University of Miami, Miami, 19th of October, 2011
Casimir effect and boundary quantum field theoriesZoltan Bajnok,
Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest
in collaboration with L. Palla and G. Takacs
F (L)A = −dE0(L)
AdL = − ~cπ2
240L4
Planar Casimir energy E0(L) from finite size effects in (integrable) boundary QFT
1
Motivation: Casimir-Polder effect
Hendrik Casimir Dirk Poldercolloidal solution: neutral atoms
force not like Van der WaalsF (L)A = − ~cπ2
240L4
not a theoretical curiosity!
2
Motivation: Casimir-Polder effect
Hendrik Casimir Dirk Poldercolloidal solution: neutral atoms
force not like Van der WaalsF (L)A = − ~cπ2
240L4
not a theoretical curiosity! Gecko legs
Motivation: Casimir-Polder effect
Hendrik Casimir Dirk Poldercolloidal solution: neutral atoms
force not like Van der WaalsF (L)A = − ~cπ2
240L4
not a theoretical curiosity! Gecko legs
Airbag trigger chip
Motivation: Casimir-Polder effect
Hendrik Casimir Dirk Poldercolloidal solution: neutral atoms
force not like Van der WaalsF (L)A = − ~cπ2
240L4
not a theoretical curiosity! Gecko legs
Airbag trigger chip
micromechanicaldevice: pieces stickfriction, levitation
Motivation: Casimir-Polder effect
Hendrik Casimir Dirk Poldercolloidal solution: neutral atoms
force not like Van der WaalsF (L)A = − ~cπ2
240L4
not a theoretical curiosity! Gecko legs
Airbag trigger chip
micromechanicaldevice: pieces stickfriction, levitation
Maritime analogy:
Aim: understand/describe planar Casimir effect
3
Aim: understand/describe planar Casimir effect
Usual explanation: energy of the vacuum: E0(L) = 12∑k(L,BC) ω(k) ∝ ∞
Aim: understand/describe planar Casimir effect
Usual explanation: energy of the vacuum: E0(L) = 12∑k(L,BC) ω(k) ∝ ∞
E0(L)− E0(∞)− 2Eplate = ∆E0(L) ;∆E0(L)
A
Aim: understand/describe planar Casimir effect
Usual explanation: energy of the vacuum: E0(L) = 12∑k(L,BC) ω(k) ∝ ∞
E0(L)− E0(∞)− 2Eplate = ∆E0(L) ;∆E0(L)
A
Lifshitz formula: QED, Parallel dielectric slabs (ε1,1, ε2)
∆E0(L)/A =∑i=‖,⊥
∫∞0
d2q8π2dζ log
[1−R1
i (ζ, q)R2i (ζ, q)e−2L
√q2+ζ2
]
R⊥(ω =√q2 + ζ2, q) =
√ω2−q2−
√εω2−q2√
ω2−q2+√εω2−q2
R‖(ω, q) = ε√ω2−q2−
√εω2−q2
ε√ω2−q2+
√εω2−q2
Aim: understand/describe planar Casimir effect
Usual explanation: energy of the vacuum: E0(L) = 12∑k(L,BC) ω(k) ∝ ∞
E0(L)− E0(∞)− 2Eplate = ∆E0(L) ;∆E0(L)
ALifshitz formula: QED, Parallel dielectric slabs (ε1,1, ε2)
∆E0(L)/A =∑i=‖,⊥
∫∞0
d2q8π2dζ log
[1−R1
i (ζ, q)R2i (ζ, q)e−2L
√q2+ζ2
]R⊥(ω =
√q2 + ζ2, q) =
√ω2−q2−
√εω2−q2√
ω2−q2+√εω2−q2
R‖(ω, q) = ε√ω2−q2−
√εω2−q2
ε√ω2−q2+
√εω2−q2
Physics can be understood in 1+1 D QFT
L
integrability helps to solve the problem even exactly→ large volume expansion in any D
Plan of talkCylinder
4
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Strip
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Strip
Semiinfinite volume������������������������
������������������������
θ
R-matrix
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Strip
Semiinfinite volume������������������������
������������������������
θ
R-matrix
Boundary Luscher correction
L
R
R
L
E0(L) =
−∫mdθ4π
cosh θK(θ)K(θ)e−2mL cosh(θ)
K(θ) = R(iπ2− θ)
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Strip
Semiinfinite volume������������������������
������������������������
θ
R-matrix
Boundary Luscher correction
L
R
R
L
E0(L) =
−∫mdθ4π
cosh θK(θ)K(θ)e−2mL cosh(θ)
K(θ) = R(iπ2− θ)
Boundary TBAE0(L) =
−∫mdθ4π
cosh θ log(1 + e−ε(θ))
Application
Casimir effect
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
5
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk multiparticle state: with n particles E(θ1, θ2, . . . , θn) =∑im cosh θi
��������
��������
��������
2vv
1 nv> > >...
