Quantum master equation for the vacuum decay dynamics

The quantum master equation required to describe the dynamics of gravity-related vacuum decay is still challenging. We aim to study this issue. Our model consists of the spacetime and scalar field with self-interaction potential. The environment is chosen as spacetime while the system is formed by the vacua of the scalar field. We demonstrate that the quantum dynamics of the vacua can be described by the Redfield equation, which can depict the evolution of both coherence and the comoving volume fraction of the vacuum. Under the Markovian limit, coherence monotonically decreases with time, leading to the initial quantum state to decohere into a classical state. This helps the understanding of the decoherence of the universe. We also highlight that in certain circumstances, the evolution of the vacuum system may display non-Markovian dynamics. In specific scenarios, the classical limit of the quantum master equation is consistent with the classical master equation. In the steady state, the dominant vacuum corresponds to the smallest cosmological constant, and various dS vacua can reach equilibrium states.


Introduction
The discovery of the gravitational waves by LIGO [1] have promoted the study to the subject of cosmological first-order phase transition. The stochastic gravitational wave background produced by this mechanism may be observed by experiments [2][3][4][5]. This may aid in understanding the physics of the early universe.
Significant theoretical progresses have been made regarding the cosmological firstorder phase transition induced by bubble nucleation. The essence of bubble nucleation is quantum tunneling. In 1980, Coleman and De Luccia studied the tunneling rate of vacuum decay related to gravity in the case where the temperature is zero [6]. They have shown that the tunneling rate is determined by the Euclidean action of the bounce solution and the initial vacuum state. The bounce is composed of a pair of instantonanti-instanton and has the O(4) symmetry [6]. Subsequently, Linde considered vacuum decay for the case of non-zero temperature [7,8]. He showed that high temperatures promote vacuum tunneling. Both the Hartle-Hawking and the Vilenkin wave function of the universe indicate that the dS universe can tunnel from nothing [9][10][11]. In 1990, several researchers showed that the transition probability from the Minkowski vacuum to the dS vacuum is not equal to zero [12,13]. Recently, Huang and Ford have pointed out that the vacuum radiation pressure fluctuations may enhance the tunneling rate [14,15]. Other important works have also been undertaken regarding vacuum decay, see [16][17][18][19] and the references therein.
Conventionally, the dynamics of the vacuum system is often described by the classical master equation (or Pauli equation) [20][21][22][23][24][25][26] Here, P i is the fraction of comoving volume of the vacuum corresponding to the cosmological constant Λ i . And Γ ij is the tunneling rate from vacuum Λ j to vacuum Λ i .
If the term 3H i P i (t) (H i is the Hubble constant of the universe Λ i ) is added to the right hand side of equation (1), then P i represents the proper volume fraction [21].
The evolution described by equation (1) is typically irreversible. Additionally, as the universe dominated by the cosmological constant exponentially expands, the multiverse landscape may be emerged in the process of bubble nucleation [27][28][29]. Therefore, the classical master equation can be used to describe the evolution of the multiverse landscape [20,21,24,29]. Throughout this study, we use units where 16πG = ℏ = k B = c = 1.
The vacuum system related to gravity may exhibit quantum natures under certain situations. For example, in the very early stages of the universe, the coherence, entanglement or other quantum natures of the vacuum system may have been significant.
Consequently, the evolution of these quantities is important. It is evident that the classical master equation (1) is inadequate for describing the evolution of these quantities or the transformation of the system from quantum to classical states. Currently, a quantum dynamic equation for the gravity related vacuum system is still under development.
In this work, we aim to develop the quantum master equation that can depict the quantum dynamics of the vacuum system. Our model consists of the FRLW spacetime and a scalar field with a self-interaction potential, where the scalar field is minimally coupled to the spacetime. We have chosen the spacetime as the environment and the vacuum states of the scalar field as the system. Starting from the Wheeler-DeWitt equation, we argued that the von Neumann equation (trace out the environment) can be utilized to describe the partial dynamics of the quantum universe. Strictly solving this equation is nearly impossible for the usual cases. We have derived the quantum master equation based on the von Neumann equation, which can be employed to represent the quantum dynamics of the vacuum system.
