I think this pretty well settles the origin question . . .
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No Big Bang? Quantum equation predicts universe has no beginning
- Thread starter Weird
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Don't complain about how long it is just read it lol.
A quantum state of the universe is specified by a wave function Ψ on the superspace of three-geometries and matter field configurations on a spacelike boundary surface Σ which we take to be closed. We consider matter models consisting of a single scalar field φ and a vector field Aμ . Schematically we write Ψ[h, χ, B⃗ ] where h stands for the metric hij (⃗x) representing theboundarythree-geometry,andχ(⃗x)andB⃗(⃗x)aretheboundaryconfigurations2 ofφand Aμ. We assume the no-boundary wave function (NBWF) as a model of the quantum state [1]. This is given by a sum over regular four-geometries gμν and matter fields φ and Aμ on a four-disk M that match (h,χ,B⃗) on Σ. In the semiclassical approximation different configurations labeled by (h,χ,B⃗) are weighted by exp(−I/h ̄) where I is the Euclidean action of a regular, complex extremal solution that matches the prescribed data on Σ.
A wave function of the universe predicts that configurations evolve classically when, with an appropriate coarse-graining, its phase varies rapidly compared to its amplitude [2]. To leading order in h ̄ both the phase and the amplitude are given by the action of the dominant saddle point. In this semiclassical approximation,
Ψ[h, χ, B⃗ ] ≈ exp{(−I[h, χ, B⃗ ]/h ̄} = exp{(−IR[h, χ, B⃗ ] + iS[h, χ, B⃗ ])/h ̄}, (2.1)
where IR [h, χ, B⃗ ] and −S [h, χ, B⃗ ] are the real and imaginary parts of the Euclidean action, evaluated at the saddle point. The probabilities of different classical configurations are pro- portional to exp(−2IR/h ̄). They are conserved along a classical trajectory as a consequence of the Wheeler-DeWitt equation [2] and therefore yield a probability measure on the phase space of classical histories.
We work in a four-dimensional bulk and take the Euclidean action I[g(x),φ(x),A(x)] to be a sum of the Einstein-Hilbert action IEH (in Planck units where h ̄ = c = G = 1)
1��4 1/2 1��3 1/2 IEH[g]=−16π d x(g) (R−2Λ)−8π d x(h) K
M ∂M and the following matter action IM
1��4 1/2 2 1��4 1/2μν IM[g,φ,A]=2 dx(g) [(∇φ) +2V(φ)]+4 dx(g) F Fμν
(2.2)
(2.3)
MM
where Fμν = ∂μAν −∂νAμ. The generalization to include a coupling between the scalar and
vector fields does not affect our results so we neglect this here for simplicity.
We take the cosmological constant Λ = 3H2 and the scalar potential V in the action (2.2)- (2.3) to be positive. This means that for positive signature metrics this is the Euclidean action of de Sitter gravity coupled to a scalar with a positive potential V and a vector field Aμ. For negative signature metrics this is minus the action of anti-de Sitter gravity coupled to a scalar with a negative potential −V and a vector field A ̃μ ≡ iAμ that is imaginary from
a de Sitter viewpoint3.
A global notion of signature is meaningless for the complex saddle point solutions that
specify the semiclassical wave function (2.1). This is because the signature changes across the saddle point geometry. Moreover the signature in the saddle point interior differs de- pending on its representation. In scalar field models an analysis of the different saddle point representations led to a useful connection, described above, between (asymptotic) Lorentzian dS histories and Euclidean AdS geometries [6]. We now generalize this to models including vector matter, given by the action (2.2)-(2.3).
The line element of a complex saddle point that specifies the amplitude of a general boundary configuration can be written as
ds2 =N2(λ)dλ2 +gij(λ,⃗x)dxidxj (2.4)
where (λ,xi) are four real coordinates on the real manifold M. Conventionally we take λ to be a ‘radial’ coordinate so that λ = 0 locates the origin or ‘South Pole’ (SP) of the saddle point and take λ = 1 to locate the boundary Σ of M. Saddle points may be represented by complex metrics – complex N and gij – but the coordinates (λ,xi) are always real.
Thefieldequationsderivedfrom(2.2)-(2.3)canbesolvedforgij(λ,⃗x),φ(λ,⃗x)andAμ(λ,⃗x)
for any complex N(λ) that is specified. Different choices of N(λ) therefore give different
representations of the same saddle point. A convenient way to exhibit these different repre-
sentations is to introduce the function τ(λ) defined by ��λ
dλ′N(λ′). (2.5)
Different choices of N(λ) correspond to different contours in the complex τ-plane. Contours startfromtheSPatλ=τ =0andendattheboundaryλ=1withτ(1)≡υ. Conversely, for any contour τ(λ) there is an N(λ) ≡ dτ(λ)/dλ. Each contour connecting τ = 0 to τ = υ is therefore a different representation of the same complex saddle point with the same action.
The semiclassical approximation to the wave function holds in the asymptotic ‘large volume’ domain where the cosmological constant dominates the dynamics. Asymptotically, a semiclassical wave function specifies an ensemble of asymptotic solutions to the equations of motion. In terms of the complex time variable
τ(λ) ≡
u ≡ eiHτ = e−Hy+iHx.
the asymptotic (small u) form of the general solution is given by
c2 ̃ ̃(2) 2 ̃(−) 2λ− ̃(3) 3 gij(u,⃗x)= u2[hij(⃗x)+hij (⃗x)u +hij (⃗x)u +hij (⃗x)u +···].
φ(u, ⃗x) = uλ− (α(⃗x) + α1(⃗x)u + · · · ) + uλ+ (β(⃗x) + β1(⃗x)u + · · · ).
Ai(u, ⃗x) = Bi(⃗x) + uCi(⃗x) + · · ·
(2.6)
(2.7a)
(2.7b) (2.7c)
0
where λ± ≡ 3 [1 ± ��1 − (2m/3)2 ], and h ̃ ij (⃗x) is real and normalized to have unit volume
2
thus determining the constant c.
The asymptotic solutions are locally determined from the asymptotic equations in terms
of the arbitrary ‘boundary values’ c2h ̃ij, α(⃗x) and Bi(⃗x), up to the u3 term in (2.7a), to order uλ+ in (2.7b) and to leading order in (2.7c). Beyond these terms in (2.7) the interior dynamics and the specific choice of quantum state become important.
For a saddle point solution to contribute to the semiclassical NBWF the observables must take real values on Σ [18, 19]. Hence the metric h, the scalar χ and the frame fields – the components of the vector field in an orthonormal frame – must all be real at τ = υ. The frame fields are given by B⃗ /b, where the ‘scale factor’ b ≡ c/u, multiplied by a real matrix specified by h ̃.
Hy Hy
a
Hy
2H
SP Hxr Hx HxaSP Hxr Hx SP Hx
FIG. 1: Left panel: The contour CA in the complex τ-plane of a de Sitter representation of a complex saddle point of the NBWF. Along the horizontal part the geometry is half a deformed Euclidean three-sphere. Along the vertical part it is asymptotically Lorentzian de Sitter space. Middle panel: The contour CB that gives the AdS representation of the same complex saddle point. The vertical part is asymptotically Euclidean AdS with a complex matter profile. The horizontal part is a complex geometry that interpolates between asymptotic AdS and de Sitter.