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle state:
��������
��������
2v>v1
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
times develop
��������
��������
2v>v1
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
times develop further
��������
��������
v2
>
v1
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
θ12
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
θ12
Free, noninteracting in particles Free, noninteracting out particles
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
θ12
Free, noninteracting in particles ← S-matrix → Free, noninteracting out particles
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
θ12
Free, noninteracting in particles ← S-matrix → Free, noninteracting out particles
|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
θ12
Free, noninteracting in particles ← S-matrix → Free, noninteracting out particles
|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out
Unitarity
12−θ
S−1(θ12) = S(θ21) Crossing
12iπ− θ
S(θ12) = S(iπ − θ12)
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
θ12
Free, noninteracting in particles ← S-matrix → Free, noninteracting out particles
|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out
Unitarity
12−θ
S−1(θ12) = S(θ21) Crossing
12iπ− θ
S(θ12) = S(iπ − θ12)
Integrability→ factorizability: |θ1, θ2, . . . , θn〉 =∏i<j S(θi − θj)|θn, θn−1, . . . , θ1〉
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
θ12
Free, noninteracting in particles ← S-matrix → Free, noninteracting out particles
|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out
Unitarity
12−θ
S−1(θ12) = S(θ21) Crossing
12iπ− θ
S(θ12) = S(iπ − θ12)
Integrability→ factorizability: |θ1, θ2, . . . , θn〉 =∏i<j S(θi − θj)|θn, θn−1, . . . , θ1〉
Minimal solutions: free boson S = ±1 sinh-Gordon S(θ) = sinh θ−i sinπpsinh θ+i sinπp , Lee-Yang p = −1
3
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
Bulk twoparticle out state: t→∞
��������
��������
v1
>
v2
θ12
Free, noninteracting in particles ← S-matrix → Free, noninteracting out particles
|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out
Unitarity
12−θ
S−1(θ12) = S(θ21) Crossing
12iπ− θ
S(θ12) = S(iπ − θ12)
Integrability→ factorizability: |θ1, θ2, . . . , θn〉 =∏i<j S(θi − θj)|θn, θn−1, . . . , θ1〉
Minimal solutions: free boson S = ±1 sinh-Gordon S(θ) = sinh θ−i sinπpsinh θ+i sinπp , Lee-Yang p = −1
3
Lagrangian: L = 12(∂φ)2 −µ(cosh bφ− 1) p = b2
8π+b2
Very large volume spectrum
12(∂φ)2−µ(cosh bφ− 1)↔S(θ) = sinh θ−i sinπp
sinh θ+i sinπp
0L
θn
θ2θ1
6
Very large volume spectrum
12(∂φ)2−µ(cosh bφ− 1)↔S(θ) = sinh θ−i sinπp
sinh θ+i sinπp
0L
θn
θ2θ1
Infinite volumeE(θ) = m cosh θ
p(θ) = m sinh θ
E(θ1, ..., θn) =∑
iE(θi)
Ε
0
m
2m
3m
Very large volume spectrum
12(∂φ)2−µ(cosh bφ− 1)↔S(θ) = sinh θ−i sinπp
sinh θ+i sinπp
0L
θn
θ2θ1
Infinite volumeE(θ) = m cosh θ
