New Vista On Excited States

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New Vista On Excited States
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Contents

• Monte Carlo Hamiltonian
• Effective Hamiltonian in low
• energy/temperature window

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• - Spectrum of excited states
• - Wave functions
• - Thermodynamical functions
• - Klein-Gordon model
• - Scalar f4 theory
• - Gauge theory
• Summary

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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

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• 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!

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

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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.

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Construction of Basis
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Box Functions
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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
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Diagonalize matrix
Spectrum of energies and wave funtions
Effective Hamiltonian
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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.
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Thermodynamical functions
Definition
Lattice
Monte Carlo Hamiltonian
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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.
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Energy spectrum
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Free energy beta x F
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Average energy U
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Specific heat C/k_B
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Scalar Model
C.Q. Huang, H. Kröger, X.Q. Luo, K.J.M.
Moriarty Phys.Lett. A299 (2002) 483.
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Energy spectrum
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Free energy F
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Average energy U
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Entropy S
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Specific heat C
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Lattice gauge theory
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• Principle
• Physical states have to be gauge invariant!
• Construct stochastic basis of
• gauge invariant states.

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Abelian U(1) gauge group. Analogy Q.M. Gauge
theory
l number of links index of irreducible
representation.
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Fourier Theorem Peter Weyl Theorem
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Transition amplitude between Bargman states
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Transition amplitude between gauge invariant
states
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Result

• Gauss law at any vertex i

Plaquette angle
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Results From Electric Term
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Spectrum 1Plaquette
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Spectrum 2 Plaquettes
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Spectrum 4 Plaquettes
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Spectrum 9 Plaquettes
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Energy Scaling Window 1 Plaquette
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Energy scaling window (fixed basis)
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Energy scaling window 4 Plaq
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4 Plaquettes a_s1
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Scaling Window Wave Functions
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Scaling Energy vs.Wave Fct
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Scaling Energy vs. Wave Fct.
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Average Energy U
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Free Energy F
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Entropy S
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Specific Heat C
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Including Magnetic Term
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IV. Outlook

• Application of Monte Carlo Hamiltonian
• - Spectrum of excited states
• Wave functions
• Hadronic structure functions (x_B, Q2) in QCD
  (?)
• - S-matrix, scattering and decay amplitudes.
• Finite density QCD (?)