Zero-range potentials: Delta-likebarrier versus self-adjoint extension
Yuri V. Grats, Pavel Spirin

Moscow State U.

20th Lomonosov Conference
August 19-25, 2021, Moscow, RFYuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Table of contents• Y.V. Grats, Phys. Atom. Nucl. 82, (2019) no.2, 153• Y.V. Grats, P.Spirin, Universe 7, (2021) no.5, 127Content:
Cosmic string: motivation
Vacuum Polarization on the flat backgroundwith static potential
Perturbative approach
Renormalized VEVs
Modelling δ 3 (r) interaction via PT
δ 3 (r) interaction via self-adjoint extension
Correspondence and renormalization of coupling
Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint extenCosmic string: motivation
1st PT order vs NP in literature:∆Tµν ∝ ξ(ξ − 1/6)
Field eqn.
√ −g 2 − ξR GFξ (x, x0 ) = −δ d (x − x0 )Potential:
γ(r) =√−g ξR = 4π(1 − β)ξ δ 2 (ρ)

Limit: (1 − β) → 0 ξ → ∞ (1 − β)ξ = const
hi22∂ + λ δ (ρ) GFλ (x, x0 ) = −δ d (x − x0 )
ρ = (x, y)hi23∂ + λ δ (r) GFλ (x, x0 ) = −δ d (x − x0 ), r = (x, y, z)
Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Green’s function: perturbation theory
Field equation:
L(x, ∂) GF (x, x0 ) = −δ d(x − x0 )Operator perturbation: L = L0 + δL
LG = −1
G = −L−1
G0 = −L0 −1Solution
h as seriesi−1= G0 + G0 δL G0 + ...
G = −L0 (1 − G0 δL)
For zero-order equation
L0 (x, ∂) = ∂2GF0 (x − x0 ) =Z0d dp eip (x−x )(2π)d p 2 + iOperator deviation:
δL(x, ∂) = U(x)Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Renormalized VEVs
Dimensional regularization: D = d − 2δ
GFren (x, x | d) = lim GFreg (x, x | D)
δ→0Renormalization:
GFren (x, x | d) = lim
δ→0hi

GFreg (x, x | D) − GFdiv (x, x | D)

Renormalized VEV hϕ2 (x)i:hϕ2 (x)iren = i GFren (x, x | d)Renormalized energy-momentum tensor:
Tµν ren = i Dµν GFren (x, x0 | d) x0 =x
1

ν
ν
ν 0Dµ = (1−2ξ) ∂ ∂µ − −2ξ δµν ∂ λ ∂λ0 −ξ ∇ν ∂µ +∇0 ∂µ02Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Static potentials
Fourier-transform: U(q) = 2π δ(q 0 ) U(q)The generic form: (q̃ µ = (0, q))
Z 4Zd pQd q iq xeU(q)VEV = −i(2π)3(2π)4 [p 2 +iε][(p+ q̃) 2 +iε]
VEVQ

hϕ2 i
hT00 ihT0i ihTij i1
p20 + [2ξ − 1/2] p · q̃(ξ − 1) p0 q̃i − p0 pi(ξ −1) q̃i pj −pi pj +[2ξ −1/2] p· q̃ δij −ξ(pi q̃j + q̃i q̃j )Z01
20 U(r )φ (r) ren =dr32π 3|r − r 0 |3Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Static potentials
Renormalized EMT:
Z03 1
3 0 U(r )Tµν (r) ren =ξ−Zµν
dr32π 36|r − r 0 |5xi xj − 2x(i x0j) + x0i x0j− 3δij
Z00 = 2Zij = 5|r − r 0 |2
Conservation: ∂ ν Tµν = ∂ i Tij ∝ ∂ i Zij |r − r 0 |−5 = 0
Spherically-symmetric potential: r0 < R  rZR1r02 U(r0 )φ2 (r) ren = − 2dr0 28π rr − r020 ZR02
202
11000 r (3r + r )T0 (r) ren = 2 ξ −dr U(r ) 24π r6(r − r02 )30
Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Trying δ 3 (r) potential as perturbation
Klein-Gordon eqn:[λ] = cm|λ|  “r0 “ 2
∂ + λ δ 3 (r) φ(t, r) = 0
Splitting of the field operator:
L0 (x) = ∂ 2 ,δL(x) = λ δ 3 (r)δL“  “L0Renormalized VEV:
λ32π 3 r3Renormalized enegry-momentum tensor:
λ3 3νTµ (t, r, ϑ, ϕ) ren =(6ξ − 1) diag 1, −1, ,64π 3 r52 2
ξ and λ are independent!
hφ2 (r)iren = −Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint extenδ 3 (r) interaction via self-adjoint extension
 2
∂t − ∆ + λ δ 3 (r) φ(t, r) = 0H ≡ −∆ + λ δ 3 (x)Factorization:

φ± (t, r) = e∓iωt uω (r) ,Huω = ω 2 uω

Hamiltonian decomposition∞MOd22 dl(l + 1)Ḣl = − 2 −+ḢlI
H → Ḣ =drr drr2| {z }l=0angularSelf-adjointness
R3 → R3 \0 ,Ḣ0 → Ḣ0,αuω → uω,α+ initial condition depending on parameter αα → +∞ as λ → 0 (free Hamiltonian)Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Eigenfunctions and eigenvalues
Attraction (α < 0):
1 sin(ωr + δ)√continuous : uω,α =E = ω22πr
ωp|α| −4π|α|r
discrete :uα =e
E = −(4πα)2r
Repulsion (α > 0):
1 sin(ωr + δ)ω√continuous : uω,α =tan δ =2πr4παωno negative energy modes!
Hadamard function D(1) (x, x0 ) = hφ(x)φ(x0 )+ ↔i/2
ZhiX(1)iω(t0 −t)∗0∗0
D = Re dω euω,α (x ) uω,α (x) + ulm (x ) ulm (x)l>0,mYuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Vacuum polarization induced by the zero-width wellRenormalizedHadamard function
Zhiiω(t0 −t)∗0∗0(1)uω,α (x ) uω,α (x)−uω,∞ (x ) uω,∞ (x)Dα = Re dω e
Scalar-field VEV
hφ2 iren = Dα(1) (x, x) =
where

Z∞

J (β) =dz

J (β),4π 2 r2sin βz + z cos βz
1 + z2β = 8παr

0Similar results: D.Fermi, L.Pizzocchero, Symmetry (2018) 10, 38

J (β) = e−β E1 (β) ,Z∞ −teE1 (β) ≡dttβYuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Renormalized EMT
Introducing Jk (β) ≡ β k dk J (β)/dβ ki1 h 11tTt ren = 2 4− 2ξ J + 2ξ − J1 − ξJ24π r h22 i111− 2ξ J1Trr ren = 2 4 4ξ − J +4π r h221  i111θTθ ren = 2 4−4ξ J + 3ξ − J1 + −ξ J2
4π r
224Total trace:2J (β) − 2J1 (β) + J2 (β)
Sp Tµν ren = (1 − 6ξ)
8π 2 r4Splitting:1 νννTµ ren = T̃µ + ξ −T̄ (ξ) ,6 µYuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint extenSmall αr
For αr  1 E1 (x) = − ln x − γ +∞X(−1)k+1 xk
k=1k · k!

Scalar-field VEV| ln(8πeγ αr)|hφ iren =,4π 2 r2
Energy-momentum-tensor:
| ln αr| h1νTµ (t, r, ϑ, ϕ) ren =ξ−
diag
1,−2,2,2−2π 2 r46i1−
diag 1, 1, −1, −112Conservation and trace:∂ν Tµν ren = 0
Sp T ren=02

ξ=ξ4

Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Large αr
For αr  1h 1 i
11+Ohφ iren =32 π 3 αr3αr
Naive relation with λ: 1 
11−→ −λ− = + O 2αα r0αEnergy-momentum-tensor:
(1 − 6ξ)3 3νhTµ (t, r, ϑ, ϕ)iren = 6 3 5 diag 1, −1, ,+ ...
2π r α2 2
Conformal tensor:

11 2 2νT̃µ (t, r, θ, ϕ) ren =diag 1, , − , −+ ...
512α2 π 4 r63 3 32

Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Correspondence of two approaches

Figure: Plot of J for all real arguments

β>0Yuri V. Grats, Pavel Spirin

α>0Zero-range potentials: Delta-like barrier versus self-adjoint exten

Renormalization
111= +,λren
λ 4π

 → 0+λren α = −1λren < 0λren =4πλ
λ + 4πλ = −4π + δδ = 16π 2 2 /|λren |Figure: Plot of λren /4π versus λ/4π
Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Conclusions
Static-potential-induced vacuum polarization ofa massless scalar field is computed within thePT framework;
Zero-order consists of tadpoles and vanishes inregularization;
The Laplacian’s SAE gives the exact result for
the repulsive 3D Aharonov–Bohm barrier
In order to have a correspondence, we have toimpose the coupling renormalization
In the PT-framework formulae one also has tosubstitute λ by λren
The case of 2D point potential (the real cosmicstring background), is of the future interest
Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten

Acknowledgment for attention

Thank you!

Yuri V. Grats, Pavel Spirin

Zero-range potentials: Delta-like barrier versus self-adjoint exten