p(θ) = m sinh θ
E(θ1, ..., θn) =∑
iE(θi)
Ε
0
m
2m
3m
Finite volume: free particles
eip(θ)L = 1 Quantizationp(θ)→ p(k) = 2πk
L
|θ1, . . . θn〉 → |k1, . . . , kn〉
0
1
2
3
4
5
0 5 10 15 20 25 30
Very large volume spectrum
12(∂φ)2−µ(cosh bφ− 1)↔S(θ) = sinh θ−i sinπp
sinh θ+i sinπp
0L
θn
θ2θ1
Infinite volumeE(θ) = m cosh θ
p(θ) = m sinh θ
E(θ1, ..., θn) =∑
iE(θi)
Ε
0
m
2m
3m
Finite volume: free particles
eip(θ)L = 1 Quantizationp(θ)→ p(k) = 2πk
L
|θ1, . . . θn〉 → |k1, . . . , kn〉
0
1
2
3
4
5
0 5 10 15 20 25 30
Very large volume, interacting particles
one particle eip(θ)L = 1;θ → p(k) = 2πkL
two particles eip(θ1)LS(θ12) = 1
p(θ1)L+ ϕ(θ12) = 2πn1 ; S = eiϕ
n particles p(θi)L+∑
j ϕ(θij) = 2πni
0
1
2
3
4
5
0 5 10 15 20 25 30
Very large volume spectrum
12(∂φ)2−µ(cosh bφ− 1)↔S(θ) = sinh θ−i sinπp
sinh θ+i sinπp
0L
θn
θ2θ1
Infinite volumeE(θ) = m cosh θ
p(θ) = m sinh θ
E(θ1, ..., θn) =∑
iE(θi)
Ε
0
m
2m
3m
Finite volume: free particles
eip(θ)L = 1 Quantizationp(θ)→ p(k) = 2πk
L
|θ1, . . . θn〉 → |k1, . . . , kn〉
0
1
2
3
4
5
0 5 10 15 20 25 30
Very large volume, interacting particles
one particle eip(θ)L = 1;θ → p(k) = 2πkL
two particles eip(θ1)LS(θ12) = 1
p(θ1)L+ ϕ(θ12) = 2πn1 ; S = eiϕ
n particles p(θi)L+∑
j ϕ(θij) = 2πni
0
1
2
3
4
5
0 5 10 15 20 25 30
Momentum quantization S(0) = −1������
2
L
π
���� ����������������2
L
π∼
Groundstate energy in finite volume
Groundstate energy E0(L) =
7
Groundstate energy in finite volume
Groundstate energy E0(L) =0
L
Groundstate energy in finite volume
Groundstate energy E0(L) =
− limR→∞1R log(Tr (e−H(L)R)) = − limR→∞
1R log(e−E0(L)R)
0L
R
Groundstate energy in finite volume
Groundstate energy E0(L) =
− limR→∞1R log(Tr (e−H(L)R)) = − limR→∞
1R log(e−E0(L)R)
0L
R
− limR→∞1R logZ(L,R) = − limR→∞
1R log(Tr(e−H(R)L))
L
R
Groundstate energy in finite volume
Groundstate energy E0(L) =
− limR→∞1R log(Tr (e−H(L)R)) = − limR→∞
1R log(e−E0(L)R)
0L
R
− limR→∞1R logZ(L,R) = − limR→∞
1R log(Tr(e−H(R)L))
L
R
Dominant contribution for large L: one particle term
Tr(e−H(R)L) = 1 +∑k e−m cosh θk(R)L +O(e−2mL)
Groundstate energy in finite volume
Groundstate energy E0(L) =
− limR→∞1R log(Tr (e−H(L)R)) = − limR→∞
1R log(e−E0(L)R)
0L
R
− limR→∞1R logZ(L,R) = − limR→∞
1R log(Tr(e−H(R)L))
L
R
Dominant contribution for large L: one particle term
Tr(e−H(R)L) = 1 +∑k e−m cosh θk(R)L +O(e−2mL)
one particle quantization m sinh θ = 2πkR
∑k → Rm
2π
∫dθ cosh θ
E0(L) = −m∫dθ cosh θ e−mL cosh θ +O(e−2mL)
Groundstate energy in finite volume
Groundstate energy E0(L) =
− limR→∞1R log(Tr (e−H(L)R)) = − limR→∞
1R log(e−E0(L)R)
0L
R
− limR→∞1R logZ(L,R) = − limR→∞
1R log(Tr(e−H(R)L))
L
R
Dominant contribution for large L: one particle term
Tr(e−H(R)L) = 1 +∑k e−m cosh θk(R)L +O(e−2mL)
one particle quantization m sinh θ = 2πkR
∑k → Rm
2π
∫dθ cosh θ
E0(L) = −m∫dθ cosh θ e−mL cosh θ +O(e−2mL)
