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Page 1: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

New Vista On Excited States New Vista On Excited States

Page 2: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

ContentsContents

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

Page 3: 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

Page 4: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

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

Page 5: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

• 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!

Page 6: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

- Basis constructed via perturbation theory: Examples: Tamm-Dancoff, Discrete Light

Cone Field Theory, …. BIASED CHOICE!

Page 7: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

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].

Page 8: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Construction of BasisConstruction of Basis

t

T

0 X

4X

fiX

2

T 3X 5X2X1X 6X

7X.. . . . . .

.

.inX

Page 9: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Box FunctionsBox Functions

Page 10: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Monte Carlo HamiltonianMonte Carlo Hamiltonian

M ijT =< x i∣e−HT / ℏ∣x j i , j∈1,2, .. . , N

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

M ijT =M0 ijT

∫ [dx ]exp[−SV ]exp[−S0]∣xj,oxi,T

∫ [dx ]exp[−S0 ]∣xj,0xi,T

=M0 ij T exp[−SV ]¿

¿

Page 11: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Diagonalize matrix

M T =U D T U

Uik=< x i∣E

eff k¿

¿Dk T =exp[−Eeff kT /ℏ ]

Spectrum of energies and wave funtions

Effective Hamiltonian

H eff =∑ k∣Eeff k ⟩ E

eff k ⟨ Eeff k∣

Page 12: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

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.

P y =∫ [dx ]exp [−S ]∣0

y

∫ dy∫ [dx ]exp [−S ]∣0y

Page 13: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Thermodynamical functions:

Definition: Z β =Tr [exp−βH ] ,

U β =−∂ logZ∂ β

U β =N s

2a t

1N t

⟨ ∂∂at

S⟩Lattice:

Monte Carlo Hamiltonian: Z eff β =∑

n=1

N

exp[−βEeff n ] ,

U eff β =−1Zeff β

∑n=1

N

Eeff

nexp[−βEeff

n ]

Page 14: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

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.

Page 15: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy spectrumEnergy spectrum

Page 16: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Free energy beta x F

Page 17: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Average energy U

Page 18: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Specific heat C/k_B

Page 19: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Scalar ModelScalar Model

C.Q. Huang, H. Kröger,X.Q. Luo, K.J.M. Moriarty

Phys.Lett. A299 (2002) 483.

Page 20: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy spectrumEnergy spectrum

Page 21: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Free energy FFree energy F

Page 22: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Average energy UAverage energy U

Page 23: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Entropy SEntropy S

Page 24: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Specific heat CSpecific heat C

Page 25: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

LLatticeattice gauge theory gauge theory

Page 26: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Principle: Physical states have to be gauge invariant!

Construct stochastic basis of gauge invariant states.

∣U ⟩≡∣U 12 ,U 23, . .. ⟩

∣U inv ⟩=ZN∫ dg1dg2dg3 .. .∣g1U 12g2−1 , g2U 23g3

−1 ,. . . ⟩

Page 27: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

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.

[ P , X ]=ℏ/i⇔ [ E , U ]=− U⟨p∣x ⟩=exp ipx /2πℏ⇔⟨l∣U ⟩=U l

Page 28: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Fourier Theorem – Peter Weyl Theorem

∑l=0,±1,±2, . ..

∣l ⟩ ⟨ l∣=1, ⟨ l'∣l⟩=δl ' , l

∑l=0,±1,±2, . ..

⟨U '∣l ⟩ ⟨ l∣U ⟩=δ U '−U

∫ dU ∣U ⟩ ⟨U ∣=1, ⟨U '∣U ⟩=δ U '−U

∫dU ⟨ l'∣U ⟩ ⟨U∣l ⟩=δl ' , l

∫ dU ∣U ⟩ ⟨U ∣=1, ⟨U '∣U ⟩=δ U '−U

Page 29: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Transition amplitude between Transition amplitude between Bargmann statesBargmann states

