fantastic symmetries and where to find them 4... · 2019. 6. 4. · lecture 4: noether symmetries...
TRANSCRIPT
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FANTASTIC SYMMETRIESAND WHERE TO FIND THEM
Maria Clara Nucci
University of Perugia & INFN-Perugia, Italy
XXIst International ConferenceGeometry, Integrability and Quantization
June 3-8, 2019, Varna, Bulgaria
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Lecture 4: Noether symmetries II.
The Ostrogradsky’s method for constructing Lagrangians forequations of order greater than two.
Ghost-free quantization via symmetry preservation:Pais–Uhlenbeck model and its “ghosts“.
Ghost-free quantization via symmetry preservation: Higgsmodel with a complex ghost pair.
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Generating ghostsSome simple linear equations of classical mechanics yield seriousproblems when quantization a la Dirac is undertaken since stateswith negative norm, commonly called ghosts, appear. We present a“ghostbuster” based on the preservation of Lie symmetries of theoriginal classical equations.
A. Pais and G. E. Uhlenbeck, On Field Theories with Non-LocalizedAction, Phys. Rev. 79 145-165 (1950)
fourth-order field-theoretic model of Pais-Uhlenbeck:....z +
(Ω21 + Ω2
2
)z + Ω2
1Ω22z = 0
Karl Jansenn, Julius Kuti and Chuan Liu, The Higgs model with acomplex ghost pair, Phys. Lett. B 309 119-126 (1993)
1
M4x (vi)+
2
M2
(cos(2Θ) +
ω2
2M2
)x (iv)−
(1 +
2ω2
M2cos(2Θ)
)x ′′+ω2x = 0
![Page 4: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/4.jpg)
Generating ghostsSome simple linear equations of classical mechanics yield seriousproblems when quantization a la Dirac is undertaken since stateswith negative norm, commonly called ghosts, appear. We present a“ghostbuster” based on the preservation of Lie symmetries of theoriginal classical equations.
A. Pais and G. E. Uhlenbeck, On Field Theories with Non-LocalizedAction, Phys. Rev. 79 145-165 (1950)
fourth-order field-theoretic model of Pais-Uhlenbeck:....z +
(Ω21 + Ω2
2
)z + Ω2
1Ω22z = 0
Karl Jansenn, Julius Kuti and Chuan Liu, The Higgs model with acomplex ghost pair, Phys. Lett. B 309 119-126 (1993)
1
M4x (vi)+
2
M2
(cos(2Θ) +
ω2
2M2
)x (iv)−
(1 +
2ω2
M2cos(2Θ)
)x ′′+ω2x = 0
![Page 5: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/5.jpg)
Generating ghostsSome simple linear equations of classical mechanics yield seriousproblems when quantization a la Dirac is undertaken since stateswith negative norm, commonly called ghosts, appear. We present a“ghostbuster” based on the preservation of Lie symmetries of theoriginal classical equations.
A. Pais and G. E. Uhlenbeck, On Field Theories with Non-LocalizedAction, Phys. Rev. 79 145-165 (1950)
fourth-order field-theoretic model of Pais-Uhlenbeck:....z +
(Ω21 + Ω2
2
)z + Ω2
1Ω22z = 0
Karl Jansenn, Julius Kuti and Chuan Liu, The Higgs model with acomplex ghost pair, Phys. Lett. B 309 119-126 (1993)
1
M4x (vi)+
2
M2
(cos(2Θ) +
ω2
2M2
)x (iv)−
(1 +
2ω2
M2cos(2Θ)
)x ′′+ω2x = 0
![Page 6: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/6.jpg)
Higher order Lagrangians
L(t, x , x , x , . . . , x (n)
)0 =
∂L
∂x− d
dt
(∂L
∂x
)+ . . .+ (−1)n
dn
dtn
(∂L
∂x (n)
)H =??
1801− 1861
Mikhail Vasilevich Ostrogradsky, Memoire sur le calcul des varia-tions des integrales multiples, Journal fur die reine und angewandteMathematik 15 (1836) 332-354
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Higher order Lagrangians
L(t, x , x , x , . . . , x (n)
)0 =
∂L
∂x− d
dt
(∂L
∂x
)+ . . .+ (−1)n
dn
dtn
(∂L
∂x (n)
)H =??
