The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda J. Mateos Guilarte

Introduction. Scalar field The Casimir effect fluctuations The TGTG formula and for vacuum energy

Double-delta systems. one-dimensional scattering

δ − δ spectrum.

δ − δ vacuum energy. Jose M. Munoz˜ Castaneda˜ 1, Juan Mateos Guilarte2,3 SUSY δ − δ spectrum. SUSY δ − δ 1Institut für Theoretische Physik, Universität Leipzig, Germany. vacuum energy. 2 Conclusions and Departamento de Física Fundamental, Universidad de Salamanca, Spain outlook 3IUFFyM, Universidad de Salamanca, Spain

What is Quantum Field Theory? , Benasque, SPAIN, 2011 The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda J. Mateos Guilarte

Introduction.

Scalar field fluctuations

The TGTG formula for vacuum energy

Double-delta systems. δ − δ spectrum. Sanctissimum est meminisse cui te dedeas. δ − δ vacuum energy.

SUSY δ − δ spectrum.

SUSY δ − δ vacuum energy.

Conclusions and outlook The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda Introduction J. Mateos Guilarte

Introduction. Vacuum energy for arbitrary geometry: M. Bordag et al “New developments in the Scalar field • fluctuations Casimir effect” (Phys.Rept. 353 (2001) 1-205) and “Advances in the Casimir effect”

The TGTG formula (Oxford Univ. Pr., 2009). E. Elizalde and A. Romeo J.Math.Phys. 30 (1989) 1133.O. for vacuum energy Kenneth and I. Klich Phys.Rev.Lett. 97 (2006) 160401, and Phys. Rev. B 78, 014103 Double-delta (2008). T. Emig, R.L. Jaffe et al J.Phys.A A41 (2008) 164001, Phys.Rev. D77 (2008) systems. 025005, Phys.Rev.Lett. 99 (2007) 170403. D. V. Vassilevich Phys.Rept. 388 (2003) δ − δ spectrum. 279-360. δ − δ vacuum energy.

SUSY δ − δ •Experimental results:J.N. Munday, F. Capasso, and V Adrian Parsegian, “Measured spectrum. long-range repulsive Casimir-Lifshitz forces”, Nature 457, 170-173 (2009). SUSY δ − δ vacuum energy. Conclusions and •Boundary conditions: M. Asorey, A. Ibort and G. Marmo, Int.J.Mod.Phys. A20 (2005) outlook 1001-1026. The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda The scalar Casimir effect J. Mateos Guilarte One real field. Introduction. • Fluctuations of 1D scalar fields on classical backgrounds Scalar field fluctuations Z ∞ 1 µ 1 2 L = ∂µΦ∂ Φ − U(x)Φ (x, t) , lim U(x) = 0 , dx U(x) < +∞ The TGTG formula 2 2 x±∞ −∞ for vacuum energy Z ∞ dω iωt 00 2 Double-delta Φ(t, x) = e φω(x) , −φ (x) + U(x)φω(x) = ω φω(x) systems. ω ∞ 2π δ − δ spectrum.  2 2 2  (U) 0 0 −ω − d /dx + U(x) Gω (x, x ) = δ(x − x ) δ − δ vacuum energy. • Fluctuation vacuum energy. SUSY δ − δ spectrum. N Z ∞ dk  dδ dδ  L SUSY δ − δ X X X 1 + − EV = ω − ω0 = ωj + k + , ρS = vacuum energy. 2 2π dk dk 0 2π j=1 −∞ Conclusions and outlook   Z L 1 dδ+ dδ− 1 h (U) (0) i ρS(k) − ρS0 = + = dx Im Gξ (x, x) − Gξ (x, x) , ξ = iω 4π dk dk π −L The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda The scalar Casimir effect J. Mateos Guilarte Two real fields Introduction. • Two real scalar fields fluctuations with SUSY QM Fourier components: Scalar field fluctuations   1 † µ 1 † φ+ L = ∂Φµ∂ Φ − Φ UΦ; Φ ≡ ; The TGTG formula 2 2 φ− for vacuum energy  U 0   0 d + dW  Double-delta U ≡ + ; Q = i dx dx ; systems. 0 U− 0 0 δ − δ spectrum. 2 ! − d + U 0   δ − δ vacuum † dx2 + H+ 0 {Q, Q } = H = 2 = energy. 0 − d + U 0 H− dx2 − SUSY δ − δ spectrum. 2  2 d W dW ± 2 ± U±(x) = ± + ; H±φ (x) = ω φ (x) SUSY δ − δ dx2 dx ω ω vacuum energy.