Right panel: The contour CC that provides the standard AdS representation of real asymptotically Euclidean AdS saddle points that are not associated with classical histories.
Reality of the observables on the boundary selects two classes of saddle points, corre- sponding respectively to asymptotically Euclidean AdS and Lorentzian de Sitter histories [18]. The saddle points associated with de Sitter histories are found by tuning the phases of the fields so that b, φ and Aμ all tend to real functions along a vertical line x = xr in the complex τ-plane [2, 6]. Writing c = |c|eiθc, α = |α|eiθα and Bi = |B|eiθB the expansions (2.7a) - (2.7c) imply
θc =xr, θα =−λ−xr, θB =0 (2.8)
By choosing a particular contour connecting τ = 0 to τ = υ = xr + iyυ we obtain a concrete representation of the saddle point geometry. The contour CA in Fig 1(a) provides an example. Along the horizontal part of this the geometry is the Euclidean geometry of half a deformed four-sphere. Along the vertical part from (xr,0) to (xr,yυ) the geometry tends to a Lorentzian space that is locally de Sitter,
ds2 ≈ −dy2 + 1 e2Hyh ̃ij(⃗x)dxidxj. (2.9) 4H2
Thus the contour CA gives the familiar representation of the no-boundary saddle points 7
that are associated with asymptotically real, Lorentzian, de Sitter histories. The ensemble of histories of this kind predicted by the NBWF is known explicitly in a homogeneous and isotropic minisuperspace approximation with linear perturbations, where it consists of universes with an early period of scalar field driven inflation [2]. The saddle point action acquires a rapidly varying phase factor along the vertical part of the contour CA so the wave function takes a WKB form (2.1) and predicts classical cosmological evolution.
There is, however, an alternative representation of the same class of saddle points given bythecontourCB inFig1(b). Alongthex=xa =xr −π/2H lineonehas
ds2 ≈ −dy2 − 1 e2Hyh ̃ij(⃗x)dxidxj. (2.10) 4H2
This is negative signature, Euclidean, asymptotically local AdS space. The asymptotic form of the matter fields along x = xa follows from (2.7b) –(2.7c) and is given by
φ(y,⃗x) ≈ |α(⃗x)|e−iλ−π/2e−λ−y ≡ α ̃e−λ−y, Ai(y,⃗x) ≈ Bi(⃗x) ≡ −iB ̃i (2.11)
where B ̃i is the boundary value of A ̃μ ≡ iAμ. Therefore the condition that the scalar be real at the endpoint υ, at x = xr, means it is complex along the AdS branch of the contour, with an asymptotic phase given by (2.11). Similarly, the requirement that Ai be real at the endpoint υ means that the usual AdS vector field A ̃i = iAi is purely imaginary in the asymptotic AdS region of de Sitter saddle points4.
To summarize, in the ‘AdS representation’ of Fig 1(b) the saddle point geometry consists of a Euclidean, asymptotically locally AdS geometry with complex matter field profiles in the ‘radial’ direction y. This is then joined smoothly onto the Lorentzian, asymptotically de Sitter geometry through a transition region – corresponding to the horizontal part of the contour CB – where the geometry is complex. The complex structure of the semiclassical NBWF thus provides a natural connection5 between (asymptotically) Lorentzian de Sitter histories and Euclidean AdS geometries.
As mentioned earlier there is a second set of saddle points with real observables on the boundary [18]. These are the real AdS domain wall solutions. They are found by taking φ
and A ̃i real at the SP, and by taking b2 to be real and negative for real u. A convenient representation of this class is provided by a contour CC that runs along the imaginary axis in the complex τ-plane to an endpoint υ (cf Fig 1(c)). As we will see below these saddle points do not describe classical universes and are of little interest in cosmology. Instead they are associated with Euclidean asymptotic AdS configurations.
The saddle point action is given by an integral over time along a contour in the complex τ-plane connecting the SP to the endpoint υ. The result is independent of the choice of contour. We first consider the action I1 of the first class of saddle points associated with Lorentzian histories. The asymptotic contribution to the saddle point action coming from the integral along the vertical branch of the contour CA is purely imaginary. This is because the integrand is real and dτ = idy. The real part IR, which governs the probabilities of the de Sitter histories, tends to a constant along the x = xr while the imaginary part S(υ) ∝ e3Hyυ .
The AdS representation based on the contour CB provides a different way to calculate IR. To see this we first consider the action integral Ih along the horizontal branch of the contour in Fig 1(b). Using the Hamiltonian constraint this can be written as,
1��xr��31��23 ⃗2 ij�� Ih(υa,υ)=8π dx dxg2 6H − R+8πV(φ)+4π(∇φ) +4πF Fij (3.1)
xa
where 3R is the scalar three curvature of gij. The expansions (2.7a) - (2.7c) imply that
the leading contribution to Ih from the vector term is O(u) and therefore negligible in the asymptotic limit. Further, the contribution to the asymptotically finite part of the action from the scalar and gravitational terms in (3.1) vanishes as a consequence of the asymptotic Einstein equations [6]. Hence the asymptotically constant term is the same on both ends of the horizontal branch. This means the tree level probabilities of different dS histories can be obtained from the asymptotically AdS region of the saddle points.
The action integral Ia along the vertical x = xa branch in the AdS representation exhibits the usual volume divergences, but Ih regulates these [6]. Specifically we find [6]
Ih(υa, υ) = −Sst(υa) + Sst(υ) + O(e−Hyυ ) (3.2) where Sst are the universal gravitational and scalar AdS counterterms, which arise here as
surface contributions to the action which are kept [6]. The second term in (3.2) gives auniversal phase factor (since the signature at υ differs from that at υa), and the first term regulates the divergences of Ia. At large υa
Ia(υa) − Sst(υa) → −Ireg (3.3) aAdS
where −Ireg [h ̃ij,α ̃,B ̃i] is the regulated asymptotic AdS action. It is a function of he aAdS
asymptotic profiles of the geometry and matter fields in the AdS regime of the saddle points, which are locally given by the argument of the wave function at υ through eq. (2.11).
Thus we find that in the large volume limit, the action of a general, inhomogeneous saddle point of the NBWF associated with an asymptotically local dS history can be expressed in terms of the regulated action of a complex AdS domain wall and a sum of purely imaginary, universal surface terms,
I1[h,χ,Bi] = −Ireg [h ̃ij(⃗x),α ̃(⃗x),B ̃i(⃗x)]+Sst(υ)+O(e−Hyυ) (3.4) aAdS
The probabilities of the asymptotically de Sitter histories are governed by the real part of I1 and hence given by
Re[I1(υ)] ≡ IR(υ) = −Re[Ireg ] . (3.5) aAdS
It is straightforward to evaluate the action I2 of the second class of real, Euclidean asymptotically AdS saddle points. This is simply given by the action integral Ia along the vertical line from the SP to the endpoint υ in Fig 1(c). Since all fields are everywhere real from an AdS viewpoint, the resulting action is real. It is given by
I2[h,χ,B ̃i] = −Ireg [h ̃ij(⃗x),α ̃(⃗x),B ̃i(⃗x)]+Sst(υ) (3.6) aAdS
where the boundary values h ̃ij(⃗x), B ̃i and α ̃ are all real. The leading surface term in Sst is real and grows as the volume of AdS. Hence the amplitude of asymptotically AdS configurations is low relative to the classical, asymptotically de Sitter histories discussed above.
