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