Conclusions and outlook • Fluctuation vacuum energy.

N X (+) (−) X X EV = (ω + ω ) − 2 ω0 = 2 ωj j=1 ∞ 1 Z dk d h (+) (+) (−) (−)i + k δ+ + δ− + δ+ + δ− 2 −∞ 2π dk The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda TGTG formula for vacuum energy J. Mateos Guilarte • Compact objects in one dimension. Introduction.

Scalar field U(x) = U1(x) + U2(x) fluctuations

The TGTG formula Ui(x) smooth functions with disjoint compact supports on the real line. for vacuum energy • The T operator associated to a potential U(x). Double-delta Z systems. (U) 0 (0) 0 (0) (U) (0) 0 Gω (x, x ) = Gω (x, x ) − dx1dx2Gω (x, x1)Tω (x1, x2)Gω (x2, x ), δ − δ spectrum.

δ − δ vacuum energy. • TGTG formula for the vacuum interaction energy. Z SUSY δ − δ int i dω spectrum. E = − tr ln (1 − Mω) 0 2 2π SUSY δ − δ vacuum energy. (0) (1) (0) (2) Mω = Gω Tω Gω Tω Conclusions and Z outlook 0 (0) (1) (0) (2) 0 Mω(x, x ) = dx1dx2dx3Gω (x, x1)Tω (x1, x2)Gω (x2, x3)Tω (x3, x )

Kenneth and Klich Phys. Rev. B 78, 014103 (2008). Bordag et al “Advances in the Casimir effect” Oxford, UK: Oxford Univ. Pr. (2009). The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda Two double delta systems J. Mateos Guilarte

Introduction.

Scalar field fluctuations • Two plates or two deltas: mimicking the Casimir effect

The TGTG formula 1 µ 1 2 for vacuum energy L = ∂µΦ∂ Φ − (αδ(x + a) + βδ(x − a)) Φ (x, t) 2 2 Double-delta  2  systems. d 2 − + αδ(x + a) + βδ(x − a) φω(x) = ω φω(x) δ − δ spectrum. dx2 δ − δ vacuum energy.

SUSY δ − δ spectrum.

SUSY δ − δ • Two plates or two deltas: SUSY interacting fluctuations: vacuum energy. 1 1 1 Conclusions and L = ∂Φ† ∂µΦ − Φ†UΦ; W = (α(x + a) + β(x − a)) outlook 2 µ 2 2 1 α2 + β2 U±(x) = ±(αδ(x + a) + βδ(x − a)) + αβ(x + a)(x − a) + 2 4 The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda 2-δ potential: scattering I. J. Mateos Guilarte • The potential Introduction. U(x) = αδ(x + a) + βδ(x − a) Scalar field fluctuations • Scattering zones: Zone II : x < −a , Zone I : −a < x < a , Zone III : x > a

