![Page 1: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/1.jpg)
Structure of the WDW equationDiffeomorphism invariance and
emergence of probabilistic interpretation in QC
Renaud Parentani1
1LPT, Paris-Sud Orsay
SISSA, September 2011
Based on: PRD56, NPB513, PRD59 (1998-99) with S. Massarand conversations (2005) with T. Jacobson
Renaud Parentani Structure of the WDW equation
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The context
There is a probabilistic interpretationof the solutions of the Schroedinger eq.,but there is no consensus on the interpretationof the solutions of the WDW eq.
Several options have been proposed:
"Page-Hawking" based on the norm : |Ψ(a, φ)|2
"Vilenkin" based on the current : W = Ψ∗i↔
∂aΨ
Third quantization.
Square root approach i∂X Ψ = HΨ (Ashtekar (2008))Adding "dust" (Timan (2006))
Renaud Parentani Structure of the WDW equation
![Page 3: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/3.jpg)
The context
There is a probabilistic interpretationof the solutions of the Schroedinger eq.,but there is no consensus on the interpretationof the solutions of the WDW eq.
Several options have been proposed:
"Page-Hawking" based on the norm : |Ψ(a, φ)|2
"Vilenkin" based on the current : W = Ψ∗i↔
∂aΨ
Third quantization.
Square root approach i∂X Ψ = HΨ (Ashtekar (2008))Adding "dust" (Timan (2006))
Renaud Parentani Structure of the WDW equation
![Page 4: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/4.jpg)
The context
There is a probabilistic interpretationof the solutions of the Schroedinger eq.,but there is no consensus on the interpretationof the solutions of the WDW eq.
Several options have been proposed:
"Page-Hawking" based on the norm : |Ψ(a, φ)|2
"Vilenkin" based on the current : W = Ψ∗i↔
∂aΨ
Third quantization.
Square root approach i∂X Ψ = HΨ (Ashtekar (2008))Adding "dust" (Timan (2006))
Renaud Parentani Structure of the WDW equation
![Page 5: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/5.jpg)
Plan
Compare (mathem.) the structureof the Schr. eq. to that of the WdW eq.
Focus on Transition Amplitudes anduse molecular physics techniquesas a tool to perform the comparison.
Establish thatno exact proba. interp. could possibly be given:Diffeo-invar. + canon. quantization → no proba. interp.
the Schrod. probabilistic interpretation is anemergent property of the sol. of the WDW eq.
Renaud Parentani Structure of the WDW equation
![Page 6: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/6.jpg)
Plan
Compare (mathem.) the structureof the Schr. eq. to that of the WdW eq.
Focus on Transition Amplitudes anduse molecular physics techniquesas a tool to perform the comparison.
Establish thatno exact proba. interp. could possibly be given:Diffeo-invar. + canon. quantization → no proba. interp.
the Schrod. probabilistic interpretation is anemergent property of the sol. of the WDW eq.
Renaud Parentani Structure of the WDW equation
![Page 7: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/7.jpg)
Plan
Compare (mathem.) the structureof the Schr. eq. to that of the WdW eq.
Focus on Transition Amplitudes anduse molecular physics techniquesas a tool to perform the comparison.
Establish thatno exact proba. interp. could possibly be given:Diffeo-invar. + canon. quantization → no proba. interp.
the Schrod. probabilistic interpretation is anemergent property of the sol. of the WDW eq.
Renaud Parentani Structure of the WDW equation
![Page 8: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/8.jpg)
Why focusing on Transition Amplitudes ?
The WdW eq. also describes matter transitions,e.g. e+ e− annihilation.→ How to compute their amplitudes ?
Historically, Born’s statistical interpretation (1926)
|n0〉 →∑
n
cn,n0 |n〉
|cn,n0 |2 = Proba. to find |n〉 at late time, (1)
when starting from|n0〉.
followed from his understanding of Transition Amplit .
Follow here the same logic:study the properties of Cn(a), Trans. Amplit. in QC ,then consider their interpretation.
Renaud Parentani Structure of the WDW equation
![Page 9: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/9.jpg)
Why focusing on Transition Amplitudes ?
The WdW eq. also describes matter transitions,e.g. e+ e− annihilation.→ How to compute their amplitudes ?
Historically, Born’s statistical interpretation (1926)
|n0〉 →∑
n
cn,n0 |n〉
|cn,n0 |2 = Proba. to find |n〉 at late time, (1)
when starting from|n0〉.
followed from his understanding of Transition Amplit .
Follow here the same logic:study the properties of Cn(a), Trans. Amplit. in QC ,then consider their interpretation.
Renaud Parentani Structure of the WDW equation
![Page 10: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/10.jpg)
Why focusing on Transition Amplitudes ?
The WdW eq. also describes matter transitions,e.g. e+ e− annihilation.→ How to compute their amplitudes ?
Historically, Born’s statistical interpretation (1926)
|n0〉 →∑
n
cn,n0 |n〉
|cn,n0 |2 = Proba. to find |n〉 at late time, (1)
when starting from|n0〉.
followed from his understanding of Transition Amplit .