1801− 1861
Mikhail Vasilevich Ostrogradsky, Memoire sur le calcul des varia-tions des integrales multiples, Journal fur die reine und angewandteMathematik 15 (1836) 332-354
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Enter Ostrogradsky
Ostrogradsky defines the momenta as
p1 =∂L
∂x− d
dt
(∂L
∂x
)+ . . .+ (−1)n−1
dn−1
dtn−1
(∂L
∂x (n)
)p2 =
∂L
∂x− d
dt
(∂L
∂...x
)+ . . .+ (−1)n−2
dn−2
dtn−2
(∂L
∂x (n)
)...
pn =∂L
∂x (n)
the canonical coordinates according to
q1 = x , q2 = x , . . . , qn = x (n−1)
and finally the Hamiltonian function is
H = −L + p1q2 + p2q3 + . . .+ pn−1qn + pnx(n)
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Pais-Uhlenbeck modelUsing the prescription of Ostrogradsky:
H = −12γ
p22γ2−(Ω21 + Ω2
2
)q22 + Ω2
1Ω22q
21
+ p1q2
q1 = q2 p1 = γΩ1Ω2q1
q2 = −p2γ
p2 = −γ(Ω21 + Ω2
2
)q2 − p1.
Carl M. Bender and Philip D. Mannheim, Giving up the ghost, J.Phys. A: Math. Theor. 41 304018 (2008)
“Ghost states are quantum states having negative norm. If aquantum theory has ghost states, it is fundamentally unacceptablebecause the norm of a quantum state is interpreted as a probability,and a negative probability is forbidden on physical grounds.”
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Is PT quantum theory necessary?
Carl M. Bender and Philip D. Mannheim, Giving up the ghost, J.Phys. A: Math. Theor. 41 304018 (2008)
“When it is quantized correctly according to the rules of PTquantum mechanics, the energy spectrum is real and boundedbelow and all of the quantum states have positive norm.”
“To make sense of the Pais-Uhlenbeck model as a consistentand physical quantum theory, it is necessary to interpret thePais-Uhlenbeck Hamiltonian as a PT quantum theory.”
Maybe PT is not necessary MCN & Leach, JMathPhys 2009.
![Page 11: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/11.jpg)
Is PT quantum theory necessary?
Carl M. Bender and Philip D. Mannheim, Giving up the ghost, J.Phys. A: Math. Theor. 41 304018 (2008)
“When it is quantized correctly according to the rules of PTquantum mechanics, the energy spectrum is real and boundedbelow and all of the quantum states have positive norm.”
“To make sense of the Pais-Uhlenbeck model as a consistentand physical quantum theory, it is necessary to interpret thePais-Uhlenbeck Hamiltonian as a PT quantum theory.”
Maybe PT is not necessary MCN & Leach, JMathPhys 2009.
![Page 12: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/12.jpg)
Is PT quantum theory necessary?
Carl M. Bender and Philip D. Mannheim, Giving up the ghost, J.Phys. A: Math. Theor. 41 304018 (2008)
“When it is quantized correctly according to the rules of PTquantum mechanics, the energy spectrum is real and boundedbelow and all of the quantum states have positive norm.”
“To make sense of the Pais-Uhlenbeck model as a consistentand physical quantum theory, it is necessary to interpret thePais-Uhlenbeck Hamiltonian as a PT quantum theory.”
Maybe PT is not necessary MCN & Leach, JMathPhys 2009.
![Page 13: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/13.jpg)
Is PT quantum theory necessary?
Carl M. Bender and Philip D. Mannheim, Giving up the ghost, J.Phys. A: Math. Theor. 41 304018 (2008)
“When it is quantized correctly according to the rules of PTquantum mechanics, the energy spectrum is real and boundedbelow and all of the quantum states have positive norm.”
