Mathematica and Quantum Corrections to Kink...
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Mathematica and Quantum Corrections to kink Masses A. Alonso M. González J. Mateos Basic Notions Application to scalar field models The Asymptotic Approach The Mathematica Program Mathematica and Quantum Corrections to Kink Masses Alberto Alonso Izquierdo 1,3 Miguel Ángel González León 1,3 Juan Mateos Guilarte 2,3 1 Departamento de Matemática Aplicada (Universidad de Salamanca) 2 Departamento de Física Fundamental (Universidad de Salamanca) 3 IUFFyM (Universidad de Salamanca) Encuentros de Invierno de Geometría, Mecánica y Teoría de Control, Zaragoza 2010
Transcript of Mathematica and Quantum Corrections to Kink...
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Mathematicaand
Quantum Corrections to Kink Masses
Alberto Alonso Izquierdo1,3
Miguel Ángel González León1,3
Juan Mateos Guilarte2,3
1Departamento de Matemática Aplicada (Universidad de Salamanca)
2Departamento de Física Fundamental (Universidad de Salamanca)
3IUFFyM (Universidad de Salamanca)
Encuentros de Invierno deGeometría, Mecánica y Teoría de Control,
Zaragoza 2010
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
THE AIM OF THE TALK
Question:How to compute the quantum correction to the kink mass in a(1+1)-scalar field theory model?
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1 + ~2∆M2 + . . .
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
ONE-LOOP QUANTUM MASS CORRECTION
∆M1 = 12 ~ tr(H 1
2 [φS])
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1 + ~2∆M2 + . . .
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
ONE-LOOP QUANTUM MASS CORRECTION
∆M1 = 12 ~ tr(H 1
2 [φS])
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
ONE-LOOP QUANTUM MASS CORRECTION
∆M1 = 12 ~ tr(H 1
2 [φS])
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
ONE-LOOP QUANTUM MASS CORRECTION
∆M1 = 12 ~ tr(H 1
2 [φS])
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
DESCRIPTION OF THE PROBLEM
PHYSICAL SYSTEM
CLASSICAL PHYSICS(~ = 0)
QUANTUM PHYSICS(~ 1)
• (1+1)-D Scalar field Model:S =
∫dx2 [ 1
2∂µφ∂µφ− U(φ)
]• Euler-Lagrange Equations
∂µ∂µφ = −∂U
∂φSOLUTION: φS
• Classical Mass:
Mcl =
∫ ∞−∞dx[
12 (∂φ
∂x)2 + U(φ)
]
Quantification ofthe classical system
SERIES EXPANSIÓN OF THE MASS
MQ(~) ≈ Mcl + ~∆M1
SEMICLASSICAL
APPROXIMATION:
• Solution stability:H[φS](ψn) = ω2
nψn
H[φS] = − d2
dx2 + ∂2U∂φ2 [φS]
ωn > 0⇒ Stable solution
∆M1 = 12 ~∑
r ωr
SEMICLASSICAL MASS
Msc = Mcl + 12 ~ tr(H 1
2 [φS])
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response will be∞SOLUTION:
We need a reference point
Zero-Point Renormalization
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response will be∞SOLUTION:
We need a reference point
Zero-Point Renormalization
We have to measure the quantum correction with respect to the minimum energysolution φV .
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response is∞−∞
SOLUTION:
We need a procedure
Mode NumberCut-off Regularization
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response is∞−∞
SOLUTION:
We need a procedure
Mode NumberCut-off Regularization
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])|M.C.
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])|M.C.
