casimir e ect and boundary quantum eld theories zolt an bajnokbajnok.web.elte.hu/miami.pdfzolt an...

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




240L4

not a theoretical curiosity! Gecko legs




240L4


Airbag trigger chip




240L4


Airbag trigger chip

micromechanicaldevice: pieces stickfriction, levitation




240L4


Airbag trigger chip

micromechanicaldevice: pieces stickfriction, levitation

Maritime analogy:

Aim: understand/describe planar Casimir effect

3


Usual explanation: energy of the vacuum: E0(L) = 12∑k(L,BC) ω(k) ∝ ∞



E0(L)− E0(∞)− 2Eplate = ∆E0(L) ;∆E0(L)

A



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



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


Infinite volume

θ12

S-matrix


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines


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


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π

∫cosh θ log(1 + e−ε(θ))dθ


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π


Strip


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π


Strip

Semiinfinite volume��

��

θ

R-matrix


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π


Strip


��

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)


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π


Strip


��

θ

R-matrix


L

R

R

L

E0(L) =

−∫mdθ4π


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



Bulk multiparticle state: with n particles E(θ1, θ2, . . . , θn) =∑im cosh θi

��

��

��

2vv

1 nv> > >...



Bulk twoparticle state:

��

��

2v>v1



Bulk twoparticle in state: t→ −∞

��

��

v1 > v

2




��

��

v1 > v

2

times develop

��

��

2v>v1




��

��

v1 > v

2

times develop further

��

��

v2

>

v1




��

��

v1 > v

2

Bulk twoparticle out state: t→∞

��

��

v1

>

v2




��

��

v1 > v

2


��

��

v1

>

v2

θ12




��

��

v1 > v

2


��

��

v1

>

v2

θ12

Free, noninteracting in particles Free, noninteracting out particles




��

��

v1 > v

2


��

��

v1

>

v2

θ12

Free, noninteracting in particles ← S-matrix → Free, noninteracting out particles




��

��

v1 > v

2


��

��

v1

>

v2

θ12


|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out




��

��

v1 > v

2


��

��

v1

>

v2

θ12


|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)




��

��

v1 > v

2


��

��

v1

>

v2

θ12


|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out

Unitarity

12−θ


12iπ− θ

S(θ12) = S(iπ − θ12)

Integrability→ factorizability: |θ1, θ2, . . . , θn〉 =∏i<j S(θi − θj)|θn, θn−1, . . . , θ1〉




��

��

v1 > v

2


��

��

v1

>

v2

θ12


|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out

Unitarity

12−θ


12iπ− θ

S(θ12) = S(iπ − θ12)


Minimal solutions: free boson S = ±1 sinh-Gordon S(θ) = sinh θ−i sinπpsinh θ+i sinπp , Lee-Yang p = −1

3




��

��

v1 > v

2


��

��

v1

>

v2

θ12


|θ1, θ2〉in = S(θ1 − θ2)|θ1, θ2〉out

Unitarity

12−θ


12iπ− θ

S(θ12) = S(iπ − θ12)


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



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



sinh θ+i sinπp

0L

θn

θ2θ1


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



sinh θ+i sinπp

0L

θn

θ2θ1


p(θ) = m sinh θ

E(θ1, ..., θn) =∑

iE(θi)

Ε

0

m

2m

3m



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



sinh θ+i sinπp

0L

θn

θ2θ1


p(θ) = m sinh θ

E(θ1, ..., θn) =∑

iE(θi)

Ε

0

m

2m

3m



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 E0(L) =0

L



− limR→∞1R log(Tr (e−H(L)R)) = − limR→∞

1R log(e−E0(L)R)

0L

R




1R log(e−E0(L)R)

0L

R

− limR→∞1R logZ(L,R) = − limR→∞

1R log(Tr(e−H(R)L))

L

R




1R log(e−E0(L)R)

0L

R



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)




1R log(e−E0(L)R)

0L

R



L

R



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)




1R log(e−E0(L)R)

0L

R



L

R



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


8


Infinite volume

θ12

S-matrix


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π



Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π



Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π


Strip


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π


Strip


��

θ

R-matrix


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π


Strip


��

θ

R-matrix


L

R

R

L

E0(L) =

−∫mdθ4π


K(θ) = R(iπ2− θ)