“To make sense of the Pais-Uhlenbeck model as a consistentand physical quantum theory, it is necessary to interpret thePais-Uhlenbeck Hamiltonian as a PT quantum theory.”
Maybe PT is not necessary MCN & Leach, JMathPhys 2009.
![Page 14: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/14.jpg)
Is PT quantum theory necessary?
Carl M. Bender and Philip D. Mannheim, Giving up the ghost, J.Phys. A: Math. Theor. 41 304018 (2008)
“When it is quantized correctly according to the rules of PTquantum mechanics, the energy spectrum is real and boundedbelow and all of the quantum states have positive norm.”
“To make sense of the Pais-Uhlenbeck model as a consistentand physical quantum theory, it is necessary to interpret thePais-Uhlenbeck Hamiltonian as a PT quantum theory.”
Maybe PT is not necessary MCN & Leach, JMathPhys 2009.
![Page 15: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/15.jpg)
Is PT quantum theory necessary?
Carl M. Bender and Philip D. Mannheim, Giving up the ghost, J.Phys. A: Math. Theor. 41 304018 (2008)
“When it is quantized correctly according to the rules of PTquantum mechanics, the energy spectrum is real and boundedbelow and all of the quantum states have positive norm.”
“To make sense of the Pais-Uhlenbeck model as a consistentand physical quantum theory, it is necessary to interpret thePais-Uhlenbeck Hamiltonian as a PT quantum theory.”
Maybe PT is not necessary MCN & Leach, JMathPhys 2009.
![Page 16: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/16.jpg)
Begin with the Lagrangian
L = 12
z2 −
(Ω21 + Ω2
2
)z2 + Ω2
1Ω22z
2
+d
dtF (t, z , z)
Six-dimensional Lie point symmetry algebra:
Γ1 = ∂t , Γ2 = z∂z , Γ3 = cos(Ω1t)∂z , Γ4 = − sin(Ω1t)∂z ,
Γ5 = cos(Ω2t)∂z , Γ6 = − sin(Ω2t)∂z ,
and five Noether point symmetries with five first integrals
Γ1 =⇒ I1 = 12(−Ω2
1Ω22z
2 − Ω21z
2 − Ω22z
2 − 2z...z + z2)
Γ3 =⇒ I3 = (Ω22z + z) sin(Ω1t)Ω1 + (Ω2
2z +...z ) cos(Ω1t)
Γ4 =⇒ I4 = (Ω22z + z) cos(Ω1t)Ω1 − (Ω2
2z +...z ) sin(Ω1t)
Γ5 =⇒ I5 = (Ω21z + z) sin(Ω2t)Ω2 + (Ω2
1z +...z ) cos(Ω2t)
Γ6 =⇒ I6 = (Ω21z + z) cos(Ω2t)Ω2 − (Ω2
1z +...z ) sin(Ω2t).
Iaut = 12(I 23 + I 24 + I 25 + I 26 )
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Begin with the Lagrangian
L = 12
z2 −
(Ω21 + Ω2
2
)z2 + Ω2
1Ω22z
2
+d
dtF (t, z , z)
Six-dimensional Lie point symmetry algebra:
Γ1 = ∂t , Γ2 = z∂z , Γ3 = cos(Ω1t)∂z , Γ4 = − sin(Ω1t)∂z ,
Γ5 = cos(Ω2t)∂z , Γ6 = − sin(Ω2t)∂z ,
and five Noether point symmetries with five first integrals
Γ1 =⇒ I1 = 12(−Ω2
1Ω22z
2 − Ω21z
2 − Ω22z
2 − 2z...z + z2)
Γ3 =⇒ I3 = (Ω22z + z) sin(Ω1t)Ω1 + (Ω2
2z +...z ) cos(Ω1t)
Γ4 =⇒ I4 = (Ω22z + z) cos(Ω1t)Ω1 − (Ω2
2z +...z ) sin(Ω1t)
Γ5 =⇒ I5 = (Ω21z + z) sin(Ω2t)Ω2 + (Ω2
1z +...z ) cos(Ω2t)
Γ6 =⇒ I6 = (Ω21z + z) cos(Ω2t)Ω2 − (Ω2
1z +...z ) sin(Ω2t).