• Hessian Operator: H[φS] = − d2
dx2 +∂2U∂φ2 [φS]
WARNING MESSAGE!!!:
The response is∞
SOLUTION:
The mass coupling constant
is infinite
Mass Renormalization
We have to introduce the counterterms (well stablished procedure in Physics)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])|M.C.+Ect[φS]− Ect[φV ]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
RENORMALIZATION AND REGULARIZATION OF ∆M
Quantum correction to the mass:
∆M = 12 ~ tr(H 1
2 [φS])− 12 ~ tr(H 1
2 [φV ])|M.C. + Ect[φS]− Ect[φV ]
• Hessian operator associated to the solution φS:
H[φS] = − d2
dx2 +∂2U∂φ2 [φS]︸ ︷︷ ︸
V[φS]≡V(x)
• Hessian operator associated to the vacuum φV :
H[φV ] = − d2
dx2 +∂2U∂φ2 [φV ]︸ ︷︷ ︸V[φV ]≡V0
• Counterterms:
Ect[φS]− Ect[φV ] = ~ δm∫
dx[∂2U∂φ2 [φS]− ∂2U
∂φ2 [φV ]]
δm =
∫dk4π
1√k2 + d2U
dφ2 [φV ]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =12
(φ2 − 1)2
• Partial Diferential Equation:∂2φ
∂t2 −∂2φ
∂x2 + 2φ(φ2 − 1) = 0
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =12
(φ2 − 1)2
• Partial Diferential Equation:∂2φ
∂t2 −∂2φ
∂x2 + 2φ(φ2 − 1) = 0
SOLUTIONS:
VACUUM SOLUTION:φV = ±1
KINK SOLUTION:φK = tanh x−vt√
1−v2
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
• KINK ENERGY DENSITY:
ε(x) = sech4 x + x0 − vt√1− v2
• KINK CLASSICAL MASS:
Mcl =
∫ ∞−∞
dx sech4(x + x0) =43
CLASSICAL KINK
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
• KINK ENERGY DENSITY:
ε(x) = sech4 x + x0 − vt√1− v2
• KINK CLASSICAL MASS:
Mcl =
∫ ∞−∞
dx sech4(x + x0) =43
QUANTUM KINK
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
HOMOGENEOUS SOLUTION
φV = ±1
- Hessian Operator:H[φV ] = − d2
dx2 + 4Spec Hd[φV ] = 4 1
2
Spec Hc[φV ] = k2 + 4k∈R
- Spectral Density:ρv(k) = L
2π
KINK SOLUTION:φK(x) = ± tanh x
- Hessian Operator:H[φK ] = − d2
dx2 + 4− 6 sech2xSpec Hd[φK ] = 0 ∪ 3 ∪ 4 1
2
Spec Hc[φK ] = q2 + 4q∈R
- Spectral Density:ρK(q) = L
2π + 12π
dδ(q)dq , δ(q) = −2 arctan 3q
2−q2
∆M =~m
2
(√
3 +1
2π
∫ ∞−∞
dq√
q2 + 4dδ(q)
dq
)+ Ect[φK] − Ect[φv] + ∆R =
=
√3~m
2−
~m
2π
∫ ∞−∞
dq3√
q2 + 4(q2 + 2)
q4 + 5q2 + 4+
3~m
2π
∫ ∞−∞
dk√k2 + 4
+~m
4π〈V(x) − 4〉 =
=
√3 ~m
2−
~m√
3+
~m
4π
∫ ∞−∞
dx(−6sech2x) = ~m
(1
2√
3−
3
π
)≈ −0.666255~m
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
HOMOGENEOUS SOLUTION
φV = ±1
- Hessian Operator:H[φV ] = − d2
dx2 + 4Spec Hd[φV ] = 4 1
2
Spec Hc[φV ] = k2 + 4k∈R
- Spectral Density:ρv(k) = L
2π
KINK SOLUTION:φK(x) = ± tanh x
- Hessian Operator:H[φK ] = − d2
dx2 + 4− 6 sech2xSpec Hd[φK ] = 0 ∪ 3 ∪ 4 1
2
Spec Hc[φK ] = q2 + 4q∈R
- Spectral Density:ρK(q) = L
2π + 12π
dδ(q)dq , δ(q) = −2 arctan 3q
2−q2
∆M =~m
2
(√
3 +1
2π
∫ ∞−∞
dq√
q2 + 4dδ(q)
dq
)+ Ect[φK] − Ect[φv] + ∆R =
=
√3~m
2−
~m
2π
∫ ∞−∞
dq3√
q2 + 4(q2 + 2)
q4 + 5q2 + 4+
3~m
2π
∫ ∞−∞
dk√k2 + 4
+~m
4π〈V(x) − 4〉 =
=
√3 ~m
2−
~m√
3+
~m
4π
∫ ∞−∞
dx(−6sech2x) = ~m
