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Ann. Phys. (Berlin), 1-14 (2016)/ DOI 10.1002/andp.201600011

physik

On decoherence in quantum gravity

Dmitriy Podolskiy1'* and Robert Lanza2

Received 13 January 2016, revised 24 June 2016, accepted 27 July 2016 Published online 26 September 2016

It was previously argued that the phenomenon of quantum gravitational decoherence described by the Wheeler-DeWitt equation is responsible for the emergence of the arrow of time. Here we show that the characteristic spatiotemporal scales of quantum gravitational decoherence are typically logarithmically larger than a characteristic curvature radius R-1/2 of the background space-time. This largeness is a direct consequence of the fact that gravity is a non-renormalizable theory, and the corresponding effective field theory is nearly decoupled from matter degrees of freedom in the physical limit MP ^ to. Therefore, as such, quantum gravitational decoherence is too ineffective to guarantee the emergence of the arrow of time and the "quantum-to-classical" transition to happen at scales of physical interest. We argue that the emergence of the arrow of time is directly related to the nature and properties of physical observer.

1 Introduction

Quantum mechanical decoherence is one of the cornerstones of the quantum theory [1, 2]. Macroscopic physical systems are known to decohere during vanishingly tiny fractions of a second, which, as generally accepted, effectively leads to emergence of a deterministic quasi-classical world which we experience. The theory of decoherence has passed extensive experimental tests, and dynamics of the decoherence process itself was many times observed in the laboratory [3-15]. The analysis of decoherence in non-relativistic quantum mechanical systems is apparently based on the notion of time, the latter itself believed to emerge due to decoherence between different WKB branches of the solutions of the Wheeler-DeWitt equation describing quantum gravity [2, 16-19]. Thus, to claim understanding of decoherence "at large", one has to first understand decoherence in quantum gravity. The latter is clearly problematic, as no consistent and complete theory of quantum gravity has emerged yet.

Although it is generally believed that when describing dynamics of decoherence in relativistic field theories and gravity one does not face any fundamental difficulties and gravity decoheres quickly due to interaction with matter [20-23], we shall demonstrate here by simple estimates that decoherence of quantum gravitational degrees of freedom might in some relevant cases (in particular, in a physical situation realized in the very early Universe) actually be rather ineffective. The nature of this ineffectiveness is to a large degree related to the non-renormalizability of gravity. To understand how the latter influences the dynamics of decoherence, one can consider theories with a Landau pole such as the scalar field theory in d = 4 dimensions. This theory is believed to be trivial [24], since the physical coupling Xphys vanishes in the continuum limit.1 When d > 5, where the triviality is certain [25, 26], critical exponents of k$4 theory and other theories from the same universality class coincide with the ones predicted by the mean field theory. Thus, such theories are effectively free in the continuum limit, i.e., Xphys ~ ^ 0 when the UV cutoff A ^to. Quantum mechanical decoherence of the field states in such QFTs can only proceed through the interaction with other degrees of freedom. If such degrees of freedom are not in the menu, decoherence is not simply slow, it is essentially absent.

In effective field theory formulation of gravity dimen-sionless couplings are suppressed by negative powers of

* Corresponding author E-mail:

Dmitriy_Podolskiy@hms.harvard.edu

1 Harvard Medical School, 77 Avenue Louis Pasteur, Boston, MA, 02115

2 Wake Forest University, 1834 Wake Forest Rd., Winston-Salem, NC, 27106

This is an open access article undertheterms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits useand distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

1 There exist counter-arguments in favor of the existence of a genuine strong coupling limit for d = 4 [42].

the Planck mass MP, which plays the role of UV cutoff and becomes infinite in the decoupling limit MP ^ to. Decoherence times for arbitrary configurations of quantum gravitational degrees of freedom also grow with growing MP although, as we shall see below, only logarithmically slowly and become infinite at complete decoupling. If we recall that gravity is almost decoupled from physical matter in the real physical world, ineffectiveness of quantum gravitational decoherence does not seem any longer so surprising. While matter degrees of freedom propagating on a fixed or slightly perturbed background space-time corresponding to a fixed solution branch of the WdW equation decohere very rapidly, decoherence of different WKB solution branches remains a question from the realm of quantum gravity. Thus, we would like to argue that in order to fit the ineffectiveness of quantum gravitational decoherence and a nearly perfectly decohered world which we experience in experiments, some additional physical arguments are necessary based on properties of observer, in particular, her/his ability to process and remember information.

This paper is organized as follows. We discuss de-coherence in non-renormalizable quantum field theories and relation between non-renormalizable QFTs and classical statistical systems with first order phase transition in Section 2. We discuss decoherence in non-renormalizable field theories in Section 3 using both first- and second-quantized formalisms. Section 4 is devoted to the discussion of decoherence in dS spacetime. We also argue that meta-observers in dS space-time should not be expected to experience effects of deco-herence. Standard approaches to quantum gravitational decoherence based on analysis of WdW solutions and master equation for the density matrix of quantum gravitational degrees of freedom are reviewed in Section 5. Finally, we argue in Section 6 that one of the mechanisms responsible for the emergence of the arrow of time is related to ability of observers to preserve information about experienced events.

2 Preliminary notes on non-renormalizable field theories

To develop a quantitative approach for studying de-coherence in non-renormalizable field theories, it is instructive to use the duality between quantum field theories in d space-time dimensions and statistical physics models in d spatial dimensions. In other words, to gain some intuition regarding behavior of non-renormalizable quantum field theories, one can first analyze the behavior of their statistical physis counterparts

Si Ap)

Figure 1 One- and two-loop contributions to £(p) in EFT.

describing behavior of classical systems with appropriate symmetries near the phase transition.