Iaut = 12(I 23 + I 24 + I 25 + I 26 )
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Begin with the Lagrangian
L = 12
z2 −
(Ω21 + Ω2
2
)z2 + Ω2
1Ω22z
2
+d
dtF (t, z , z)
Six-dimensional Lie point symmetry algebra:
Γ1 = ∂t , Γ2 = z∂z , Γ3 = cos(Ω1t)∂z , Γ4 = − sin(Ω1t)∂z ,
Γ5 = cos(Ω2t)∂z , Γ6 = − sin(Ω2t)∂z ,
and five Noether point symmetries with five first integrals
Γ1 =⇒ I1 = 12(−Ω2
1Ω22z
2 − Ω21z
2 − Ω22z
2 − 2z...z + z2)
Γ3 =⇒ I3 = (Ω22z + z) sin(Ω1t)Ω1 + (Ω2
2z +...z ) cos(Ω1t)
Γ4 =⇒ I4 = (Ω22z + z) cos(Ω1t)Ω1 − (Ω2
2z +...z ) sin(Ω1t)
Γ5 =⇒ I5 = (Ω21z + z) sin(Ω2t)Ω2 + (Ω2
1z +...z ) cos(Ω2t)
Γ6 =⇒ I6 = (Ω21z + z) cos(Ω2t)Ω2 − (Ω2
1z +...z ) sin(Ω2t).
Iaut = 12(I 23 + I 24 + I 25 + I 26 )
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Begin with the Lagrangian
L = 12
z2 −
(Ω21 + Ω2
2
)z2 + Ω2
1Ω22z
2
+d
dtF (t, z , z)
Six-dimensional Lie point symmetry algebra:
Γ1 = ∂t , Γ2 = z∂z , Γ3 = cos(Ω1t)∂z , Γ4 = − sin(Ω1t)∂z ,
Γ5 = cos(Ω2t)∂z , Γ6 = − sin(Ω2t)∂z ,
and five Noether point symmetries with five first integrals
Γ1 =⇒ I1 = 12(−Ω2
1Ω22z
2 − Ω21z
2 − Ω22z
2 − 2z...z + z2)
Γ3 =⇒ I3 = (Ω22z + z) sin(Ω1t)Ω1 + (Ω2
2z +...z ) cos(Ω1t)
Γ4 =⇒ I4 = (Ω22z + z) cos(Ω1t)Ω1 − (Ω2
2z +...z ) sin(Ω1t)
Γ5 =⇒ I5 = (Ω21z + z) sin(Ω2t)Ω2 + (Ω2
1z +...z ) cos(Ω2t)
Γ6 =⇒ I6 = (Ω21z + z) cos(Ω2t)Ω2 − (Ω2
1z +...z ) sin(Ω2t).
Iaut = 12(I 23 + I 24 + I 25 + I 26 )
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Right HamiltonianThat implies
Iaut = 12
[(Ω2
2z + z)2Ω21 + (Ω2
2z +...z )2 + (Ω2
1z + z)2Ω22 + (Ω2
1z +...z )2]
We make the obvious transformations
q1 = z + Ω22z , q2 = z + Ω2
1z ,
p1 =...z + Ω2
2z , p2 =...z + Ω2
1z ,
and consequently we obtain the Hamiltonian
H ≡ Iaut = 12 [p21 + p22 + Ω2
1q21 + Ω2
2q22 ]
and the corresponding canonical equations
q1 = p1, q2 = p2, p1 = −Ω21q1, p2 = −Ω2
2q2.
This is the right Hamiltonian for the quantization of thefourth-order field-theoretic model of Pais-Uhlenbeck.
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Right HamiltonianThat implies
Iaut = 12
[(Ω2
2z + z)2Ω21 + (Ω2
2z +...z )2 + (Ω2
1z + z)2Ω22 + (Ω2
1z +...z )2]
We make the obvious transformations
q1 = z + Ω22z , q2 = z + Ω2
1z ,
p1 =...z + Ω2
2z , p2 =...z + Ω2
1z ,
and consequently we obtain the Hamiltonian
H ≡ Iaut = 12 [p21 + p22 + Ω2
1q21 + Ω2
2q22 ]
and the corresponding canonical equations
q1 = p1, q2 = p2, p1 = −Ω21q1, p2 = −Ω2
2q2.