(1
2√
3−
3
π
)≈ −0.666255~m
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
HOMOGENEOUS SOLUTION
φV = ±1
- Hessian Operator:H[φV ] = − d2
dx2 + 4Spec Hd[φV ] = 4 1
2
Spec Hc[φV ] = k2 + 4k∈R
- Spectral Density:ρv(k) = L
2π
KINK SOLUTION:φK(x) = ± tanh x
- Hessian Operator:H[φK ] = − d2
dx2 + 4− 6 sech2xSpec Hd[φK ] = 0 ∪ 3 ∪ 4 1
2
Spec Hc[φK ] = q2 + 4q∈R
- Spectral Density:ρK(q) = L
2π + 12π
dδ(q)dq , δ(q) = −2 arctan 3q
2−q2
∆M =~m
2
(√
3 +1
2π
∫ ∞−∞
dq√
q2 + 4dδ(q)
dq
)+ Ect[φK] − Ect[φv] + ∆R =
=
√3~m
2−
~m
2π
∫ ∞−∞
dq3√
q2 + 4(q2 + 2)
q4 + 5q2 + 4+
3~m
2π
∫ ∞−∞
dk√k2 + 4
+~m
4π〈V(x) − 4〉 =
=
√3 ~m
2−
~m√
3+
~m
4π
∫ ∞−∞
dx(−6sech2x) = ~m
(1
2√
3−
3
π
)≈ −0.666255~m
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL φ4
HOMOGENEOUS SOLUTION
φV = ±1
- Hessian Operator:H[φV ] = − d2
dx2 + 4Spec Hd[φV ] = 4 1
2
Spec Hc[φV ] = k2 + 4k∈R
- Spectral Density:ρv(k) = L
2π
KINK SOLUTION:φK(x) = ± tanh x
- Hessian Operator:H[φK ] = − d2
dx2 + 4− 6 sech2xSpec Hd[φK ] = 0 ∪ 3 ∪ 4 1
2
Spec Hc[φK ] = q2 + 4q∈R
- Spectral Density:ρK(q) = L
2π + 12π
dδ(q)dq , δ(q) = −2 arctan 3q
2−q2
∆M =~m
2
(√
3 +1
2π
∫ ∞−∞
dq√
q2 + 4dδ(q)
dq
)+ Ect[φK] − Ect[φv] + ∆R =
=
√3~m
2−
~m
2π
∫ ∞−∞
dq3√
q2 + 4(q2 + 2)
q4 + 5q2 + 4+
3~m
2π
∫ ∞−∞
dk√k2 + 4
+~m
4π〈V(x) − 4〉 =
=
√3 ~m
2−
~m√
3+
~m
4π
∫ ∞−∞
dx(−6sech2x) = ~m
(1
2√
3−
3
π
)≈ −0.666255~m
• R. Dashen, B. Hasslacher, A. Neveu, Phys. Rev. D 10 (1974) 4130
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL sinh4 φ
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =14
(sinh2 φ− 1)2
• Partial Diferential Equation:∂2φ∂t2 −
∂2φ∂x2 + 1
2 sinh(2φ)(sinh2 φ− 1) = 0
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL sinh4 φ
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =14
(sinh2 φ− 1)2
• Partial Diferential Equation:∂2φ∂t2 −
∂2φ∂x2 + 1
2 sinh(2φ)(sinh2 φ− 1) = 0
SOLUTIONS:VACUUM SOLUTION:φV = ±arcsinh 1
Hessian OperatorH[φV ]:
H[φV ] = − d2
dx2 + 4
KINK SOLUTION:φK = arctanh[ 1√
2tanh x−vt√
1−v2]
Hessian OperatorH[φK ]:
H[φK ] = − d2
dx2 +2sech2x(9 + sech2x)
(1 + sech2x)2
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODEL sinh4 φ
• Action Functional:S =
∫dxdt
[ 12∂φ∂t
∂φ∂t −
12∂φ∂x
∂φ∂x − U(φ)
]• Potential:
U =14
(sinh2 φ− 1)2
• Partial Diferential Equation:∂2φ∂t2 −
∂2φ∂x2 + 1
2 sinh(2φ)(sinh2 φ− 1) = 0
SOLUTIONS:VACUUM SOLUTION:φV = ±arcsinh 1
Hessian OperatorH[φV ]:
H[φV ] = − d2
dx2 + 4
KINK SOLUTION:φK = arctanh[ 1√
2tanh x−vt√
1−v2]
Hessian OperatorH[φK ]:
H[φK ] = − d2
dx2 +2sech2x(9 + sech2x)
(1 + sech2x)2
UNKOWN SPECTRUM: ∆M NO COMPUTABLE
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODELS
Question:Which (1+1) scalar field models we can generalize the Dashen - Hasslacher -Neveau computation to in order to compute the quantum correction ∆M to thekink mass?
