A model of spinfoam coupled with an environment

Abstract

In this paper, an open quantum system theory for spinfoams is developed. This new formalism aims at deriving an effective Lindblad equation to compute the reduced dynamics of a quantum gravitational field. The system parameters are determined from numerical ab initio calculations, based on the spinfoam formalism. This theoretical proposal is illustrated by means of examples. The decoherence effect can induce the relaxation of the quantum gravitational state toward a sate of a small area. This is analogue to the well-known Purcell relaxation of QED, for which the qubits are replaced by the spin-network representation of the gravitational field. Some thermodynamic properties of these systems are computed, and several issues with the thermal time hypothesis are underlined. Moreover, the results suggest that further approximations can be performed to study reduced dynamics of quantum space-time.

Appendices

A Some mathematical definitions

This appendix is devoted to some mathematical definitions, which are omitted in Sect. 2 for conciseness. Further technical details on this subject can be found in Refs. [2, 5, 19, 20].

Definition 4

A spin-network $\psi$ is a triple $(\varGamma ,\chi , \imath$)consisting of:

• 1. a 1-dimensional oriented complex $\varGamma$, represented by a graph whose vertices are points of the space-time manifold and edges are paths connecting the points.

• 2. a labeling $\chi$ of each edge e by a unitary irreducible representation $\chi _e$ of ${\mathfrak {H}}$, where ${\mathfrak {H}} \subseteq {\mathfrak {G}}$ is a subgroup of the gauge group of the theory. Elements of ${\mathfrak {H}}$ associated with edges encode the structure of space at the classical level.

• 3. a labeling $\imath$ of each vertex v by an intertwiner

  $$\begin{aligned} \imath _v:\chi _{e_1} \otimes \cdot \cdot \cdot \otimes \chi _{e_n} \rightarrow \chi _{e'_1} \otimes \cdot \cdot \cdot \otimes \chi _{e'm} \end{aligned}$$

  where $e_1,\ldots ,e_n$ are the edges incoming to v and $e'_1,\ldots ,e'_m$ are the edges outgoing from v.

It is usual to consider ${\mathfrak {G}} = SL(2,{\mathbb {C}})$ and ${\mathfrak {H}} = SU(2)$ in 3+1D gavity, or ${\mathfrak {G}} = {\mathfrak {H}} = SU(2)$ in 3D Euclidan gravity [5]. For reasons that will become clearer below, it is preferred to substitute the names vertex and edge by node and link.

Definition 5

A spinfoam F is a triple $({{\tilde{\varGamma }}},{{\tilde{\chi }}},{{\tilde{\imath }}}$)consisting of:

• 1. a 2-dimensional oriented complex ${{\tilde{\varGamma }}}$.

• 2. a labeling ${{\tilde{\chi }}}$ of each face f by a unitary irreducible representation ${{\tilde{\chi }}}_e$ of ${\mathfrak {G}}$.

• 3. a labeling ${{\tilde{\imath }}}$ of each vertex e by an intertwiner

  $$\begin{aligned} {{\tilde{\imath }}}_e:{{\tilde{\chi }}}_{f_1} \otimes \cdot \cdot \cdot \otimes {{\tilde{\chi }}}_{f_n} \rightarrow {{\tilde{\chi }}}_{f'_1} \otimes \cdot \cdot \cdot \otimes {{\tilde{\chi }}}_{f'_m} \end{aligned}$$

  where $f_1,\ldots ,f_n$ are faces incoming to e and $f_1',\ldots ,f_m'$ are faces outgoing from e.

Notice that here, vertices are D dimensional objects, edges are D-1 dimensional objects, and faces are D-2 dimensional objects.

Definition 6

Let $\psi$ be a spin-network $\psi =(\varGamma ,\chi ,\imath )$. A spinfoam $F:\emptyset \rightarrow \psi$ is a triple $({{\tilde{\varGamma }}},{{\tilde{\chi }}}, {{\tilde{\i }}})$ verifying the definition 5, and:

• 1. for any link l of $\varGamma$, $\chi _l = P({{\tilde{\chi }}}_f)$ if f is incoming to l, and $(\chi _l)^* = P({{\tilde{\chi }}}_f)$ otherwise.

• 2. for any node n of $\varGamma$, $\imath _n = P({{\tilde{\imath }}}_e)$.

P is a map from unitary irreducible representations of ${\mathfrak {G}}$ to unitary irreducible representations of ${\mathfrak {H}}$. It is an input of the theory.

This definition gives the transition amplitude from the vacuum to a given spin-network. The evolution from a spin-network to another one is given by:

Definition 7

A spinfoam $F: \psi \rightarrow \psi '$ is defined by $F:\emptyset \rightarrow \psi ^* \otimes \psi '$, where $^*$ denotes the dual of a spin-network.