This is the right Hamiltonian for the quantization of thefourth-order field-theoretic model of Pais-Uhlenbeck.
![Page 22: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/22.jpg)
Right HamiltonianThat implies
Iaut = 12
[(Ω2
2z + z)2Ω21 + (Ω2
2z +...z )2 + (Ω2
1z + z)2Ω22 + (Ω2
1z +...z )2]
We make the obvious transformations
q1 = z + Ω22z , q2 = z + Ω2
1z ,
p1 =...z + Ω2
2z , p2 =...z + Ω2
1z ,
and consequently we obtain the Hamiltonian
H ≡ Iaut = 12 [p21 + p22 + Ω2
1q21 + Ω2
2q22 ]
and the corresponding canonical equations
q1 = p1, q2 = p2, p1 = −Ω21q1, p2 = −Ω2
2q2.
This is the right Hamiltonian for the quantization of thefourth-order field-theoretic model of Pais-Uhlenbeck.
![Page 23: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/23.jpg)
Right HamiltonianThat implies
Iaut = 12
[(Ω2
2z + z)2Ω21 + (Ω2
2z +...z )2 + (Ω2
1z + z)2Ω22 + (Ω2
1z +...z )2]
We make the obvious transformations
q1 = z + Ω22z , q2 = z + Ω2
1z ,
p1 =...z + Ω2
2z , p2 =...z + Ω2
1z ,
and consequently we obtain the Hamiltonian
H ≡ Iaut = 12 [p21 + p22 + Ω2
1q21 + Ω2
2q22 ]
and the corresponding canonical equations
q1 = p1, q2 = p2, p1 = −Ω21q1, p2 = −Ω2
2q2.
This is the right Hamiltonian for the quantization of thefourth-order field-theoretic model of Pais-Uhlenbeck.
![Page 24: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/24.jpg)
Preservation of symmetriesThe application of the Legendre transformation gives
L = 12 [q21 + q22 − (Ω2
1q21 + Ω2
2q22)]
and corresponding Lagrange equations
q1 = −Ω21q1, q2 = −Ω2
2q2
which admit a seven-dimensional Lie point symmetry algebra
Γ1 = ∂t , Γ2 = q1∂q1 , Γ3 = cos(Ω1t)∂q1 , Γ4 = − sin(Ω1t)∂q1 ,
Γ5 = cos(Ω2t)∂q2 , Γ6 = − sin(Ω2t)∂q2 , Γ7 = q2∂q2
and five Noether point symmetries with five first integrals
Γ1 =⇒ I1 = 12(Ω2
1q21 + Ω2
2q22 + q21 + q22)
Γ3 =⇒ I3 = sin(Ω1t)Ω1q1 + cos(Ω1t)q1
Γ4 =⇒ I4 = cos(Ω1t)Ω1q1 − sin(Ω1t)q1
Γ5 =⇒ I5 = sin(Ω2t)Ω2q2 + cos(Ω2t)q2
Γ6 =⇒ I6 = cos(Ω2t)Ω2q2 − sin(Ω2t)q2.