Consider for example a large class of non-renormalizable QFTs, which includes theories with global discrete and continuous symmetries in the number of space-time dimensions higher than the upper critical dimension djp: d > dip. Euclidean versions of such theories are known to describe a vicinity of the 1st order phase transition on the lattice [27], and their continuum limits do not formally exist2: even at close proximity of the critical temperature T = Tc physical

(T - Tc)-1/2

correlation length of the theory f never becomes infinite.

One notable example of such a theory is the k(02 — v2)2 scalar statistical field theory, describing behavior of the order parameter 0 in the nearly critial system with discrete Z2 symmetry. This theory is trivial [25, 26] in d > dUp = 4.3 Triviality roughly follows from the observation that the effective dimensionless coupling falls off as k/f d—4, when the continuum limit f is approached.

What does it mean physically? First, the behavior of the theory in d > 4 is well approximated by mean field. This can be readily seen when applying Ginzburg criterion for the applicability of mean field approximation [28]: at d > 4 the mean field theory description is applicable arbitrarily close to the critical temperature. This is also easy to check at the diagrammatic level: the two-point function of the field 0 has the following form in momentum representation

(0(- p)0(p)) ~ (V + m0 + £(p)) 1,

where m0 = a(T — Tc), and at one loop level (see Fig. 1) 2 /a(T — Tcd/2—1

£( p) - ci g A2 + C2 g A2

2 Similarly, Euclidean Z2, O(2) and SU(N) gaugefield theories all known to possess a first order phase transition on the lattice at

d > dUp = 4.

3 Most probably, it is trivial even in d = 4 [24], where it features a Landau pole (although there exist arguments in favor of a nontrivial behavior at strong coupling, see for example [42]).

Figure 2 A possible configuration of order parameter in the Z2 statistial model in d > 5 spatial dimensions. The left panel represents the configuration of the field at scales slightly larger than the critical radius Rcrit ~ f. that coinsides with the size of bubbles of the true vacuum with broken Z2 symmetry; + corresponds to bubbles with the vacuum +®0 inside, and — to the bubbles with the vacuum —®0. At much larger scales of the order of Rir given by the expression (2) (®) =0 in average, as the contribution of multiple bubbles with ® = +<£0 is compensated by the contribution of bubbles with ® = —®0.

where g = kAd-4 is the dimensionless coupling. The first term in the r.h.s. of (1) represents the mean field correction leading to the renormalization/redefinition of Tc. The second term is strongly suppressed at d > 4 in comparison to the first one. The same applies to any high order corrections in powers of k as well as corrections from any other local terms ~ <6,<8,..., pF<n,... in the effective Lagrangian of the theory.

As we see, the behavior of the theory is in fact simple despite its non-renormalizability; naively, since the coupling constant k has a dimension [Z]d-4, one expects uncontrollable power-law corrections to observables and coupling constants of the theory. Nevertheless, as (1) implies, the perturbation theory series can be re-summed in such a way that only mean field terms survive. Physics-wise, it is also clear why one comes to this conclusion. At d > 4 Z2-invariant statistical physics models do not possess a second order phase transition, but of course do possess a first order one.4 Behavior of the theory in the vicinity of the first order phase transition can always be described in the mean field approximation, in terms of the homogeneous order parameter ® = {<}.

Our argument is not entirely complete as there is a minor culprit. Assume that an effective field theory with the EFT cutoff A coinciding with the physical cutoff is considered. Near the point of the 1st order phase transition, when the very small spatial scales (much smaller than the correlation length f of the theory) are probed, it is almost guaranteed that the probed physics is the one of the broken phase. The first order phase transition proceeds through the nucleation of bubbles of a critical size

4 This is equivalent to the statement that trvial theories do not admit continuum limit.

R ~ (T — Tc)—l/2 ~ f, thus very small scales correspond to physics inside a bubble of the true vacuum (0) = ±v, and the EFT of the field 50 = 0 — (0) is a good description of the behavior of the theory at such scales. As the spatial probe scale increases, such description will inevitably break down at the IR scale

RIR ~ m exp

« d-4 A 2

Const.

ConSt.A (d-4)(d/2-1

gd/2-1 yd-4

where m~ f—1 ~ *Jkv ^ 0 in the pre-critical limit. This scale is directly related to the nucleation rate of bubbles: at scales much larger than the bubble size R one has to take into account the stochastic background of the ensemble of bubbles of true vacuum on top of the false vacuum, and deviation of it from the the single-bubble background (0) = ±v leads to the breakdown of the effective field theory description, see Fig. 2. Spatial homogeneity is also broken at scales m-1 < l ^ Rir by this stochastic background, and this large-scale spatial inhomogeneity is one of the reasons of the EFT description breakdown.

Finally, if the probe scale is much larger than f ~ (T — Tc)-1/2 (say, roughly, of the order of Rr or larger), the observer probes a false vacuum phase with (0) = 0. Z2 symmetry dictates the existence of two true minima (0) = ±v, and different bubbles have different vacua among the two realized inside them. If one waits long enough, the process of constant bubble nucleation will lead to self-averaging of the observed (0). As a result, the "true" (0) measured over very long spatial scales is always zero.

The main conclusion of this Section is that despite the EFT breakdown at both UV (momenta p > A) and IR (momenta p < R—R ) scales, the non-renormalizable statistical X04 theory perfectly remains under control: one can effectively use a description in terms of EFT at small scales R—R < p < A and a mean field at large scales. In all cases, the physical system remains nearly completely described in terms of the homogeneous order parameter ® = (0) or a "master field", as its fluctuations are almost decoupled. Let us now see what this conclusion means for the quantum counterparts of the discussed statistical physics systems.