A spinfoam model is defined with a specific choice of (${\mathfrak {G}},{\mathfrak {H}},\imath ,P,Z$).

Proof of proposition 1

This appendix is devoted to the proof of proposition 1.

Proof

We provide here a simple proof with a minimum of details. Further information is given in Reference [17].

We are looking for a master equation $d_t \rho _e = \mathcal {L}_e \rho = [\mathcal {L}_{e,0} + \epsilon \mathcal {L}_{e,1} + o(\epsilon ^2) ]\rho _e$ that provides the evolution of the system up to an error of $\epsilon ^2$. To determine this equation, we introduce the map:

$$\begin{aligned} \rho (t) = K( \rho _e(t) )= \sum _{m\ge 0} \epsilon ^m K_m (\rho _e(t)). \end{aligned}$$

Then, by definition we have:

$$\begin{aligned} \frac{d \rho }{dt} = \mathcal {L}_0 K(\rho _e) + \epsilon \mathcal {L}_1 K(\rho _e) = K(\mathcal {L}_e \rho _e). \end{aligned}$$

Using the expansion of K in powers of $\epsilon$, we deduce that:

$$\begin{aligned} \mathcal {L}_0 K_0(\rho _e) = K_0(\mathcal {L}_{e,0} \rho _e) ~~(m=0), \end{aligned}$$

(23)

and,

$$\begin{aligned} \mathcal {L}_0 K_1(\rho _e) + \mathcal {L}_1 K_0(\rho _e) = K_0(\mathcal {L}_{e,1} \rho _e) ~~(m=1). \end{aligned}$$

(24)

Since we are interested in dynamics restricted in $\mathcal {D}_0$, we define $K_0$ as a projector on $\mathcal {D}_0$: $K_0 (\rho ) = P_0 \rho P_0 ^\dagger$, with $P_0 = \sum _{n=1}^{dim \mathcal {H}_0} | n \rangle \langle n |$, with $\{| n \rangle \}$ a basis of $\mathcal {H}_0$. From the definition of $\mathcal {D}_0$, we deuce that Eq. (23) gives $\mathcal {L}_{e,0}=0$. We also introduce the Kraus map of a solution of the unperturbed system when $t \rightarrow \infty$: $U_0\rho (0)=\lim _{t \rightarrow \infty }e^{t\mathcal {L}_0}\rho (0) \equiv \sum _\mu M_\mu \rho (0) M_\mu ^\dagger$, with $\{ M_\mu \}$ an ensemble of operators such that $\sum _\mu M_\mu ^\dagger M_\mu = {\mathbb {I}}_{\mathcal {H}} \}$.

To arrive at the desired result, we apply the super-operator $U_0$ on Eq. (24). Since $U_0.\mathcal {L}_0 =0$, we have:

$$\begin{aligned} U_0 \mathcal {L}_1 K_0(\rho _e) = U_0 K_0(\mathcal {L}_{e,1} \rho _e) \end{aligned}$$

(25)

Moreover, $U_0$ leaves $K_0$ unchanged. Therefore,

$$\begin{aligned}&U_0 \mathcal {L}_1 K_0(\rho _e) = K_0(\mathcal {L}_{e,1} \rho _e) \end{aligned}$$

(26)

$$\begin{aligned}&\quad \Rightarrow ~ K_0(U_0 \mathcal {L}_1 K_0(\rho _e)) = \mathcal {L}_{e,1} \rho _e \end{aligned}$$

(27)

In the second line, we have used $K_0^2 = K_0$ and the fact that $\mathcal {L}_{e,1}$ is restricted to $\mathcal {H}_0$. The structure of the operator $\mathcal {L}_1$ is by assumptions:

$$\begin{aligned} \mathcal {L}_1 \rho _e \equiv \sum _{n = 1}^{dim \mathcal {H}_0} \left( P_n \rho _e P_n ^\dagger -\frac{1}{2} P_n^\dagger P_n \rho _e - \frac{1}{2} \rho _e P_n^\dagger P_n \right) ~~,~~ P_n = | n \rangle \langle n |. \end{aligned}$$

Then,

$$\begin{aligned} \mathcal {L}_{e,1} \rho _e \equiv \sum _{n,m,\mu } |M_{\mu ,nm}|^2 \left( | n \rangle \langle m | \rho _e | m \rangle \langle n | -\frac{1}{2}| m \rangle \langle n | | n \rangle \langle m | \rho _e - \frac{1}{2} \rho _e | m \rangle \langle n | | n \rangle \langle m | \right) \end{aligned}$$