The representation of both classes of saddle points in terms of Euclidean AdS ‘domain wall’ geometries with real or complex matter profiles leads to a holographic formulation of the NBWF by applying the Euclidean version of AdS/CFT. This relates the asymptoticAdS factor exp(−Ireg /h ̄) in the wave function to the partition function of a Euclidean dual aAdS
field theory in three dimensions [15, 16],
reg ̃ ̃ ̃ ̃��3�� ̃�� ̃i��
exp(−IaAdS[hij,α ̃,Bi]/h ̄)=ZQFT[hij,α ̃,Bi]=⟨exp d x h α ̃O+BiJ ⟩QFT (4.1)
The dual QFT is defined on the conformal boundary represented here by the three-metric h ̃ij. The operator O is constructed from scalars in the boundary theory and J⃗ is a current. The brackets ⟨· · · ⟩ on the right hand side denote the functional integral average involving the boundary field theory action minimally coupled to the metric conformal structure rep- resented by h ̃ij. For the spherical domain walls corresponding to homogeneous histories this is the round three-sphere, but in general α ̃,B ̃i and h ̃ij are functions of the boundary coordinates ⃗x. The three-divergence of the current J⃗ vanishes ensuring the invariance of the right hand sided of (4.1) under gauge transformations of B ̃i matching the invariance of the AdS action on the left [15]. Applying (4.1) to (3.4) and (3.6) yields
Ψ(υ) = 1 exp(−Sst(υ)/h ̄) (4.2) Zε [h ̃ij , α ̃, B ̃i]
where ε ∼ 1/|Hb| is a UV cutoff [6]. The surface terms are imaginary for the complex saddle points associated with Lorentzian histories, and real for the saddle points corresponding to Euclidean AdS configurations. The action Sct is independent of the vector field. This together with the gauge invariance of ZQFT ensures that the wave function satisfies the ∇⃗ · E⃗ = 0 constraint of Maxwell theory.
Equation (4.2) is an example of a semiclassical dS/CFT duality. The arguments of the wave function enter as external sources in the dual partition function that turn on deforma- tions (except for the scale factor b = c/u which enters as a UV cutoff). The dependence of the partition function on the values of the external sources yields a holographic no-boundary measure on the space of asymptotic configurations. For boundary configurations given by sufficiently small values of the matter sources and sufficiently mild deformations of the round three sphere one expects the integral defining the partition function to converge, yielding a non-zero amplitude. The holographic form (4.2) of the NBWF thus involves a range of different deformations of a single underlying CFT. The sources α ̃ and B ̃μ of the deformations are real for saddle points associated with asymptotically AdS configurations. By contrast the sources are complex for saddle points associated with Lorentzian, classical, asymptoti- cally de Sitter histories. In the latter regime of superspace the dual form of the NBWF thus involves complex deformations of a Euclidean CFT.
QFT
As an illustration we compute the asymptotic NBWF of vector perturbations in a ho- mogeneous and isotropic, expanding background [20]. Here we have in mind that the scalar field and Λ are responsible for the background evolution. The details of this do not mat- ter, because the vector is conformally invariant and hence decouples to quadratic order. We evaluate the semiclassical vector wave function first using the dS representation of the background saddle point and then using its AdS representation.
The line element of a homogeneous and isotropic saddle point can be written as
ds2 = N2(λ)dλ2 + a2(λ)γijdxidxj (5.1)
where γij is the metric on a unit round three sphere. Homogeneous and isotropic minisuper- space is spanned by the boundary value b of the scale factor and the value χ of the scalar field. Neglecting the back reaction of the Maxwell field on the background, we write
Ψ(b, χ, B⃗ ) = Ψ0(b, χ)ψV (b, χ, B⃗ ), (5.2)
where Ψ0(b, χ) is a background wave function given by the action of a saddle point solution (a(τ),φ(τ)) that matches (b,χ) at the boundary of the disk, and is regular elsewhere [2, 3]. We assume the phase of Ψ0 varies rapidly so that it predicts a classical background. The wave function ψV of vector perturbations is defined by the remaining integral over Aμ,
C
We evaluate (5.3) in the steepest descents approximation. The Euclidean action I(2) of V
vector perturbations around saddle points of the form (5.1) is given by
(2) ��√��a��ij k k��Nk i i��
IV = γ 2N γ Ai,λAj,λ +A0(2A ;k0 −A0;k + 2aA (2Ak +A ;ik −Ak;i ) (5.4)
Working in Coulomb gauge A0 = ∇iAi = 0 and in terms of conformal time η defined as adη = Ndλ, the solutions take the following form,
⃗�� (2)
ψV (b,χ,B) ≡
where the integral is over all regular histories on a disk which match B⃗ on its boundary.
δAμ exp(−IV [a(τ),φ(τ),Aμ(τ,⃗x)]/h ̄). (5.3)
Ai(η, Ω) =
�� �� (p)��n
nlmp
lm
fnlmp(η) Si
(5.5)
�� (p)��n
lm
where B
are the mode coefficients of the boundary configuration B⃗ and iθυ is the imaginary part of η at the boundary surface6. The asymptotic expansion of (5.6) in terms of the complex time variable u defined in (2.6) is of the form (2.7c),
Ai = �� B ��1 − 2ine−ixr u + O(u2)�� S (5.7) i
where we have used that η = iθυ − 2ie−ixr u + O(u2) in the large volume regime. Evaluated
where Si
sphere, with ∆Si = −(n2 − 2)Si. From here onwards we denote the indices collectively by. The solutions for f are proportional to e±nη. The no-boundary condition of regularity at the SP selects the solution that decays as η → −∞. Hence we get
Ai(η, Ω) = �� Be−inθυ enηS (5.6) i
are the transverse eigenfunctions of the vector Laplacian ∆ on the three-
on solutions the vector action (5.4) reduces to a surface term, (2) 1��3 1/2ij
(5.8)
(5.9)
Substituting (5.6) we get
IV = 2 d x(γ) γ Ai,ηAj I(2) = �� nB2
V
2
This is real and positive, indicating that the vector perturbations are in their quantum
mechanical ground state.
We could equally well have computed the vector wave function using the AdS represen-
tation of the saddle point. The saddle point action of a vector field A ̃μ with boundary value B ̃i in a homogeneous and isotropic, Euclidean, asymptotically AdS background is obtained in a similar manner and given by
I(2) = �� nB ̃2 (5.10)
Eq. (3.5) implies this gives the wave function of vector perturbations in a classical, homo- geneous and isotropic, asymptotically de Sitter background when B ̃i is taken to be purely imaginary. Substituting B ̃i = iBi in (5.10) and using (3.5) indeed yields (5.9).