The TGTG formula 0 for vacuum energy • Delta conditions: continuity of ψ and finite step discontinuity of ψ Double-delta Z ±a+δ systems. 0 0 ψ(±a<) = ψ(±a>) , ψ (±a<) − ψ (±a>) = lim dx U(x)ψ(x) δ − δ spectrum. δ→0 ±a−δ 0 0 0 0 δ − δ vacuum ψ (−a<) − ψ (−a>) = αψ(−a) , ψ (a<) − ψ (a>) = βψ(a) energy. + SUSY δ − δ • Scattering states (right-to-left and left-to-right), ∀ k ∈ R spectrum.  −ikx ikx  −ikx SUSY δ − δ e ρ + e , x ∈ II e σ , x ∈ II  r  l vacuum energy. r ikx −ikx l ikx −ikx ψk(x) = Are + Bre , x ∈ I ψk(x) = Ale + Ble , x ∈ I Conclusions and  ikx  ikx −ikx outlook e σr , x ∈ III e ρl + e , x ∈ III The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda 2-δ potential: scattering II. J. Mateos Guilarte • Left-to-right (diestro) scattering amplitudes Introduction. −2iak 4iak
Scalar field ie βe (2k − iα) + α(2k + iβ) ρr = − , fluctuations ∆(k; α, β, a) The TGTG formula 2 2iak for vacuum energy 4k 2k(2k + iβ) 2ikβe σr = , Ar = , Br = − Double-delta ∆(k; α, β, a) ∆(k; α, β, a) ∆(k; α, β, a)) systems.

δ − δ spectrum. • Right-to-left(zurdo) scattering amplitudes

δ − δ vacuum −2iak 4iak
energy. ie αe (2k − iβ) + β(2k + iα) ρl = − , SUSY δ − δ ∆(k; α, β, a) spectrum. 4k2 2ikαe2iak 2k(2k + iα) SUSY δ − δ σl = , Al = − , Bl = , vacuum energy. ∆(k; α, β, a) ∆(k; α, β, a) ∆(k; α, β, a)

Conclusions and outlook • Denominator of scattering amplitudes:   ∆(k; α, β, a) = αβ −1 + e4iak + 4k2 + 2ik(α + β)

• Phase shifts and spectral density √ 1 d(δ + δ ) 2iδ± + − e = σ ± ρlρr , ρS(k) = + ρS 2π dk 0 The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda 2-δ potential: Bound states. J. Mateos Guilarte Study the roots of ∆ in the positive imaginary axis in terms of Λ = αa and Γ = βa. 3 Introduction. possibilities: no bound states, one bound state and two bound states. Scalar field fluctuations

The TGTG formula •NO BOUND STATES (real vacuum energy): for vacuum energy α, β > 0 Double-delta systems. α+β α · β < 0 and − 2a < αβ < 0 δ − δ spectrum.

δ − δ vacuum energy. • ONE BOUND STATE (complex vacuum energy): SUSY δ − δ α+β spectrum. α, β < 0 and αβ < −2a SUSY δ − δ α + β < 0 and α · β < 0 vacuum energy. α+β Conclusions and α + β > 0 and − 2a < αβ < 0 outlook

• TWO BOUND STATES (complex vacuum energy): α+β α, β < 0 and − 2a < αβ < 0 The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda 2-δ potential: Bound states II. J. Mateos Guilarte •The plane (α · a, β · a) is divided in three zones by the hyperbola α+β = −2a: Introduction. αβ

Scalar field fluctuations

The TGTG formula for vacuum energy

Double-delta systems.

δ − δ spectrum.

δ − δ vacuum energy.

SUSY δ − δ spectrum.

SUSY δ − δ vacuum energy.

Conclusions and outlook The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda The ultra-strong limit I. J. Mateos Guilarte β = α → ∞ Introduction. •For α = β (Ar = Bl and Br = Al): Scalar field fluctuations 2  4iak 2  4iak ∆(k; α, α, a) = α −1 + e + 4k + 4ikα;∆ 2 ≡ −1 + e The TGTG formula for vacuum energy Double-delta iαe−2iak e4iak(2k − iα) + α(2k + iα)
systems. ρ = − , δ − δ spectrum. ∆(k; α, α, a) 2 2iak δ − δ vacuum 4k 2k(2k + iα) 2ikαe σ = , Ar = , Br = − energy. ∆(k; α, α, a) ∆(k; α, α, a) ∆(k; α, α, a) SUSY δ − δ spectrum. • Ultra-strong limit for arbitrary k > 0 SUSY δ − δ −2iak vacuum energy. lim ρ = −e ; lim σ = lim Ar = lim Br = 0 α→∞ α→∞ α→∞ α→∞ Conclusions and outlook