The quantum master equation is valid for any number of vacuum states. For simplicity, we numerically simulated a system with two types of dS vacuum states. We show that the coherence monotonically decrease over time under the Markovian limit, resulting in the transition from an initial quantum state to a final classical state. This may contribute to our understanding of the decoherence of the universe. Additionally, we noted that the vacuum system's evolution can show non-Markovian behavior in certain cases. The absolute value of the flux between vacuum states also decreases over time, eventually approaching to zero. This indicates that in the steady state, the vacuum system is in an equilibrium state, and the detailed balance is preserved. In the steady state, we found that the dominant vacuum corresponds to the smallest cosmological constant. We demonstrated that in certain cases, the classical limit of the quantum master equation is consistent with the classical master equation.
2 Partial dynamics of the quantum universe General relativity mandates that any isolated physical system must adhere to the generalized covariant principle. Taking the 3+1 decomposition for the spacetime, general covariance dictates that any isolated system must satisfy the subsequent constraints [30]: Equations (2) and (3) are the Hamiltonian constraint and the diffeomorphism constraint, respectively. H tot represents the total Hamiltonian of the isolated system.
These equations contain all the classical dynamic information of the system. The Dirac quantization procedure transform these constrains into [30] H tot |Ψ⟩ = 0; H a |Ψ⟩ = 0.
Here, |Ψ⟩ represents the wave function of the universe. Equation (4) is the Wheeler-DeWitt equation. In the following, we have sometimes omitted the operator hat for convenience. Readers can easily distinguish between c-numbers and q-numbers based on the related content.
In equations (4) and (5), there is no time variable, making it difficult to extract quantum dynamical information about the entire universe. This is known as the time problem in quantum gravity, which has not yet been perfectly solved [30][31][32]. For a homogenous and isotropic spacetime, the diffeomorphism constraints (3) and (5) are trivial and do not need to be considered. Equation (4) can be written equivalently Here, ρ tot is the density matrix of the universe.
The universe can be divided into two parts: the system and the environment. If we are just interested in the dynamical information of the system, we need to trace out the environment [33]. By using ρ to represent the reduced density matrix of the system, then one can obtain the von Neumann equation [34,35] Here, Tr B represents the partial trace for the environment. The von Neumann equation (7) forms the foundation of the theory of open quantum systems. Although the general covariance requires that dρ tot /dt = 0 for any isolated system, this does not imply that dρ/dt must also be equal to zero! Typically, as one part of the universe, the system interacts with the environment and can not be seen as an isolated one. Thus, the general covariance does not constrain the (effective) Hamiltonian of the system to be equal to zero [36]. Or in other words, the general covariance does not constrain dρ/dt to be equal to zero. Therefore, one can use equation (7) to describe the quantum dynamics of the system [36].
Looking at this from another perspective, the system interacts with the environment.
It will evolve under the influence of the environment. Energy may flow between the system and the environment. Some of the physical quantities, such as the coherence and the entanglement entropy, will change over time. This means that dρ/dt ̸ = 0, even for the systems related to gravity [36]. Extracting all the quantum dynamic information of the universe seems to be blocked by the general covariance according to the Wheeler-DeWitt equation (4), yet we can still obtain partial dynamical information of the quantum universe according to equation (7).
Starting from the Wheeler-DeWitt equation (4), one can also obtain the von-Neumann equation (7) in another way. The action of the Brown-Kuchař dust field is [33,37,38] Here, ρ and T represent the rest mass density and the dust field, respectively. g µν is the metric of the spacetime and g is the determinant of the metric. Using the canonical time gauge fixing condition T = t [38,39] (this gauge condition sets the dust field as the clock), one can obtain the Wheeler-DeWitt equation [38] (P T + H tot )|Ψ⟩ = 0.
Here, P T = −i∂/∂t is the Hamiltonian operator of the dust field and H tot is the total Hamiltonian of the universe (except for the dust field). Equation (9) can be equivalently written as dρ tot /dt = −i[H tot , ρ tot ]. After tracing out the environment, one can obtain the von Neumann equation (7) [33]. Therefore, one can also interpret the time variable in equation (7) as the dust field.