Answer:List of the scalar field models with computed ∆M:
1 Model φ4 and the kink.
2 Model sine-Gordon and the soliton.
Conclusion:Dashen, Hasslacher and Neveu are lucky guys.
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODELS
Question:Which (1+1) scalar field models we can generalize the Dashen - Hasslacher -Neveau computation to in order to compute the quantum correction ∆M to thekink mass?
Answer:List of the scalar field models with computed ∆M:
1 Model φ4 and the kink.
2 Model sine-Gordon and the soliton.
Conclusion:Dashen, Hasslacher and Neveu are lucky guys.
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODELS
Question:Which (1+1) scalar field models we can generalize the Dashen - Hasslacher -Neveau computation to in order to compute the quantum correction ∆M to thekink mass?
Answer:List of the scalar field models with computed ∆M:
1 Model φ4 and the kink.
2 Model sine-Gordon and the soliton.
Conclusion:Dashen, Hasslacher and Neveu are lucky guys.
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
QUANTUM CORRECTION: MODELS
Task:Approach a procedure to compute approximately the semiclassical mass of akink in a (1+1) dimensional scalar field model.
Clue:We will estimate the trace of the differential operator without the knowledge ofthe exact spectrum.
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
Quantum Correction
∆M = 12 ~[tr (H(φK))
12 − tr (H(φV))
12
]+ Ect(φK)− Ect(φV)
Generalized Zeta Function: Heat Function
ζH(s) = TrH−s =
∞∑n=0
(ω2n)−s hH(β) = Tr e−βH =
∞∑n=0
e−βω2n
Mellin Transform
ζH(s) =1
Γ(s)
∫ ∞0
dβ βs−1 hH(β)
Quantum Correction
∆M =~2
lims→− 1
2
1Γ(s)
∫ ∞0
dβ βs−1[h∗H[φK ](β)− h∗H[φV ](β)] + lims→ 1
2
Ect[ζH[φV ](s)] + ∆R
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
Quantum Correction
∆M = 12 ~[tr (H(φK))
12 − tr (H(φV))
12
]+ Ect(φK)− Ect(φV)
Generalized Zeta Function: Heat Function
ζH(s) = TrH−s =
∞∑n=0
(ω2n)−s hH(β) = Tr e−βH =
∞∑n=0
e−βω2n
Mellin Transform
ζH(s) =1
Γ(s)
∫ ∞0
dβ βs−1 hH(β)
Quantum Correction
∆M =~2
lims→− 1
2
1Γ(s)
∫ ∞0
dβ βs−1[h∗H[φK ](β)− h∗H[φV ](β)] + lims→ 1
2
Ect[ζH[φV ](s)] + ∆R
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Heat function: hH(β) = tr(e−βH) =
∫Ω
dx KH( x , x , β)︸ ︷︷ ︸Trace of a HEAT KERNEL
H
KH( x , y ;β) = 〈x| e−βH |y〉 =
∫dkξk(y)∗ξk(x)e−βω
2k
verifies the PDE[∂∂β− ∂2
∂x2 + V(x)]
KH(x, y;β) = 0; KH(x, y; 0) = δ(x− y)
Plugging the Series Expansion into the PDE:
KH[φK ](x, y;β) = KH[φV ](x, y;β)
∞∑n=0
an(x, y)βn a0(x, y) = 1
provides us with the Recurrence Relation in x− y variables:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Heat function: hH(β) = tr(e−βH) =
∫Ω
dx KH( x , x , β)︸ ︷︷ ︸Trace of a HEAT KERNEL
H
KH( x , y ;β) = 〈x| e−βH |y〉 =
∫dkξk(y)∗ξk(x)e−βω
2k
verifies the PDE[∂∂β− ∂2
∂x2 + V(x)]
KH(x, y;β) = 0; KH(x, y; 0) = δ(x− y)
Plugging the Series Expansion into the PDE:
KH[φK ](x, y;β) = KH[φV ](x, y;β)
∞∑n=0
an(x, y)βn a0(x, y) = 1
provides us with the Recurrence Relation in x− y variables:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Heat function: hH(β) = tr(e−βH) =