3 Decoherence in relativistic non-renormalizable field theories

We first focus on the quantum field theory with global Z2-symmetry. All of the above (possibility of EFT descriptions at both R—R < E < A and E ^ R—R, breakdown of EFT at E ~ A and E ~ R—R with Rir given by the expression (2)) can be applied to the quantum theory, but there is an important addition concerning decoherence, which we shall now discuss in more details.

3.1 Master field and fluctuations

As we discussed above, for the partition function of the Z2—invariant statistical field theory describing a vicinity of a first order phase transition TT—Tc ^ 1 one approximately has

Z=JD0 exp (~jd^x Q(30)2 ± 2m202 + 4X04 + .. ^«

d® exp (t1 Vdm2®2 — 4 VdX®4 — Vdi® , (4)

where Vd is the d—volume of the system, and d > 5 as in the previous Section. Physically, the spatial fluctuations of the order parameter 0 are suppressed, and the system is well described by statistical properties of the homogeneous order parameter ® ~ (0).

The Wick rotated quantum counterpart of the statistical physics model (3) is determined by the expression for the quantum mechanical "amplitude"

to; ®, t) d® exp (iVa—1 T (t4m2®2 — 4X®4J) =

= i D® expfiVd—1i dt(t 1 m2®2 — 1 X®4)) ,

j®(t>)=®0 V J to V 2 4 //

written entirely in terms of the "master field" ® (as usual, Vd—1 = f dd—1 x is the volume of (d — 1)-dimensional space). In other words, in the first approximation the non-renormalizable X04 theory in d > 5 dimensions can be described in terms of a master field ®, roughly homogeneous in space-time. As usual, the wave function of the field can be described as

^ (®, t) ~ A(®0, t0; ®, t),

where ®0 and /0 are fixed, while ® and t are varied, and the density matrix is given by

pt) = Tr^t)Vt),

where the trace is taken over the degrees of freedom not included into ® and namely, fluctuations of the field 50 above the master field configuration The contribution of the latter can be described using the prescription

A ~ j d®V50 exp (iVd-i t(t 2 m2®2 - ix®4j)x

x exp ^i J ddx ^i(350)2 t 1 m2502 - 2X$>2502

-X®3 80 - X&503 - ...

In the "mean field" approximation (corresponding to the continuum limit) X ^ 0 fluctuations 50 are completely decoupled from the master field ®, making (5) a good approximation of the theory. To conclude, one physical consequence of the triviality of statistical physics models describing vicinity of a first order phase transition is that in their quantum counterparts decoherence of entangled states of the master field ® does not proceed.

3.2 Decoherence in the EFT picture

When the correlation length f

is large but finite,

decoherence takes a finite but large amount of time, essentially, as we shall see, determined by the magnitude of f. This time scale will now be estimated by two different methods.

As non-renormalizable QFTs admit an EFT description (which eventually breaks down), dynamics of deco-herence in such theories strongly depends on the probe scale, coarse-graining effectively performed by the observer. Consider a spatio-temporal coarse-graining scale l > A—1 and assume that all modes of the field 0 with energies/momenta l— 1 < p ^ A represent the "environment", and interaction with them leads to the deco-herence of the observed modes with momenta p < l—1.

If also p > R—R, EFT expansion near (0) is applicable. In practice, similar to Kenneth Wilson's prescription for renormalization group analysis, we separate the field 0 into the fast, 0 f, and slow, 0s, components, considering 0 f as an environment, and since translational invariance holds "at large", 0s and 0 f are linearly separable.5

The density matrix p(t, 0s , 0's) of the "slow" field or master field configurations is related to the Feynman-Vernon influence functional SI [0^ 02] of the theory [21] according to

p(t,(ps,<t>'s) = J d0od0op(i, 0o, 00)x

r 0s r 0S / /

J 00 J^'o

d02exp(iS[0i] - iS[02]

+iSi[01,02]),

S[01,2] = J ddx ( 1(301,2)2 - 1 m202h2 - 1 ¿04,2)

SI = -2k j ddxAP(x, x)(02 - 0|)+

j ddxddy02(x)(Ap(x, y))202(y)

j ddxddy02(x)(A- (x, y))20|(y)+

5 A note should betaken at this point regarding the momentum representation of the modes. As usual, 0f is defined as integral over Fourier modes of thefield with small momenta. As explained above, the quantum theory with existing continuum limit is a

Wick-rotated counterpart of the statistical physics model describing a second order phase transition. In the vicinity of a second order phase transition broken and unbroken symmetry phases are continuously intermixed together, which leads to the translational invariance of correlation functions of the order parameter 0.Inthe case of the first order phase transition, such invariance is strictly speaking broken in the presence of stochastic background of nucleating bubbles of the broken symmetry phase, see the discussion in the previous Section. Therefore, the problem "at large" rewritten in terms of 0f and 0s becomes of Caldeira-Legetttype [43]. If we focus our attention on the physics at scales smaller than the bubble size, translational invariance does approximately hold, and we can consider 0s and 0f as linearly separable (if they are not, we simply diagonalize the part of the Hamiltonian quadratic in 0).

j ddxddy02(x)(AD(x, y))202(y) + ...,

where 012 are the Schwinger-Keldysh components of the field 0s, and are Feynman, negative frequency

Wightman and Dyson propagators of the "fast" field 0 f, respectively.6 It is easy to see that the expression (9) is essentially the same as (7), that is of no surprise since an observer with an IR cutoff cannot distinguish between ® and 0s.

The part of the Feynman-Vernon functional (10) that is interesting for us can be rewritten as

Si=i ^ AAf^i- ^u».- yK0?( y>- 0!(y)>-

r (ii)

-k2 dd+1 ^dd+1(0f(x) - 0f(x)V(x- y)

x (02 (y) + 02 (y)) + •••

(note that non-trivial effects including the one of deco-herence appear in the earliest only at the second order in k).