•There are no quantum fluctuations between plates in the ultra-strong limit for arbitrary k > 0. The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda The ultra-strong limit II. J. Mateos Guilarte Unitary QFT between plates Introduction. • Non-trivial ultra-strong limit: Scalar field fluctuations π + ∆2(k, a) = 0 ⇒ kn = n , n ∈ Z The TGTG formula 2a for vacuum energy ∆2 is the Dirichlet spectral function obtained by Asorey, Munoz-Castaneda et al. Double-delta systems. • For kn ∈ ker(∆2) and α = β = ∞: δ − δ spectrum. 2iak δ − δ vacuum e n energy. Ar(kn) = 1/2 = Bl(kn) , Br(kn) = − = Al(kn) 2 SUSY δ − δ spectrum.   2iakn 1 iknx 2iakn −iknx ⇒ ψn(x) ≡ ψ(x, kn) = −e ψl(x, kn) = e − e e SUSY δ − δ 2 vacuum energy. + Dirichlet boundary conditions are satisfied by ψ(x, kn) for all n ∈ . Conclusions and Z + outlook ψ2m+1 ∼ cos (k2m+1x) and ψ2m ∼ sin (k2mx), m ∈ Z . •Zeta function prescription for regularized vacuum energy: ∞ −s  −2s 1 X  2 2 2 1 π Ed(s) = n π /(2a) = ζ(2s) , s ∈ C. 2 2 2a n=1 π Physical limit s = −1: Ed = 4 ζ(−1) = −π/(48a). The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda TGTG calculation I. J. Mateos Guilarte • Vacuum interaction between two deltas: first calculated by Bordag et al in , J. Phys. A Introduction.

Scalar field 25, 4483 (1992). Direct calculation for identical deltas. fluctuations

The TGTG formula for vacuum energy Double-delta • Euclidean propagator of a single delta at x = 0 (k2 = ξ2): systems.  −k|x−y| −k(|x|+|y|) δ − δ spectrum.  G+(x, y) = − e − α e , sgn(x) = sgn(y) G(α)(x, y) = ξ (2k) 2k+α 2k δ − δ vacuum ξ − e−k|x−y| energy.  Gξ (x, y) = 2k+α , sgn(x) 6= sgn(y) SUSY δ − δ spectrum. • T-operator of a single delta potential placed at x = 0: SUSY δ − δ vacuum energy. 2kα T(α)(x, y) = δ(x)δ(y) Conclusions and ξ 2k + α outlook

• One delta at x = −a with weight α and another at x = a with weight β:  αβe−4ka  tr ln(1 − M(α,β)) = ln 1 − ξ (2k + α)(2k + β) The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda TGTG calculation II. J. Mateos Guilarte •Casimir energy and force Introduction. Z ∞  −4ka  Scalar field 1 αβe fluctuations Eint(α, β; a) = dk ln 1 − 2π 0 (2k + α)(2k + β) TGTG The formula 1 dE (a) αβ Z ∞ 4k for vacuum energy F (α, β; a) = − int = − dk int 4ak Double-delta 2 da 4π 0 e (2k + α)(2k + β) − αβ systems. δ − δ spectrum. •Ultra-strong limit α = β = ∞: δ − δ vacuum Z ∞ energy. ∞ 1  −4ka π 1 Eint (a) = dk ln 1 − e = − SUSY δ − δ 2π 0 24 2a spectrum. α2 Z ∞ 4k π 1 SUSY δ − δ F∞(a) = − dk = vacuum energy. int 4ak 2 2 2 4π 0 e (2k + α) − α 24 4a Conclusions and outlook The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda TGTG calculation III. J. Mateos Guilarte Some plots of the energy. Introduction.