∫Ω
dx KH( x , x , β)︸ ︷︷ ︸Trace of a HEAT KERNEL
H
KH( x , y ;β) = 〈x| e−βH |y〉 =
∫dkξk(y)∗ξk(x)e−βω
2k
verifies the PDE[∂∂β− ∂2
∂x2 + V(x)]
KH(x, y;β) = 0; KH(x, y; 0) = δ(x− y)
Plugging the Series Expansion into the PDE:
KH[φK ](x, y;β) = KH[φV ](x, y;β)
∞∑n=0
an(x, y)βn a0(x, y) = 1
provides us with the Recurrence Relation in x− y variables:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Heat function: hH(β) = tr(e−βH) =
∫Ω
dx KH( x , x , β)︸ ︷︷ ︸Trace of a HEAT KERNEL
H
KH( x , y ;β) = 〈x| e−βH |y〉 =
∫dkξk(y)∗ξk(x)e−βω
2k
verifies the PDE[∂∂β− ∂2
∂x2 + V(x)]
KH(x, y;β) = 0; KH(x, y; 0) = δ(x− y)
Plugging the Series Expansion into the PDE:
KH[φK ](x, y;β) = KH[φV ](x, y;β)
∞∑n=0
an(x, y)βn a0(x, y) = 1
provides us with the Recurrence Relation in x− y variables:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
We have to take the limit y→ x
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Recurrence Relations:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
y→ x (Delicated limit) (k)An(x) = limy→x
∂kan(x, y)
∂xk
(0)An(x) = an(x, x) (k)A0(x) = δk0
(k)An(x) =1
n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
• Coefficients of the recurrence relationships:(0)A5(x)(0)A4(x) (1)A4(x) (2)A4(x)(0)A3(x) (1)A3(x) (2)A3(x) (3)A3(x) (4)A3(x)(0)A2(x) (1)A2(x) (2)A2(x) (3)A2(x) (4)A2(x) (5)A2(x) (6)A2(x)(0)A1(x) (1)A1(x) (2)A1(x) (3)A1(x) (4)A1(x) (5)A1(x) (6)A1(x) (7)A1(x) (8)A1(x)(0)A0(x) (1)A0(x) (2)A0(x) (3)A0(x) (4)A0(x) (5)A0(x) (6)A0(x) (7)A0(x) (8)A0(x) (9)A0(x) (10)A0(x)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Recurrence Relations:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
y→ x (Delicated limit) (k)An(x) = limy→x
∂kan(x, y)
∂xk
(0)An(x) = an(x, x) (k)A0(x) = δk0
(k)An(x) =1
n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
• Coefficients of the recurrence relationships:(0)A5(x)(0)A4(x) (1)A4(x) (2)A4(x)(0)A3(x) (1)A3(x) (2)A3(x) (3)A3(x) (4)A3(x)(0)A2(x) (1)A2(x) (2)A2(x) (3)A2(x) (4)A2(x) (5)A2(x) (6)A2(x)(0)A1(x) (1)A1(x) (2)A1(x) (3)A1(x) (4)A1(x) (5)A1(x) (6)A1(x) (7)A1(x) (8)A1(x)(0)A0(x) (1)A0(x) (2)A0(x) (3)A0(x) (4)A0(x) (5)A0(x) (6)A0(x) (7)A0(x) (8)A0(x) (9)A0(x) (10)A0(x)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Recurrence Relations:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
y→ x (Delicated limit) (k)An(x) = limy→x
∂kan(x, y)
∂xk
(0)An(x) = an(x, x) (k)A0(x) = δk0
(k)An(x) =1
n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
• Coefficients of the recurrence relationships:(0)A5(x)(0)A4(x) (1)A4(x) (2)A4(x)(0)A3(x) (1)A3(x) (2)A3(x) (3)A3(x) (4)A3(x)(0)A2(x) (1)A2(x) (2)A2(x) (3)A2(x) (4)A2(x) (5)A2(x) (6)A2(x)(0)A1(x) (1)A1(x) (2)A1(x) (3)A1(x) (4)A1(x) (5)A1(x) (6)A1(x) (7)A1(x) (8)A1(x)(0)A0(x) (1)A0(x) (2)A0(x) (3)A0(x) (4)A0(x) (5)A0(x) (6)A0(x) (7)A0(x) (8)A0(x) (9)A0(x) (10)A0(x)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• Recurrence Relations:
(n + 1) an+1(x, y) + (x− y)∂an+1(x, y)