An important observation to make is that since the considered non-renormalizable theory becomes trivial in the continuum limit, see (5), the kernels ^ and v can be approximated as local, i.e., ¡x(x — y) ~ /j,0S(x — y), v(x — y) ~ v0S(x — y). This is due to the fact that fluctuations 50 ~ 0f are (almost) decoupled from the master field ® ~ 0s in the continuum limit, their contribution to (9) is described by the (almost) Gaussian functional. Correspondingly, if one assumes factorization and Gaus-sianity of the initial conditions for the modes of the "fast" field 0f, the Markovian approximation is valid for the functional (9), (10).

A rather involved calculation (see [21]) then shows that the density matrix (8) is subject to the master equation

3p(t,(ps ,(p's )

= - / d x[ Hr (x, t), p] + ...,

Hi ~ 2k2vo(02(T, x) - 0,2(t, x))2

Here, we kept only the leading terms in k ~ f 4— d as higher loops as well as other non-renormalizable interactions provide contributions to the FVfunctional, which are subdominant (and vanishing!) in the continuum limit f ^ <x).

annalen der

where only terms of the Hamiltonian density HI, which lead to the exponential decay of non-diagonal matrix elements of p are kept explicitly while... denote oscillatory terms.

The decoherence time can easily be estimated as follows. If only "quasi"-homogeneous master field is kept in (12), the density matrix is subject to the equation

dp(t, O, ') 1 2 r, ,

P '' = — ^2V0Vd-il(® - O')2(O + O')2,p]

= -2 x2v0 Vd-i[(® - ®')2®2,p],

where O = ¿(O + O'). We expect that O is close to (but does not necesserily coincides with) the minimum of the potential V(O), which will be denoted O0 in what follows. For O ~ O', i.e., diagonal matrix elements of the density matrix the decoherence effects are strongly suppressed. For the matrix elements with O = O' the decoherence rate is determined by

from observations. In this sense, (15) holds to all orders in k, and it can be expected that

tD > Consta • (£/S£),

where 8% ~|O — O'| universally for all O, O' of physical interest.

According to the expressions (15), (16) decay of nondiagonal elements of the density matrix p(t, O , O ) would take much longer than %/c (where c is the speed of light) for |®i — O2| < |Oi + O2|. It still takes about ~ %/c for matrix elements with |O1 — O2|~ |O1 + O2| to decay, a very long time in the limit % ^ro.

Finally, if O = O0, i.e., the "vacuum" is excited, O returns to minimum after a certain time and fluctuates near it. It was shown in [21] that the field O is subject to the Langevin equation

d® 2 2^0 + m2(® - ®o) ~ (t), dt

r = 2k voVd-i(® -

1 k2vo Vd-i(® - ®')2®2-

<£ (t)> = 0,

(14) <£(i)£(t)> = voS(t- {),

Thus, the decoherence time scale in this regime is

k2vo Vd-1(® - ®')2®o

It is possible to further simplify this expression. First of all, one notes that krenorm will be entering the final answer instead of the bare coupling k. As was discussed above (and shown in details in [25, 26]), the dimension-less renormalized coupling grenorm is suppressed in the continuum limit as C%™s/', where % is the physical correlation length. Second, the physical volume V satisfies the relation V < %d—1 (amounting to the statement that the continuum limit corresponds to correlation length being of the order of the system size). Finally, O0 ~ mp ~ %d—6, i.e., every quantity in (15) can be presented in terms of the physical correlation length % only. This should not be surprising. As was argued in the previous Sections, the mean field theory description holds effectively in the limit A ^ro (or % ^ ro), which is characterized by uncoupling of fluctuations from the mean field O. Self-coupling of fluctuations 80 is also suppressed in the same limit, thus the physical correlation length % becomes a single parameter defining the theory. The only effect of taking into account next orders in powers of k (or other interactions!) in the effective action (9) and the Feynman-Vernon functional (10) is the redefinition of %, which ultimately has to be determined

where the random force is due to the interaction between the master field ® and the fast modes 80, determined by the term §k®§802 in the effective action. (The Eq. (17) was derived be application of Hubbard-Stratonovich transformation to the effective action for the fields ® and 80 and assuming that ® is close to ®0.) The average

<®> - ®o ~ (®init - ®o) exp -

m2 2^o®o

(t - 5nit)

so the master field rolling towards the minimum of its potential plays a role of "time" in the theory. The roll towards the minimum O0 is very slow, as the rolling time ~ w|o ^ ^ %d—3 is large in the continuum limit % ^

m2 Jkm ^ b ^

0. Once the field reaches the minimum, there is no "time", as the master field O providing the function of a clock is minimized. The decoherence would naively be completely absent for the superposition state of vacua ±O0 as follows from (14). However, the physical vaccuum as seen by a coarse-grained observer is subject to the Langevin equation (17) even in the closest vicinity of O = ±O0, and the fluctuations ((O — O0)2) are never zero; one roughly has

((® - ®o)2

which should be substituted in the estimate (15) for matrix elements with O ~ O' ~ O0.

What was discussed above holds for coarse-graining scales p > Rir1, where Rir is given by the expression (2). If the coarse-graining scale is p < Rir1, the EFT description breaks down, since at this scale the effective dimen-sionless coupling between different modes becomes of the order 1, and the modes contributing to 0s and 0f can no longer be considered weakly interacting. However, we recall that at probe scales l > Rir the unbroken phase mean field description is perfectly applicable (see above). This again implies extremely long decoherence time scales.

