Why there is something rather than nothing (out of everything)?
Why there is something rather than nothing (out of everything)?
A.O.Barvinsky
Theory Department, Lebedev Physics Institute, Leninsky Prospect 53, 119991 Moscow, Russia
The path integral over Euclidean geometries for the recently suggested density matrix of the
Universe is shown to describe a microcanonical ensemble in quantum cosmology. This ensemble
corresponds to a uniform (weight one) distribution in phase space of true physical variables, but
in terms of the observable spacetime geometry it is peaked about complex saddle-points of the
Lorentzian path integral. They are represented by the recently obtained cosmological instantons
limited to a bounded range of the cosmological constant. Inflationary cosmologies generated by these
instantons at late stages of expansion undergo acceleration whose low-energy scale can be attained
within the concept of dynamically evolving extra dimensions. Thus, together with the bounded
range of the early cosmological constant, this cosmological ensemble suggests the mechanism of
constraining the landscape of string vacua and, simultaneously, a possible solution to the dark energy
problem in the form of the quasi-equilibrium decay of the microcanonical state of the Universe.
PACS numbers: 04.60.Gw, 04.62.+v, 98.80.Bp, 98.80.Qc
Euclidean quantum gravity (EQG) is a lame duck in
modern particle physics and cosmology. After its sum-
mit in early and late eighties (in the form of the cosmo-
logical wavefunction proposals [1, 2] and baby universes
boom [3]) the interest in this theory gradually declined,
especially, in cosmological context, where the problem of
quantum initial conditions was superseded by the con-
cept of stochastic inflation [4]. EQG could not stand the
burden of indefiniteness of the Euclidean gravitational
action [5] and the cosmology debate of the tunneling vs
no-boundary proposals [6].
Thus, a recently suggested EQG density matrix of the
Universe [7] is hardly believed to be a viable candidate
for the initial state of the Universe, even though it avoids
the infrared catastrophe of small cosmological constant
Λ, generates an ensemble of universes in the limited
range of Λ, and suggests a strong selection mechanism
for the landscape of string vacua [7, 8]. Here we want to
justify this result by deriving it from first principles of
Lorentzian quantum gravity applied to a microcanonical
ensemble of closed cosmological models.
Thermal properties in quantum cosmology [9] are in-
corporated by a mixed physical state, which is dynam-
ically more preferable than a pure state of the Hartle-
Hawking type. This follows from the path integral for
the EQG statistical sum [7, 8]. It can be cast into the
form of the integral over a minisuperspace of the lapse
function N(τ) and scale factor a(τ) of spatially closed
FRW metric ds2 = N2(τ) dτ2 + a2(τ) d2Ω(3),
e−Γ =
periodic
D[ a,N ] e−ΓE [ a,N ], (1)
e−ΓE [ a,N ] =
periodic
Dφ(x) e−SE [ a,N ;φ(x) ]. (2)
Here ΓE [ a, N ] is the Euclidean effective action of all
inhomogeneous “matter” fields which include also met-
ric perturbations on minisuperspace background Φ(x) =
(φ(x), ψ(x), Aµ(x), hµν (x), ...). SE [a,N ;φ(x)] is the clas-
sical Eucidean action, and the integration runs over pe-
riodic fields on the Euclidean spacetime with a compact-
ified time τ (of S1 × S3 topology).
For free matter fields φ(x) conformally coupled to grav-
ity (which are assumed to be dominating in the sys-
tem) the effective action has the form [7] ΓE [ a,N ] =
dτ NL(a, a′) + F (η), a′ ≡ da/Ndτ . Here NL(a, a′) is
the effective Lagrangian of its local part including the
classical Einstein term (with the cosmological constant
Λ = 3H2) and the contribution of the conformal anomaly
of quantum fields and their vacuum (Casimir) energy,
L(a, a′) = −aa′2−a+H2a3+B
. (3)
F (η) is the free energy of their quasi-equilibrium excita-
tions with the temperature given by the inverse of the
conformal time η =
dτ N/a. This is a typical boson or
fermion sum F (η) = ±
1∓ e−ωη
over field oscil-
lators with energies ω on a unit 3-sphere. We work in
units of mP = (3π/4G)
1/2, and B is the constant deter-
mined by the coefficient of the Gauss-Bonnet term in the
overall conformal anomaly of all fields φ(x).
Semiclassically the integral (1) is dominated by the
saddle points — solutions of the Friedmann equation
−H2 − C
, (4)
modified by the quantum B-term and the radiation term
C/a4 with the constant C satisfying the bootstrap equa-
tion C = B/2 + dF (η)/dη. Such solutions represent
garland-type instantons which exist only in the limited
range 0 < Λmin < Λ < 3m
P/B [7, 8] and eliminate the
infrared catastrophe of Λ = 0. The period of these quasi-
thermal instantons is not a freely specifiable parameter,
but as a function of Λ follows from this bootstrap. There-
fore this is not a canonical ensemble.