Follow here the same logic:study the properties of Cn(a), Trans. Amplit. in QC ,then consider their interpretation.
Renaud Parentani Structure of the WDW equation
![Page 11: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/11.jpg)
Why focusing on Transition Amplitudes ?
The WdW eq. also describes matter transitions,e.g. e+ e− annihilation.→ How to compute their amplitudes ?
Historically, Born’s statistical interpretation (1926)
|n0〉 →∑
n
cn,n0 |n〉
|cn,n0 |2 = Proba. to find |n〉 at late time, (1)
when starting from|n0〉.
followed from his understanding of Transition Amplit .
Follow here the same logic:study the properties of Cn(a), Trans. Amplit. in QC ,then consider their interpretation.
Renaud Parentani Structure of the WDW equation
![Page 12: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/12.jpg)
Why using molecular physics techniques ?
Appropriate for dynamical systems containinglight and heavy (i.e. fast and slow) deg. of freedom.
Transition amplitudes are governed byfrequency ratios and not by coupling constants(called Non-Adiabatic Transition Amplitudes NATA )
certain NATA are exponentially suppressed w.r.t. others,thereby introducing a hierarchy of NATA .
use this hierarchy toorganize solutions of the WDW eq.get algebraic relations betweenSchrod’s cn(t) and WDW’s Cn(a).
Long tradition: Born ’26, Heisenberg ’35, Gottfried ’66 + ’98All considering the statistical interpret. of QM.
Renaud Parentani Structure of the WDW equation
![Page 13: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/13.jpg)
Why using molecular physics techniques ?
Appropriate for dynamical systems containinglight and heavy (i.e. fast and slow) deg. of freedom.
Transition amplitudes are governed byfrequency ratios and not by coupling constants(called Non-Adiabatic Transition Amplitudes NATA )
certain NATA are exponentially suppressed w.r.t. others,thereby introducing a hierarchy of NATA .
use this hierarchy toorganize solutions of the WDW eq.get algebraic relations betweenSchrod’s cn(t) and WDW’s Cn(a).
Long tradition: Born ’26, Heisenberg ’35, Gottfried ’66 + ’98All considering the statistical interpret. of QM.
Renaud Parentani Structure of the WDW equation
![Page 14: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/14.jpg)
Why using molecular physics techniques ?
Appropriate for dynamical systems containinglight and heavy (i.e. fast and slow) deg. of freedom.
Transition amplitudes are governed byfrequency ratios and not by coupling constants(called Non-Adiabatic Transition Amplitudes NATA )
certain NATA are exponentially suppressed w.r.t. others,thereby introducing a hierarchy of NATA .
use this hierarchy toorganize solutions of the WDW eq.get algebraic relations betweenSchrod’s cn(t) and WDW’s Cn(a).
Long tradition: Born ’26, Heisenberg ’35, Gottfried ’66 + ’98All considering the statistical interpret. of QM.
Renaud Parentani Structure of the WDW equation
![Page 15: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/15.jpg)
Why using molecular physics techniques ?
Appropriate for dynamical systems containinglight and heavy (i.e. fast and slow) deg. of freedom.
Transition amplitudes are governed byfrequency ratios and not by coupling constants(called Non-Adiabatic Transition Amplitudes NATA )
certain NATA are exponentially suppressed w.r.t. others,thereby introducing a hierarchy of NATA .
use this hierarchy toorganize solutions of the WDW eq.get algebraic relations betweenSchrod’s cn(t) and WDW’s Cn(a).
Long tradition: Born ’26, Heisenberg ’35, Gottfried ’66 + ’98All considering the statistical interpret. of QM.
Renaud Parentani Structure of the WDW equation
![Page 16: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/16.jpg)
Why using molecular physics techniques ?
Appropriate for dynamical systems containinglight and heavy (i.e. fast and slow) deg. of freedom.
Transition amplitudes are governed byfrequency ratios and not by coupling constants(called Non-Adiabatic Transition Amplitudes NATA )
certain NATA are exponentially suppressed w.r.t. others,thereby introducing a hierarchy of NATA .
use this hierarchy toorganize solutions of the WDW eq.get algebraic relations betweenSchrod’s cn(t) and WDW’s Cn(a).
Long tradition: Born ’26, Heisenberg ’35, Gottfried ’66 + ’98All considering the statistical interpret. of QM.
Renaud Parentani Structure of the WDW equation
![Page 17: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/17.jpg)
Why using molecular physics techniques ?
Appropriate for dynamical systems containinglight and heavy (i.e. fast and slow) deg. of freedom.
Transition amplitudes are governed byfrequency ratios and not by coupling constants(called Non-Adiabatic Transition Amplitudes NATA )
certain NATA are exponentially suppressed w.r.t. others,thereby introducing a hierarchy of NATA .
use this hierarchy toorganize solutions of the WDW eq.get algebraic relations betweenSchrod’s cn(t) and WDW’s Cn(a).
Long tradition: Born ’26, Heisenberg ’35, Gottfried ’66 + ’98All considering the statistical interpret. of QM.
Renaud Parentani Structure of the WDW equation
![Page 18: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/18.jpg)
The Schroed. equation in cosmology.