Scalar field fluctuations

The TGTG formula for vacuum energy

Double-delta systems.

δ − δ spectrum.

δ − δ vacuum energy.

SUSY δ − δ spectrum.

SUSY δ − δ vacuum energy.

Conclusions and outlook

re(Eint) = •; im(Eint) = • The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda The SUSY δ − δ spectrum. J. Mateos Guilarte Scattering amplitudes and bound states. Introduction. • Scattering amplitudes for the U+(x) problem: Scalar field fluctuations e−2ia(k+q) e4iaq(k + q − iα)(k − q + iβ) − (k − q − iα)(k + q + iβ)
The TGTG formula ρr = for vacuum energy ∆ −2ia(k+q) 4iaq
Double-delta e e (k − q + iα)(k + q − iβ) − (k + q + iα)(k − q − iβ) systems. ρl = ∆ δ − δ spectrum. 4kqe−2iak δ − δ vacuum σr = σl = − energy. ∆ 2iaq −2iaq SUSY δ − δ ∆ = e (k − q + iα)(k − q + iβ) − e (k + q + iα)(k + q + iβ) spectrum. 2 2 2 2 2 SUSY δ − δ E = ω = q + (α − β) = k + (α + β) . U−(x) ⇒ (α, β) 7→ (−α, −β) vacuum energy.

Conclusions and outlook • Identical deltas: α = β. Bound states given by (k = iµ) q α + µ cot(qa) − tan(qa) = − α + µ q • Ultra-strong limit: α = β → ∞. Non zero zero quantum fluctuations between plates ⇔ sin(2aq) = 0 The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda The single δ step I. J. Mateos Guilarte Scattering amplitudes and propagator. Introduction. • Single δ step potential (sδ potential) centered at x = 0: Scalar field fluctuations 2 U(x) = Usδ(x; α, s) = αδ(x) + s (1 − θ(x)) The TGTG formula for vacuum energy Scattering amplitudes and Wronskian Double-delta systems. 2q 2k σr = ; σl = ; ρr = −1 + σr; ρl = −1 + σl; δ − δ spectrum. ∆sδ ∆sδ

δ − δ vacuum 4ikq energy. W(ψr, ψl) = ; ∆sδ ≡ k + q + iα ∆sδ SUSY δ − δ spectrum. k and q are the momenta in the zones x > 0 and q < 0: E = ω2 = k2 = q2 + s2 SUSY δ − δ vacuum energy.

Conclusions and • Reduced propagator: outlook ik|x−y| ik(x+y)   ++ e e 2k Gω (x, y) = − − −1 + 2ik 2ik ∆sδ iq|x−y| iq(x+y)   −− e e 2q Gω (x, y) = − − −1 + 2iq 2iq ∆sδ i(kx−qy) i(ky−qx) +− e −+ e Gω (x, y) = i ; Gω (x, y) = i ∆sδ ∆sδ The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda The single δ step I. J. Mateos Guilarte T-operator. Introduction. • For 1 dimensional systems Tω(x, y) = V(x)δ(x − y) − V(x)Gω(x, y)V(y). Calculation Scalar field for the sδ potential: fluctuations

The TGTG formula  2  ++ iα for vacuum energy Tω (x, y) = 1 + δ(x)δ(y); ∆sδ Double-delta systems.  2  2 −− iα iαs −iqy −iqx
δ − δ spectrum. Tω (x, y) = 1 + δ(x)δ(y) + e δ(x) + e δ(y) ; ∆sδ ∆sδ δ − δ vacuum 2 2 energy. +− iα iαs −iqy Tω (x, y) = δ(x)δ(y) + e δ(x); SUSY δ − δ ∆sδ ∆sδ spectrum. 2 2 −+ iα iαs −iqx SUSY δ − δ Tω (x, y) = δ(x)δ(y) + e δ(y) vacuum energy. ∆sδ ∆sδ Conclusions and outlook