Ground state energy exactly: Al. Zamolodchikov ’90
E0(L) = −m∫ dθ
2π cosh(θ) log(1 + e−ε(θ))
ε(θ) = mL cosh θ −∫ dθ′
2πϕ′(θ − θ′) log(1 + e−ε(θ
′))0
L
Plan of talkCylinder
8
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Strip
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Strip
Semiinfinite volume������������������������
������������������������
θ
R-matrix
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Strip
Semiinfinite volume������������������������
������������������������
θ
R-matrix
Boundary Luscher correction
L
R
R
L
E0(L) =
−∫mdθ4π
cosh θK(θ)K(θ)e−2mL cosh(θ)
K(θ) = R(iπ2− θ)
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
Luscher correction
0L
R
L
R
E0(L) =
−m∫
dθ2π
cosh θ e−m cosh(θ)L
Exact groundstate
0L
R
L
R
E0(L) =
−m2π
∫cosh θ log(1 + e−ε(θ))dθ
Strip
Semiinfinite volume������������������������
������������������������
θ
R-matrix
Boundary Luscher correction
L
R
R
L
E0(L) =
−∫mdθ4π
cosh θK(θ)K(θ)e−2mL cosh(θ)
K(θ) = R(iπ2− θ)
Boundary TBAE0(L) =
−∫mdθ4π
cosh θ log(1 + e−ε(θ))
Application
Casimir effect
Integrable boundary field theory: Bootstrap
9
Integrable boundary field theory: Bootstrap
Boundary multiparticle state: with n particles����������������
����������������
v1 > v
2 > ... > vn > 0
Integrable boundary field theory: Bootstrap
Boundary one particle state:��������
��������
v1
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
times develop��������
��������
v1
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
times develop further��������
��������
v1
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Free in particle Free out particle
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Free in particle ← R-matrix → Free out particle
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Free in particle ← R-matrix → Free out particle
|θ〉B = R(θ)| − θ〉B
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Free in particle ← R-matrix → Free out particle
|θ〉B = R(θ)| − θ〉B
UnitarityR∗(θ) = R−1(θ) = R(−θ)
Boundary crossing unitarity
R(iπ2 + θ) = S(2θ)R(iπ2 − θ)
������������������������
������������������������
−θ
����������������������������������������������������
2θ
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Free in particle ← R-matrix → Free out particle
|θ〉B = R(θ)| − θ〉B
UnitarityR∗(θ) = R−1(θ) = R(−θ)
Boundary crossing unitarity
R(iπ2 + θ) = S(2θ)R(iπ2 − θ)
������������������������
������������������������
−θ
����������������������������������������������������
2θ
sinh-Gordon S(θ) = sinh θ−i sinπpsinh θ+i sinπp
= [−p] = −(−p)(1 + p) ; (p) =sinh( θ
2+ iπp
2 )sinh( θ
2− iπp
2 )
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Free in particle ← R-matrix → Free out particle
|θ〉B = R(θ)| − θ〉B
UnitarityR∗(θ) = R−1(θ) = R(−θ)
Boundary crossing unitarity
R(iπ2 + θ) = S(2θ)R(iπ2 − θ)
������������������������
������������������������
−θ