V,aAdS
2
We have generalized the holographic formulation of the NBWF to models including vector matter. We first derived an ‘AdS representation’ of the NBWF saddle points associated with classical, asymptotically dS universes with scalar and vector matter. In this representation, the saddle point geometry consists of a Euclidean AdS domain wall which joins smoothly onto an asymptotically real, Lorentzian, de Sitter geometry through a transition region where the spatial metric changes signature. The scalar and vector matter profiles along the AdS domain wall are complex. The relative probabilities of different classical boundary configurations (h, χ, B⃗ ) are given by the regularized AdS domain wall action.
Applying Euclidean AdS/CFT then yields a holographic form of the semiclassical NBWF
in terms of the partition function of a dual, Euclidean CFT with complex sources specified
by the asymptotic scalar and vector boundary values in the AdS regime of the saddle points.
The dual description of the no-boundary measure on classical, Lorentzian configurations thus
involves a generalization of Euclidean AdS/CFT to CFTs with complex deformations. The
phases of the sources are independent of the specific quantum state and of the dynamics
in the interior. They are fully specified by the requirement that the fields be real at the
final de Sitter boundary. The phase of the scalar source in the dual is e−iλ−π/2, where
λ− = 3 [1 − ��1 − (2m/3)2 ], and the vector field source is purely imaginary7 . 2
From a holographic viewpoint it is natural to consider the wave function on an extended domain that includes real deformations of the dual CFT [18, 19]. Partition functions with real deformations specify the amplitude of real, asymptotically AdS boundary configurations. In the bulk their amplitude is given by real Euclidean AdS domain walls. These are of little relevance in cosmology, since the lack of a rapidly varying phase factor in the action means they are not associated with classical histories. In fact the AdS volume contribution to the action heavily suppresses their contribution to the NBWF. Nevertheless, the inclusion of both sets of saddle points in the configuration space - as suggested by holography - yields an appealing, signature-invariant formulation of the wave function defined on an extended configuration space with three-metrics of both signatures [18]. One signature corresponds to deformations with real sources and the other to complex sources.
From a bulk viewpoint the relation between (asymptotic) Euclidean AdS and Lorentzian de Sitter exhibited here is reminiscent of a ‘symmetry’ of the action (2.2)-(2.3) under a reversal of the signature of the metric [17]. A signature reversal relates the action of a dS theory of gravity coupled to a scalar with a positive potential V and a vector field Aμ to the action of an AdS theory of gravity coupled to a scalar with potential −V and a vector field A ̃μ = iAμ that is imaginary from a dS viewpoint. This is not unlike the transition from supersymmetry in AdS to pseudo-supersymmetry in dS, where the vectors in the de Sitter theory exhibit a similar behavior. In that context as well it has been argued that the dS and AdS theories should really be viewed as two real domains of a single underlying complexified theory [21, 22]. The complex structure of the wave function in our scheme provides a natural setup for a unified description of this kind. This is made explicit in its holographic form which involves complex deformations of a single underlying dual conformal field theory.
Our analysis so far applies only to toy models of de Sitter gravity or, in AdS terms, to consistent truncations of AdS supergravity theories that retain only low mass scalars, and vectors. A full realization of holographic cosmology in string theory remains an open prob- lem. A particularly intriguing issue has to do with the tower of irrelevant operators featuring in the usual AdS/CFT duals. These correspond to tachyonic fields in the de Sitter domain of the theory. The requirement of an asymptotic de Sitter structure in dS/CFT amounts to a final boundary condition on these fields. More generally the asymptotic dS structure implies that the asymptotic geometry and fields behave classically [18]. The no-boundary condition of regularity on the saddle points implements the final boundary condition on tachyons since it excludes classical histories in which these are not set to zero [2, 24].
The dual partition function is not concerned with the classical evolution itself - which is governed by the phase factor of the wave function (2.1) - but merely calculates the probability measure on classical phase space in the no-boundary state [6, 16]. Given that a probabilistic interpretation of the wave function of the universe is restricted to its classical domain anyway [2] this suggests dS/CFT should be viewed as a statement about the coarse-grained, classical predictions of the wave function only8. But it would be very interesting to understand the origin and nature of the late-time classicality constraint from a dual viewpoint.
A quantum state of the universe is specified by a wave function Ψ on the superspace of three-geometries and matter field configurations on a spacelike boundary surface Σ which we take to be closed. We consider matter models consisting of a single scalar field φ and a vector field Aμ . Schematically we write Ψ[h, χ, B⃗ ] where h stands for the metric hij (⃗x) representing theboundarythree-geometry,andχ(⃗x)andB⃗(⃗x)aretheboundaryconfigurations2 ofφand Aμ. We assume the no-boundary wave function (NBWF) as a model of the quantum state [1]. This is given by a sum over regular four-geometries gμν and matter fields φ and Aμ on a four-disk M that match (h,χ,B⃗) on Σ. In the semiclassical approximation different configurations labeled by (h,χ,B⃗) are weighted by exp(−I/h ̄) where I is the Euclidean action of a regular, complex extremal solution that matches the prescribed data on Σ.
A wave function of the universe predicts that configurations evolve classically when, with an appropriate coarse-graining, its phase varies rapidly compared to its amplitude [2]. To leading order in h ̄ both the phase and the amplitude are given by the action of the dominant saddle point. In this semiclassical approximation,
Ψ[h, χ, B⃗ ] ≈ exp{(−I[h, χ, B⃗ ]/h ̄} = exp{(−IR[h, χ, B⃗ ] + iS[h, χ, B⃗ ])/h ̄}, (2.1)
where IR [h, χ, B⃗ ] and −S [h, χ, B⃗ ] are the real and imaginary parts of the Euclidean action, evaluated at the saddle point. The probabilities of different classical configurations are pro- portional to exp(−2IR/h ̄). They are conserved along a classical trajectory as a consequence of the Wheeler-DeWitt equation [2] and therefore yield a probability measure on the phase space of classical histories.
We work in a four-dimensional bulk and take the Euclidean action I[g(x),φ(x),A(x)] to be a sum of the Einstein-Hilbert action IEH (in Planck units where h ̄ = c = G = 1)
1��4 1/2 1��3 1/2 IEH[g]=−16π d x(g) (R−2Λ)−8π d x(h) K
M ∂M and the following matter action IM
1��4 1/2 2 1��4 1/2μν IM[g,φ,A]=2 dx(g) [(∇φ) +2V(φ)]+4 dx(g) F Fμν
(2.2)
(2.3)
MM
where Fμν = ∂μAν −∂νAμ. The generalization to include a coupling between the scalar and
vector fields does not affect our results so we neglect this here for simplicity.