For definiteness,consider some matter fields governed by an hermitian HM
in an expanding compact RW sp-time described by a(t).
The S. eq. isi∂t |ψ(t)〉 = HM |ψ(t)〉. (2)
Because of the expansion da/dt > 0,
HM = HM(a(t))
is t-dependent. This is crucial for what follows.
Renaud Parentani Structure of the WDW equation
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Structure of the S. eq. in Cosmology. 1
To reveal its structure, and/or to solve it, introducethe instantaneous eigenstates of Hm(a):
HM(a) |χn(a)〉 = En(a) |χn(a)〉〈χn(a)|χm(a)〉 = δn,m (3)
decompose |ψ(t)〉 in this basis :
|ψ(t)〉 =∑
n
cn(t) e−iR t dt ′En(t ′) |χn(a(t))〉, (4)
compute the cn(t) by injecting (4) in i∂t |ψ〉 = HM |ψ〉.
Renaud Parentani Structure of the WDW equation
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Structure of the S. eq. in Cosmology. 1
To reveal its structure, and/or to solve it, introducethe instantaneous eigenstates of Hm(a):
HM(a) |χn(a)〉 = En(a) |χn(a)〉〈χn(a)|χm(a)〉 = δn,m (3)
decompose |ψ(t)〉 in this basis :
|ψ(t)〉 =∑
n
cn(t) e−iR t dt ′En(t ′) |χn(a(t))〉, (4)
compute the cn(t) by injecting (4) in i∂t |ψ〉 = HM |ψ〉.
Renaud Parentani Structure of the WDW equation
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Structure of the S. eq. in Cosmology. 2
This gives a N × N matricial eq.
∂tcn =∑
m
〈∂tχm|χn〉 e−iR t dt ′(Em−En)cm. (5)
where
〈χm|∂tχn〉 =〈χm|∂tHM |χn〉En(a) − Em(a)
, n 6= m (6)
NB. When da/dt = 0, one has
∂tcn ≡ 0.
Hence all NA transitions are induced by da/dt .
Renaud Parentani Structure of the WDW equation
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Structure of the S. eq. in Cosmology. 2
This gives a N × N matricial eq.
∂tcn =∑
m
〈∂tχm|χn〉 e−iR t dt ′(Em−En)cm. (5)
where
〈χm|∂tχn〉 =〈χm|∂tHM |χn〉En(a) − Em(a)
, n 6= m (6)
NB. When da/dt = 0, one has
∂tcn ≡ 0.
Hence all NA transitions are induced by da/dt .
Renaud Parentani Structure of the WDW equation
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Structure of the S. eq. in Cosmology. 2 Remarks.
The amplitudes dn(a) defined in the correspondingcontracting universe, da/dt → −da/dt ,obey the same equation with i → −i .
The Schrod. eq. can be written in terms of a:
∂acn =∑
m 6=n
〈∂aχm|χn〉 e−iR a da′(dt/da′)(Em−En) cm(a). (7)
The cosmic time t only appears through t(a).
Renaud Parentani Structure of the WDW equation
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Structure of the S. eq. in Cosmology. 2 Remarks.
The amplitudes dn(a) defined in the correspondingcontracting universe, da/dt → −da/dt ,obey the same equation with i → −i .
The Schrod. eq. can be written in terms of a:
∂acn =∑
m 6=n
〈∂aχm|χn〉 e−iR a da′(dt/da′)(Em−En) cm(a). (7)
The cosmic time t only appears through t(a).
Renaud Parentani Structure of the WDW equation
![Page 25: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/25.jpg)
Structure of the S. eq. in Cosmology. 2 Remarks.
The amplitudes dn(a) defined in the correspondingcontracting universe, da/dt → −da/dt ,obey the same equation with i → −i .
The Schrod. eq. can be written in terms of a:
∂acn =∑
m 6=n
〈∂aχm|χn〉 e−iR a da′(dt/da′)(Em−En) cm(a). (7)
The cosmic time t only appears through t(a).
Renaud Parentani Structure of the WDW equation
![Page 26: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/26.jpg)
NATA: Non-Adiabatic Transition Amplitudes.
Starting with c1(t = −∞) = 1, the amplitude to find the systemin the state n is cn(+∞).
To first order in NA, it is
cn(+∞) ≃∫ +∞
−∞
dt〈χn|∂tχ1〉 e−iR t−∞
dt ′(E1(t ′)−En(t ′)) (8)
When cn ≪ 1, can be evaluated by a saddle point approx.
The sp time t∗ is complex, and hence cn is expon. damped:
cn(+∞) ≃ C e−iR t∗
−∞dt ′(E1−En)
NB. C → 1 in the adiabatic limit, see refs.
Renaud Parentani Structure of the WDW equation
![Page 27: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/27.jpg)
NATA: Non-Adiabatic Transition Amplitudes.
Starting with c1(t = −∞) = 1, the amplitude to find the systemin the state n is cn(+∞).
To first order in NA, it is
cn(+∞) ≃∫ +∞
−∞
dt〈χn|∂tχ1〉 e−iR t−∞
dt ′(E1(t ′)−En(t ′)) (8)
When cn ≪ 1, can be evaluated by a saddle point approx.