Equation (9) indicates that if the Hamiltonian of the dust field is non-zero, the clock will influence the evolution of the universe. This is unsatisfactory as it is generally expected in physical theory that the clock should have no impact on the system. One can use the constraint Tr(H tot ρ tot ) = 0 to eliminate the influence of the clock on the evolution of the universe. It is easily to prove that Tr(H tot ρ tot ) is conserved. Therefore, this constraint can typically be met by selecting the initial state of the universe appropriately [33].
In 2015, Maeda used equation (9) to study the quantum evolution of the dark energy dominated universe. He showed that the evolution of the wave packet is consistent with the classical trajectory of the universe [40]. In [33], based on equation (7), we also obtained the same conclusion for the radiation or the non-relativistic matter dominated universe. These works indicate that it is reasonable to study the quantum dynamics of the subsystem of the universe based on equation (7).
To sum up, equation (7) can be derived from the Wheeler-DeWitt equation by introducing the Brown-Kuchař dust field. All of the above arguments indicate that it should be reasonable to use equation (7) to describe the quantum evolution of the open gravitational system. The more general form of this equation can be found in [36].
Next, based on equation (7), we will derive the quantum master equation for describing the vacuum decay quantum dynamics. 3 The quantum master equation of the vacuum system

The Hamiltonian of the model
The universe is composed of spacetime and matter fields. For simplicity, we assume that the matter is a scalar field. Then the total action of the universe is [6] S tot = S g + S ϕ + S ∂M Here, S tot is the total action of the universe. S g and S ϕ are the actions of the spacetime and scalar field, respectively. Matter and spacetime is for simplicity assumed to be minimally coupled. And S ∂M is the Gibbons-Hawking surface term. R is the Ricci scalar of the 4-dimensional spacetime and ϕ denotes the scalar field. V (ϕ) is the scalar potential. V (ϕ) has some minimal values. Or in other words, the scalar field has some non-degeneracy vacua, as shown in figure 1.
And the Ricci scalar is [41] whereȧ = da/dt. Combining equations (10), (11) and (12), the action of the spacetime becomes In equation (13), the boundary term generated by the partial integral for the coordinate time variable is eliminated by the Gibbons-Hawking surface term when going from the first to the second step. Combining equations (10) and (11), the action of the scalar field becomes Noted that 2π 2 a 3 is the area of the 3-dimensional spherical surface which the sphere radius is a. From equation (13) and (14), one can obtain the Lagrangian of the spacetime and the scalar field are and respectively.
From equations (15) and (16), one can obtain the conjugate momentum of the scale factor a and the scalar field ϕ as P a = ∂L g /∂ȧ = −24π 2 aȧ and P ϕ = ∂L ϕ /∂φ = 2π 2 a 3φ , respectively. Then one can obtain the Hamiltonian of the spacetime The term 12π 2 a in (17) leads the classical dynamics of the closed (k = 1) universe to be different from the flat (k = 0) universe. And the Hamiltonian of the scalar field in the FRLW spacetime is Equation (18) shows that every term in H ϕ is coupled with the scale factor a. Or in other words, the free Hamiltonian of the scalar field is zero. The coupling between the scalar field and the spacetime is non-linear. This is caused by the minimal coupling of the scalar field with the spacetime. H ϕ represents the interaction between the scalar field and spacetime and is therefore sometimes referred to as the interaction Hamiltonian.
The total Hamiltonian of the universe is The general covariance constrains H tot = H g + H ϕ = 0 . This is the Friedmann equation. Solving this equation gives the classical dynamical information of the universe.
However, general covariance does not restrict H ϕ from being zero. Thus we expected that equation (7) can be used to describe the quantum evolution of the scalar field.
In addition, equation (19) shows that H ϕ cannot be considered a small quantity when compared to H g .
We are interested in the quantum dynamics of the scalar field, so we have chosen the scalar field as the system and the spacetime as the environment. Carrying out the canonical quantization procedure to the universe, P a and P ϕ are replaced byP a = −i∂/∂a andP ϕ = −i∂/∂ϕ, respectively. The quantum dynamics of the scalar field is then determined by the von Neumann equation (7). After tracing out the spacetime, one can obtain the quantum master equation of the scalar field. This is the task of the next subsection.