We take the cosmological constant Λ = 3H2 and the scalar potential V in the action (2.2)- (2.3) to be positive. This means that for positive signature metrics this is the Euclidean action of de Sitter gravity coupled to a scalar with a positive potential V and a vector field Aμ. For negative signature metrics this is minus the action of anti-de Sitter gravity coupled to a scalar with a negative potential −V and a vector field A ̃μ ≡ iAμ that is imaginary from
a de Sitter viewpoint3.
A global notion of signature is meaningless for the complex saddle point solutions that
specify the semiclassical wave function (2.1). This is because the signature changes across the saddle point geometry. Moreover the signature in the saddle point interior differs de- pending on its representation. In scalar field models an analysis of the different saddle point representations led to a useful connection, described above, between (asymptotic) Lorentzian dS histories and Euclidean AdS geometries [6]. We now generalize this to models including vector matter, given by the action (2.2)-(2.3).
The line element of a complex saddle point that specifies the amplitude of a general boundary configuration can be written as
ds2 =N2(λ)dλ2 +gij(λ,⃗x)dxidxj (2.4)
where (λ,xi) are four real coordinates on the real manifold M. Conventionally we take λ to be a ‘radial’ coordinate so that λ = 0 locates the origin or ‘South Pole’ (SP) of the saddle point and take λ = 1 to locate the boundary Σ of M. Saddle points may be represented by complex metrics – complex N and gij – but the coordinates (λ,xi) are always real.
Thefieldequationsderivedfrom(2.2)-(2.3)canbesolvedforgij(λ,⃗x),φ(λ,⃗x)andAμ(λ,⃗x)
for any complex N(λ) that is specified. Different choices of N(λ) therefore give different
representations of the same saddle point. A convenient way to exhibit these different repre-
sentations is to introduce the function τ(λ) defined by ��λ
dλ′N(λ′). (2.5)
Different choices of N(λ) correspond to different contours in the complex τ-plane. Contours startfromtheSPatλ=τ =0andendattheboundaryλ=1withτ(1)≡υ. Conversely, for any contour τ(λ) there is an N(λ) ≡ dτ(λ)/dλ. Each contour connecting τ = 0 to τ = υ is therefore a different representation of the same complex saddle point with the same action.
The semiclassical approximation to the wave function holds in the asymptotic ‘large volume’ domain where the cosmological constant dominates the dynamics. Asymptotically, a semiclassical wave function specifies an ensemble of asymptotic solutions to the equations of motion. In terms of the complex time variable
τ(λ) ≡
u ≡ eiHτ = e−Hy+iHx.
the asymptotic (small u) form of the general solution is given by
c2 ̃ ̃(2) 2 ̃(−) 2λ− ̃(3) 3 gij(u,⃗x)= u2[hij(⃗x)+hij (⃗x)u +hij (⃗x)u +hij (⃗x)u +···].
φ(u, ⃗x) = uλ− (α(⃗x) + α1(⃗x)u + · · · ) + uλ+ (β(⃗x) + β1(⃗x)u + · · · ).
Ai(u, ⃗x) = Bi(⃗x) + uCi(⃗x) + · · ·
(2.6)
(2.7a)
(2.7b) (2.7c)
0
where λ± ≡ 3 [1 ± ��1 − (2m/3)2 ], and h ̃ ij (⃗x) is real and normalized to have unit volume
2
thus determining the constant c.
The asymptotic solutions are locally determined from the asymptotic equations in terms
of the arbitrary ‘boundary values’ c2h ̃ij, α(⃗x) and Bi(⃗x), up to the u3 term in (2.7a), to order uλ+ in (2.7b) and to leading order in (2.7c). Beyond these terms in (2.7) the interior dynamics and the specific choice of quantum state become important.
For a saddle point solution to contribute to the semiclassical NBWF the observables must take real values on Σ [18, 19]. Hence the metric h, the scalar χ and the frame fields – the components of the vector field in an orthonormal frame – must all be real at τ = υ. The frame fields are given by B⃗ /b, where the ‘scale factor’ b ≡ c/u, multiplied by a real matrix specified by h ̃.
Hy Hy
a
Hy
2H
SP Hxr Hx HxaSP Hxr Hx SP Hx
FIG. 1: Left panel: The contour CA in the complex τ-plane of a de Sitter representation of a complex saddle point of the NBWF. Along the horizontal part the geometry is half a deformed Euclidean three-sphere. Along the vertical part it is asymptotically Lorentzian de Sitter space. Middle panel: The contour CB that gives the AdS representation of the same complex saddle point. The vertical part is asymptotically Euclidean AdS with a complex matter profile. The horizontal part is a complex geometry that interpolates between asymptotic AdS and de Sitter.
Right panel: The contour CC that provides the standard AdS representation of real asymptotically Euclidean AdS saddle points that are not associated with classical histories.
Reality of the observables on the boundary selects two classes of saddle points, corre- sponding respectively to asymptotically Euclidean AdS and Lorentzian de Sitter histories [18]. The saddle points associated with de Sitter histories are found by tuning the phases of the fields so that b, φ and Aμ all tend to real functions along a vertical line x = xr in the complex τ-plane [2, 6]. Writing c = |c|eiθc, α = |α|eiθα and Bi = |B|eiθB the expansions (2.7a) - (2.7c) imply
θc =xr, θα =−λ−xr, θB =0 (2.8)
By choosing a particular contour connecting τ = 0 to τ = υ = xr + iyυ we obtain a concrete representation of the saddle point geometry. The contour CA in Fig 1(a) provides an example. Along the horizontal part of this the geometry is the Euclidean geometry of half a deformed four-sphere. Along the vertical part from (xr,0) to (xr,yυ) the geometry tends to a Lorentzian space that is locally de Sitter,
ds2 ≈ −dy2 + 1 e2Hyh ̃ij(⃗x)dxidxj. (2.9) 4H2
Thus the contour CA gives the familiar representation of the no-boundary saddle points 7
that are associated with asymptotically real, Lorentzian, de Sitter histories. The ensemble of histories of this kind predicted by the NBWF is known explicitly in a homogeneous and isotropic minisuperspace approximation with linear perturbations, where it consists of universes with an early period of scalar field driven inflation [2]. The saddle point action acquires a rapidly varying phase factor along the vertical part of the contour CA so the wave function takes a WKB form (2.1) and predicts classical cosmological evolution.
There is, however, an alternative representation of the same class of saddle points given bythecontourCB inFig1(b). Alongthex=xa =xr −π/2H lineonehas
ds2 ≈ −dy2 − 1 e2Hyh ̃ij(⃗x)dxidxj. (2.10) 4H2
This is negative signature, Euclidean, asymptotically local AdS space. The asymptotic form of the matter fields along x = xa follows from (2.7b) –(2.7c) and is given by
φ(y,⃗x) ≈ |α(⃗x)|e−iλ−π/2e−λ−y ≡ α ̃e−λ−y, Ai(y,⃗x) ≈ Bi(⃗x) ≡ −iB ̃i (2.11)
where B ̃i is the boundary value of A ̃μ ≡ iAμ. Therefore the condition that the scalar be real at the endpoint υ, at x = xr, means it is complex along the AdS branch of the contour, with an asymptotic phase given by (2.11). Similarly, the requirement that Ai be real at the endpoint υ means that the usual AdS vector field A ̃i = iAi is purely imaginary in the asymptotic AdS region of de Sitter saddle points4.