The sp time t∗ is complex, and hence cn is expon. damped:
cn(+∞) ≃ C e−iR t∗
−∞dt ′(E1−En)
NB. C → 1 in the adiabatic limit, see refs.
Renaud Parentani Structure of the WDW equation
![Page 28: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/28.jpg)
WDW Equation
We now replace [−i∂t + HM ] |ψ(t)〉 = 0 by the WDW eq.
[HG + HM ] |Ψ(a)〉 = 0 (9)
where
HG =−G2π2
a − a2 + Λa4
2Ga. (10)
πa = −i∂a is the momentum conjugated to a.
With (9, 10) we have displaced the Heisenberg cutso as to include (a, πa) in the quantum description.
Eqs. (9, 10) follow from S =∫
d4x√
gR + LM when
3 geometries are compact, and
matter distribution is sufficiently homogeneous.
N.B. HG + HM = 0 is the Friedmann eq.
Renaud Parentani Structure of the WDW equation
![Page 29: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/29.jpg)
WDW Equation
We now replace [−i∂t + HM ] |ψ(t)〉 = 0 by the WDW eq.
[HG + HM ] |Ψ(a)〉 = 0 (9)
where
HG =−G2π2
a − a2 + Λa4
2Ga. (10)
πa = −i∂a is the momentum conjugated to a.
With (9, 10) we have displaced the Heisenberg cutso as to include (a, πa) in the quantum description.
Eqs. (9, 10) follow from S =∫
d4x√
gR + LM when
3 geometries are compact, and
matter distribution is sufficiently homogeneous.
N.B. HG + HM = 0 is the Friedmann eq.
Renaud Parentani Structure of the WDW equation
![Page 30: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/30.jpg)
WDW Equation
We now replace [−i∂t + HM ] |ψ(t)〉 = 0 by the WDW eq.
[HG + HM ] |Ψ(a)〉 = 0 (9)
where
HG =−G2π2
a − a2 + Λa4
2Ga. (10)
πa = −i∂a is the momentum conjugated to a.
With (9, 10) we have displaced the Heisenberg cutso as to include (a, πa) in the quantum description.
Eqs. (9, 10) follow from S =∫
d4x√
gR + LM when
3 geometries are compact, and
matter distribution is sufficiently homogeneous.
N.B. HG + HM = 0 is the Friedmann eq.
Renaud Parentani Structure of the WDW equation
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Conserved Wronskian
The unique conserved quantity associatedwith the Schrod. eq. is the norm
N = 〈ψ(t)|ψ(t)〉 ≡ Cst . =∑
n
|cn(t)|2. (11)
Instead the unique one associatedwith the WDW eq. is the Wronskian
W = 〈Ψ(a)| i↔
∂a|Ψ(a)〉 ≡ Cst . (12)
Question: Can W be written as W =∑
n |Cn(a)|2 ?
Renaud Parentani Structure of the WDW equation
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Conserved Wronskian
The unique conserved quantity associatedwith the Schrod. eq. is the norm
N = 〈ψ(t)|ψ(t)〉 ≡ Cst . =∑
n
|cn(t)|2. (11)
Instead the unique one associatedwith the WDW eq. is the Wronskian
W = 〈Ψ(a)| i↔
∂a|Ψ(a)〉 ≡ Cst . (12)
Question: Can W be written as W =∑
n |Cn(a)|2 ?
Renaud Parentani Structure of the WDW equation
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Conserved Wronskian
The unique conserved quantity associatedwith the Schrod. eq. is the norm
N = 〈ψ(t)|ψ(t)〉 ≡ Cst . =∑
n
|cn(t)|2. (11)
Instead the unique one associatedwith the WDW eq. is the Wronskian
W = 〈Ψ(a)| i↔
∂a|Ψ(a)〉 ≡ Cst . (12)
Question: Can W be written as W =∑
n |Cn(a)|2 ?
Renaud Parentani Structure of the WDW equation
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First attempt
Decompose
|Ψ(a)〉 =∑
n
Cn(a) Ψn(a) |χn(a)〉, (13)
where |χn(a)〉 is the same as for the Schrod. eq.and Ψn(a) is the WKB solution of
[HG + En(a)] Ψn(a) = 0 (14)
The unit positive Wronskian WKB solution is
Ψn(a) =e−i
R ada pn(a)
√
2pn(a)(15)
where the momentum pn(a) > 0 solves HG + En(a) = 0.
These solutions correspond to expanding universes.Their complex conjugated describe contracting universes.
Renaud Parentani Structure of the WDW equation
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First attempt
Decompose
|Ψ(a)〉 =∑
n
Cn(a) Ψn(a) |χn(a)〉, (13)
where |χn(a)〉 is the same as for the Schrod. eq.and Ψn(a) is the WKB solution of
[HG + En(a)] Ψn(a) = 0 (14)
The unit positive Wronskian WKB solution is
Ψn(a) =e−i
R ada pn(a)
√
2pn(a)(15)
where the momentum pn(a) > 0 solves HG + En(a) = 0.