The emergent physical picture is the one of entangled states with coherence surviving during a very long time (at least ~ f/c) on spatial scales of the order of at least f. The largeness of the correlation length f in statistical physics models describing the vicinity of a first order phase transition implied a large scale correlation at the spatial scales ~ f .As was suggested above, the de-coherence is indeed very ineffective in such theories. We shall see below that the physical picture presented here has a very large number of analogies in the case of deco-herence in quantum gravity.

where dT = and H8$ is the Hamiltonian of fluctua-

tions 8$,

X| + 1(v8$)2 + V($'8$)

where V(®,80) = ±2m2502 + |k®202, and the full state of the field is ^(®,80) ~ (®, 80) ~

exp (iS0(®))0(®, 80) (again, we naturally assume that the initial state was a factorized Gaussian). It was previously shown (see [17] and references therein) that the "time"-like affine parameter t in (18) coincides in fact with the physical time t.

Writing down the expression for the density matrix of the master field ®

p(t, 0> 1,®2) = Tr,,№(®1, 80H*(®2, 80))

= po D8$ f(T, $1, 8$)f *(t, $2, 8$), (19)

3.3 Decoherence in functional Schrodinger picture

Let us now perform a first quantization analysis of the theory and see how decoherence emerges in this analysis. As the master field ® is constant in space-time, the field state approximately satisfies the Schrodinger equation

po = exp (/So($i) - iSo($2)) ■

Sd(®) = 2 Vd— 1(®)2 t 2 Vd— 1m2®2 — 4 Vd— 1k®4,

one can then repeat the analysis of [17]. Namely, one takes a Gaussian ansatz for ^ (t, ®, 80) (again, this is validated by the triviality of the theory)

H®|¥ (®)) = № (®)),

where the form of the Hamiltonian H® follows straightforwardly from (5):

fl$ = -1 Vd-i$ ± 1 Vd-in?$ + 4 Vd-ik$4.

The physical meaning of E0 is the vacuum energy of the scalar field, which one can safely choose to be 0.

Next, one looks for the quasi-classical solution of the Schrodinger equation of the form %(®) ~ exp(iS0(®)). The wave function of fluctuations 80 (or 0 f using terminology of the previous Subsection) in turn satisfies the Schrodinger equation

.df($,$ f)

= H8$f ($,$f),

f (t, $, 8$) = N(t) exp ^- j dd-1 p8$(p)ß(p, t)8$(p)j ,

where N and ß satisfy the equations

.d log N(t )

= Tr ß.

,9ß( p, t)

= -ß2(p, t) + 0)2(p, t),

a2(p, t) = p2 + m2 + Ik®2 + •••,

and the trace denotes integration over modes with different momenta:

C dd-1 p

(18) Trß = Vd-1j jd~-pß( p,T.

The expression for N(t) can immediately be found using the Eq. (20) and the normalization condition

VS0\f (t, S0)\2 = 1

(if N(t) = | N(t) | exp(/% (t)), the former completely determines the absolute value |N(t)|, while the latter — the phase %(t)). Then, after taking the Gaussian functional integration in (19), the density matrix can be rewritten in terms of the real part of p, t) as

p(®1, ®2)

PoVdet(Re(®1)) det(Re(®2)) Vdet(fl(®1) + fl*(®2))

x exp I -1

i J dt • (Reß(®1) - Reß(®2))

Assuming the closeness of O1 and O2 and following [17] we expand

ß(®2) ~ ß(^) + ß'(®)A + 1 ß"(Ö)A2 + ...,

where again O = 2(O1 + O2), A = i(O1 — O2), and keep terms proportional to A2 only. A straightforward but lengthy calculation shows that the exponentially decaying term in the density matrix has the form

exp (- D) = exp I - Tr

(Reß(®))2

where D is the decoherence factor, and the decoherence time can be directly extracted from this expression.

To do so, we note that is subject to the Eq. (21). When ® = ®0, one has Œ2 = a2, and Œ does not have any dynamics according to (21). However, if ®12 = ®0, Œ2 = a2. As the dynamics of ® is slow (see Eq. (17)), one can consider a as a function of the constant field ® and integrate the Eq. (21) directly. As the time t enters the solution of this equation only in combination at, one immediately sees that the factor (22) contains a term ~ t in the exponent, defining the decoherence time. The latter coincides with the expression (15) derived in the previous Section as should have been expected.

Thus, the main conclusion of this Section is that the characteristic decoherence time scale in non-renormalizable field theories akin to the k04 theory in number of dimensions higher than 4 is at least of the order of the physical correlation length f of the theory, which is taken to be large in the continuum limit. Thus, decoherence in the nearly continuum limit is very ineffective for such theories.

4 Decoherence of OFTs on curved space-times

Before proceeding to the discussion of the case of gravity, it is instructive to consider how the dynamics of decoherence of a QFT changes once the theory is set on a curved space-time. As we shall see in a moment, even when the theory is renormalizable (the number of space-time dimensions d = dup), the setup features many similarities with the case of a non-renormalizable field theory in the flat space-time discussed in the previous Section.

Consider a scalar QFT with potential V(0) = 1 m202 + 4k04 in 4 curved space-time dimensions. Again, we assume the nearly critical " T ^ Tc' case, and that is why the renormalized quadratic term 1 m202 determining the correlation length of the theory % ~ mreJiorm is set to vanish (compared to the cutoff scale A, again for definite-ness A ~ MP).

The scale % is no longer the only relevant one in the theory. The structure of the Riemann tensor of the spacetime (the latter is assumed to be not too curved) introduces new infrared scales for the theory, and the dynamics of decoherence in the theory depends on relation between these scales and the mass scale m. Without a much loss of generality and for the sake of simplicity, one can consider a dS4 space-time characterized by a single such scale (cosmological constant) related to the Ricci curvature of the background space-time. It is convenient to write

V (0) = Vo + ! m202 + ! k04,

assuming that the V0 term dominates in the energy density.