The quantum master equation
Equation (7) determined the evolution of the system. Given the initial reduced density matrix, one can obtain the quantum dynamical information of the system in any time.
Thus one can introduce the dynamical map V (t, t 0 ) [35] so that where ρ(t) and ρ(t 0 ) are the reduced density matrix of the system at the moment t and t 0 (t > t 0 ), respectively. Then, the time derivative of the reduced density matrix can be expressed as The dynamical map V (t, t 0 ) is a linear super-operator.
Assuming that there is no entanglement between the system and the environment in the initial state, and the set of orthonormal operators {F i , i = 1, 2, 3, ...; Tr(F † i F j ) = δ ij } constitutes a complete basis of the Liouville space. Then V (t, t 0 )ρ can be expressed as [35] in the Liouville space. Here, c ij are c-number coefficients. And N is the dimension of the Hilbert space of the system. Bringing equation (21) into (20), one can obtain the general form of the quantum master equation [35] dρ dt where Here, {A, B} ≡ AB +BA. In equation (23), H F L is the sum of the free Hamiltonian and the Lamb shift Hamiltonian of the system [35,43]. In equation (24), D(ρ) represents the dissipator. D(ρ) leads to non-unitary quantum evolution in the system.
In the model where the total Hamiltonian is defined by equation (19), the free Hamiltonian of the system (scalar field) is zero. If we neglect the Lamb shift Hamiltonian, we will have H F L = 0. Moreover, dissipation, irreversible non-equilibrium evolution, decoherence and other non-unitary related behaviors are all described by the dissipator D(ρ). These are induced by the interaction between the system and the environment. If we define A = 2π 2 a 3 and K = P 2 ϕ /2, the interaction Hamiltonian operator H ϕ becomes Here, K can be seen as the kinetic energy operator of the scalar field and V is the potential energy operator. Equation (25) indicates that all the non-unitary behaviors in the system are induced by the operators K and V (A −1 and A are operators of the environment. After tracing out the environment, they are absorbed into the c-number coefficients of the dissipator ) . Thus, it is reasonable to infer that the dissipator is constructed using the operators K and V , that is Here, a ij are c-number coefficients. To Summarize these arguments, the form of the quantum master equation should be It is easy to prove that the operators K and V are Hermitian operators. However, we still distinguish K (V ) and K † (V † ) because we are interested in the coarse-grained dynamical information of the system, and coarse-graining may destroy the Hermiticity of the operator. The detailed contents about the coarse-graining will be presented in the next section. The quantum master equation (27) determines the quantum dynamics of the scalar field. It is obtained above based on some arguments rather than rigorous derivation. Therefore, before coarse-graining this quantum master equation, it is useful to derive it in a more rigorous way.
For convenience, we temporarily work in the interaction picture. In the interaction picture, the von Neumann equation (7) becomes [34,35] where ρ is the reduced density matrix of the system in the interaction picture. H ϕ is the interaction Hamiltonian in the interaction picture. And ρ tot is the density matrix of the universe in the interaction picture. It should be noted that as the free Hamiltonian of the scalar field is zero, the free Hamiltonian of the total system is H g . For the same reason, K = K and V = V where K and V are the kinetic and the potential energy operators of the scalar field in the interaction picture, respectively. In addition, Thus, one can obtain Equation (28) can be equivalently written as [34,35] where ρ tot (0) is the initial state of the universe in the interaction picture. It is common to set Tr B ( H ϕ (t) ρ tot (0)) = 0 when studying the dynamics of the open quantum system [34,35]. Noted that the classical master equation is a Markovian master equation. That is, to the diagonal element of ρ(t). Thus, it seems reasonable to introduce the Markov approximation (replacing ρ tot (s) with ρ tot (t) [35]) in equation (33). Then equation (33) becomes the Redfield equation [34,35] From equation (33) to (34), although the Markov approximation is introduced, this does not mean that (34) represents a Markovian quantum master equation [35].
In this work, we consider a specific case where the entanglement between the system and the environment is small and can be neglected. Then the density matrix of the universe can be written as Here, ρ B (t) represents the density matrix of the environment in the interaction picture.