These solutions correspond to expanding universes.Their complex conjugated describe contracting universes.
Renaud Parentani Structure of the WDW equation
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First attempt
inserting (13) in the WDW eq. gives a second order matricialequation which mixes
corrections to WKB corrections with
n → n′ matter transitions.
Moreover the conserved Wronskian W reads
W =∑
n
|Cn(a)|2 +∑
n
(C∗n i
↔
∂aCn) |Ψn(a)|2. (16)
The first term is OK, the second unwanted.
Question: How to sort this out ?
Renaud Parentani Structure of the WDW equation
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First attempt
inserting (13) in the WDW eq. gives a second order matricialequation which mixes
corrections to WKB corrections with
n → n′ matter transitions.
Moreover the conserved Wronskian W reads
W =∑
n
|Cn(a)|2 +∑
n
(C∗n i
↔
∂aCn) |Ψn(a)|2. (16)
The first term is OK, the second unwanted.
Question: How to sort this out ?
Renaud Parentani Structure of the WDW equation
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First attempt
inserting (13) in the WDW eq. gives a second order matricialequation which mixes
corrections to WKB corrections with
n → n′ matter transitions.
Moreover the conserved Wronskian W reads
W =∑
n
|Cn(a)|2 +∑
n
(C∗n i
↔
∂aCn) |Ψn(a)|2. (16)
The first term is OK, the second unwanted.
Question: How to sort this out ?
Renaud Parentani Structure of the WDW equation
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WDW equation: The correct decomposition
The "correct" decomposition
|Ψ(a)〉 =∑
n
[Cn(a) Ψn(a) + Dn(a) Ψ∗n(a)] |χn(a)〉, (17)
introduces 2N arbitrary fonctions Cn(a),Dn(a).
Eq. (17) gives an under-constrained system.
Exploit this and impose
〈χn|i→
∂a |Ψ〉 = pn [Cn Ψn −Dn Ψ∗n] . (18)
Now the Cn (Dn) instantanenously weighexpanding (contracting) solutions.
Insert (17) in the WdW eq. and use (18) to get
Renaud Parentani Structure of the WDW equation
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WDW equation: The correct decomposition
The "correct" decomposition
|Ψ(a)〉 =∑
n
[Cn(a) Ψn(a) + Dn(a) Ψ∗n(a)] |χn(a)〉, (17)
introduces 2N arbitrary fonctions Cn(a),Dn(a).
Eq. (17) gives an under-constrained system.
Exploit this and impose
〈χn|i→
∂a |Ψ〉 = pn [Cn Ψn −Dn Ψ∗n] . (18)
Now the Cn (Dn) instantanenously weighexpanding (contracting) solutions.
Insert (17) in the WdW eq. and use (18) to get
Renaud Parentani Structure of the WDW equation
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WDW equation: The correct decomposition
The "correct" decomposition
|Ψ(a)〉 =∑
n
[Cn(a) Ψn(a) + Dn(a) Ψ∗n(a)] |χn(a)〉, (17)
introduces 2N arbitrary fonctions Cn(a),Dn(a).
Eq. (17) gives an under-constrained system.
Exploit this and impose
〈χn|i→
∂a |Ψ〉 = pn [Cn Ψn −Dn Ψ∗n] . (18)
Now the Cn (Dn) instantanenously weighexpanding (contracting) solutions.
Insert (17) in the WdW eq. and use (18) to get
Renaud Parentani Structure of the WDW equation
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WDW equation: The correct decomposition
The "correct" decomposition
|Ψ(a)〉 =∑
n
[Cn(a) Ψn(a) + Dn(a) Ψ∗n(a)] |χn(a)〉, (17)
introduces 2N arbitrary fonctions Cn(a),Dn(a).
Eq. (17) gives an under-constrained system.
Exploit this and impose
〈χn|i→
∂a |Ψ〉 = pn [Cn Ψn −Dn Ψ∗n] . (18)
Now the Cn (Dn) instantanenously weighexpanding (contracting) solutions.
Insert (17) in the WdW eq. and use (18) to get
Renaud Parentani Structure of the WDW equation
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Structure of the WDW equation, 1
∂aCn =∑
m 6=n
Mnm e−iR a(pn−pm)da Cm
+∑
m
Nnm e−iR a(pn+pm)da Dm. (19)
+ the same eq. with Cn → Dn and i → −iwhere
Mnm = 〈∂aψm|ψn〉pn + pm
2√
pnpm(20)
Nnm = 〈∂aψm|ψn〉pn−pm
2√
pnpm+ δnm
∂apn
2pn(21)
Eqs. (19,20,21) are exact
Eq.(19) is the WDW eq.
Renaud Parentani Structure of the WDW equation
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Structure of the WDW equation, 1
∂aCn =∑
m 6=n
Mnm e−iR a(pn−pm)da Cm
+∑
m
Nnm e−iR a(pn+pm)da Dm. (19)
+ the same eq. with Cn → Dn and i → −iwhere
Mnm = 〈∂aψm|ψn〉pn + pm
2√
pnpm(20)
Nnm = 〈∂aψm|ψn〉pn−pm
2√
pnpm+ δnm
∂apn
2pn(21)
Eqs. (19,20,21) are exact
Eq.(19) is the WDW eq.