• Kernel of operator Mω: Z Z (1) (2) (0) Mω(x, y) = dzNω (x, z)Nω (z, y); Nω(x, y) = dzGω (x, z)Tω(z, y) The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda The single δ step II. J. Mateos Guilarte N-operator. Introduction. • N-operator can be computed: Scalar field fluctuations  2 2  ++ 1 1 iα αs ikx The TGTG formula Nω (x, y) = − + − e δ(y) for vacuum energy 2ik 2 ∆sδ (k + q)∆sδ  2 2  Double-delta −− 1 1 iα αs −ikx systems. Nω (x, y) = − + − e 2ik 2 ∆sδ (q − k)∆sδ δ − δ spectrum. 2kαs2  αs2 δ − δ vacuum + e−iqx δ(y) − eikxe−iqy 2 2 energy. (q − k )∆sδ 2k∆sδ SUSY δ − δ  2 2  spectrum. +− 1 1 iα αs ikx Nω (x, y) = − + − e δ(y) SUSY δ − δ 2ik 2 ∆sδ (k + q)∆sδ vacuum energy. 2 αs ikx −iqy Conclusions and − e e outlook 2k∆sδ  2 2  −+ 1 1 iα αs −ikx Nω (x, y) = − + − e 2ik 2 ∆sδ (q − k)∆sδ 2kαs2  + e−iqx δ(y) 2 2 (q − k )∆sδ The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda SUSY δ − δ vacuum energy. J. Mateos Guilarte Calculation prescription. Introduction. • Prescription for vacuum energy calculation between two delta steps: Scalar field fluctuations U(x) = Usδ(x + a; α, s) + Usδ(−(x − a); β, s) The TGTG formula for vacuum energy

Double-delta systems. For each delta step the N operator is easily obtained:

δ − δ spectrum. (α) sδ (β) sδ Nω (x, y) = Nω (x + a, y + a; α, s); Nω (x, y) = Nω (−x + a, −y + a; β, s) δ − δ vacuum energy.

SUSY δ − δ spectrum. Operator Mω and power traces are computed using the general expression for the SUSY δ − δ kernel: vacuum energy. Z sδ sδ Conclusions and Mω(x, y; s, a, α, β) = dz Nω (x + a, z + a; α, s)Nω (−z + a, −y + a; β, s) outlook

First order approximation to the vacuum energy (euclidean): Z ∞ Z int 1 dξ  2  E0 ' dx Miξ(x, x; s, a, α, β) + O Miξ 2 s 2π

SUSY restriction for α = β: s2 = α2. The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda Conclusions and outlook. J. Mateos Guilarte

Introduction.

Scalar field fluctuations The TGTG formula New paths for obtaining quantum field theories in bounded domains has been explored. for vacuum energy •

Double-delta • Potentials concentrated on hypersurfaces allow to use the TGTG method to compute the systems. vacuum energy. δ − δ spectrum. • Integral expressions for the vacuum energy have been obtained for arbitrary values of δ − δ vacuum energy. the delta weights in the double delta potential. For the case of the SUSY double delta the SUSY δ − δ vacuum energy can easily be obtained from the calculations done. spectrum.

SUSY δ − δ vacuum energy. • Which boundary conditions can be implemented using potentials concentrated on Conclusions and 0 outlook hypersurfaces? ⇒ δ + δ systems. • Extension of the TGTG to curved backgrounds: double δ in a kink background (double δ in a Posch-Teller background). The Casimir effect, and 1D scattering

Jose M. Mu˜noz Casta˜neda J. Mateos Guilarte

Introduction.

Scalar field fluctuations

The TGTG formula for vacuum energy

Double-delta systems. δ − δ spectrum. MANOLO, ANIVERSARIA DIES FELIX! δ − δ vacuum energy.

SUSY δ − δ spectrum.

SUSY δ − δ vacuum energy.

Conclusions and outlook