Download Cosmological particle creation, conformal invariance and A. D.

Cosmological particle creation,
conformal invariance and
A. D. Sakharov’s induced gravity
Victor Berezin, Vyacheslav Dokuchaev and Yury Eroshenko
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow
Ginzburg Centennial Conference on Physics, May 29 – June 3, 2017 Moscow
Outline
I
Introduction
II
Particle creation — phenomenology
III
Conformal symmetry
IV
Scalar field
V
The “good” hydrodynamical variables and equations of motion
VI
Vacuum equations
“Empty” vacuum and
“Dynamical” vacuum = Dirac sea
I. Introduction
Particle creation — quantum phenomenon
Renormalization. Counter-terms.
A. D. Sakharov — induced gravity
Ya. B. Zel’dovich, A. A. Starobinsky, L. Parker, S. Fulling,
V. N. Lukash, A. A. Grib, S. G. Mamaev, V. A. Mostepanenko
Back reaction — problems
Process of creation
Global geometry: horizons, energy dominance violation
Phenomenological description of particle creation:
Eulerian coordinates, “creation law”,
Weyl tensor — renormalizability
Conformal gravity. History
Nowadays — Higgs Mechanism
Creatioon of universe from “nothing” — A. Vilenkin
Initial states is conformally invariant — R. Penrose, G. ‘t Hooft
Sign convention
signature (+, −, −, −)
∂Γµ
νσ
∂x λ
R µνµσ
R µνλσ =
Rνσ =
−
∂Γµ
νλ
∂x σ
+ Γµκλ Γκνσ − Γµκσ Γκνλ
R = g νσ Rνσ
Γλµν — metric connections
Units
~=c =1
[G ] = [`]2 =
1
[m]2
= [t]2
II. Particle creation — phenomenology
Hydrodynamics of perfect fluid with varying number of particles
Lagrange description is not adequate =⇒ worldline is varied
without taking into account the process of its creation =⇒ Euler
description
Standard action:
(J. R. Ray 1972)
Z
√
ε(X , n) −g dx +
Z
µ√
Shydro = −
+
λ2 X,µ u
Z
Z
−g dx +
√
λ0 (u µ uµ − 1) −g dx
√
λ1 (nu µ );µ −g dx
ε(X , n) — invariant energy density
n — invariant particle number density
u µ — four-velocity of particle flux
X — auxiliary variable (numeration of worldlines)
λ0 , λ1 , λ2 — Lagrange multipliers
(nu µ );µ = 0 — particle number conservation law
Dynamical variables: n, u µ , X
Our proposal: to include into the formalism the “creation law”
(nu µ );µ = Φ(inv ) 6= 0
(V. A. Berezin 1987)
Fundamental result for (cosmological) particle creation
Φ = βC 2
C µνλσ
Shydro
(Ya. B. Zel’dovich, A. A. Starobinsky 1977)
— Weyl tensor
Z
Z
√
√
= − ε(X , n) −g dx + λ0 (u µ uµ − 1) −g dx
Z
Z
√
µ√
+ λ2 X,µ u −g dx + λ1 ((nu µ );µ − βC 2 ) −g dx
λ1 is defined up to an additive constant
λ1 → λ1 + γ0 , γ0 = const
Z
Z
√
√
µ
2 √
γ0 ((nu );µ − βC ) −g dx = γ0 ((n −g u µ ),µ − βC 2 −g ) dx
Z
→
−γ0 β
√
C 2 −g dx
→
conformal Weyl gravity
III. Conformal invariance — the postulate
“+” Additional symmetry
1
2
Creation of the universe from “nothing”
A.Vilenkin, Ya. B. Zel’dovich & L.P.Grishchuk, S.W.Hawking...
R. Penrose, G ’t Hooft
“−”
1
2
Theory was rejected — absence of massive particles
H. Weyl, A. Einstein 1921
However, the Higgs mechanism was unknown in 1921!