Renaud Parentani Structure of the WDW equation
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Structure of the WDW equation, 1
∂aCn =∑
m 6=n
Mnm e−iR a(pn−pm)da Cm
+∑
m
Nnm e−iR a(pn+pm)da Dm. (19)
+ the same eq. with Cn → Dn and i → −iwhere
Mnm = 〈∂aψm|ψn〉pn + pm
2√
pnpm(20)
Nnm = 〈∂aψm|ψn〉pn−pm
2√
pnpm+ δnm
∂apn
2pn(21)
Eqs. (19,20,21) are exact
Eq.(19) is the WDW eq.
Renaud Parentani Structure of the WDW equation
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Structure of the WDW equation, 2
Moreover, as an identity , one has
W = 〈Ψ(a)|i↔
∂a|Ψ(a)〉=
∑
n
|Cn(a)|2 − |Dn(a)|2 ≡ Const . (22)
for any Hermitian matter hamiltonian HM ,and any state.
Let’s describe the consequences of Eq. 19 and 22
Renaud Parentani Structure of the WDW equation
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Structure of the WDW equation, 2
Moreover, as an identity , one has
W = 〈Ψ(a)|i↔
∂a|Ψ(a)〉=
∑
n
|Cn(a)|2 − |Dn(a)|2 ≡ Const . (22)
for any Hermitian matter hamiltonian HM ,and any state.
Let’s describe the consequences of Eq. 19 and 22
Renaud Parentani Structure of the WDW equation
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Theorem 1
When, for a given matter Hamiltonian HM ,the Schrod. eq. gives a N × N first order matricial eq.
the WDW eq. gives a 2N × 2N first order matricial eq.
NB. A similar doubling is also found innon-relativistic molecular/atomic physics.
It is a general conseq. of QM, when moving the H-cut.
Renaud Parentani Structure of the WDW equation
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Theorem 1
When, for a given matter Hamiltonian HM ,the Schrod. eq. gives a N × N first order matricial eq.
the WDW eq. gives a 2N × 2N first order matricial eq.
NB. A similar doubling is also found innon-relativistic molecular/atomic physics.
It is a general conseq. of QM, when moving the H-cut.
Renaud Parentani Structure of the WDW equation
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Theorem 2
Neglecting only the Cn → Dn transitions, one gets
∂aCn =∑
m 6=n
Mnm e−iR a(pn−pm)da Cm (23)
and∑
n
|Cn(a)|2 ≡ Cst .
+ the same eq. with Cn → Dn and i → −i .
WDW eq. thus gives two separate (and unitary ) eqs:one for the expanding sector, one for the contracting.
Eq. (23) is not the Schrod. equation.There is no background sp-time, no cosmic time (yet)
Renaud Parentani Structure of the WDW equation
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Theorem 2
Neglecting only the Cn → Dn transitions, one gets
∂aCn =∑
m 6=n
Mnm e−iR a(pn−pm)da Cm (23)
and∑
n
|Cn(a)|2 ≡ Cst .
+ the same eq. with Cn → Dn and i → −i .
WDW eq. thus gives two separate (and unitary ) eqs:one for the expanding sector, one for the contracting.
Eq. (23) is not the Schrod. equation.There is no background sp-time, no cosmic time (yet)
Renaud Parentani Structure of the WDW equation
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Theorem 2
Neglecting only the Cn → Dn transitions, one gets
∂aCn =∑
m 6=n
Mnm e−iR a(pn−pm)da Cm (23)
and∑
n
|Cn(a)|2 ≡ Cst .
+ the same eq. with Cn → Dn and i → −i .
WDW eq. thus gives two separate (and unitary ) eqs:one for the expanding sector, one for the contracting.
Eq. (23) is not the Schrod. equation.There is no background sp-time, no cosmic time (yet)
Renaud Parentani Structure of the WDW equation
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Theorem 3. The recovery of cosmic time
Neglecting the Cn → Dn transitions, andto first order in En − En around the mean matter state n,
Cn(a) ≡ cn(t(a)), (24)
where
t(a) =
∫ a
da′∂Ep(a′)|E=En , (25)
is the HJ time to reach a when matter is in the n state.
The identity (24) is Heisenberg-’35 result.It follows from
Mnm ≡ MSmn
e−iR a da′(pn−pm) ≡ e−i
R t dt′(En−Em)
Renaud Parentani Structure of the WDW equation
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Theorem 3. The recovery of cosmic time
Neglecting the Cn → Dn transitions, andto first order in En − En around the mean matter state n,
Cn(a) ≡ cn(t(a)), (24)
where
t(a) =
∫ a
da′∂Ep(a′)|E=En , (25)
is the HJ time to reach a when matter is in the n state.
