Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an...
Transcript of Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an...
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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
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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
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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
![Page 4: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/4.jpg)
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
![Page 5: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/5.jpg)
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
![Page 6: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/6.jpg)
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:
![Page 7: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/7.jpg)
Aim: understand/describe planar Casimir effect
3
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Aim: understand/describe planar Casimir effect
Usual explanation: energy of the vacuum: E0(L) = 12∑k(L,BC) ω(k) ∝ ∞
![Page 9: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/9.jpg)
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
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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
![Page 11: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/11.jpg)
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
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Plan of talkCylinder
4
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Plan of talkCylinder
Infinite volume
θ12
S-matrix
![Page 14: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/14.jpg)
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
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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
![Page 16: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/16.jpg)
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
![Page 17: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/17.jpg)
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
![Page 18: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/18.jpg)
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
![Page 19: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/19.jpg)
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)
![Page 20: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/20.jpg)
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
![Page 21: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/21.jpg)
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
5
![Page 22: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/22.jpg)
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> > >...
![Page 23: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/23.jpg)
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle state:
��������
��������
2v>v1
![Page 24: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/24.jpg)
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
![Page 25: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/25.jpg)
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
times develop
��������
��������
2v>v1
![Page 26: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/26.jpg)
Integrable field theory: Bootstrap
pi = m sinh θiEi = m cosh θi
Bulk twoparticle in state: t→ −∞
��������
��������
v1 > v
2
times develop further
��������
��������
v2
>
v1
![Page 27: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/27.jpg)
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
![Page 28: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/28.jpg)
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
![Page 29: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/29.jpg)
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
![Page 30: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/30.jpg)
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
![Page 31: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/31.jpg)
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
![Page 32: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/32.jpg)
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)
![Page 33: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/33.jpg)
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〉
![Page 34: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/34.jpg)
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
![Page 35: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/35.jpg)
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
![Page 36: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/36.jpg)
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
![Page 37: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/37.jpg)
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
![Page 38: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/38.jpg)
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
![Page 39: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/39.jpg)
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
![Page 40: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/40.jpg)
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∼
![Page 41: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/41.jpg)
Groundstate energy in finite volume
Groundstate energy E0(L) =
7
![Page 42: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/42.jpg)
Groundstate energy in finite volume
Groundstate energy E0(L) =0
L
![Page 43: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/43.jpg)
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
![Page 44: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/44.jpg)
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
![Page 45: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/45.jpg)
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)
![Page 46: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/46.jpg)
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)
![Page 47: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/47.jpg)
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
![Page 48: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/48.jpg)
Plan of talkCylinder
8
![Page 49: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/49.jpg)
Plan of talkCylinder
Infinite volume
θ12
S-matrix
![Page 50: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/50.jpg)
Plan of talkCylinder
Infinite volume
θ12
S-matrix
Large volumes
0L
θn
θ2θ1
Bethe-Yang lines
![Page 51: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/51.jpg)
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
![Page 52: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/52.jpg)
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
![Page 53: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/53.jpg)
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
![Page 54: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/54.jpg)
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
![Page 55: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/55.jpg)
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)
![Page 56: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/56.jpg)
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
![Page 57: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/57.jpg)
Integrable boundary field theory: Bootstrap
9
![Page 58: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/58.jpg)
Integrable boundary field theory: Bootstrap
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Integrable boundary field theory: Bootstrap
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Integrable boundary field theory: Bootstrap
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|θ〉B = R(θ)| − θ〉B
![Page 68: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/68.jpg)
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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)
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−θ
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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 )
![Page 70: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/70.jpg)
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
![Page 71: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/71.jpg)
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±
![Page 72: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/72.jpg)
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
![Page 73: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/73.jpg)
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =
10
![Page 74: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/74.jpg)
Boundary Luscher correction
Groundstate energy for large L from IR reflection: E0(L) =L
![Page 75: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/75.jpg)
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
![Page 76: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/76.jpg)
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
![Page 77: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/77.jpg)
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
![Page 78: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/78.jpg)
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 + . . .
![Page 79: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/79.jpg)
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
![Page 80: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/80.jpg)
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
![Page 81: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/81.jpg)
Casimir effect: Boundary finite size effect
11
![Page 82: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/82.jpg)
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‖))
![Page 83: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/83.jpg)
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〉
![Page 84: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/84.jpg)
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)
![Page 85: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/85.jpg)
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
![Page 86: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/86.jpg)
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
![Page 87: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/87.jpg)
Conclusion
12
![Page 88: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/88.jpg)
Conclusion
Usual derivation:
summing up zero freqencies
E0(L) = 12∑k(L) ω(k(L)) ∝ ∞
Complicated finite volume problem
+ divergencies
![Page 89: Casimir e ect and boundary quantum eld theories Zolt an Bajnokbajnok.web.elte.hu/Miami.pdfZolt an Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, Budapest](https://reader036.fdocuments.us/reader036/viewer/2022071412/610834817d66f42eb23bec17/html5/thumbnails/89.jpg)
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