outlook