![Page 25: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/25.jpg)
Higgs modelx (vi) =−2M2
(cos(2Θ) + ω2
2M2
)x (iv)−M4
(1 + 2ω2
M2 cos(2Θ))x ′′−M4ω2x
L =1
2M4
(− 2 cos(2Θ)M2ω2x ′2 − 2 cos(2Θ)M2x ′′2 −M4ω2x2
−M4x ′2 − ω2x ′′2 + x ′′′2)
x (vi) = −(ω21 +ω2
2 +ω23)x (iv)− (ω2
1ω22 +ω2
1ω23 +ω2
2ω23)x ′′−ω2
1ω22ω
23x
L = 12
(x ′′′2−(ω2
1+ω22+ω2
3)x ′′2+(ω21ω
22+ω2
1ω23+ω2
2ω23)x ′2−ω2
1ω22ω
23x
2)
Eight-dimensional Lie point symmetry algebra:
Γ1 = sin(ω1t)∂x , Γ2 = cos(ω1t)∂x , Γ3 = sin(ω3t)∂x , Γ4 = cos(ω3t)∂x ,
Γ5 = sin(ω2t)∂x , Γ6 = cos(ω2t)∂x , Γ7 = x∂x , Γ8 = ∂t ,
and seven Noether point symmetries with seven first integrals:
Γ1, Γ2, Γ3, Γ4, Γ5, Γ6, Γ8
![Page 26: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/26.jpg)
Higgs modelx (vi) =−2M2
(cos(2Θ) + ω2
2M2
)x (iv)−M4
(1 + 2ω2
M2 cos(2Θ))x ′′−M4ω2x
L =1
2M4
(− 2 cos(2Θ)M2ω2x ′2 − 2 cos(2Θ)M2x ′′2 −M4ω2x2
−M4x ′2 − ω2x ′′2 + x ′′′2)
x (vi) = −(ω21 +ω2
2 +ω23)x (iv)− (ω2
1ω22 +ω2
1ω23 +ω2
2ω23)x ′′−ω2
1ω22ω
23x
L = 12
(x ′′′2−(ω2
1+ω22+ω2
3)x ′′2+(ω21ω
22+ω2
1ω23+ω2
2ω23)x ′2−ω2
1ω22ω
23x
2)
Eight-dimensional Lie point symmetry algebra:
Γ1 = sin(ω1t)∂x , Γ2 = cos(ω1t)∂x , Γ3 = sin(ω3t)∂x , Γ4 = cos(ω3t)∂x ,
Γ5 = sin(ω2t)∂x , Γ6 = cos(ω2t)∂x , Γ7 = x∂x , Γ8 = ∂t ,
and seven Noether point symmetries with seven first integrals:
Γ1, Γ2, Γ3, Γ4, Γ5, Γ6, Γ8
![Page 27: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/27.jpg)
Higgs modelx (vi) =−2M2
(cos(2Θ) + ω2
2M2
)x (iv)−M4
(1 + 2ω2
M2 cos(2Θ))x ′′−M4ω2x
L =1
2M4
(− 2 cos(2Θ)M2ω2x ′2 − 2 cos(2Θ)M2x ′′2 −M4ω2x2
−M4x ′2 − ω2x ′′2 + x ′′′2)
x (vi) = −(ω21 +ω2
2 +ω23)x (iv)− (ω2
1ω22 +ω2
1ω23 +ω2
2ω23)x ′′−ω2
1ω22ω
23x
L = 12
(x ′′′2−(ω2
1+ω22+ω2
3)x ′′2+(ω21ω
22+ω2
1ω23+ω2
2ω23)x ′2−ω2
1ω22ω
23x
2)
Eight-dimensional Lie point symmetry algebra:
Γ1 = sin(ω1t)∂x , Γ2 = cos(ω1t)∂x , Γ3 = sin(ω3t)∂x , Γ4 = cos(ω3t)∂x ,
Γ5 = sin(ω2t)∂x , Γ6 = cos(ω2t)∂x , Γ7 = x∂x , Γ8 = ∂t ,
and seven Noether point symmetries with seven first integrals:
Γ1, Γ2, Γ3, Γ4, Γ5, Γ6, Γ8
![Page 28: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/28.jpg)
Higgs modelx (vi) =−2M2
(cos(2Θ) + ω2
2M2
)x (iv)−M4
(1 + 2ω2
M2 cos(2Θ))x ′′−M4ω2x
L =1
2M4
(− 2 cos(2Θ)M2ω2x ′2 − 2 cos(2Θ)M2x ′′2 −M4ω2x2
−M4x ′2 − ω2x ′′2 + x ′′′2)
x (vi) = −(ω21 +ω2
2 +ω23)x (iv)− (ω2
1ω22 +ω2
