More magics at the λ2 = −2 opaque point. The amplitudes between plates are: Ar(
k; µ1, λ1, µ2, −2, a) = 2ik(λ2. 1 − 4)µ2. ∆(k; µ1, λ1, µ2, −2, a). Br(k; µ1, λ1, µ2, −2, ...
Dirac δ configurations, boundary conditions, quantum fluctuations Jose M. Mu˜ noz Casta˜ neda J. Mateos Guilarte Scalar field fluctuations Doubledelta/delta0 configuration. Opaque couplings: boundary conditions The TGTG formula for vacuum energy
Dirac δ configurations, boundary conditions, and quantum fluctuations of scalar (Higgs) fields Jose M. Mu˜ noz Casta˜ neda1 , Juan Mateos Guilarte2,3 1 2
Institut für Theoretische Physik, Universität Leipzig, Germany.
Departamento de Física Fundamental, Universidad de Salamanca, Spain 3
IUFFyM, Universidad de Salamanca, Spain
MSQSA
,
Benasque, SPAIN, 2012
Dirac δ configurations, boundary conditions, quantum fluctuations
Outline
Jose M. Mu˜ noz Casta˜ neda J. Mateos Guilarte Scalar field fluctuations
1
Scalar field fluctuations
2
Double-delta/delta0 configuration.
3
Opaque couplings: boundary conditions
4
The TGTG formula for vacuum energy
Doubledelta/delta0 configuration. Opaque couplings: boundary conditions The TGTG formula for vacuum energy
•~=c=1
Dirac δ configurations, boundary conditions, quantum fluctuations Jose M. Mu˜ noz Casta˜ neda J. Mateos Guilarte Scalar field fluctuations Doubledelta/delta0 configuration. Opaque couplings: boundary conditions The TGTG formula for vacuum energy
The scalar Casimir effect One real field.
• Fluctuations of 1D scalar fields on classical backgrounds Z ∞ 1 1 ∂µ Φ∂ µ Φ − U(x)Φ2 (x, t) , lim U(x) = 0 , dx U(x) < +∞ x±∞ 2 2 −∞ Z ∞ dω iωt 2 Φ(t, x) = e φω (x) , −φ00 ω (x) + U(x)φω (x) = ω φω (x) ∞ 2π (U) −ω 2 − d2 /dx2 + U(x) Gω (x, x0 ) = δ(x − x0 )
L=
• Fluctuation vacuum energy. EV =
X
ω−
X
ω0 =
N X
ωj +
j=1
ρS (k) − ρS0
1 = 4π
dδ+ dδ− + dk dk
=
1 π
Z
1 2 L −L
Z
∞ −∞
dk dδ+ dδ− k + , 2π dk dk
ρS0 =
L 2π
h i (U) (0) dx Im Gξ (x, x) − Gξ (x, x) , ξ = iω
Dirac δ configurations, boundary conditions, quantum fluctuations Jose M. Mu˜ noz Casta˜ neda J. Mateos Guilarte Scalar field fluctuations Doubledelta/delta0
Double-delta/delta0 spectral problem • Two delta/delta0 potentials: mimicking two soft plates in the Casimir effect 1 1 L = ∂µ Φ∂ µ Φ − µ1 δ(x + a) + λ1 δ 0 (x + a) + µ2 δ(x − a) + λ2 δ 0 (x − a) Φ2 (x, t) 2 2 d2 − 2 + µ1 δ(x + a) + λ1 δ 0 (x + a) + µ2 δ(x − a) + λ2 δ 0 (x − a) φω (x) = ω 2 φω (x) dx
configuration. Opaque couplings: boundary conditions The TGTG formula for vacuum energy
ˆ • The spectrum of L:
ˆ ω (x) = ω 2 φω (x) Lφ 2 ˆ = − d + µ1 δ(x + a) + λ1 δ 0 (x + a) + µ2 δ(x − a) + λ2 δ 0 (x − a) L dx2
• Delta/delta0 matching conditions: finite step discontinuity of ψ and twisted discontinuity of ψ 0 at x = ±a 1+λ1 /2 0 0 0 1−λ1 /2 ψ(−a< ) ψ(−a> ) 1−λ1 /2 µ1 0 0 0 ψ 0 (−a< ) 2 1+λ1 /2 ψ (−a> ) 1−λ1 /4 1+λ2 /2 ψ(a< ) = ψ(a> ) 0 0 0 1−λ2 /2 0 0 ψ (a< ) ψ (a> ) 1−λ2 /2 µ2 0 0 1+λ /2 1−λ2 /4 2
2
Dirac δ configurations, boundary conditions, quantum fluctuations Jose M. Mu˜ noz Casta˜ neda J. Mateos Guilarte Scalar field fluctuations
2-δ/δ 0 potential: scattering I. • The potential U(x) = µ1 δ(x + a) + λ1 δ 0 (x + a) + µ2 δ(x − a) + λ2 δ 0 (x − a) P. Kurasov, J. Math. Anal. Appl.201 (1996) 297 M. Gadella, J. Negro, L.M. Nieto, Phys. Lett. A373 (2009) 1310
Doubledelta/delta0 configuration.