Contrary to the construction above, the density ma-
trix that we advocate here is given by the canonical path
integral of Lorentzian quantum gravity. Its kernel in the
http://arxiv.org/abs/0704.0083v2
representation of 3-metrics and matter fields denoted be-
low as q reads
ρ(q+, q−) = e
q(t±)= q±
D[ q, p,N ] e
dt (p q̇−NµHµ)
, (5)
where the integration runs over histories of phase-space
variables (q(t), p(t)) interpolating between q± at t± and
the Lagrange multipliers of the gravitational constraints
Hµ = Hµ(q, p) — lapse and shift functionsN(t) = N
µ(t).
The measure D[ q, p,N ] includes the gauge-fixing factor
containing the delta function δ(χ) =
µ δ(χ
µ) of gauge
conditions χµ and the ghost factor [10, 11] (condensed
index µ includes also continuous spatial labels). It is
important that the integration range of Nµ
−∞ < N < +∞, (6)
is such that it generates in the integrand the delta-
functions of these constraints δ(H) =
µ δ(Hµ). As a
consequence the kernel (5) satisfies the set of quantum
Dirac constraints — Wheeler-DeWitt equations
q, ∂/i∂q
ρ( q, q− ) = 0, (7)
and the density matrix (5) can be regarded as an operator
delta-function of these constraints
ρ̂ ∼ “
δ(Ĥµ)
”. (8)
This notation should not be understood literally because
this multiple delta-function is not well defined, for the
operators Ĥµ do not commute and form a quasi-algebra
with nonvanishing structure functions. Moreover, ex-
act operator realization Ĥµ is not known except the
first two orders of a semiclassical ~-expansion [12]. For-
tunately, we do not need a precise form of these con-
straints, because we will proceed with their path-integral
solutions well adjusted to the semiclassical perturbation
theory. This strategy does not extend beyond typical
field-theoretic considerations when the path integral is
regarded more fundamental than the Schrodinger equa-
tion marred with the problems of divergent equal-time
commutators, operator ordering, etc.
The very essence of our proposal is the interpretation
of (5) and (8) as the density matrix of a microcanonical
ensemble in spatially closed quantum cosmology. A sim-
plest analogy is the density matrix of an unconstrained
system having a conserved Hamiltonian Ĥ in the micro-
canonical state with a fixed energy E, ρ̂ ∼ δ(Ĥ − E).
Major distinction of (8) from this case is that spatially
closed cosmology does not have freely specifiable con-
stants of motion like the energy or other global charges.
Rather it has as constants of motion the Hamiltonian and
momentum constraints Hµ, all having a particular value
— zero. Therefore, the expression (8) can be considered
as a most general and natural candidate for the quantum
state of the closed Universe. Below we confirm this fact
by showing that in the physical sector the correspond-
ing statistical sum is just a uniformly distributed (with
a unit weight) integral over entire phase space of true
physical degrees of freedom. Thus, this is a sum over
Everything. However, in terms of the observable quanti-
ties, like spacetime geometry, this distribution turns out
to be nontrivially peaked around a particular set of uni-
verses. Semiclassically this distribution is given by the
EQG density matrix and the saddle-point instantons of
the above type [7].
From the normalization of the density matrix in the
physical Hilbert space the statistical sum follows as the
path integral
1 = Trphys ρ̂ =
q, ∂/i∂q
ρ(q, q′)
periodic
D[ q, p,N ] e i
dt(p q̇−NµHµ), (9)
where the integration runs over periodic in time histo-
ries of q = q(t). Here µ
q, ∂/i∂q
= µ̂ is the mea-
sure which distinguishes the physical inner product in
the space of solutions of the Wheeler-DeWitt equations
(ψ1|ψ2) = 〈ψ1|µ̂|ψ2〉 from that of the space of square-
integrable functions, 〈ψ1|ψ2〉 =
dq ψ∗1ψ2. This measure
includes the delta-function of unitary gauge conditions
and an operator factor built with the aid of the relevant
ghost determinant [12].
On the other hand, in terms of the physical phase space
variables the Faddeev-Popov path integral equals [10, 11]
D[ q, p,N ] e i
dt (p q̇−NµHµ)
DqphysDpphys e
dt (pphys q̇phys−Hphys(t))
= Trphys
T e−i
dt Ĥphys(t)
, (10)
where T denotes the chronological ordering. Here the
physical Hamiltonian Hphys(t) and its operator realiza-
tion Ĥphys(t) are nonvanishing only in unitary gauges ex-
plicitly depending on time [12], ∂tχ
µ(q, p, t) 6= 0. In static
gauges, ∂tχ
µ = 0, they identically vanish, because in spa-
tially closed cosmology the full Hamiltonian reduces to
the combination of constraints.