We pointed out that equation (35) Combining equations (32) and (36), one can obtain We introduce the following definitions: It is obvious that a ij depends on the state of the environment. Thus different initial states of the environment correspond to different a ij . Equations (39) and (40) show that a 12 = a 21 . Using these definitions, equation (37) can be written as Equation (42) is the quantum master equation in the interaction picture. Noted that K = K and V = V . In addition, the free Hamiltonian of the scalar field is zero.
Thus, transforming equation (42) into the Schrödinger picture, it becomes equation (27). Equations (38), (39), (40) and (41) show that a ij may change with time. Thus, equation (27) may be a non-Markovian quantum master equation [44,45]. The dynamics of the system may exhibit certain non-Markovian properties. In the case where a ij is unchanged with time, equation (27) is a Markovian quantum master equation and the related dynamical map V forms the dynamical semigroup. Because we have not introduced the secular approximation, equation (27) can also be referred to as the Redfield equation. The quantum dynamics of the scalar field is described by the Redfield equation (27).
Equation (27) determines the quantum evolution of the scalar field, but is difficult to solve. In this work, we focus on a specific situation in which the probability of the scalar field being in the vacuum states is high. Thus all the other non-vacuum states are not important and can be neglected. This condition can be easily achieved by controlling the parameters in the potential V (ϕ). In this case, the dynamics of the scalar field is reduced to the dynamics of the vacua. And the physical Hilbert space of the system is spanned by the vacuum states.
Equation (27) not only can describe the evolution of the comoving volume fraction of the vacuum but also can be used to study the evolution of specific quantum properties of the system, such as coherence. Equation (27) can also describe the evolution of the quantum superposition state. However, the classical master equation (1) can only describe the evolution of the comoving volume fraction.
To summarize, in this subsection, we derived the quantum master equation (27) using two different methods. In the first method, in order to obtain (27), we neglected the Lamb shift Hamiltonian without the justification. However, in the second method, we did not make this approximation and still obtained the same equation. This suggests that neglecting the Lamb shift Hamiltonian is a conservative approach.

Coarse graining
For simplicity, we consider a special case where there are only two potential minimal states, both of which are greater than zero, as shown in figure 2. Thus, the potential minimal states correspond to the dS spaceime. It is well known that the temperature of the dS spacetime is [46] T = 1 2π where Λ is the cosmological constant. Λ is equal to the minimal value of the potential V (ϕ). Equation (43) relates to the unit 16πG = 1. If we set 8πG = 1, then T = |2 > |1 > Λ/12π 2 . In addition, the Euclidean action of the dS spacetime is [20,25] Here, H = Λ/6 is the Hubble constant (If we set 8πG = 1, then H = Λ/3 and S E (Λ) = −8π 2 /H 2 ). Combining equations (43) and (44), one can obtain that the thermal entropy of the dS spacetime is Here, U and Z are the internal energy and the partition function of the dS spacetime, respectively.
The relationship between the thermal entropy and the number of microstates is S = lnΩ, where Ω represents the number of microstates. Thus, equation (45) shows that the number of microstates of the dS spacetime is All these microstates correspond to the same vacuum energy Λ. However, in this work, we are not interested in the differences between these microstates. Thus, we need to coarse-grain the physical Hilbert space. The microstates of the dS spacetime are coarsegrained into the same state. The coarse-grained physical Hilbert space is spanned by two coarse-grained vacuum states {|1⟩, |2⟩; ⟨µ|ν⟩ = δ µν }. The state |µ⟩ (µ, ν = 1, 2) represents the vacuum state in which the cosmological constant is Λ µ . And the degree of degeneracy of the state |µ⟩ is Ω = e −2S E (Λµ) .
The degree of degeneracy is induced by coarse graining. We take a simple example to illustrate this point. Assuming that there are three states, the distribution of these states in the steady state is uniform. That is (P 1 , P 2 , P 3 ) = (1/3, 1/3, 1/3) where P i is the fraction of the ith state. If we do not distinguish between the second and third states, then there are only two states, and the steady state distribution becomes (P 1 , P 2 ) = (1/3, 2/3). Noted that P 2 = 2 × 1/3. The factor "2" represents the degree of degeneracy induced by the coarse graining. Thus, coarse graining induces the degree of degeneracy for the states.