∂x+ (V(x)− V0)an(x, y) =
∂2an(x, y)
∂x2
y→ x (Delicated limit) (k)An(x) = limy→x
∂kan(x, y)
∂xk
(0)An(x) = an(x, x) (k)A0(x) = δk0
(k)An(x) =1
n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
• Coefficients of the recurrence relationships:(0)A5(x)(0)A4(x) (1)A4(x) (2)A4(x)(0)A3(x) (1)A3(x) (2)A3(x) (3)A3(x) (4)A3(x)(0)A2(x) (1)A2(x) (2)A2(x) (3)A2(x) (4)A2(x) (5)A2(x) (6)A2(x)(0)A1(x) (1)A1(x) (2)A1(x) (3)A1(x) (4)A1(x) (5)A1(x) (6)A1(x) (7)A1(x) (8)A1(x)(0)A0(x) (1)A0(x) (2)A0(x) (3)A0(x) (4)A0(x) (5)A0(x) (6)A0(x) (7)A0(x) (8)A0(x) (9)A0(x) (10)A0(x)
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• OUTLINE:
∆M~
= − 12√π− 1
8π
N∑n=2
cn · (V0)−n+1 · γ[n− 1,V0]
↓cn =
∫ ∞−∞
an(x, x) dx
↓an(x, x) = (0)An(x)
↓(k)An(x) =
1n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• OUTLINE:
∆M~
= − 12√π− 1
8π
N∑n=2
cn · (V0)−n+1 · γ[n− 1,V0]
↓cn =
∫ ∞−∞
an(x, x) dx
↓an(x, x) = (0)An(x)
↓(k)An(x) =
1n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• OUTLINE:
∆M~
= − 12√π− 1
8π
N∑n=2
cn · (V0)−n+1 · γ[n− 1,V0]
↓cn =
∫ ∞−∞
an(x, x) dx
↓an(x, x) = (0)An(x)
↓(k)An(x) =
1n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Quantum correction: Asymptotic approach
• OUTLINE:
∆M~
= − 12√π− 1
8π
N∑n=2
cn · (V0)−n+1 · γ[n− 1,V0]
↓cn =
∫ ∞−∞
an(x, x) dx
↓an(x, x) = (0)An(x)
↓(k)An(x) =
1n + k
[(k+2)An−1(x)−
k∑j=0
(kj
)∂ j(V − V0)
∂xj(k−j)An−1(x)
]
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Outline
1 Basic Notions
2 Application to scalar field models
3 The Asymptotic Approach
4 The Mathematica Program
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Constructing an algorithm
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Constructing an algorithm
Go to the Mathematica Program
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Testing the algorithm
Exact quantum correction to the kink mass∆M = −0.666255~m
Estimated quantum correction to the kink mass∆M = −0.666619~m
Relative Error: 0.07%
Virtue of the ApproachApplicable to every model
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Testing the algorithm
Exact quantum correction to the kink mass∆M = −0.666255~m
Estimated quantum correction to the kink mass∆M = −0.666619~m
Relative Error: 0.07%
Virtue of the ApproachApplicable to every model
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Testing the algorithm
Exact quantum correction to the kink mass∆M = −0.666255~m
Estimated quantum correction to the kink mass∆M = −0.666619~m
Relative Error: 0.07%
Virtue of the ApproachApplicable to every model
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
Testing the algorithm
Exact quantum correction to the kink mass∆M = −0.666255~m
Estimated quantum correction to the kink mass∆M = −0.666619~m
Relative Error: 0.07%
Virtue of the ApproachApplicable to every model
Mathematica andQuantum
Corrections to kinkMasses
A. AlonsoM. González
J. Mateos
Basic Notions
Application toscalar field models
The AsymptoticApproach
The MathematicaProgram
End of the talk