• Scattering zones:
Opaque couplings: boundary conditions
• Scattering waves (right-going and left-going), ∀ k ∈ R+
The TGTG formula for vacuum energy
ψkr (x)
Zone II : x < −a ,
−ikx ρ + eikx r e = Ar eikx + Br e−ikx ikx e σr
, x ∈ II , x∈I , x ∈ III
Zone I : −a < x < a ,
;
ψkl (x)
Zone III : x > a
−ikx σ l e = Al eikx + Bl e−ikx ikx e ρl + e−ikx
, x ∈ II , x∈I , x ∈ III
• Right-going waves (diestro) scattering amplitudes ρr
=
Ar =
2e2iak (k(λ21 +4)−2iµ1 )(2kλ2 +iµ2 ) − ∆(k;µ1 ,λ1 ,µ2 ,λ2 ,a) 2e−2iak (2kλ1 +iµ1 )(k(λ22 +4)+2iµ2 ) − ∆(k;µ1 ,λ1 ,µ2 ,λ2 ,a)
−
k(λ21 − 4)(k(λ22 + 4) + 2iµ2 ) , ∆(k; µ1 , λ1 , µ2 , λ2 , a)
, σr =
Br = −
(λ21 − 4)(λ22 − 4)k2 ∆(k; µ1 , λ1 , µ2 , λ2 , a)
2e2iak k(−4 + λ21 )(2kλ2 + iµ2 ) ∆(k; µ1 , λ1 , µ2 , λ2 , a)
Dirac δ configurations, boundary conditions, quantum fluctuations Jose M. Mu˜ noz Casta˜ neda J. Mateos Guilarte Scalar field fluctuations Doubledelta/delta0 configuration.
2-δ/δ 0 potential: scattering II. • Left-going waves (zurdo) scattering amplitudes ρl
=
Al = −
2e−2iak (k(λ21 +4)+2iµ1 )(2kλ2 −iµ2 ) + ∆(k;µ1 ,λ1 ,µ2 ,λ2 ,a) 2e2iak (2kλ1 −iµ1 )(k(λ22 +4)−2iµ2 ) + ∆(k;µ1 ,λ1 ,µ2 ,λ2 ,a)
2e2iak k(λ22 − 4)(2kλ1 − iµ1 ) , ∆(k; µ1 , λ1 , µ2 , λ2 , a)
, σl =
Bl = −
(λ21 − 4)(λ22 − 4)k2 ∆(k; µ1 , λ1 , µ2 , λ2 , a)
k(λ22 − 4)(k(λ21 + 4) + 2iµ1 ) ∆(k; µ1 , λ1 , µ2 , λ2 , a)
Opaque couplings: boundary conditions The TGTG formula for vacuum energy
• Denominator of scattering amplitudes: ∆(k; µ1 , λ1 , µ2 , λ2 , a)
= +
4e4iak (2kλ1 − iµ1 )(2kλ2 + iµ2 ) + k λ21 + 4 + 2iµ1 k λ22 + 4 + 2iµ2
• Phase shifts and spectral density e2iδ± = σ ±
√
ρl ρr
,
ρS (k) =
1 d(δ+ + δ− ) + ρS0 2π dk
(1)
Dirac δ configurations, boundary conditions, quantum fluctuations Jose M. Mu˜ noz Casta˜ neda J. Mateos Guilarte Scalar field fluctuations Doubledelta/delta0
2-δ/δ 0 potential: Bound states. Study the roots of ∆ in the positive imaginary axis of the k-complex plane.
4e
−4aκ
∆(iκ; µ1 , λ1 , µ2 , λ2 , a) = 0 , κ ∈ R+ κ λ22 + 4 + 2µ2 = 0 (2κλ1 − µ1 )(2κλ2 + µ2 ) + κ λ21 + 4 + 2µ1
• Switch-off the δ 0 ’s: λ1 = λ2 = 0: e−4aκ =
configuration. Opaque couplings: boundary conditions The TGTG formula for vacuum energy
4κ2 µ1 + µ2 +2 κ+1 µ1 µ2 µ1 µ2
3 possibilities of intersections on the positive real κ-axis of the two families of curves: 0, no bound states, 1, one bound state or 2, two bound states. •NO BOUND STATES : µ1 , µ 2 > 0 µ1 · µ2 < 0 and − 2a
