SLIDE 1
QFT Dynamics from CFT Data
Zuhair U. Khandker
University of Illinois, Urbana-Champaign Boston University
with N. Anand, V. Genest, E. Katz, C. Hussong, M. Walters
Non-Perturbative Methods in Quantum Field Theory, ICTP, Sep 4th 2019
QFT Dynamics from CFT Data Zuhair U. Khandker University of - - PowerPoint PPT Presentation
QFT Dynamics from CFT Data Zuhair U. Khandker University of - - PowerPoint PPT Presentation
QFT Dynamics from CFT Data Zuhair U. Khandker University of Illinois, Urbana-Champaign Boston University with N. Anand, V. Genest, E. Katz, C. Hussong, M. Walters Non-Perturbative Methods in Quantum Field Theory, ICTP, Sep 4 th 2019 Preface
SLIDE 2
SLIDE 3
Basic Strategy QFT
SLIDE 4
Basic Strategy QFT
Write QFT as deformation of UV CFT. Use CFT data to organize QFT calculation.
CFT (UV) QFT (IR)
+ X λiO(relevant)
i
SLIDE 5
Free Fields Minimal / Integrable Perturbative Supersymmetric Bootstrap-able e.g.
CFT (UV) QFT (IR)
+ X λiO(relevant)
i
Basic Strategy
SLIDE 6
CFT (UV) QFT (IR)
+ X λiO(relevant)
i
UV CFT Data: Δ’s + OPE coefficients
Input
IR QFT Observables:
- Spectrum
- Correlation Functions
(real-time, infinite-volume)
Output
Goal: Extract QFT dynamics from CFT data
Basic Strategy
SLIDE 7
Novel Feature of Conformal Truncation
No Wick rotation, no lattice, no compactification Formulated so that entire computation takes place in real time and infinite volume, allowing access to dynamics
SLIDE 8
Novel Feature of Conformal Truncation
No Wick rotation, no lattice, no compactification Formulated so that entire computation takes place in real time and infinite volume, allowing access to dynamics
Conformal truncation is a specific implementation of Hamiltonian truncation.
SLIDE 9
Hamiltonian Truncation
1. Identify a basis of QFT states 2. Write Hamiltonian in chosen basis 3. Truncate in some way 4. Diagonalize numerically 5. Look for convergence w/ truncation level
m1 m2
|b1i, |b2i, |b3i, . . .
H = H11 H12 · · · H21 H22 · · · . . . . . . . . .
evals + evecs (infinite)
SLIDE 10
Hamiltonian Truncation
1. Identify a basis of QFT states 2. Write Hamiltonian in chosen basis 3. Truncate in some way 4. Diagonalize numerically 5. Look for convergence w/ truncation level
m1 m2
|b1i, |b2i, |b3i, . . .
H = H11 H12 · · · H21 H22 · · · . . . . . . . . .
evals + evecs (infinite) Heart of any truncation scheme. How to discretize QFT???
SLIDE 11
Conformal Truncation Basis
Use UV CFT operators O∆(xµ) to construct basis |b1i, |b2i, |b3i, . . .
CFT (UV) QFT (IR)
+ X λiO(relevant)
i
SLIDE 12
Conformal Truncation Basis
Use UV CFT operators O∆(xµ) to construct basis |b1i, |b2i, |b3i, . . .
Λ2
P 2
1
P 2
2
P 2
kmax
· · ·
P 2
O∆(x)
- !
|∆, ~ P, P 2i = Z ddx e−iP ·x O∆(x)|0i
Final basis states
(k = 1, . . . , kmax)
- !
|∆, ~ P, P 2
k i
Think: [H, ~ P] = 0.
Note: Still real time and infinite volume
SLIDE 13
Truncation Parameters:
Λ2
P 2
1
P 2
2
P 2
kmax
· · · P 2
O∆(x)
- !
|∆, ~ P, P 2i = Z ddx e−iP ·x O∆(x)|0i
(k = 1, . . . , kmax)
- !
|∆, ~ P, P 2
k i
∆max , kmax ∆max kmax
SLIDE 14
Why Truncate in ?