To summarize, in the ‘AdS representation’ of Fig 1(b) the saddle point geometry consists of a Euclidean, asymptotically locally AdS geometry with complex matter field profiles in the ‘radial’ direction y. This is then joined smoothly onto the Lorentzian, asymptotically de Sitter geometry through a transition region – corresponding to the horizontal part of the contour CB – where the geometry is complex. The complex structure of the semiclassical NBWF thus provides a natural connection5 between (asymptotically) Lorentzian de Sitter histories and Euclidean AdS geometries.
As mentioned earlier there is a second set of saddle points with real observables on the boundary [18]. These are the real AdS domain wall solutions. They are found by taking φ
and A ̃i real at the SP, and by taking b2 to be real and negative for real u. A convenient representation of this class is provided by a contour CC that runs along the imaginary axis in the complex τ-plane to an endpoint υ (cf Fig 1(c)). As we will see below these saddle points do not describe classical universes and are of little interest in cosmology. Instead they are associated with Euclidean asymptotic AdS configurations.
The saddle point action is given by an integral over time along a contour in the complex τ-plane connecting the SP to the endpoint υ. The result is independent of the choice of contour. We first consider the action I1 of the first class of saddle points associated with Lorentzian histories. The asymptotic contribution to the saddle point action coming from the integral along the vertical branch of the contour CA is purely imaginary. This is because the integrand is real and dτ = idy. The real part IR, which governs the probabilities of the de Sitter histories, tends to a constant along the x = xr while the imaginary part S(υ) ∝ e3Hyυ .
The AdS representation based on the contour CB provides a different way to calculate IR. To see this we first consider the action integral Ih along the horizontal branch of the contour in Fig 1(b). Using the Hamiltonian constraint this can be written as,
1��xr��31��23 ⃗2 ij�� Ih(υa,υ)=8π dx dxg2 6H − R+8πV(φ)+4π(∇φ) +4πF Fij (3.1)
xa
where 3R is the scalar three curvature of gij. The expansions (2.7a) - (2.7c) imply that
the leading contribution to Ih from the vector term is O(u) and therefore negligible in the asymptotic limit. Further, the contribution to the asymptotically finite part of the action from the scalar and gravitational terms in (3.1) vanishes as a consequence of the asymptotic Einstein equations [6]. Hence the asymptotically constant term is the same on both ends of the horizontal branch. This means the tree level probabilities of different dS histories can be obtained from the asymptotically AdS region of the saddle points.
The action integral Ia along the vertical x = xa branch in the AdS representation exhibits the usual volume divergences, but Ih regulates these [6]. Specifically we find [6]
Ih(υa, υ) = −Sst(υa) + Sst(υ) + O(e−Hyυ ) (3.2) where Sst are the universal gravitational and scalar AdS counterterms, which arise here as
surface contributions to the action which are kept [6]. The second term in (3.2) gives auniversal phase factor (since the signature at υ differs from that at υa), and the first term regulates the divergences of Ia. At large υa
Ia(υa) − Sst(υa) → −Ireg (3.3) aAdS
where −Ireg [h ̃ij,α ̃,B ̃i] is the regulated asymptotic AdS action. It is a function of he aAdS
asymptotic profiles of the geometry and matter fields in the AdS regime of the saddle points, which are locally given by the argument of the wave function at υ through eq. (2.11).
Thus we find that in the large volume limit, the action of a general, inhomogeneous saddle point of the NBWF associated with an asymptotically local dS history can be expressed in terms of the regulated action of a complex AdS domain wall and a sum of purely imaginary, universal surface terms,
I1[h,χ,Bi] = −Ireg [h ̃ij(⃗x),α ̃(⃗x),B ̃i(⃗x)]+Sst(υ)+O(e−Hyυ) (3.4) aAdS
The probabilities of the asymptotically de Sitter histories are governed by the real part of I1 and hence given by
Re[I1(υ)] ≡ IR(υ) = −Re[Ireg ] . (3.5) aAdS
It is straightforward to evaluate the action I2 of the second class of real, Euclidean asymptotically AdS saddle points. This is simply given by the action integral Ia along the vertical line from the SP to the endpoint υ in Fig 1(c). Since all fields are everywhere real from an AdS viewpoint, the resulting action is real. It is given by
I2[h,χ,B ̃i] = −Ireg [h ̃ij(⃗x),α ̃(⃗x),B ̃i(⃗x)]+Sst(υ) (3.6) aAdS
where the boundary values h ̃ij(⃗x), B ̃i and α ̃ are all real. The leading surface term in Sst is real and grows as the volume of AdS. Hence the amplitude of asymptotically AdS configurations is low relative to the classical, asymptotically de Sitter histories discussed above.
The representation of both classes of saddle points in terms of Euclidean AdS ‘domain wall’ geometries with real or complex matter profiles leads to a holographic formulation of the NBWF by applying the Euclidean version of AdS/CFT. This relates the asymptoticAdS factor exp(−Ireg /h ̄) in the wave function to the partition function of a Euclidean dual aAdS
field theory in three dimensions [15, 16],
reg ̃ ̃ ̃ ̃��3�� ̃�� ̃i��
exp(−IaAdS[hij,α ̃,Bi]/h ̄)=ZQFT[hij,α ̃,Bi]=⟨exp d x h α ̃O+BiJ ⟩QFT (4.1)
The dual QFT is defined on the conformal boundary represented here by the three-metric h ̃ij. The operator O is constructed from scalars in the boundary theory and J⃗ is a current. The brackets ⟨· · · ⟩ on the right hand side denote the functional integral average involving the boundary field theory action minimally coupled to the metric conformal structure rep- resented by h ̃ij. For the spherical domain walls corresponding to homogeneous histories this is the round three-sphere, but in general α ̃,B ̃i and h ̃ij are functions of the boundary coordinates ⃗x. The three-divergence of the current J⃗ vanishes ensuring the invariance of the right hand sided of (4.1) under gauge transformations of B ̃i matching the invariance of the AdS action on the left [15]. Applying (4.1) to (3.4) and (3.6) yields
Ψ(υ) = 1 exp(−Sst(υ)/h ̄) (4.2) Zε [h ̃ij , α ̃, B ̃i]
where ε ∼ 1/|Hb| is a UV cutoff [6]. The surface terms are imaginary for the complex saddle points associated with Lorentzian histories, and real for the saddle points corresponding to Euclidean AdS configurations. The action Sct is independent of the vector field. This together with the gauge invariance of ZQFT ensures that the wave function satisfies the ∇⃗ · E⃗ = 0 constraint of Maxwell theory.