The identity (24) is Heisenberg-’35 result.It follows from
Mnm ≡ MSmn
e−iR a da′(pn−pm) ≡ e−i
R t dt′(En−Em)
Renaud Parentani Structure of the WDW equation
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A remark on NATAmplitudes
For the Schrod. eq., cn is given by the overlap
cn(t) = 〈χn(t)|e+iR t dt ′En |ψ(t)〉. (26)
Instead, for the WDW eq., Cn is given by
Cn(a) = 〈χn(a)|Ψn(a)∗ i↔
∂a|Ψ(a)〉 (27)
This is the Vilenkin current ,and NOT the Page-Hawking overlap 〈χn(a)|Ψ(a)〉.
Renaud Parentani Structure of the WDW equation
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A remark on NATAmplitudes
For the Schrod. eq., cn is given by the overlap
cn(t) = 〈χn(t)|e+iR t dt ′En |ψ(t)〉. (26)
Instead, for the WDW eq., Cn is given by
Cn(a) = 〈χn(a)|Ψn(a)∗ i↔
∂a|Ψ(a)〉 (27)
This is the Vilenkin current ,and NOT the Page-Hawking overlap 〈χn(a)|Ψ(a)〉.
Renaud Parentani Structure of the WDW equation
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Theorem 4. The Cn → Dm NATransitions.
Far from turning points (big bounce),the transitions Cn → Dm are exponentially suppressedw.r.t. to the Cn → Cm.
Because, typically
Cn → Cm ∼ e−(pn−pm) Im∆aspC
Cn → Dm ∼ e−(pn+pm) Im∆aspD (28)
pn+pm scales with the total matter energy .
The Lesson:The WDW eq. predicts a hierarchy of NAT ,and governs it .
Renaud Parentani Structure of the WDW equation
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Theorem 4. The Cn → Dm NATransitions.
Far from turning points (big bounce),the transitions Cn → Dm are exponentially suppressedw.r.t. to the Cn → Cm.
Because, typically
Cn → Cm ∼ e−(pn−pm) Im∆aspC
Cn → Dm ∼ e−(pn+pm) Im∆aspD (28)
pn+pm scales with the total matter energy .
The Lesson:The WDW eq. predicts a hierarchy of NAT ,and governs it .
Renaud Parentani Structure of the WDW equation
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Theorem 4. The Cn → Dm NATransitions.
Far from turning points (big bounce),the transitions Cn → Dm are exponentially suppressedw.r.t. to the Cn → Cm.
Because, typically
Cn → Cm ∼ e−(pn−pm) Im∆aspC
Cn → Dm ∼ e−(pn+pm) Im∆aspD (28)
pn+pm scales with the total matter energy .
The Lesson:The WDW eq. predicts a hierarchy of NAT ,and governs it .
Renaud Parentani Structure of the WDW equation
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The Cn → Dm NAT in molecular phys.
Consider a molecule of mass M and CM R
Hatom =P2
2M+ V (R) + He(R),
where He(R) governs the electronic deg. of freedom.
There are two levels of quantization:
The LZ way: treat R classically as a given function of time R(t).The residual elect. d.o.f. then obey the t-dep. Schrod. eq.
i∂t |ψe〉 = He(R(t))|ψe〉
The WdW way: treat both R and el. d.o.f on the same footing .Then, at fixed total energy E , one has
[
− ∂2R
2M+ V (R) + Hel(R)
]
|ΨE(R)〉 = E |ΨE(R)〉
Renaud Parentani Structure of the WDW equation
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The Cn → Dm NAT in molecular phys.
Consider a molecule of mass M and CM R
Hatom =P2
2M+ V (R) + He(R),
where He(R) governs the electronic deg. of freedom.
There are two levels of quantization:
The LZ way: treat R classically as a given function of time R(t).The residual elect. d.o.f. then obey the t-dep. Schrod. eq.
i∂t |ψe〉 = He(R(t))|ψe〉
The WdW way: treat both R and el. d.o.f on the same footing .Then, at fixed total energy E , one has
[
− ∂2R
2M+ V (R) + Hel(R)
]
|ΨE(R)〉 = E |ΨE(R)〉
Renaud Parentani Structure of the WDW equation
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The Cn → Dm NAT in atomic phys.
Decomposing |ΨE(R)〉 =∑
n(Cn Ψn + Dn Ψ∗
n)|χn(R)〉
neglecting Cn → Dm and to first order in Een − Ee
m, one gets
Cn(R) ≡ cn(t(R)),
where cn are the Landau-Z tr. amplitudes.Hence the |Cn|2 give transition probablities .
The Cn(R) are given by Cn ≡ 〈χn|Ψ∗
n i↔
∂R |ΨE〉.,i.e. by the "Vilenkin" current.
Taking into account the NAT Cn → Dm, one gets, as an identity,∑
n |Cn(R)|2 − |Dn(R)|2 ≡ Const.This is a re-expression of unitarity .
The H-Page norm |〈χn|ΨE〉|2 = |CnΨn(R) + DnΨ∗
n(R)|2 is
highly interferent, and
answers another question.
Renaud Parentani Structure of the WDW equation
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Conclusions. 1
The WDW eq. determines the NATA.There is NO ambiguity in computing these amplitudes.
It predicts that there is a hierarchy of NA-regimes.
Renaud Parentani Structure of the WDW equation
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Conclusions. 2. The S. regime, the mildest one.