At spatio-temporal probe scales much smaller than

the horizon size H—1 ~ one can choose the state of

0 V»0

the field to be the Bunch-Davies (or Allen-Mottola) vacuum or an arbitrary state from the same Fock space. Procedures of renormalization, construction the effective action of the theory and its Feynman-Vernon influence function are similar to the ones for QFT in Minkowski space-time. Thus, so is the dynamics of decoherence due to tracing out unobservable UV modes;the decoherence time scale is again of the order of the physical correlation length of the theory:

tD ~ % ~ ^enorm'

in complete analogy with the estimate (16). This standard answer is replaced by

tD ~ h-

when the mass of the field becomes smaller than the Hubble scale, m2 ^ H^, and the naive correlation length f exceeds the horizon size of dS4. (The answer (23) is correct up to a logarithmic prefactor ~ log(H ).)

It is interesting to analyze the case m2 ^ HQ in more details. The answer (4) is only applicable for a physical observer living inside a single Hubble patch. How does the decoherence of the field 0 look like from the point of view of a meta-observer, who is able to probe the superhorizon large scale structure of the field 0?7 It is well-known [29, 30] that the field 0 in the planar patch of dS4 coarse-grained at the spatio-temporal scale of cos-mological horizon HQ"1 is (approximately) subject to the Langevin equation

3H0dj- = -m2j - kj3 + f (t),

(f (t) f (')) = 34h24 á(t - f),

where average is taken over the Bunch-Davies vacuum, very similar to (17), but with the difference that the amplitude of the white noise and the dissipation coefficient are correlated with each other. The corresponding Fokker-Planck equation

d P (t,j) dt

3H ' dj

9 V \ H3 92 P U P (t>j)) + dJP

describes behavior of the probability P(t, 0) to measure a given value of the field 0 at a given moment of time at a given point of coarse-grained space. Its solution is nor-malizable and has an asymptotic behavior

P (t ^ œ,0)

8n 2 V (j)

As correlation functions of the coarse-grained field 0 are calculated according to the prescription

jn(t, x)

JnP(j, t),

(note that two-, three, etc. point functions of 0 are zero, and only one-point correlation functions are non-trivial) what we are dealing with in the case (26) is nothing but

7 This question is not completely meaningless, since a setup is possible in which the value of V suddenly jumps to zero, so that the background space-time becomes Minkowski in the limit MP ^ 0, andthefield structure inside a single Minkowski lightcone becomes accessible for an observer. If her probe/coarse-graining scale is l > H0-1, this is the question which we are tryingto address.

Figure 3 The hierarchy of decoherence scales for a metaobserver in dSD space. RH ~ H-1 represents the Hubble radius, at comov-ing scales < H0-1 the correct physical description is the one in terms of interacting OFT in a de Sitter-invariant vacuum state; the freeze-out of modes leaving the horizon, vanishing of the decaying mode and decoherence of the background field ("master field" proceeds at comoving scales RH < l < Rdecoherence, where the latter is by a few efoldings larger than the former, see the next Section; at RH < l < Rdecoherencethe field ® and related observables are subject to the Langevin equation (24) and represent a stochastic time-dependent background of Hubble patches; at comoving scales > Rir given by (27) the stochastic field ® reaches the equilibrium solution (26) of the Fokker-Planck equation (25), and the notion of time is not well defined; the correct description of the theory is in terms of the mean/master field with partition function given by (26).

(25) a mean field theory with a free energy F = V (0) calculated as an integral of the mean field 0 over the 4—volume

~ H— of a single Hubble patch. As we have discussed in the previous Section, decoherence is not experienced as a physical phenomenon by the meta-observer at all. In fact, the coarse-graining comoving scale lc separating the two distinctly different regimes of a weakly coupled theory with a relatively slow decoherence and a mean field theory with entirely absent decoherence is of the order

Rir ~ H0 1 exp(SdS),

where SdS = is the de Sitter entropy (compare this

expression with (2)).

Overall, the physical picture which emerges for the scalar quantum field theory on dS4 background is not very different from the one realized for the non-renormalizable k04 field theory in Minkowski spacetime, see Fig. 3:

• for observers with small coarse-graining (comoving) scale l < H—1 the decoherence time scale is at most H—1, which is rather large physically (of the order of cosmological horizon size for a given Hubble patch),

• for a meta-observer with a coarse-graining (comoving) scale l > where is given by

(27), the decoherence is absent entirely, and the underlying theory is experienced as a mean field by such meta-observers.

Another feature of the present setup which is consistent with the behavior of a non-renormalizable field theory in a flat space-time is the breakdown of the effective field theory for the curvature perturbation in the IR [31] (as well as IR breakdown of the perturbation theory on a fixed dS4 background) [32], compare with the discussion in Section 3. The control on the theory can be recovered if the behavior of observables in the EFT regime is glued to the IR mean field regime of eternal inflation [33].

5 Decoherence in quantum gravity

Given the discussions of the previous two Sections, we are finally ready to muse on the subject of decoherence in quantum gravity, emergence of time and the cosmo-logical arrow of time, focusing on the case of d = 3 + 1 dimensions. The key observation for us is that the critical number of dimensions for gravity is djp = 2, thus it is tempting to hypothesize that the case of gravity might have some similarities with the non-renormalizable theories discussed in Section 3.