The path integral (10) is gauge-independent on-shell
[10, 11] and coincides with that in the static gauge.
Therefore, from Eqs.(9)-(10) with Ĥphys = 0, the sta-
tistical sum of our microcanonical ensemble equals
e−Γ = Trphys Iphys =
dqphys dpphys
= sum over Everything. (11)
This ultimate equipartition, not modulated by any non-
trivial density of states, is a result of general covariance
and closed nature of the Universe lacking any freely speci-
fiable constants of motion. The volume integral of entire
physical phase space, whose structure and topology is
not known, is very nontrivial. However, below we show
that semiclassically it is determined by EQG methods
and supported by instantons of [7] spanning a bounded
range of the cosmological constant.
Integration over momenta in (9) yields a Lagrangian
path integral with a relevant measure and action
e−Γ =
D[ q,N ] eiSL[ q, N ]. (12)
Integration runs over periodic fields (not indicated ex-
plicitly but assumed everywhere below) even despite the
Lorentzian signature of the underlying spacetime. Sim-
ilarly to the procedure of [7, 8] leading to (1)-(2) in
the Euclidean case, we decompose [ q,N ] into a min-
isuperspace [ aL(t), NL(t) ] and the “matter” φL(x) vari-
ables, the subscript L indicating their Lorentzian na-
ture. With a relevant decomposition of the measure
D[ q,N ] = D[ aL, NL ]×DφL(x), the microcanonical sum
takes the form
e−Γ =
D[ aL, NL ] e
iΓL[ aL, NL ], (13)
eiΓL[ aL, NL ] =
DφL(x) e
iSL[ aL, NL;φL(x)], (14)
where ΓL[ aL, NL ] is a Lorentzian effective action. The
stationary point of (13) is a solution of the effective equa-
tion δΓL/δNL(t) = 0. In the gauge NL = 1 it reads as
a Lorentzian version of the Euclidean equation (4) and
the bootstrap equation for the radiation constant C with
the Wick rotated τ = it, a(τ) = aL(t), η = i
dt/aL(t).
However, with these identifications C turns out to be
purely imaginary (in view of the complex nature of the
free energy F (i
dt/aL)). Therefore, no periodic solu-
tions exist in spacetime with a real Lorentzian metric.
On the contrary, such solutions exist in the Euclidean
spacetime. Alternatively, the latter can be obtained with
the time variable unchanged t = τ , aL(t) = a(τ), but
with the Wick rotated lapse function
NL = −iN, iSL[ aL, NL;φL] = −SE[ a,N ;φ ]. (15)
In the gauge N = 1 (NL = −i) these solutions exactly
coincide with the instantons of [7]. The corresponding
saddle points of (13) can be attained by deforming the
integration contour (6) of NL into the complex plane to
pass through the point NL = −i and relabeling the real
Lorentzian t with the Euclidean τ . In terms of the Eu-
clidean N(τ), a(τ) and φ(x) the integrals (13) and (14)
take the form of the path integrals (1) and (2) in EQG,
iΓL[ aL, NL] = −ΓE [ a, N ]. (16)
However, the integration contour for the Euclidean N(τ)
runs from −i∞ to +i∞ through the saddle point N = 1.
This is the source of the conformal rotation in Euclidean
quantum gravity, which is called to resolve the problem of
unboundedness of the gravitational action and effectively
renders the instantons a thermal nature, even though
they originate from the microcanonical ensemble. This
mechanism implements the justification of EQG from
canonical quantization of gravity [14] (see also [15] in
black hole context).
To show this we calculate (1) in the one-loop approx-
imation with the measure inherited from the canonical
path integral (5) D[ a,N ] = DaDN µ[ a,N ] δ[χ ] DetQ.
Here µ[ a,N ] is a local measure determined by the La-
grangian NL(a, a′), (3), in the local part of ΓE [ a,N ],
µ1−loop =
∂2(NL)
∂ȧ ∂ȧ
N a2a′2
D = a a′2(a2 −B +B a′2). (17)
The Faddeev-Popov factor δ[χ ] DetQ contains a gauge
condition χ = χ(a,N) fixing the one-dimensional dif-
feomorphism, τ → τ̄ = τ − f/N , which for infinitesi-
mal f = f(τ) has the form ∆fN ≡ N̄(τ) − N(τ) = ḟ ,
∆fa ≡ ā(τ) − a(τ) = ȧ f/N , and Q = Q(d/dτ) is a
ghost operator determined by the gauge transformation
of χ(a,N), ∆fχ = Q(d/dτ) f(τ).
