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The Casimir Effect and One-Dimensional Scattering

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The Casimir Effect and One-Dimensional Scattering 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 1 µ 1 2 L = @µΦ@ Φ − U(x)Φ (x; t) ; lim U(x) = 0 ; dx U(x) < +1 The TGTG formula 2 2 x±∞ −∞ for vacuum energy Z 1 d! i!t 00 2 Double-delta Φ(t; x) = e φ!(x) ; −φ (x) + U(x)φ!(x) = ! φ!(x) systems. ! 1 2π δ − δ spectrum. 2 2 2 (U) 0 0 −! − d =dx + U(x) G! (x; x ) = δ(x − x ) δ − δ vacuum energy. • Fluctuation vacuum energy. SUSY δ − δ spectrum. N Z 1 dk dδ dδ L SUSY δ − δ X X X 1 + − EV = ! − !0 = !j + 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 = i! 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 y µ 1 y 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 y dx2 + H+ 0 fQ; Q g = H = 2 = energy. 0 − d + U 0 H− dx2 − SUSY δ − δ spectrum. 2 2 d W dW ± 2 ± U±(x) = ± + ; H±φ (x) = ! φ (x) SUSY δ − δ dx2 dx ! ! vacuum energy. Conclusions and outlook • Fluctuation vacuum energy. N X (+) (−) X X EV = (! + ! ) − 2 !0 = 2 !j j=1 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 G! (x; x ) = G! (x; x ) − dx1dx2G! (x; x1)T! (x1; x2)G! (x2; x ); δ − δ spectrum. δ − δ vacuum energy. • TGTG formula for the vacuum interaction energy. Z SUSY δ − δ int i d! spectrum. E = − tr ln (1 − M!) 0 2 2π SUSY δ − δ vacuum energy. (0) (1) (0) (2) M! = G! T! G! T! Conclusions and Z outlook 0 (0) (1) (0) (2) 0 M!(x; x ) = dx1dx2dx3G! (x; x1)T! (x1; x2)G! (x2; x3)T! (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) φ!(x) = ! φ!(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 = @Φy @µΦ − ΦyUΦ; 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 and finite step discontinuity of Double-delta Z ±a+δ systems. 0 0 (±a<) = (±a>) ; (±a<) − (±a>) = lim dx U(x) (x) δ − δ spectrum. δ!0 ±a−δ 0 0 0 0 δ − δ vacuum (−a<) − (−a>) = αψ(−a) ; (a<) − (a>) = β (a) energy. + SUSY δ − δ • Scattering states (right-to-left and left-to-right), 8 k 2 R spectrum. 8 −ikx ikx 8 −ikx SUSY δ − δ e ρ + e ; x 2 II e σ ; x 2 II <> r <> l vacuum energy. r ikx −ikx l ikx −ikx k(x) = Are + Bre ; x 2 I k(x) = Ale + Ble ; x 2 I Conclusions and > ikx > ikx −ikx outlook :e σr ; x 2 III :e ρl + e ; x 2 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 p 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 ! 1 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.
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