G ’t Hooft 2014
δStot
=0 ⇒
δΩ
Conformal multiplier Ω may be considered as an independent
variable
gµν = Ω2 ĝµν ,
Stot = Sgrav + Smatter
By definition:
1
2
Z
δSmatter =
1
2
Z
δSmatter =
Let
δg µν = −
√
Tµν −g δg µν dx
T̂µν
p
−ĝ δĝ µν dx
2 µν
2
ĝ δΩ = − g µν δΩ
3
Ω
Ω
Then, if
0 = δSmatter
δSgrav
=0 ⇒
δΩ
Z
δΩ √
= − Tµν g µν
−g dx
Ω
⇒
Tr (Tµν ) = 0
Let
δg µν = Ω2 δĝ µν
⇒
T̂µν = Ω2 Tµν
(T̂νµ = Ω4 Tνµ , T̂ µν = Ω6 T µν )
IV. What are those fields responsible for particle crreation?
The simplest case — scalar field
Z 1 2 2 √
1 µ
Sscalar =
χ χµ − m χ
−g dx
2
2
Conformal transformation (standard):
δg µν = Ω2 δĝ µν ,
Sscalar
1
χ̂
Ω
⇒
Z
1 µ
1
1 χ̂2
1 2 2 2 √
µ
µ
=
χ̂ χ̂µ − χ̂µ Ω +
Ωµ Ω − m Ω χ
−g dx
2
Ω
2 Ω2
2
χµ = χ,µ
χµ = g µν χν
χ=
χ̂µ = ĝ µν χ̂ν
Ωµ = g µν Ων
How to make Sscalar to be conformally covariant?
R 2
The known prescription: to add to Lagrangian 12
χ
R — scalar curvature, constructed on metric gµν ⇒
√
R
1
1 µ
χ χµ + χ2 − m2 χ2 −g dx
2
12
2
!
Z
R̂ 2 1 2 2 2 p
1 µ
χ̂ χ̂µ + χ − m Ω χ̂
=
−ĝ dx
2
12
2
Z
1
Ωλ p
−ĝ dx
−
χ̂2
2
Ω |λ
Z
Sscalar =
The last term is the surface integral
m2 = 0 — all is OK?
But! The correct sign ⇐⇒ the incorrect sign
Our choice:
1
2
R̂ 2
The “correct” sign for R̂: − 12
χ̂
The “incorrect” sign for kinetic term:
− 12 χ̂µ χ̂µ
“Explanatory note”
Our scalar field χ is not “real” (fundamental). The already
created particles are quanta of this field (phenomenology!)
What remains — is a “vacuum” part of the scalar field,
containing conformal anomaly, responsible for particle creation
The wrong sign sign of kinetic term =⇒
absence of the law energy bound
(C -field by Hoyle and Narlikar)
Scalar filed χ is not a dynamical variable =⇒ there is no
variation
How could the field χ know about the conformal transformation?
It is always possible to choose χ̂ = 1` ϕ, ϕ = Ω
1
(χ = Ω1 χ̂, [`] = [m]
, ~ = c = 1) =⇒
!
Z
1
1 µ
R̂ 2 1 2 4 p
Sscalar = − 2
ϕ ϕµ + ϕ − m ϕ
−ĝ dx
`
2
12
2
It appears ϕ4 ! (d = dim M = 4)!
Conformal covariance p
p
ϕ(new ) = Ω̃ϕ(old),
−g (old) = Ω̃4 −g (new )
Cosmological term or quintessence: 3m2 = Λ
Independent variations of ϕ and ĝ µν
Energy-momentum tensor
1
1
1
scalar
= − 2 ϕµ ϕµ + 2 ϕσ ϕσ ĝµν − 2 m2 ϕ4 ĝµν
T̂µν
` 2`
2`
1
1
2
σ
− 2 ϕ (R̂µν − ĝµν R̂) − 2 (ϕϕν )|µ − (ϕϕ )|σ ĝµν
6`
2
1
1
2
scalar
Tr Tµν
= − 2 ϕϕσ|σ + 2 R̂ϕ2 − 2 m2 ϕ4
`
6`
`
V. The “good” hydrodynamical variables for hydrodynamical action
(nu µ );µ = βC 2
ds 2 = ϕ2 dŝ 2
⇒
⇒
p
√
C 2 −g = Ĉ 2 −ĝ ,
uµ =
1 µ
û ,
ϕ
√
û µ ûµ = 1,
−g = ϕ4
uµ = ϕûµ
p
√
√
(nu µ );µ −g = (nu µ −g ),µ = (ϕ3 nu µ −ĝ ),µ
Z
p
n̂ = nϕ3 −ĝ ⇒ N = n̂ d 3 x − inv
⇒
√
λ0 (u µ uµ − 1) −g dx
Z
Z
√
µ√
+ λ2 X,µ u −g dx + λ1 (nu µ );µ −g dx =⇒
Z Z
p
p
n̂
= − ε X , 3√
ϕ4 −ĝ dx + λ0 (û µ ûµ − 1)ϕ4 −ĝ dx
ϕ −ĝ
Z
Z
p
p µ 3
+ λ2 X,µ û ϕ −ĝ dx + λ1 (n̂û µ ),µ − β Ĉ 2 −ĝ dx
Z
Shydro = −
Shydro
√
ε(X , n) −g dx +
p
−ĝ
Z
ε+p =n
hydro
T̂µν
=
∂ε
∂n
=⇒
|λ|σ 1
λσ
(ε+p)ϕ ûµ ûν −pϕ ĝµν −4βϕ (λ1 Ĉµσνλ ) + λ1 Ĉµλνσ R̂
2
4
4
2
hydro
Tr Tµν
= (ε − 3p)ϕ4
total
Tr Tµν
=−
1
2
1
ϕϕσ|σ + 2 R̂ϕ2 − 2 m2 ϕ4 + (ε − 3p)ϕ4
2
`
6`
`
Variation of ϕ:
δStotal
δϕ
=0
=⇒
1 σ
1
(ϕ |σ − R̂ϕ + 2m2 ϕ3 ) + (ε − 3p)ϕ3 = 0
2
`
6
As it should be!