⟨U12fi ,U23

fi ,U43fi ,U14

fi ∣exp [−HelecT /ℏ ]∣U12in ,U23

in ,U43in ,U14

in ⟩

¿ ∏ij=12,23 ,43,14 { ∑

nij=0,±1,±2, . .exp[−g2ℏT

2anij2 ]cos [nijα ij

fi−aijin ]}

Page 30: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Transition amplitude between Transition amplitude between gauge invariant statesgauge invariant states

inv ⟨U12fi ,U23

fi ,U 43fi ,U14

fi ∣exp [−HelecT /ℏ ]∣U12in ,U23

in ,U 43in ,U14

in ⟩inv

¿ 12π 4

∫0

dβ1.. .∫0

dβ4

∏ij=12,23,43,14{ ∑

nij=0,±1,±2, ..exp[−g2ℏ T

2anij2 ]cos [nijα ij

fi−aijinβi−β j ]}

Page 31: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Result:Result:

• Gauss’ law at any vertex i:

∑jnij=0

inv ⟨U12fi ,U23

fi ,U 43fi ,U14

fi ∣exp [−HelecT /ℏ ]∣U12in ,U23

in ,U 43in ,U14

in ⟩inv=

∏ { ∑nplaq=0,±1,±2, ..

exp[−g2ℏT2a

4nplaq2 ]cos [nplaq θplaq

fi −θplaqin ]}

θplaq=α 12α 23α 34α 41Plaquette angle:

Page 32: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Electric Hamiltonian…Electric Hamiltonian…Lattice results versus Lattice results versus

analytical results analytical results

Page 33: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy Spectrum 2x2Energy Spectrum 2x2

Page 34: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy Spectrum 3x3Energy Spectrum 3x3

Page 35: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy Spectrum 4x4Energy Spectrum 4x4

Page 36: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy Spectrum 10x10Energy Spectrum 10x10

4x44x4

Page 37: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Scaling of energy levels 2x2Scaling of energy levels 2x2

Page 38: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Scaling of wave functions 2x2Scaling of wave functions 2x2

Page 39: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Scaling of excited states: energy - Scaling of excited states: energy - wave fct. 2x2wave fct. 2x2

Page 40: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Scaling of exited states: energy - Scaling of exited states: energy - wave fct. 2x2wave fct. 2x2

Page 41: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy scaling: 3x3, a_s=1Energy scaling: 3x3, a_s=1

Page 42: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy scaling: 3x3, a_s=0.05Energy scaling: 3x3, a_s=0.05

Page 43: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy scaling: 6x6Energy scaling: 6x6

Page 44: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Wave fct scaling: 6x6Wave fct scaling: 6x6

Page 45: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Wave fct scaling: ground state + 1st Wave fct scaling: ground state + 1st excited state: 6x6excited state: 6x6

Page 46: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Wave fct scaling. 2Wave fct scaling. 2ndnd excited state: excited state: 6x6 6x6

Page 47: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Wave fct scaling: 3Wave fct scaling: 3rdrd excited state: excited state: 6x66x6

Page 48: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Spectrum: 8x8Spectrum: 8x8

Page 49: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Spectrum -Degeneracy: 8x8Spectrum -Degeneracy: 8x8

Page 50: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Spectrum - Error estimate: 8x8Spectrum - Error estimate: 8x8

Page 51: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Energy scaling: 8x8Energy scaling: 8x8

Page 52: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Thermodynamics: Average energy Thermodynamics: Average energy U: 2x2U: 2x2

Page 53: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Free energy FFree energy F

Page 54: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Entropy SEntropy S

Page 55: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Specific heat CSpecific heat C

Page 56: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Including Magnetic Term…Including Magnetic Term…

Page 57: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

Comparison of electric and...Comparison of electric and...

Page 58: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

... full Hamiltonian: 2x2, a_s=a_t=1... full Hamiltonian: 2x2, a_s=a_t=1

Page 59: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

a_s=1, a_t=0.05a_s=1, a_t=0.05

Page 60: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window
Page 61: New Vista On Excited States. Contents Monte Carlo Hamiltonian: Effective Hamiltonian in low energy /temperature window

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. OutlookIV. Outlook