����������������������������������������������������
2θ
sinh-Gordon S(θ) = sinh θ−i sinπpsinh θ+i sinπp
= [−p] = −(−p)(1 + p) ; (p) =sinh( θ
2+ iπp
2 )sinh( θ
2− iπp
2 )
reflection factor R(θ) =(
12
) (1+p2
) (1− p
2
) [32− ηp
π
] [32− Θp
π
]Ghoshal-Zamolodchikov ’94
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Free in particle ← R-matrix → Free out particle
|θ〉B = R(θ)| − θ〉B
UnitarityR∗(θ) = R−1(θ) = R(−θ)
Boundary crossing unitarity
R(iπ2 + θ) = S(2θ)R(iπ2 − θ)
������������������������
������������������������
−θ
����������������������������������������������������
2θ
sinh-Gordon S(θ) = sinh θ−i sinπpsinh θ+i sinπp
= [−p] = −(−p)(1 + p) ; (p) =sinh( θ
2+ iπp
2 )sinh( θ
2− iπp
2 )
reflection factor R(θ) =(
12
) (1+p2
) (1− p
2
) [32− ηp
π
] [32− Θp
π
]Ghoshal-Zamolodchikov ’94
Lagrangian: L = 12(∂φ)2−µ(cosh bφ− 1)− δ
[µB+e
b
2φ + µB−e
− b
2φ] √
µsin b2 cosh b2(η ±Θ) = µB±
Integrable boundary field theory: Bootstrap
Boundary one particle in state: t→ −∞��������
��������
v1
Boundary one pt out state: t→∞��������
��������
v1
������������������������
������������������������
θ
Free in particle ← R-matrix → Free out particle
|θ〉B = R(θ)| − θ〉B
UnitarityR∗(θ) = R−1(θ) = R(−θ)
Boundary crossing unitarity
R(iπ2 + θ) = S(2θ)R(iπ2 − θ)
������������������������
������������������������
−θ
����������������������������������������������������
2θ
sinh-Gordon S(θ) = sinh θ−i sinπpsinh θ+i sinπp
= [−p] = −(−p)(1 + p) ; (p) =sinh( θ
2+ iπp
2 )sinh( θ
2− iπp
2 )
reflection factor R(θ) =(
12
) (1+p2
) (1− p
2
) [32− ηp
π
] [32− Θp
π
]Ghoshal-Zamolodchikov ’94
Lagrangian: L = 12(∂φ)2−µ(cosh bφ− 1)− δ
[µB+e
b
2φ + µB−e
− b
2φ] √
µsin b2 cosh b2(η ±Θ) = µB±
Integrability→factorizability: |θ1, θ2, . . . , θn〉B =∏i<j S(θi−θj)S(θi+θj)
∏iR(θi)|−θ1,−θ2, . . . ,−θn〉B
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =
10
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =L
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =
− limR→∞1R log(Tr(e−H
B(L)R)) = − limR→∞1R log(e−E0(L)R)
L
R
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =
− limR→∞1R log(Tr(e−H
B(L)R)) = − limR→∞1R log〈B|e−H(R)L|B〉
L
R
R
L
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =
− limR→∞1R log(Tr(e−H
B(L)R)) = − limR→∞1R log〈B|e−H(R)L|B〉
L
R
R
L
Boundary state |B〉 = exp{∫∞−∞
dθ4πR(iπ2 − θ)A+(−θ)A+(θ)
}|0〉
������������������������������������������������
+ +1
������������������������������������������������
++ 1
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =
− limR→∞1R log(Tr(e−H
B(L)R)) = − limR→∞1R log〈B|e−H(R)L|B〉
L
R
R
L
Boundary state |B〉 = exp{∫∞−∞
dθ4πR(iπ2 − θ)A+(−θ)A+(θ)
}|0〉
������������������������������������������������
+ +1
������������������������������������������������
++ 1
Dominant contribution for large L: two particle term
〈B|e−H(R)L|B〉 = 1 +∑kR(iπ2 − θ)R(iπ2 + θ)e−2m cosh θkL + . . .