The conformal mode σ of the perturbations δa = σa0
and δN = σN0 on the saddle-point background (labeled
below by zero, a = a0 + δa, N = N0 + δN) origi-
nates from imposing the background gauge χ(a,N) =
δN − (N0/a0) δa. In this gauge Q = a(d/dτ)a−1, and
the quadratic part of ΓE takes the form [13]
δ2σΓE = −
3πm2P
, (18)
where D is given by (17). As is known from [7] for the
background instantons a20(τ) ≥ a2− > B (a− is the turn-
ing point with the smallest value of a0(τ)), so that D > 0
everywhere except the turning points where D degener-
ates to zero. Therefore δ2σΓE < 0 for real σ, but the
Gaussian integration runs along the imaginary axes and
yields the functional determinant of the positive operator
— the kernel of the quadratic form (18)
e−Γ1−loop = e−Γ0 DetQ0
D/a′2
= e−Γ0×Det
)]−1/2
In view of periodic boundary conditions for both oper-
ators their determinants cancel each other (their zero
modes to be eliminated because they correspond to the
conformal Killing symmetry of FRW instantons) [13].
Therefore, the contribution of the conformal mode re-
duces to the selection of instantons with a fixed time pe-
riod, effectively endowing them with a thermal nature.
As suggested in [7, 8, 16] these instantons serve as
initial conditions for inflationary universes which evolve
according to the Lorentzian version of Eq.(4) and, at late
stages, have two branches of a cosmological acceleration
with Hubble scales H2
= (m2P /B)(1±(1−2BH2)1/2). If
the initial Λ = 3H2 is a composite inflaton field decaying
at the end of inflation, then one of the branches under-
goes acceleration with H2+ = 2m
P/B. This is determined
by the new quantum gravity scale suggested in [8] – the
upper bound of the instanton Λ-range, Λmax = 3m
P /B.
To match the model with inflation and the dark energy
phenomenon, one needs B of the order of the inflation
scale in the very early Universe and B ∼ 10120 now, so
that this parameter should effectively be a growing func-
tion of time.
This picture seems to fit into string theory at its low-
energy field-theoretic level. Then, with a bounded range
of Λ, it might constrain the landscape of string vacua
[7, 8]. Moreover, string theory has a qualitative mecha-
nism to promote the constant B to the level of a mod-
uli variable indefinitely growing with the evolving size
R(t) of extra dimensions. Indeed B as a coefficient in
the overall conformal anomaly of 4-dimensional quantum
fields basically counts their number N , B ∼ N . In the
Kaluza-Klein (KK) theory and string theory the effective
4-dimensional fields arise as KK and winding modes with
the masses [17]
m2n,w =
R2 (19)
(enumerated by the KK and winding numbers), which
break their conformal symmetry. These modes remain
approximately conformally invariant as long as their
masses are much smaller than the spacetime curvature,
m2n,w ≪ H2+ ∼ m2P /N . Therefore the number of confor-
mally invariant modes changes with R. Simple estimates
show that for pure KK modes (w = 0, n ≤ N) their num-
ber grows with R as N ∼ (mPR)2/3, whereas for pure
winding modes (n = 0, w ≤ N) their number grows with
the decreasing R as N ∼ (mPα′/R)2/3. Thus, the effect
of indefinitely growing B is possible for both scenarios
with expanding or contracting extra dimensions. In the
first case this is the growing tower of superhorizon KK
modes which make the horizon scale H+ = mP
2/B ∼
mP /(mPR)
1/3 indefinitely decreasing with R → ∞. In
the second case this is the tower of superhorizon winding
modes which make this acceleration scale decrease with
the decreasing R as H+ ∼ mP (R/mPα′)1/3. This effect
is flexible enough to accommodate the present day ac-
celeration scale H0 ∼ 10−60mP (though, by the price of
fine-tuning an enormous coefficient of expansion or con-
traction of R). This gives a new dark energy mechanism
driven by the conformal anomaly and transcending the
inflationary and matter-domination stages starting with
the state of the microcanonical distribution.
To summarize, within a minimum set of assumptions
(the equipartition in the physical phase space (11)),
we seem to have the mechanism of constraining the
landscape of string vacua and get the full evolution of
the Universe as a quasi-equilibrium decay of its initial
microcanonical state. Thus, contrary to anticipations
of Sidney Coleman, “there is Nothing rather than
Something” [3], one can say that Something (rather
than Nothing) comes from Everything.
The author thanks O.Andreev, C.Deffayet, A.Kamen-
shchik, J.Khoury, H.Tye, A.Tseytlin, I.Tyutin and
B.Voronov for thought provoking discussions and espe-
cially Andrei Linde, this work having appeared as an un-
intended response to his discontent with EQG initial con-
ditions. The work was supported by the RFBR grant 05-
02-17661, the grant LSS 4401.2006.2 and SFB 375 grant
at the Ludwig-Maximilians University in Munich.
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