One can perform the analysis of decoherence of quantum gravity following the strategy represented in Section 3.2, i.e., studying EFT of the second-quantized gravitational degrees of freedom, constructing the Feynman-Vernon functional for them and extracting the characteristic decoherence scales from it (see for example [34]). However, it is more convenient to follow the strategy outlined in Section 3.3. Namely, we would like to apply the Born-Oppenheimer approximation [17] to the Wheeler-de Witt equation

fl V =

16n /2. +VP) MP R - f

Mp d hijd hki

is explicitly separated from the wave functions of the multipoles fn [17], so that the full state is factorized: ^ = f (a) ]"[n fn. Similar to the case discussed in Section 3.3, the latter then satisfy the functional Schrodinger equations

ßfn jy

= f nfn

(compare to (18)). In other words, as gravity propagates in d = 4 > djp = 2 space-time dimensions, we assume a almost complete decoupling of the multipoles fn from each other. Their Hamiltonian Hn is expected to be Gaussian with possible dependence on a: fn's are analogous to the states f described by (18) in the case of a non-renormalizable field theory in the flat space-time. (We note though that this assumption of fn decoupling might, generally speaking, break down in the vicinity of horizons such as black hole horizons, where the effective dimensionality of space-time approaches 2, the critical number of dimensions for gravity.)

The affine parameter t along the WKB trajectory is again defined according to the prescription

- = — S —

dt daS da

and starts to play a role of physical time [17]. One is motivated to conclude that the emergence of time is related to the decoherence between different WKB branches of the WdW wave function ^, and such emergence can be quantitatively analyzed.

It was found in [17] by explicit calculation that the density matrix for the scale factor a behaves as

p (a1, a2) ~ exp(— D)

with the decoherence factor for a single WKB branch of the WdW solution is given by

describing behavior of the relevant degrees of freedom (gravity + a free massive scalar field with mass m and the Hamiltonian Hm). As usual, gravitational degrees of freedom include functional variables of the ADM split: scale factor a, shift and lapse functions Nfl and the transverse traceless tensor perturbations hj. The WdW equation (28) does not contain time at all; similar to the case of the Fokker-Planck equation (25) for inflation [29] the scale factor a replaces it. Time emerges only after a particular WKB branch of the solution ^ is picked, and the WKB piece f (a) ~ exp(iS0) of the wave function ^

D ~ T7f(ai + a2)(ai - a2). Ml

We note the analogy of this expression with the expression (22) derived in the the Section 3.3: decoherence vanishes in the limit a1 = a2 (or a1 = — a2) and is suppressed by powers of cutoff MP (m/ MP can roughly be considered as a dimensionless effective coupling between matter and gravity). In particular, decoherence is completely absent in the decoupling limit MP ^ ro.

To estimate the involved time scales, let us consider for definiteness the planar patch of dS4 with a(t) ~ exp (H01). It immediately follows from (30) that the

single WKB branch decoherence only becomes effective after

Hotd > log

m3(a1 — a2)

Hubble times, a logarithmically large number of efold-ings in the regime of physical interest, when MP » OTphys ^ 0 (see also discussion of the decoherence of cosmic fluctuations in [35], where a similar logarithmic amplification with respect to a single Hubble time is found). Similarly, the decoherence scale between the two WKB branches of the WdW solution (corresponding to expansion and contraction of the inflating space-time)

~ c1elSo + c2e

can be shown to be somewhat smaller [ 17, 34]: one finds

for the decoherence factor

d~ mH0a

and the decoherence time (derived from the bound D(d) > 1) is given by

Hotd > log' Mp

latter is exponentially larger than the characteristic scale of curvature radius ~ RH of the background;we roughly expect

/m ~ Rh exp

Const.

still representing a logarithmically large number of efold-ings. Taking for example m ~ 100 GeV and H0 ~ 10-42 GeV one finds H0d ~ 300. Even for inflaitonary energy scale H0 ~ 1016 GeV the decoherence time scale is given by H0td ~ 3 inflationary efoldings, still a noticeable number. Interestingly, it also takes a few efoldings for the modes leaving the horizon to freeze and become quasi-classical.

Note that (a) H0 does not enter the expression (31) at all, and it can be expected to hold for other (relatively spatially homogeneous) backgrounds beyond dSd, (b) (31) is proportional to powers of effective dimensionless coupling between matter and gravity, which gets suppressed in the "continuum"/decoupling limit by powers of cutoff, (c) decoherence is absent for the elements of the density matrix with a1 = ±a2. These analogies allow us to expect that a set of conclusions similar to the ones presented in Sections 3 and 23 would hold for gravity on other backgrounds as well:

• we expect the effective field theory description of

gravity to break down in the IR at scales l ~ lIR8; the

• at very large probe scales / > /IR gravitational decoherence is absent;a meta-observer testing theory at such scales is dealing with the "full" solution of the Wheeler-de Witt equation, not containing time in analogy with eternal inflation sale (27) in dS spacetime filled with a light scalar field,

• at probe scales / < RH purely gravitational decoherence is slow, as it typically takes tD > RH for the WdW wave function ф ~ c1 exp(iS0[a]) + c2 exp(— iS0[a]) to decohere, if time is measured by the clock associated with the matter degrees of freedom.

Finally, it should be noted that gravity differs from non-renormalizable field theories described in Sections 2, 3 in several respects, two of which might be of relevance for our analysis: (a) gravity couples to a// matter degrees of freedom, the fact which might lead to a suppression of the corresponding effective coupling entering in the decoherence factor (3o) and (b) it effectively couples to macroscopic configurations of matter fields without any screening effects (this fact is responsible for a rapid de-coherence rate calculated in the classic paper [20]). Regarding the point (a), it has been previously argued that the actual scale at which effective field theory for gravity breaks down and gravity becomes strongly coupled is suppressed by the effective number of matter fields N (see for example [36], where the strong coupling scale is estimated to be of the order MP/vN, rather than the Planck mass MP). It is in fact rather straightforward to extend the arguments presented above to the case of N scalar fields with Z2 symmetry. One immediately finds that the time scale of decoherence between expanding and contracting branches of the WdW solution is given by

Hold > log

mN1/2 H02

(to be compared with the Eq. (32)), while the single branch decoherence proceeds at time scales of the order

m3 N3/2(a1 — a>)

Space-like interval / = / ds connectingtwo causally unconnected events.

