new vista on excited states
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
New Vista On Excited States New Vista On Excited States
ContentsContents
• 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 Critical review of Lagrangian vs Hamiltonian LGT 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 BASISSTOCHASTIC 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 BasisConstruction of Basis
…
t
T
0 X
4X
fiX
2
T 3X 5X2X1X 6X
…
7X.. . . . . .
.
.inX
Box FunctionsBox Functions
Monte Carlo HamiltonianMonte Carlo Hamiltonian
NjixexTM jHT
iij ,...,2,1,)( /
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
]exp[)(
]exp[][
]exp[]exp[][)()(
)0(
,0,0
,,)0( 0
Vij
Txixj
TxioxjV
ijij
STM
Sdx
SSdxTMTM
Diagonalize matrix
UTDUTM )()(
]/exp[)( TETD
ExU
keff
k
keff
iik
Spectrum of energies and wave funtions
Effective Hamiltonian
keff
keff
kk
effeff EEEH
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.
y
y
Sdxdy
SdxyP
0
0
]exp[][
]exp[][)(
Thermodynamical functions:
Definition:
Z
U
HTrZ
log)(
)],[exp()(
SaNa
NU
ttt
s
1
2)(
Lattice:
Monte Carlo Hamiltonian:
]exp[)(
1)(
,]exp[)(
1
1
neff
N
n
neff
effeff
N
n
neffeff
EEZ
U
EZ
Klein Gordon ModelKlein Gordon ModelX.Q.Luo, H. Jirari, H. Kröger, K.J.M. Moriarty, Non-perturbative Methods and Lattice QCD, World Scientific Singapore (2001), p.100.
Energy spectrumEnergy spectrum
Free energy beta x F
Average energy U
Specific heat C/k_B
Scalar ModelScalar Model
C.Q. Huang, H. Kröger,X.Q. Luo, K.J.M. Moriarty
Phys.Lett. A299 (2002) 483.
Energy spectrumEnergy spectrum
Free energy FFree energy F
Average energy UAverage energy U
Entropy SEntropy S
Specific heat CSpecific heat C
LLatticeattice gauge theory gauge theory
Principle:
Physical states have to be gauge invariant!
Construct stochastic basis of
gauge invariant states.
,...,...
,...,
13232
12121321
2312
gUggUgdgdgdgZU
UUU
Ninv
Abelian U(1) gauge group. Abelian U(1) gauge group. Analogy: Q.M. – Gauge theoryAnalogy: Q.M. – Gauge theory
l = number of links = index of irreducible representation.
lUUlipxxp
UUEiXP
)(2/)exp(
ˆˆ,ˆ/ˆ,ˆ
Fourier Theorem – Peter Weyl Theorem
lll
llll ,,...2,1,0
,1
)(,...2,1,0
UUUllUl
)(,1 UUUUUUdU
lllUUldU ,
)(,1 UUUUUUdU
Transition amplitude between Transition amplitude between Bargman statesBargman states
14,43,23,12 ,..2,1,0
22
1443231214432312
)(cos2
exp
,,,/exp,,,
ij n
inij
fiijijij
ininininelecfifififi
ij
anna
Tg
UUUUTHUUUU
Transition amplitude between Transition amplitude between gauge invariant statesgauge invariant states
14,43,23,12 ,..2,1,0
22
2
0
4
2
0
1
4
1443231214432312
)(cos2
exp
...2
1
,,,/exp,,,
ij nji
inij
fiijijij
inv
ininininelecfifififi
inv
ij
anna
Tg
dd
UUUUTHUUUU
Result:Result:
• Gauss’ law at any vertex i:
0j
ijn
,..2,1,0
22
1443231214432312
)(cos42
exp
,,,/exp,,,
plaqn
inplaq
fiplaqplaqplaq
inv
ininininelecfifififi
inv
nna
Tg
UUUUTHUUUU
41342312 plaqPlaquette angle:
Results From Electric Term…Results From Electric Term…
Spectrum 1PlaquetteSpectrum 1Plaquette
Spectrum 2 PlaquettesSpectrum 2 Plaquettes
Spectrum 4 PlaquettesSpectrum 4 Plaquettes
Spectrum 9 PlaquettesSpectrum 9 Plaquettes
Energy Scaling Window: 1 PlaquetteEnergy Scaling Window: 1 Plaquette
Energy scaling window (fixed basis)Energy scaling window (fixed basis)
Energy scaling window: 4 PlaqEnergy scaling window: 4 Plaq
4 Plaquettes: a_s=14 Plaquettes: a_s=1
Scaling Window: Wave FunctionsScaling Window: Wave Functions
Scaling: Energy vs.Wave FctScaling: Energy vs.Wave Fct
Scaling: Energy vs. Wave Fct.Scaling: Energy vs. Wave Fct.
Average Energy UAverage Energy U
Free Energy FFree Energy F
Entropy SEntropy S
Specific Heat CSpecific Heat C
Including Magnetic Term…Including Magnetic Term…
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.
- Finite density QCD (?)
IV. OutlookIV. Outlook