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =
− limR→∞1R log(Tr(e−H
B(L)R)) = − limR→∞1R log〈B|e−H(R)L|B〉
L
R
R
L
Boundary state |B〉 = exp{∫∞−∞
dθ4πR(iπ2 − θ)A+(−θ)A+(θ)
}|0〉
������������������������������������������������
+ +1
������������������������������������������������
++ 1
Dominant contribution for large L: two particle term
〈B|e−H(R)L|B〉 = 1 +∑kR(iπ2 − θ)R(iπ2 + θ)e−2m cosh θkL + . . .
quantization condition: m sinh θk = 2πR
∑k → Rm
4π
∫dθ cosh θ
E0(L) = −∫ m cosh θdθ
4π R(iπ2 − θ)R(iπ2 + θ) e−2mL cosh θ; Z.B, L. Palla, G. Takacs ’04-’08
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =
− limR→∞1R log(Tr(e−H
B(L)R)) = − limR→∞1R log〈B|e−H(R)L|B〉
L
R
R
L
Boundary state |B〉 = exp{∫∞−∞
dθ4πR(iπ2 − θ)A+(−θ)A+(θ)
}|0〉
������������������������������������������������
+ +1������������������������������������������������
++ 1
Dominant contribution for large L: two particle term
〈B|e−H(R)L|B〉 = 1 +∑kR(iπ2 − θ)R(iπ2 + θ)e−2m cosh θkL + . . .
quantization condition: m sinh θk = 2πR
∑k → Rm
4π
∫dθ cosh θ
E0(L) = −∫ m cosh θdθ
4π R(iπ2 − θ)R(iπ2 + θ) e−2mL cosh θ; Z.B, L. Palla, G. Takacs ’04-’08
Ground state energy exactly: E0(L) = −m∫ dθ
4π cosh(θ) log(1 + e−ε(θ))
ε(θ) = 2mL cosh θ − log(R(iπ2 − θ)R(iπ2 + θ))
−∫ dθ′
2πϕ′(θ − θ′) log(1 + e−ε(θ
′)) LeClair, Mussardo, Saleur, SkorikL
R R
Casimir effect: Boundary finite size effect
11
Casimir effect: Boundary finite size effect
Extension to higher dimensions: ~k‖ label
k
k
L
Dispersion ω =√m2 + ~k2
‖ + k2⊥ =
√m2
eff + k2⊥
rapidity ω = meff(k‖) cosh θ, k⊥ = meff(k‖) sinh θ
Reflection R(θ,meff(k‖))
Casimir effect: Boundary finite size effect
Extension to higher dimensions: ~k‖ label
k
k
L
Dispersion ω =√m2 + ~k2
‖ + k2⊥ =
√m2
eff + k2⊥
rapidity ω = meff(k‖) cosh θ, k⊥ = meff(k‖) sinh θ
Reflection R(θ,meff(k‖))
Bstate: |B〉 =
{1 +
∫ dD−1k‖(2π)D−1
dθ4πR(iπ2 − θ,meff(k‖))A+(−θ,−~k‖)A+(θ,~k‖) + ...