∆max
Holographic Intuition:
CFTd
AdSd+1
O∆(x) ← → Φ(x, z)
M 2
AdS ∼ ∆2
Large ∆ operators = heavy objects in AdS
(expect to decouple)
SLIDE 15
Why Truncate in ?
∆max
(1+1)d λφ4-theory
Experimental Evidence:
µ2
i (∆max) = A +
B (∆max)#
1 (∆max)#
small parameter:
!
SLIDE 16
Hamiltonian Matrix Elements
CFT Spectrum − → basis OPE Coefficients − → H matrix elements HQF T = HCF T + Z d~ x Orel(~ x)
h∆, P|H|∆0, P 0i = (~ P ~ P 0) Z ddx ddx0 ei(P ·xP 0·x0) hO(x)Orel(0)O0(x0)i Fourier transform of CFT 3PF H matrix element
Quantization scheme: Lightcone
SLIDE 17
Technology
CFT Spectrum − → basis OPE Coefficients − → H matrix elements
- 1. How to enumerate all primary operators in a
CFT (even just free CFT)?
- 2. How to efficiently compute OPE coefficients
(even just free CFT)?
- 3. How to Fourier transform general-spin CFT 3PFs?
specifically, Wightman functions
SLIDE 18
Conformal Truncation Deliverables
- Spectrum: bound states, onset of critical behavior, etc.
- Real-time, infinite-volume correlation functions:
hO(x)O(0)i = Z dµ2 ρO(µ) Z ddp (2π)d e−ip·xθ(p0)(2π)δ(p2 µ2)
IO(µ) ≡ Z µ2 dµ02 ρO(µ0)
ρO(µ) K¨ all´ en-Lehmann spectral density
e.g.,
SLIDE 19
Conformal Truncation Deliverables
ρO(µ) K¨ all´ en-Lehmann spectral density µ IO(µ)
UV
IR Encodes RG
IO(µ) ≡ Z µ2 dµ02 ρO(µ0)
SLIDE 20
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 21
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 22
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 23
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 24
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 25
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 26
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- ● ● ●
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 27
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- ●● ● ●
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 28
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- ●● ● ●
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 29
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
¯ λ ≡ λ m2
- ● ●
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
SLIDE 30
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
- ● ● ●
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
¯ λ ≡ λ m2
SLIDE 31
Example: (1+1)d λφ4-theory
Tµ
µ : Spectral Density vs. ¯
λ
- ● ● ●
- 2
4 6 8 0.00 0.05 0.10 0.15
μ / +-
Δ = λ π =
¯ λ ≡ λ m2
CFT!
SLIDE 32
Convergence
∆max
▲▲ ▲ ▲ ▲ ▲ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ■ ■ ■ ■■■ ■ ■ ■ ■ ■ ■
- ●● ●
- ▲ Δ =
◆ Δ = ■ Δ =
- Δ =
2 4 6 8 10 12 14 0.00 0.05 0.10 0.15
μ / +-
(@ fixed λ)
SLIDE 33
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ■
■ ■■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 34
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ■
■ ■■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 35
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ■
■ ■■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 36
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ■
■ ■■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆◆◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 37
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ■
■ ■■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 38
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ■
■ ■■■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 39
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ● ●
- ■
■ ■■■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 40
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ●● ●
- ■
■ ■■■■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 41
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ●● ●
- ■
■ ■ ■■■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 42
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ● ●
- ■
■ ■ ■■■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 43
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ● ●
- ■
■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
SLIDE 44
Example: (1+1)d λφ4-theory
¯ λ ≡ λ m2
φ2n : Spectral Density vs. ¯ λ
- ● ●
- ■
■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
μ / ϕ
Δ = λ π =
Universal Behavior!
SLIDE 45
Example: (1+1)d λφ4-theory
- ● ●
- ■ ■ ■
■ ■ ■ ■ ■ ■ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
- ϕ
■ ϕ ◆ ϕ 0.0 0.5 1.0 1.5 2.0 2.5 0.00 0.01 0.02 0.03 0.04 0.05
μ / ϕ
Δ = λ π =
¯ λ ≡ λ m2
Ising Prediction ρε
IR Zoom-In
SLIDE 46
Summary of Conformal Truncation
It’s a Hamiltonian truncation method formulated directly in real time and infinite volume, allowing access to nonperturbative dynamics. Tries to harness small parameter:
1 (∆max)#