The quantum master equation (27) also needs to be coarse-grained. More specifically, the operators K and V need to be coarse-grained. The coefficients a ij are determined by the environment. Coarse graining for the system does not change these coefficients. In addition, V = µν V µν |µ⟩⟨ν|. As different vacuum states are orthogonal to each other, ⟨µ|ν⟩ = δ µν , thus the coarse-grained potential operator is V 11 (V 22 ) is the first (second) minimal value of the potential V (ϕ). As a comparison, the potential operator without coarse-graining is Here, the number of ). Thus, the dimension of operator (48) is e −2(S E (V 11 )+S E (V 22 )) . However, the dimension of the coarse-grained operator V is 2.
If the dS spacetime has no microstate, then the system is equivalent to a two level system. Generally, for the two level system, the kinetic energy operator is approximately proportional to the Pauli matrix σ x . And the proportional coefficient is e −S cl [47]. We use S cl to represent the Euclidean action of the instanton, which is half of the Euclidean action of the bounce (a pair of instanton-anti-instanton) solution [48]. Thus, we have K µν represents the tunneling amplitude from the state |ν⟩ to the state |µ⟩ [47]. Equation (49) shows that the kinetic energy operator does not induce tunneling within the same state. In actuality, tunneling within the same state contributes to the vacuum energy.
However, this effect is typically minor when compared to the minimal values of the classical potential. Thus one often neglect this effect [47].
Considering that dS spacetime has microstates and the number of microstates are given by equation (46), then the dimension of the operator K should be e −2(S E (V 11 )+S E (V 22 )) .
Thus, equation (49) should be generalized to Here, 0 is the zero matrix with the dimension e −2S E (V 11 ) . And Ξ is defined as The dimension of the matrix Ξ is e −2S E (V 22 ) . The physical significance of equation (50) is that if the two states correspond to the same dS spacetime, then the tunneling amplitude between these two states is zero, otherwise, the tunneling amplitude is e −S cl .
In order to show how to coarse-grain the kinetic energy operator (50). We first examine the coarse-graining of the classical master equation (1). We consider a simple example in which the system has N ′ states and the transition matrix Γ is Noted that the diagonal elements of the transition matrix do not have an influence on the evolution of the system. Thus the diagonal elements are not important. The dimension of the transition matrix (52) is N ′ . One can easily prove that the steady state distribution is (P 1 , P 2 , ..., P N ′ ) = (1/N ′ , 1/N ′ , ..., 1/N ′ ). If we do not distinguish the second to the N ′ th states (these states are seen as the same state), then the steady state distribution becomes (P 1 , P 2 ) = (1/N ′ , (N ′ − 1)/N ′ ).
After coarse graining the second to the N ′ th states into one state, the degree of degeneracy of this state is N ′ − 1. Then, the transition rate from the first state to this coarse-grained state is (N ′ − 1)ω. Thus, the transition matrix (52) should be coarse-grained into In equation (53) This example shows that if the coarse graining of the system contributes a degree of degeneracy D e to a state, then the transition rate from other states to this coarsegrained state will enhance D e times. In addition, we point out that the transition matrix (52) is Hermitian but (53) is not, indicating that the coarse graining can destroy the Hermiticity.
The coarse graining of the kinetic energy operator K is independent of the coefficients a ij . In this case where a 12 = a 21 = a 22 = 0, the quantum master equation (27) becomes Then, the relationship between the transition rate Γ µν and the tunneling amplitude Equation (55) can be equivalently written as This equation indicates that if the transition rate Γ µν enhances D e times, then the tunneling amplitude K µν should be enhanced √ D e times. Therefore, the kinetic energy operator (50) should be coarse-grained into The dimension of the coarse-grained kinetic energy operator (57) is 2. Noted that for , the factor e −S E (V 11 ) is the square root of the degree of degeneracy of the vacuum state |1⟩. And for K 21 = e −S cl −S E (V 22 ) , the factor e −S E (V 22 ) is the square root of the degree of degeneracy of the vacuum state |2⟩. Thus, the coarse-grained kinetic energy operator is not a Hermitian operator, K ̸ = K † . This is the reason that we distinguish between K and K † in the quantum master equation (27).