For the decoherence between expanding and contracting WdW branches discussed in this Section and for the

emergence of cosmological arrow of time, it is important that most of the matter fields are in the corresponding vacuum states (with the exception of light scalars, they are not redshifted away), and the effective N remains rather low, so our estimations remained affected only extremely weakly by N dependence. As for the point (b), macroscopic configurations of matter (again, with the exception of light scalars with m ^ H0) do not yet exist at time scales of interest.

6 discussion

We have concluded the previous Section with an observation that quantum gravitational decoherence responsible for the emergence of the arrow of time is in fact rather ineffective. If the typical curvature scale of the space-time is ~ R, it takes at least

matter degrees of freedom at time scale of the order

log ( » 1

Ei — E2

efoldings for the quasi-classical WdW wave-function 0 ~ c1 exp(/Sb[a]) + c2 exp(-/S0[a]) describing a superposition of expanding and contracting regions to decohere into separate WKB branches. Whichever matter degrees of freedom we are dealing with, we expect the estimate (34) to hold and remain robust.

Once the decoherence happened, the direction of the arrow of time is given by the vector dt = daS0da; at smaller spatio-temporal scales than (34) the decoherence factor remains small, and the state of the system represents a quantum foam, the amplitudes c1>2 determining probabilities to pick an expanding/contracting WKB branch, correspondingly. Interestingly, the same picture is expected to be reproduced once the probe scale of an observer becomes larger than characteristic curvature scale R. As we explained above, the ineffectiveness of gravitational decoherence is directly related to the fact that gravity is a non-renormalizable theory, which is nearly completely decoupled from the quantum dynamics of the matter degrees of freedom.

If so, a natural question emerges why do we then experience reality as a quasi-classical one with the arrow of time strictly directed from the past to the future and quantum mechanical matter degrees of freedom decohered at macroscopic scales? Given one has an answer to the first part of the question, and the quantum gravitational degrees of freedom are considered as quasi-classical albeit perhaps stochastic ones, its second part is very easy to answer. Quasi-classical stochastic gravitational background radiation leads to a decoherence of

where E1>2 two rest energies of two quantum states of the considered configuration of matter (see for example [37, 38]). This decoherence process happens extremely quickly for macroscopic configurations of total mass much larger than the Planck mass MP ~ 10-8 kg. Thus, the problem, as was mentioned earlier, is with the first part of the question.

As there seems to be no physical mechanism in quantized general relativity leading to quantum gravitational decoherence at spatio-temporal scales smaller than (34), an alternative idea would be to put the burden of fixing the arrow of time on the observer. In particular, it is tempting to use the idea of [39, 40], where it was argued that quasi-classical past ^ future trajectories are associated with the increase of quantum mutual information between the observer and the observed system and the corresponding increase of the mutual entanglement entropy. Vice versa, it should be expected that quasi-classical trajectories future ^ past are associated with the decrease of the quantum mutual information. Indeed, consider an observer A, an observed system B and a reservoir R such that the state of the combined system ABR is pure, i.e., R is a purification space of the system AB. It was shown in [39] that

AS(A) + AS(B) - AS(R) - AS(A : B) = 0,

where AS( A) = S(pA, t) — S(pA, 0) is the difference of the von Neumann entropies of the observer subsystem described by the density matrix pA, estimated at times t and 0, while A S( A : B) is the quantum mutual information difference, trivially related to the difference in quantum mutual entropy for subsystems A and B. It immediately follows from (35) that an apparent decrease of the von Neumann entropy AS(B) < 0 is associated with the decrease in the quantum mutual information A S(A : B) < 0, very roughly, erasure of the quantum correlations between A and B (encoded the memory of the observer A during observing the evolution of the system B).

As the direction of the arrow of time is associated with the increase of von Neumann entropy, the observer A is simply unable to recall behavior of the subsystem A associated with the decrease of its von Newmann entropy in time. In other words, if the physical processes representing "probing the future" are possible to physically happen, and our observer is capable to detect them, she will not be able to store the memory about such processes. Once the quantum trajectory returns to the starting point

("present"), any memory about observer's excursion to the future is erased.

It thus becomes clear discussion of the emergence of time (and physics of decoherence in general) demands somewhat stronger involvement of an observer than usually accepted in literature. In particular, one has to prescribe to the observer not only the infrared and ultraviolet "cutoff" scales defining which modes of the probed fields should be regarded as environmental degrees of freedom to be traced out in the density matrix, but also a quantum memory capacity. In particular, if the observer does not possess any quantum memory capacity at all, the accumulation of the mutual information between the observer and the observed physical system is impossible, and the theorem of [39, 40] does not apply: in a sense, the "brainless" observer does not experience time and/or decoherence of any degrees of freedom (as was earlier suggested in [41]).

It should be emphasized that the argument of [39] applies only to quantum mutual information;such processes are possible that the classical mutual information Sci (A : B) increases, whereas the quantum mutual information S(A : B) decreases: recall that the quantum mutual information S( A : B) is the upper bound of Sci( A : B). Thus, the logic of the expression (35) applies to observers with "quantum memory" with exponential capacity in the number of qubits9 rather than with classical memory with polynomial capacity such as the ones described by Hopfield networks.

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