The first regime, the mildest one, is obtained by
neglecting Cn → Dm NAT,
the first order in En − En, but nothing else .
In this regimeCn(a) ≡ cn(t(a)).
Therefore |Cn(a)|2 is the proba. to find the state nin an expand. universe at a (in fact around a).
Renaud Parentani Structure of the WDW equation
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Conclusions. 2. The S. regime, the mildest one, 2.
Unitarity,∑
n
|Cn(a)|2 ≡ Cst .
relies on the conserved current W .
NO problem of interpretation, NO problem of time.
Renaud Parentani Structure of the WDW equation
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Conclusions. 2. The S. regime, the mildest one, 2.
Unitarity,∑
n
|Cn(a)|2 ≡ Cst .
relies on the conserved current W .
NO problem of interpretation, NO problem of time.
Renaud Parentani Structure of the WDW equation
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Conclusions. 3. The second intermediate regime.
Neglecting only Cn → Dm, the WDW givestwo separate 1st order eqs. in i∂a.
Unitarity , i.e.∑
n |Cn(a)|2 ≡ Cst . is obtained,as in the Schrod. regime, from the conservation of W ,even though there is no cosmic timebecause no backd sp-time common to all matter states.
Lesson:The recovery of cosmic time requires more condition thanthe recovery of unitarity .
Renaud Parentani Structure of the WDW equation
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Conclusions. 3. The second intermediate regime.
Neglecting only Cn → Dm, the WDW givestwo separate 1st order eqs. in i∂a.
Unitarity , i.e.∑
n |Cn(a)|2 ≡ Cst . is obtained,as in the Schrod. regime, from the conservation of W ,even though there is no cosmic timebecause no backd sp-time common to all matter states.
Lesson:The recovery of cosmic time requires more condition thanthe recovery of unitarity .
Renaud Parentani Structure of the WDW equation
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Conclusions. 3. The second intermediate regime.
Neglecting only Cn → Dm, the WDW givestwo separate 1st order eqs. in i∂a.
Unitarity , i.e.∑
n |Cn(a)|2 ≡ Cst . is obtained,as in the Schrod. regime, from the conservation of W ,even though there is no cosmic timebecause no backd sp-time common to all matter states.
Lesson:The recovery of cosmic time requires more condition thanthe recovery of unitarity .
Renaud Parentani Structure of the WDW equation
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Conclusions. 4. The least Adiabatic regime.
Taking into account Cn → Dm NAT, one faces the identity∑
n
|Cn(a)|2 − |Dn(a)|2 ≡ Cst .
How to interprete this equation ?
Proposal: as we just did it for cosmic time:the recovery of unitarity requires some conditions .
Hence we conclude/conjecture:
The statistical interpretation of the NATA Cn(a)is not as a fundamental property ,but should be conceived as an emergent property of QC.(Jacobson 2005, Massar & RP 1999, Vilenkin 1989)
Renaud Parentani Structure of the WDW equation
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Conclusions. 4. The least Adiabatic regime.
Taking into account Cn → Dm NAT, one faces the identity∑
n
|Cn(a)|2 − |Dn(a)|2 ≡ Cst .
How to interprete this equation ?
Proposal: as we just did it for cosmic time:the recovery of unitarity requires some conditions .
Hence we conclude/conjecture:
The statistical interpretation of the NATA Cn(a)is not as a fundamental property ,but should be conceived as an emergent property of QC.(Jacobson 2005, Massar & RP 1999, Vilenkin 1989)
Renaud Parentani Structure of the WDW equation
![Page 72: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/72.jpg)
Conclusions. 4. The least Adiabatic regime.
Taking into account Cn → Dm NAT, one faces the identity∑
n
|Cn(a)|2 − |Dn(a)|2 ≡ Cst .
How to interprete this equation ?
Proposal: as we just did it for cosmic time:the recovery of unitarity requires some conditions .
Hence we conclude/conjecture:
The statistical interpretation of the NATA Cn(a)is not as a fundamental property ,but should be conceived as an emergent property of QC.(Jacobson 2005, Massar & RP 1999, Vilenkin 1989)
Renaud Parentani Structure of the WDW equation
![Page 73: Structure of the WDW equation · of the solutions of the WDW eq. Several options have been proposed: "Page-Hawking"based on the norm: |Ψ(a,φ)|2 "Vilenkin" based on the current:](https://reader035.vdocument.in/reader035/viewer/2022071216/6047b1ada6559e123962c839/html5/thumbnails/73.jpg)
Conclusions. 4. The least Adiabatic regime.
Taking into account Cn → Dm NAT, one faces the identity∑
n
|Cn(a)|2 − |Dn(a)|2 ≡ Cst .
How to interprete this equation ?
Proposal: as we just did it for cosmic time:the recovery of unitarity requires some conditions .
Hence we conclude/conjecture:
The statistical interpretation of the NATA Cn(a)is not as a fundamental property ,but should be conceived as an emergent property of QC.(Jacobson 2005, Massar & RP 1999, Vilenkin 1989)
Renaud Parentani Structure of the WDW equation