Equation (4.2) is an example of a semiclassical dS/CFT duality. The arguments of the wave function enter as external sources in the dual partition function that turn on deforma- tions (except for the scale factor b = c/u which enters as a UV cutoff). The dependence of the partition function on the values of the external sources yields a holographic no-boundary measure on the space of asymptotic configurations. For boundary configurations given by sufficiently small values of the matter sources and sufficiently mild deformations of the round three sphere one expects the integral defining the partition function to converge, yielding a non-zero amplitude. The holographic form (4.2) of the NBWF thus involves a range of different deformations of a single underlying CFT. The sources α ̃ and B ̃μ of the deformations are real for saddle points associated with asymptotically AdS configurations. By contrast the sources are complex for saddle points associated with Lorentzian, classical, asymptoti- cally de Sitter histories. In the latter regime of superspace the dual form of the NBWF thus involves complex deformations of a Euclidean CFT.
QFT
As an illustration we compute the asymptotic NBWF of vector perturbations in a ho- mogeneous and isotropic, expanding background [20]. Here we have in mind that the scalar field and Λ are responsible for the background evolution. The details of this do not mat- ter, because the vector is conformally invariant and hence decouples to quadratic order. We evaluate the semiclassical vector wave function first using the dS representation of the background saddle point and then using its AdS representation.
The line element of a homogeneous and isotropic saddle point can be written as
ds2 = N2(λ)dλ2 + a2(λ)γijdxidxj (5.1)
where γij is the metric on a unit round three sphere. Homogeneous and isotropic minisuper- space is spanned by the boundary value b of the scale factor and the value χ of the scalar field. Neglecting the back reaction of the Maxwell field on the background, we write
Ψ(b, χ, B⃗ ) = Ψ0(b, χ)ψV (b, χ, B⃗ ), (5.2)
where Ψ0(b, χ) is a background wave function given by the action of a saddle point solution (a(τ),φ(τ)) that matches (b,χ) at the boundary of the disk, and is regular elsewhere [2, 3]. We assume the phase of Ψ0 varies rapidly so that it predicts a classical background. The wave function ψV of vector perturbations is defined by the remaining integral over Aμ,
C
We evaluate (5.3) in the steepest descents approximation. The Euclidean action I(2) of V
vector perturbations around saddle points of the form (5.1) is given by
(2) ��√��a��ij k k��Nk i i��
IV = γ 2N γ Ai,λAj,λ +A0(2A ;k0 −A0;k + 2aA (2Ak +A ;ik −Ak;i ) (5.4)
Working in Coulomb gauge A0 = ∇iAi = 0 and in terms of conformal time η defined as adη = Ndλ, the solutions take the following form,
⃗�� (2)
ψV (b,χ,B) ≡
where the integral is over all regular histories on a disk which match B⃗ on its boundary.
δAμ exp(−IV [a(τ),φ(τ),Aμ(τ,⃗x)]/h ̄). (5.3)
Ai(η, Ω) =
�� �� (p)��n
nlmp
lm
fnlmp(η) Si
(5.5)
�� (p)��n
lm
where B
are the mode coefficients of the boundary configuration B⃗ and iθυ is the imaginary part of η at the boundary surface6. The asymptotic expansion of (5.6) in terms of the complex time variable u defined in (2.6) is of the form (2.7c),Ai = �� B
��1 − 2ine−ixr u + O(u2)�� S (5.7) iwhere we have used that η = iθυ − 2ie−ixr u + O(u2) in the large volume regime. Evaluated
where Si
sphere, with ∆Si = −(n2 − 2)Si. From here onwards we denote the indices collectively by
. The solutions for f are proportional to e±nη. The no-boundary condition of regularity at the SP selects the solution that decays as η → −∞. Hence we getAi(η, Ω) = �� B
e−inθυ enηS (5.6) iare the transverse eigenfunctions of the vector Laplacian ∆ on the three-
on solutions the vector action (5.4) reduces to a surface term, (2) 1��3 1/2ij
(5.8)
(5.9)
Substituting (5.6) we get
IV = 2 d x(γ) γ Ai,ηAj I(2) = �� nB2
V
2
This is real and positive, indicating that the vector perturbations are in their quantum
mechanical ground state.
We could equally well have computed the vector wave function using the AdS represen-
tation of the saddle point. The saddle point action of a vector field A ̃μ with boundary value B ̃i in a homogeneous and isotropic, Euclidean, asymptotically AdS background is obtained in a similar manner and given by
I(2) = �� nB ̃2 (5.10)
Eq. (3.5) implies this gives the wave function of vector perturbations in a classical, homo- geneous and isotropic, asymptotically de Sitter background when B ̃i is taken to be purely imaginary. Substituting B ̃i = iBi in (5.10) and using (3.5) indeed yields (5.9).
V,aAdS
2
We have generalized the holographic formulation of the NBWF to models including vector matter. We first derived an ‘AdS representation’ of the NBWF saddle points associated with classical, asymptotically dS universes with scalar and vector matter. In this representation, the saddle point geometry consists of a Euclidean AdS domain wall which joins smoothly onto an asymptotically real, Lorentzian, de Sitter geometry through a transition region where the spatial metric changes signature. The scalar and vector matter profiles along the AdS domain wall are complex. The relative probabilities of different classical boundary configurations (h, χ, B⃗ ) are given by the regularized AdS domain wall action.
Applying Euclidean AdS/CFT then yields a holographic form of the semiclassical NBWF
in terms of the partition function of a dual, Euclidean CFT with complex sources specified
by the asymptotic scalar and vector boundary values in the AdS regime of the saddle points.
The dual description of the no-boundary measure on classical, Lorentzian configurations thus
involves a generalization of Euclidean AdS/CFT to CFTs with complex deformations. The
phases of the sources are independent of the specific quantum state and of the dynamics
in the interior. They are fully specified by the requirement that the fields be real at the
final de Sitter boundary. The phase of the scalar source in the dual is e−iλ−π/2, where
λ− = 3 [1 − ��1 − (2m/3)2 ], and the vector field source is purely imaginary7 . 2
From a holographic viewpoint it is natural to consider the wave function on an extended domain that includes real deformations of the dual CFT [18, 19]. Partition functions with real deformations specify the amplitude of real, asymptotically AdS boundary configurations. In the bulk their amplitude is given by real Euclidean AdS domain walls. These are of little relevance in cosmology, since the lack of a rapidly varying phase factor in the action means they are not associated with classical histories. In fact the AdS volume contribution to the action heavily suppresses their contribution to the NBWF. Nevertheless, the inclusion of both sets of saddle points in the configuration space - as suggested by holography - yields an appealing, signature-invariant formulation of the wave function defined on an extended configuration space with three-metrics of both signatures [18]. One signature corresponds to deformations with real sources and the other to complex sources.