To sum up, in this section, we have coarse-grained the quantum master equation.
If we consider the case where the potential has two minimal values, then the dimension of the coarse-grained physical Hilbert space is two. Thus, the dimensions of the coarsegrained reduced density matrix, kinetic energy operator and potential energy operator all are two. Coarse graining has destroyed the Hermiticity of the kinetic energy operator.
Next, we will simulate the dynamics of the vacuum system for certain special situations using the coarse-grained quantum master equation.

Simulations for certain cases
The scalar potential differs among various models. For convenience, we consider the washboard scalar potential [49] V (ϕ) = αϕ + βcosγϕ + δ, where, 0 < ϕ < 4π/γ. α, β and γ are some positive parameters. In this interval, the potential has two vacuum states. We constrain that δ > β and α/γ is small compared to V (ϕ). Under these constraints, the thin-wall approximation is valid and the energy of the vacuum states is greater than zero. Thus the potential minimal states are dS vacua. Equation (58) shows that ϕ 1 = π/γ corresponds to the first dS vacuum state |1⟩, and ϕ 2 = 3π/γ corresponds to the second dS vacuum state |2⟩. The cosmological constant of the first (second) dS vacuum is Λ 1 = V 11 = δ − β + απ/γ (Λ 2 = V 22 = δ − β + 3απ/γ). Noted that Λ 1 < Λ 2 . Thus, sometimes |1⟩ and |2⟩ are referred to as the true vacuum state and false vacuum state, respectively. As α/γ is a small quantity, then V 22 −V 11 = 2απ/γ is also small. Under this condition, the thin-wall approximation is valid and the wall of the Euclidean bounce is small in thickness [6].
Introducing the definition then the potential (58) can be written as [6] V Here, o(α/γ) is a small quantity. U (ϕ) is similar to the axion field potential [41].
According to the definition (59), one can easily show that U (ϕ 1 ) = U (ϕ 2 ). The dS vacuum is a bubble, and the surface tension of the bubble is [42] Bringing equation (59) into (61), one can obtain In [18], de Alwis and other researchers present the detailed calculation for the Euclidean action S cl . The result is where In equations (63) One can learn from equations (38)-(41) that the coefficients a ij are difficult to calculate. Thus, in this work, we consider some special cases. Even so, valuable insights into the quantum dynamics of the vacuum system can be gained. The classical master equation (1) is a Markovian master equation. Hence, we investigate the Markovian limit of the quantum master equation (27). In this case, one can neglect the variation of the coefficients a ij . Bringing equations (47), (57), (62), (63) and (64) into (27), then one can simulate the evolution of the vacuum system.

Figures 3, 4 and 5 show the evolution of some quantities of the system in the
Markovian limit. In figure 3, we set a 12 = a 21 = a 22 = 0. In this case, the quantum master equation (27) becomes (54). Noted that the transition rate Γ ij in the classical master equation (1) is [6,21,26] In equation (65), 2S cl is the Euclidean action of the bounce solution, while S(Λ j ) refers to the thermal entropy of the dS vacuum Λ j . In some references [24,26], the prefactor A j is chosen as (4π/3)H −3 j . In [21], Linde set A j = 1. The prefactor is not important for this work (its effect can be included by modify the entropy). Thus, we also set Combining equations (55), (57) and (65), one can show that a 11 = e 2S E (Λ 1 )+2S E (Λ 2 ) .
In figure 4, we set a 11 = a 12 = a 21 = 0 and a 22 = 1. In this case, the quantum master equation (27) becomes And in figure 5, we set a 11 = e 2S E (Λ 1 )+2S E (Λ 2 ) = a 12 = a 21 = a 22 . Then equation (27) becomes The initial state of the system is ρ(0). For convenience, we rewrite ρ(0) into the vector form Bringing a 12 = a 21 = a 22 = 0 into the quantum master equation (27), one can easily prove that in the steady state, the fraction of the dS vacuum Λ i is S(Λ i ) is the thermal entropy of the dS vacuum. Equation (69) is also the steady state solution of the classical master equation (1). This result is consistent with the Hartle-Hawking wave function [9]. Equation (69) shows that a larger cosmological constant corresponds to a lower fraction. Thus, in the steady state, the dominant vacuum corresponds to the smallest cosmological constant.
