Dirac configurations, boundary conditions, and quantum fluctuations ...

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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