To finish the proof, we define $\kappa _{nm} = \sum _\mu |M_{\mu ,nm}|^2$. $\square$

Numerical investigation of hypothesis 3

In this section, we present briefly the numerical investigation that leads us to the hypothesis 3. Contrary to examples in the main text, the 3D theory with gauge group ${\mathfrak {G}} = SU(2)$ is used. The transition amplitude is given by [5, 48]

$$\begin{aligned} W_{{{\tilde{\varGamma }}}} (j_l) = \mathcal {N}_{{{\tilde{\varGamma }}}} \sum _{j_f} \prod _f (-1)^ {j_f} (2 j_f +1) \prod _v (-1)^{\sum _{a=1}^6 j_{v,a}} \left\{ \begin{array}{ccc} j_{v,1} &{} j_{v,2} &{} j_{v,3} \\ j_{v,4} &{} j_{v,5} &{} j_{v,6} \end{array} \right\} , \end{aligned}$$

(28)

with $j_l$ the spin label of the link l, of a boundary spin-network. $j_f$ is the spin label of the face f of the spinfoam, and v is a vertex of the foam. (v, a) denotes the face a associated with the vertex v, $\{\ldots \}$ is Wigner’s 6j-symbol, and $\mathcal {N}_{{{\tilde{\varGamma }}}}$ is a normalization factor.

In the computation, we consider foams with $V= 2,3$ or 4 vertices glued one-by-one with at most one edge. The 2-complex is defined without bubbles. The transition amplitude can be written as $W(\psi _{out},\psi _{in})$, with $\psi _{in}$ a spin-network formed by the boundary of the firsts vertices, and $\psi _{out}$ the spin-network formed the boundary of other vertices. We can take symmetric networks, but it is not necessary for our purpose. For each in and out states, we are interested in a reduced number of degrees of freedom, given by a sub-spin-network, made of a single node and its three boundary links. The reduced Hilbert space $\mathcal {H}_r$ is defined by the formal association of its canonical basis to a set of specific sub-spin-networks.

For example, if we consider a sub-spin-network composed of 1 node and 3 adjacent links with spin numbers (1, 1, 1) and (2, 2, 2), we have: $\mathcal {H}_r = {\mathbb {C}}^2$, $e_1 \rightarrow (1,1,1)$ and $e_2 \rightarrow (2,2,2)$. For the numerical calculation presented below, sub-spin-networks are generated randomly, with dim$(\mathcal {H}_r) = 10$.

The transition amplitude between these sub-spin-networks is calculated using a large number ($\sim 10^6$) of different spin-networks generated randomly. This defines a map $\mathcal {H}_r \otimes \mathcal {H}_r \rightarrow {\mathbb {C}}$ such that $(n,m) \rightarrow W_{nm}$. The goal is to find an approximation of $W_{nm}$ with a simpler model. Here, we choose for the simplest model, a foam made of two unconnected vertices, such that $W \propto W_{v_1}(j_l)W_{v_2}(j_l')$. With a minimization algorithm (the NMinimize function of Mathematica), we find a linear superposition of spin-networks such that the transition amplitudes between sub-spin-networks of the simplified system are the closest to the values given by the initial system. To quantify the difference between models, the following cost function is used: $C= \Vert \mathbf {W}_1/\Vert \mathbf {W}_1 \Vert - \mathbf {W}_2/\Vert \mathbf {W}_2 \Vert \Vert$, where $\mathbf {W}_i$ is a vector of dimension dim$(\mathcal {H}_r)^2$ whose components are transition amplitudes $W_{nm}$ given by the model i. Vectors are normalized in the definition of the cost function in order to take into account the normalization factor $\mathcal {N}_{{{\tilde{\varGamma }}}}$ introduced in Eq. (28).

Fig. 8

a Minimum value of C found after numerical minimization as a function of the number of vertices V taken into account in the foam. b Probability density function of the value of C in the case $V=4$. The minimum value is $\min [C]=0.0056$ and the maximum value is $\max [C]=1.9998$

Figure 8 show the minimum value of C after numerical minimization of the cost function, and the probability density function to find a certain value of C. The second graph is computed with$10^5$ random spin-networks of the simplified model. A step of 0.1 is used to compute the histogram. We observe that very low values of C are reachable, but these low values are not representative of general values of C (the average value is $\approx 1.5$). This emphasizes that a specific design of $\psi _{bath}$ is required for the approximation. Due to many linear dependence between parameters, and the large number of parameters, the numerical optimization is not a difficult task, the convergence is fast. We also notice that the approximation is better for initial spinfoams of large dimension. This is explained by the fact that, the boundary of two distant vertices are weakly correlated.