1ω23 +ω2
2ω23)x ′′−ω2
1ω22ω
23x
L = 12
(x ′′′2−(ω2
1+ω22+ω2
3)x ′′2+(ω21ω
22+ω2
1ω23+ω2
2ω23)x ′2−ω2
1ω22ω
23x
2)
Eight-dimensional Lie point symmetry algebra:
Γ1 = sin(ω1t)∂x , Γ2 = cos(ω1t)∂x , Γ3 = sin(ω3t)∂x , Γ4 = cos(ω3t)∂x ,
Γ5 = sin(ω2t)∂x , Γ6 = cos(ω2t)∂x , Γ7 = x∂x , Γ8 = ∂t ,
and seven Noether point symmetries with seven first integrals:
Γ1, Γ2, Γ3, Γ4, Γ5, Γ6, Γ8
![Page 29: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/29.jpg)
Higgs modelx (vi) =−2M2
(cos(2Θ) + ω2
2M2
)x (iv)−M4
(1 + 2ω2
M2 cos(2Θ))x ′′−M4ω2x
L =1
2M4
(− 2 cos(2Θ)M2ω2x ′2 − 2 cos(2Θ)M2x ′′2 −M4ω2x2
−M4x ′2 − ω2x ′′2 + x ′′′2)
x (vi) = −(ω21 +ω2
2 +ω23)x (iv)− (ω2
1ω22 +ω2
1ω23 +ω2
2ω23)x ′′−ω2
1ω22ω
23x
L = 12
(x ′′′2−(ω2
1+ω22+ω2
3)x ′′2+(ω21ω
22+ω2
1ω23+ω2
2ω23)x ′2−ω2
1ω22ω
23x
2)
Eight-dimensional Lie point symmetry algebra:
Γ1 = sin(ω1t)∂x , Γ2 = cos(ω1t)∂x , Γ3 = sin(ω3t)∂x , Γ4 = cos(ω3t)∂x ,
Γ5 = sin(ω2t)∂x , Γ6 = cos(ω2t)∂x , Γ7 = x∂x , Γ8 = ∂t ,
and seven Noether point symmetries with seven first integrals:
Γ1, Γ2, Γ3, Γ4, Γ5, Γ6, Γ8
![Page 30: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/30.jpg)
Right Hamiltonian
Iaut = 12(Int21 + Int22 + Int23 + Int24 + Int25 + Int26 )
that implies the obvious transformations
q1 = ω22ω
23x + (ω2
2 + ω23)x ′′ + x (iv)
q2 = ω21ω
23x + (ω2
1 + ω23)x ′′ + x (iv)
q3 = ω21ω
22x + (ω2
1 + ω22)x ′′ + x (iv)
p1 = ω22ω
23x′ + (ω2
2 + ω23)x ′′′ + x (v)
p2 = ω21ω
23x′ + (ω2
1 + ω23)x ′′′ + x (v)
p3 = ω21ω
22x′ + (ω2
1 + ω22)x ′′′ + x (v)
and consequently we obtain the Hamiltonian
H =1
2(p21 + p22 + p23 + ω2
1q21 + ω2
2q22 + ω2
3q23)
This is the right Hamiltonian for the quantization of thesixth-order Higgs model.
![Page 31: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/31.jpg)
Right Hamiltonian
Iaut = 12(Int21 + Int22 + Int23 + Int24 + Int25 + Int26 )
that implies the obvious transformations
q1 = ω22ω
23x + (ω2
2 + ω23)x ′′ + x (iv)
q2 = ω21ω
23x + (ω2
1 + ω23)x ′′ + x (iv)
q3 = ω21ω
22x + (ω2
1 + ω22)x ′′ + x (iv)
p1 = ω22ω
23x′ + (ω2
2 + ω23)x ′′′ + x (v)
p2 = ω21ω
23x′ + (ω2
1 + ω23)x ′′′ + x (v)
p3 = ω21ω
22x′ + (ω2
1 + ω22)x ′′′ + x (v)
and consequently we obtain the Hamiltonian
H =1
2(p21 + p22 + p23 + ω2
1q21 + ω2
2q22 + ω2
3q23)
This is the right Hamiltonian for the quantization of thesixth-order Higgs model.