Decoherence is the process of decreasing coherence. The system is equivalent to a two level system. Thus one can also introduce the decoherence function [45] to describe the decoherence process. In figure 3(c), the initial state is chosen as u(0) = (1, 1, 1, 0). The coherence of the initial state is not zero. One can see from figure 3(c) that both the coherence and the decoherence function decrease monotonically over time and ultimately tend towards zero. This indicates that decoherence has occurred and the initial quantum state will decohere into the final classical state.
The issue of decoherence of the universe has not been completely resolved yet, although significant numbers of the studies have been performed [51][52][53][54][55][56][57][58][59][60]. In [51][52][53][54][55]    Thus, in the initial state, the fraction of the true vacuum state is larger than the false vacuum state. The flux between the true vacuum state and the false vacuum state is defined as [61,62] From figure 3(d), one can see that the value of F 12 (t) is negative. This indicates that the fraction of the true vacuum state decreases with time. As the flux approaches zero, the distribution of the vacuum state reaches a steady-state, wherein detailed balance is preserved. However, in the non-steady state, the flux is non-zero, indicating that the evolution is irreversible.   showed that vacuum decay dynamics may entail non-Markovian correlations [63].

Conclusions and discussions
There is no time variable in the Wheeler-DeWitt equation. This induced the time problem in quantum gravity. However, our interest lies in the dynamical information of the subsystem of the universe. The subsystem is not an isolated system, usually the general covariance does not constrain that the (effective) Hamiltonian of the subsystem must be zero. Thus, maybe one can use the von Neumann equation (7) to describe the quantum dynamics of the subsystem. Equation (7) can also be derived from the Wheeler-DeWitt equation by introducing the Brown-Kuchař dust field. Therefore, the time variable in equation (7) can also be interpreted as the dust field. Although the generalized covariant principle leads to the difficulty to obtain all quantum dynamical information of the universe, the von Neumann equation (7) indicates that we can still obtain partial quantum dynamical information of the universe. The von Neumann equation (7) serves as the fundamental equation in our work.
Starting from the von Neumann equation (7), after tracing out the environment, we obtained the quantum master equation (27) by two different methods. In the derivation process, we did not introduce the secular approximation. Thus equation (27) can also be referred to as the Redfield equation. We also did not introduce the Born approximation since the interaction Hamiltonian cannot be viewed as a small value. Thus, the coefficients a ij may change over time. Therefore, equation (27) may exhibit non-Markovian properties in some situations. Equation (27) can be used to describe the evolution of the comoving volume fraction of the vacua, as well as certain quantum quantities of the vacuum system such as coherence. Equation (27) can also describe the evolution of the superposition state of the vacuum. However, the classical master equation (1) can only describe the evolution of the comoving volume fraction of the vacuum.
The entropy of the dS spacetime is not zero, which indicates that the dS spacetime has micro degrees of freedom. We are not interested in the microstates. Thus we need to coarse-grain the physical Hilbert space of the system. Coarse graining contributes a degree of degeneracy to the vacuum state and destroys the Hermiticity of the kinetic energy operator. Similarly, it can also destroy the Hermiticity of the Hamiltonian operator. Therefore, if we are interested in the coarse-grained dynamical information, the evolution of the coarse-grained system may be non-unitary even for an isolated system. In other words, coarse grained information is not conserved.
Finally, we simulated the quantum master equation (27) in the Markovian limit. We found that in some cases, the classical limit of the quantum master equation is consistent with the classical master equation. We show that both the coherence and decoherence function decrease monotonically with time. This indicates that the decoherence has emerged. Consequently, the initial quantum state will decohere to the final classical state. This helps the understanding of the decoherence of the universe. We also show that the absolute value of the flux decreases with time and eventually approaches zero, indicating that the evolution in time is irreversible and in the steady state, detailed balance is maintained. Therefore, in the steady state, different vacua are in equilibrium, and the dominant vacuum corresponds to the smallest cosmological constant.