}|0〉
Casimir effect: Boundary finite size effect
Extension to higher dimensions: ~k‖ label
k
k
L
Dispersion ω =√m2 + ~k2
‖ + k2⊥ =
√m2
eff + k2⊥
rapidity ω = meff(k‖) cosh θ, k⊥ = meff(k‖) sinh θ
Reflection R(θ,meff(k‖))
Bstate: |B〉 =
{1 +
∫ dD−1k‖(2π)D−1
dθ4πR(iπ2 − θ,meff(k‖))A+(−θ,−~k‖)A+(θ,~k‖) + ...
}|0〉
Ground state energy (for free bulk):
E0(L) =∫ dD−1k‖
(2π)D−1dθ4π log(1 +R1(iπ2 + θ,meff)R2(iπ2 − θ,meff)e−2meff cosh θL)
Casimir effect: Boundary finite size effect
Extension to higher dimensions: ~k‖ label
k
k
L
Dispersion ω =√m2 + ~k2
‖ + k2⊥ =
√m2
eff + k2⊥
rapidity ω = meff(k‖) cosh θ, k⊥ = meff(k‖) sinh θ
Reflection R(θ,meff(k‖))
Bstate: |B〉 =
{1 +
∫ dD−1k‖(2π)D−1
dθ4πR(iπ2 − θ,meff(k‖))A+(−θ,−~k‖)A+(θ,~k‖) + ...
}|0〉
Ground state energy (for free bulk):
E0(L) =∫ dD−1k‖
(2π)D−1dθ4π log(1 +R1(iπ2 + θ,meff)R2(iπ2 − θ,meff)e−2meff cosh θL)
QED: Parallel dielectric slabs (ε1,1, ε2)
reflections E‖,⊥, B‖,⊥ −→ R‖,⊥ look it up in Jackson:
R⊥(ω, k‖ = q) =√ω2−q2−
√εω2−q2√
ω2−q2+√εω2−q2
R‖(ω, k‖ = q) = ε√ω2−q2−
√εω2−q2
ε√ω2−q2+
√εω2−q2
Casimir effect: Boundary finite size effect
Extension to higher dimensions: ~k‖ label
k
k
L
Dispersion ω =√m2 + ~k2
‖ + k2⊥ =
√m2
eff + k2⊥
rapidity ω = meff(k‖) cosh θ, k⊥ = meff(k‖) sinh θ
Reflection R(θ,meff(k‖))
Bstate: |B〉 =
{1 +
∫ dD−1k‖(2π)D−1
dθ4πR(iπ2 − θ,meff(k‖))A+(−θ,−~k‖)A+(θ,~k‖) + ...
}|0〉
Ground state energy (for free bulk):
E0(L) =∫ dD−1k‖
(2π)D−1dθ4π log(1 +R1(iπ2 + θ,meff)R2(iπ2 − θ,meff)e−2meff cosh θL)
QED: Parallel dielectric slabs (ε1,1, ε2)
reflections E‖,⊥, B‖,⊥ −→ R‖,⊥ look it up in Jackson:
R⊥(ω, k‖ = q) =√ω2−q2−
√εω2−q2√
ω2−q2+√εω2−q2
R‖(ω, k‖ = q) = ε√ω2−q2−
√εω2−q2
ε√ω2−q2+
√εω2−q2
Lifshitz
formula
Conclusion
12
Conclusion
Usual derivation:
summing up zero freqencies
E0(L) = 12∑k(L) ω(k(L)) ∝ ∞
Complicated finite volume problem
+ divergencies
Conclusion
Usual derivation:
summing up zero freqencies
E0(L) = 12∑k(L) ω(k(L)) ∝ ∞
Complicated finite volume problem
+ divergencies
as a boundary finite size effect
E0(L) = −∫ dp
2π log(1 +R(−p)R(p)e−2E(p)L)
Reflection factor of the IR degrees of freedom:
semi infinite settings,
easier to calculate,
no divergences