VI. Vacuum equations
Without “hats” (!):
1
hydro
Rµν − gµν R − Λgµν = 8πGTµν
2
hydro
Tµν
= ( + p)uµ uν − pgµν − 4βBµν [λ1 ]
1
Bµν [λ1 ] = (λ1 Cµσνλ );λ;σ + λ1 Cµλνσ R λσ
2
“Empty” vacuum:
=p=0
1
Λ
Gµν −
gµν = 0
16πG
16πG
1
Gµν = Rµν − gµν R
2
4α0 Bµν [λ1 ] +
In terms of ϕ and ĝµν (gµν = ϕ2 ĝµν ):
1
|λ|σ
λσ
0 = −4β (λ1 Ĉµσνλ )
+ λ1 Ĉµλνσ R̂
2
1
1
− 2 ϕ2 (R̂µν − ĝµν R̂) + 4ϕµ ϕν − ϕσ ϕσ ĝµν
6`
2
|σ
−2ϕϕν|µ + 2ϕϕ|σ ĝµν − Λϕ4 ĝµν
Trace:
|σ
ϕ|σ −
1
2
R̂ϕ − Λϕ3 = 0
6
3
Dirac sea
Weyl tensor Cµσνλ 6= 0
Ingredients: particles of both positive and negative energies
µ
(±) , p(±) , ~v(±) , u(±)
Hydrodynamical equations
(±) (±)
(±)
(±) + p(±) uµ(±) + λ2 X,µ
− n(±) λ1,µ = 0,
∂(±)
(±) (±)σ
−
λ
u
= 0
2
;σ
∂X (±)
Complete compensations
+ = −− ,
u
(+)µ
(+)
λ2
=u
=
⇒
n+ = n−
(−)µ
(−)
−λ2
(+)
λ1
⇒
⇒
X
(+)
λ1,µ
(−)
+ λ1
(+)
p+ = −p−
= X (−)
⇒
(−)
−λ1,µ
⇒
=
= const
“Dynamical vacuum” equations
4α0 Bµν +
16πG
Gµν −
1
Cµλνσ R λσ
2
1
= Rµν − gµν R
2
Λ
Bµν = Cµσνλ;λ;σ +
Gµν
1
Λ
16πG
gµν = 0
⇒
Bach tensor
⇒
Einstein tensor
⇒
Cosmological constant
Thanks to everybody!
The end
Abstract
We constructed a phenomenological model for cosmological
particle creation. It allows to take into account the back reaction
on the space-time metric not only of the energy-momentum tensor
of the already created particles, but also the very process of the
creation. Adopting the well-known fact that the particle
production rate is proportional to the square of the Weyl tensor,
we found that there is no more need to include, ad hoc, the
conformal gravity term into the action integral, it is already there.
Such a feature is in accordance with the idea of the induced
gravity, first proposed by A. D. Sakharov. Assuming, then, the
simplest possible (without self-interaction) action for the scalar
field responsible for the particle creation, we found that the
requirement for the local conformal transformation to be the
fundamental physical law, leads not only to the induced scalar
term in the action (Einstein-dilaton gravity), but also to the
appearance of the self-interaction for the scalar field absent before.
Moreover, the exponent of this self-interaction term (= 4) is due
to the dimensionality of our space-time.