From a bulk viewpoint the relation between (asymptotic) Euclidean AdS and Lorentzian de Sitter exhibited here is reminiscent of a ‘symmetry’ of the action (2.2)-(2.3) under a reversal of the signature of the metric [17]. A signature reversal relates the action of a dS theory of gravity coupled to a scalar with a positive potential V and a vector field Aμ to the action of an AdS theory of gravity coupled to a scalar with potential −V and a vector field A ̃μ = iAμ that is imaginary from a dS viewpoint. This is not unlike the transition from supersymmetry in AdS to pseudo-supersymmetry in dS, where the vectors in the de Sitter theory exhibit a similar behavior. In that context as well it has been argued that the dS and AdS theories should really be viewed as two real domains of a single underlying complexified theory [21, 22]. The complex structure of the wave function in our scheme provides a natural setup for a unified description of this kind. This is made explicit in its holographic form which involves complex deformations of a single underlying dual conformal field theory.
Our analysis so far applies only to toy models of de Sitter gravity or, in AdS terms, to consistent truncations of AdS supergravity theories that retain only low mass scalars, and vectors. A full realization of holographic cosmology in string theory remains an open prob- lem. A particularly intriguing issue has to do with the tower of irrelevant operators featuring in the usual AdS/CFT duals. These correspond to tachyonic fields in the de Sitter domain of the theory. The requirement of an asymptotic de Sitter structure in dS/CFT amounts to a final boundary condition on these fields. More generally the asymptotic dS structure implies that the asymptotic geometry and fields behave classically [18]. The no-boundary condition of regularity on the saddle points implements the final boundary condition on tachyons since it excludes classical histories in which these are not set to zero [2, 24].
The dual partition function is not concerned with the classical evolution itself - which is governed by the phase factor of the wave function (2.1) - but merely calculates the probability measure on classical phase space in the no-boundary state [6, 16]. Given that a probabilistic interpretation of the wave function of the universe is restricted to its classical domain anyway [2] this suggests dS/CFT should be viewed as a statement about the coarse-grained, classical predictions of the wave function only8. But it would be very interesting to understand the origin and nature of the late-time classicality constraint from a dual viewpoint.
Midwest sticky
Resident Smartass & midget connoisseur
I think this pretty well settles the origin question . . .
[URL=http://i188.photobucket.com/albums/z189/13SneakyPete13/gifs/84486189_zpsee8gxhyn.gif]View Image[/URL]
Best post of the day
Actually to think were alone in the universe one would or could possibly say your crazy and of course being were fairly new to the universe as in time they would with out a doubt be able to kick our ass
IMO there there helping us along the way and one day they will show there presence possibly once the world has become a NWO
Midwest sticky
Resident Smartass & midget connoisseur
I never said were alone in the universe.Im pretty sure it's mathematically impossible for this planet to be the only one wich has intelligent life. I just don't think it'll end well for us when the aliens come a callin. See pic on top of this page to see their intentions
pretty intellectually shallow to think intelligent alien (celestial) life would be humanoid or human like, Most gnostic philosophies mention aliens although they use language specific to their vernacular.
Pseudo intellects focus on the malignment of others, the operative word being pseudo.
Pseudo intellects focus on the malignment of others, the operative word being pseudo.
Midwest sticky
Resident Smartass & midget connoisseur
^^ is that directed towards me? If so you got me all wrong. Who knows what they would look like.
Mid you got to watch this
https://www.youtube.com/watch?v=MS412Wlv6Yg
https://www.youtube.com/watch?v=AaZW75jJjnw
https://www.youtube.com/watch?v=MS412Wlv6Yg
https://www.youtube.com/watch?v=AaZW75jJjnw
It is not directed at people its directed at mindsets
The first part echos the bias of the human ego, that advanced life would be like us at all, be in a similar biological container or function in a similar manner.
the second is directed at mindsets as well, prejudice of any kind is a mindset of weakness.
The first part echos the bias of the human ego, that advanced life would be like us at all, be in a similar biological container or function in a similar manner.
the second is directed at mindsets as well, prejudice of any kind is a mindset of weakness.
Midwest sticky
Resident Smartass & midget connoisseur
Gotcha. I know this probably sounds stupid but whenever I wonder what alien life would look like I always envision a jellyfish type of organism.
Bottom line i believe aliens are among us look at mad lab lol kidding but seriously though Government knows and if the public knew and us being savages can only imagine the havoc and looting that would occur
were are puppets that not need to know really anything we can only guess and assume
that's it.
DEath n taxes is all were about
were are puppets that not need to know really anything we can only guess and assume
that's it.
DEath n taxes is all were about
I linked a bunch of references
the last of one had a 35 minute radio interview with Brian Greene a brilliant quantum physicist,
his pbs special elegant universe is excellent and a must watch. Quantum physics explains what physics cannot (like these anomalies in big bang theory).
Many great minds have tackled the questions asked here and the question has been driven much farther in every regards than anyone is asking here (and that tbh is where the conversation gets interesting).
The problem with discussions here is that there is this primitive notion that ideas need to compete in order to achieve validity or add value.
The bigger picture is only revealed when you take all perspectives into consideration, since they all have a valid basis even if only personally relevant, because we are all part of the same shared experience, both in spirit, matter and time.
Even in this context the value of religion is not discounted, just not understood by a physicist.
Ask a social scientist the value of religion and you may find their answer is different, as per the study I posted in this thread.
There is a treatise in the above quoted, one that allows us to understand it from a perspective of human condition.
Some people are intrigued by the world around us so they put their mind to that and if they feel fulfilled they claim success by proxy, and it becomes a relative belief they project onto others.
Some people are intrigued by the world inside of us so they put their mind to that if they and if they feel fulfilled they claim success by proxy, and it becomes a relative belief they project onto others.
Fewer people see one in themselves but accept the different focus and fulfillment of others as being of the diversity required by our race for success.
the last of one had a 35 minute radio interview with Brian Greene a brilliant quantum physicist,
his pbs special elegant universe is excellent and a must watch. Quantum physics explains what physics cannot (like these anomalies in big bang theory).
Many great minds have tackled the questions asked here and the question has been driven much farther in every regards than anyone is asking here (and that tbh is where the conversation gets interesting).
The problem with discussions here is that there is this primitive notion that ideas need to compete in order to achieve validity or add value.
The bigger picture is only revealed when you take all perspectives into consideration, since they all have a valid basis even if only personally relevant, because we are all part of the same shared experience, both in spirit, matter and time.
Even in this context the value of religion is not discounted, just not understood by a physicist.
Ask a social scientist the value of religion and you may find their answer is different, as per the study I posted in this thread.
There is a treatise in the above quoted, one that allows us to understand it from a perspective of human condition.
Some people are intrigued by the world around us so they put their mind to that and if they feel fulfilled they claim success by proxy, and it becomes a relative belief they project onto others.
Some people are intrigued by the world inside of us so they put their mind to that if they and if they feel fulfilled they claim success by proxy, and it becomes a relative belief they project onto others.
Fewer people see one in themselves but accept the different focus and fulfillment of others as being of the diversity required by our race for success.
genedigger
Member
people believe in aliens but not a god?i dig that.thats some good shit right there.
Midwest sticky
Resident Smartass & midget connoisseur
Midwest sticky
Resident Smartass & midget connoisseur
Through The Wormhole: Does The Ocean Think?
http://www.dailymotion.com/video/x1ztmsc_through-the-wormhole-does-the-ocean-think_creation
- Status