![Page 32: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/32.jpg)
Right Hamiltonian
Iaut = 12(Int21 + Int22 + Int23 + Int24 + Int25 + Int26 )
that implies the obvious transformations
q1 = ω22ω
23x + (ω2
2 + ω23)x ′′ + x (iv)
q2 = ω21ω
23x + (ω2
1 + ω23)x ′′ + x (iv)
q3 = ω21ω
22x + (ω2
1 + ω22)x ′′ + x (iv)
p1 = ω22ω
23x′ + (ω2
2 + ω23)x ′′′ + x (v)
p2 = ω21ω
23x′ + (ω2
1 + ω23)x ′′′ + x (v)
p3 = ω21ω
22x′ + (ω2
1 + ω22)x ′′′ + x (v)
and consequently we obtain the Hamiltonian
H =1
2(p21 + p22 + p23 + ω2
1q21 + ω2
2q22 + ω2
3q23)
This is the right Hamiltonian for the quantization of thesixth-order Higgs model.
![Page 33: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/33.jpg)
Preservation of symmetries
The application of the Legendre transformation gives
L = 12 [q21 + q22 + q23 − (ω2
1q21 + ω2
2q22 + ω2
3q23)]
and corresponding Lagrange equations
q1 = −ω21q1, q2 = −ω2
2q2, q3 = −ω23q3
which admit a ten-dimensional Lie point symmetry algebra andseven Noether point symmetries with seven first integrals.
Λ1 = ∂t , Λ2 = q1∂q1 , Λ3 = cos(ω1t)∂q1 , Λ4 = − sin(ω1t)∂q1 ,
Λ5 = q2∂q2 , Λ6 = cos(ω2t)∂q2 , Λ7 = − sin(ω2t)∂q2 ,
Λ8 = q3∂q3 , Λ9 = cos(ω3t)∂q3 , Λ10 = − sin(ω3t)∂q3 .
More details in MCN, Theor.Math.Phys., 2011.
![Page 34: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/34.jpg)
Eliminating the ghosts
Find the Lie symmetries of the Lagrange equations
Find the Noether symmetries and the corresponding firstintegrals
Construct the Hamiltonian from the first integrals
Quantize preserving the symmetries
![Page 35: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/35.jpg)
Eliminating the ghosts
Find the Lie symmetries of the Lagrange equations
Find the Noether symmetries and the corresponding firstintegrals
Construct the Hamiltonian from the first integrals
Quantize preserving the symmetries
![Page 36: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/36.jpg)
Eliminating the ghosts
Find the Lie symmetries of the Lagrange equations
Find the Noether symmetries and the corresponding firstintegrals
Construct the Hamiltonian from the first integrals
Quantize preserving the symmetries
![Page 37: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/37.jpg)
Eliminating the ghosts
Find the Lie symmetries of the Lagrange equations
Find the Noether symmetries and the corresponding firstintegrals
Construct the Hamiltonian from the first integrals
Quantize preserving the symmetries
![Page 38: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/38.jpg)
Eliminating the ghosts
Find the Lie symmetries of the Lagrange equations
Find the Noether symmetries and the corresponding firstintegrals
Construct the Hamiltonian from the first integrals
Quantize preserving the symmetries
![Page 39: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/39.jpg)
THEREFORE...
WHO ARE THE TRUE GHOSTBUSTERS?
Sophus Lie and Emmy Noether
![Page 40: FANTASTIC SYMMETRIES AND WHERE TO FIND THEM 4... · 2019. 6. 4. · Lecture 4: Noether symmetries II. The Ostrogradsky’s method for constructing Lagrangians for equations of order](https://reader036.vdocument.in/reader036/viewer/2022071117/600431cbe62de46eca31e31e/html5/thumbnails/40.jpg)
THEREFORE...
WHO ARE THE TRUE GHOSTBUSTERS?
Sophus Lie and Emmy Noether