Infinite volume

θ12

S-matrix

Large volumes

0L

θn

θ2θ1

Bethe-Yang lines

Luscher correction

0L

R

L

R

E0(L) =

−m∫

dθ2π


Exact groundstate

0L

R

L

R

E0(L) =

−m2π


Strip


��

θ

R-matrix


L

R

R

L

E0(L) =

−∫mdθ4π


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


Boundary multiparticle state: with n particles��

��

v1 > v

2 > ... > vn > 0


Boundary one particle state:��

��

v1


Boundary one particle in state: t→ −∞��

��

v1



��

v1

times develop��

��

v1



��

v1

times develop further��

��

v1



��

v1

Boundary one pt out state: t→∞��

��

v1



��

v1


��

v1

��

��

θ



��

v1


��

v1

��

��

θ

Free in particle Free out particle



��

v1


��

v1

��

��

θ

Free in particle ← R-matrix → Free out particle



��

v1


��

v1

��

��

θ


|θ〉B = R(θ)| − θ〉B



��

v1


��

v1

��

��

θ


|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



��

v1


��

v1

��

��

θ


|θ〉B = R(θ)| − θ〉B



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 )



��

v1


��

v1

��

��

θ


|θ〉B = R(θ)| − θ〉B



R(iπ2 + θ) = S(2θ)R(iπ2 − θ)

��

��

−θ

��

2θ


= [−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



��

v1


��

v1

��

��

θ


|θ〉B = R(θ)| − θ〉B



R(iπ2 + θ) = S(2θ)R(iπ2 − θ)

��

��

−θ

��

2θ


= [−p] = −(−p)(1 + p) ; (p) =sinh( θ

2+ iπp

2 )sinh( θ

2− iπp

2 )


12

) (1+p2

) (1− p

2

) [32− ηp

π

] [32− Θp

π


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±



��

v1


��

v1

��

��

θ


|θ〉B = R(θ)| − θ〉B



R(iπ2 + θ) = S(2θ)R(iπ2 − θ)

��

��

−θ

��

2θ


= [−p] = −(−p)(1 + p) ; (p) =sinh( θ

2+ iπp

2 )sinh( θ

2− iπp

2 )


12

) (1+p2

) (1− p

2

) [32− ηp

π

] [32− Θp

π


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


Groundstate energy for large L from IR reflection: E0(L) =

10


Groundstate energy for large L from IR reflection: E0(L) =L



− limR→∞1R log(Tr(e−H

B(L)R)) = − limR→∞1R log(e−E0(L)R)

L

R




B(L)R)) = − limR→∞1R log〈B|e−H(R)L|B〉

L

R

R

L





L

R

R

L

Boundary state |B〉 = exp{∫∞−∞

dθ4πR(iπ2 − θ)A+(−θ)A+(θ)

}|0〉

��

+ +1

��

++ 1





L

R

R

L



}|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 + . . .





L

R

R

L



}|0〉

��

+ +1

��

++ 1



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





L

R

R

L



}|0〉

��

+ +1��

++ 1



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


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‖))



k

k

L


‖ + k2⊥ =

√m2

eff + k2⊥



Bstate: |B〉 =

{1 +

∫ dD−1k‖(2π)D−1

dθ4πR(iπ2 − θ,meff(k‖))A+(−θ,−~k‖)A+(θ,~k‖) + ...

}|0〉



k

k

L


‖ + k2⊥ =

√m2

eff + k2⊥



Bstate: |B〉 =

{1 +

∫ dD−1k‖(2π)D−1


}|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)



k

k

L


‖ + k2⊥ =

√m2

eff + k2⊥



Bstate: |B〉 =

{1 +

∫ dD−1k‖(2π)D−1


}|0〉


E0(L) =∫ dD−1k‖


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



k

k

L


‖ + k2⊥ =

√m2

eff + k2⊥



Bstate: |B〉 =

{1 +

∫ dD−1k‖(2π)D−1


}|0〉


E0(L) =∫ dD−1k‖


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

casimir e ect and boundary quantum eld theories zolt an bajnokbajnok.web.elte.hu/miami.pdfzolt an...

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