# Emergent \(\alpha \)-like fermionic vacuum structure and entanglement in the hyperbolic de Sitter spacetime

## Abstract

We report a non-trivial feature of the vacuum structure of free massive or massless Dirac fields in the hyperbolic de Sitter spacetime. Here we have two causally disconnected regions, say *R* and *L* separated by another region, *C*. We are interested in the field theory in \(R\cup L\) to understand the long range quantum correlations between *R* and *L*. There are local modes of the Dirac field having supports individually either in *R* or *L*, as well as global modes found via analytically continuing the *R* modes to *L* and vice versa. However, we show that unlike the case of a scalar field, the analytic continuation does not preserve the orthogonality of the resulting global modes. Accordingly, we need to orthonormalise them following the Gram–Schmidt prescription, prior to the field quantisation in order to preserve the canonical anti-commutation relations. We observe that this prescription naturally incorporates a spacetime independent continuous parameter, \(\theta _{\mathrm{RL}}\), into the picture. Thus interestingly, we obtain a naturally emerging one-parameter family of \(\alpha \)-like de Sitter vacua. The values of \(\theta _{\mathrm{RL}}\) yielding the usual thermal spectra of massless created particles are pointed out. Next, using these vacua, we investigate both entanglement and Rényi entropies of either of the regions and demonstrate their dependence on \(\theta _{\mathrm{RL}}\).

## 1 Introduction

The de Sitter (dS) spacetime is a maximally symmetric manifold endowed with a positive cosmological constant. It is physically interesting in many ways. First, owing to the recent observed phase of the accelerated cosmic expansion, there is a strong possibility that our current universe is dominated by a small but positive cosmological constant at large scales. Second, the high degree of homogeneity and isotropy of the current universe at large scales indicate that our early universe went through an inflationary phase described by a quasi de Sitter spacetime [1]. Being accelerated with expansion and endowed with a cosmological event horizon, the dS offers interesting thermal and other field theoretic properties, we refer our reader to e.g., [2, 3, 4, 5, 6] and also references therein.

It is interesting to investigate the long range quantum correlations between two observers in the dS space, not only causally separated but so by a large distance, say of the order of the superhorizon size. This issue was first addressed in [7] for a scalar field theory using the coordinatisation of [8, 9], known as the hyperbolic or open chart describing two casually disconnected expanding regions in dS, as denoted by regions *R* and *L* in Fig. 1. Since *R* and *L* are separated by an entire causally disconnected region *C*, the framework described by Fig. 1 offers a very natural stage to investigate such long range non-local quantum correlations. Being motivated by this, we wish to investigate in the following the entanglement properties of the Dirac fermionic vacua in the hyperbolic dS.

Let us first briefly review the case of a real scalar field [7]. One first defines orthonormal local basis mode functions having supports either in *R* or in *L*, with definite positive or negative frequency behaviour in the asymptotic past. One makes a field expansion using them and defines the local vacuum as a direct product between the vacua of *R* and *L*. However, if there is any correlation between these local states, clearly there must exist some mode functions having supports in *both * *R* and *L*. Such *global* modes are obtained by analytically continuing the local modes from one region into the another along a complex path going through *C* [8, 9]. One then makes a field quantisation using the global modes and defines a suitable global vacuum. The field quantisations in terms of the local and global modes give a Bogoliubov relation, yielding in turn an expansion of the global vacuum in terms of the local states. It follows that the states belonging to *R* and *L* are entangled. The entanglement entropy density is computed using the reduced density operator, found by tracing out the states belonging to either of the regions. Being originated from the long range correlations, the entanglement entropy thus found will not be proportional to any area. This procedure will be more explicit in the due course of the discussion.

A lot of effort has been given to explore quantum entanglement in dS so far, in various coordinatisations, e.g. [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. They not only involve the computations of the entanglement or the Rényi entropy (e.g. [29] and references therein), but also studies of other measures like Bell’s inequality, entanglement negativity and discord, in the Bunch-Davies or more general \(\alpha \)-vacua (e.g. [30, 31, 32] and references therein). We further refer our reader to e.g. [33, 34, 35, 36, 37, 38, 39, 40, 41] for discussions on various aspects of quantum correlation in dS, including their possible observational consequences.

Most of the references cited above focus on the scalar field theory and discussions on other spin fields seem sparse. In [42], the entanglement properties of a Dirac fermion in the hyperbolic dS was discussed and certain qualitative differences with a scalar field were pointed out. See [39, 43, 44, 45] for interesting aspects of fermionic entanglement in cosmological spacetimes. See also [46, 47] for discussions on fermionic entanglement in the Rindler spacetime.

In this work we wish to point out a further qualitative difference of the fermionic field theory in the hyperbolic dS from that of the scalar [7], which seems to have been missed in the earlier literature, as follows. After reviewing the construction of the local orthonormal modes and the global ones in Sect. 2, we show in Sect. 3 that those global modes are *not* orthogonal, as opposed to the scalar field theory. It follows from a simple and generic result of the canonical quantum field theory (see [48, 49] and references therein), that if one attempts to do field quantisation with such non-orthogonal global modes, the resulting Bogoliubov structure would *not * preserve the desired anti-commutation relations for the creation and annihilation operators corresponding to the field quantisation of the global modes. Thus we need to orthonormalise those global modes before making any sensible field quantisation. We argue in Sect. 3 that such orthonormalisation is *never* unique *a priori* in the present scenario, giving rise to a continuous, one parameter family of global vacua. In other words, we wish to demonstrate a natural emergence of de Sitter \(\alpha \)-like vacua for Dirac fermions in the hyperbolic de Sitter spacetime. This is the main result of this paper. Using such vacua, we investigate next the fermionic entanglement properties in Sect. 4.

We shall work with the mostly positive signature of the metric in \((3+1)\)-dimensions and will set \(c=G=\hbar =\kappa _B=1\) throughout.

## 2 The Dirac modes in hyperbolic dS

In this section we shall review the basic geometry of the hyperbolic dS, the local Dirac modes and their analytic continuation to form the global modes. The detail of the following can be seen in [8, 9, 42, 50, 51].

*R*,

*L*and

*C*, as shown in Fig. 1. The relationship between the Lorentzian (

*t*,

*r*) and Euclidean coordinates \((\tau , \rho )\) of these three regions is given by,

*r*and

*t*are dimensionless in the units of

*H*.

*R*and

*L*[42, 50, 51]. In the region

*R*, the four Dirac modes are given by,

*p*is continuos and positive. The temporal parts \(u_p\), \(v_p\) appearing in Eq. (4) are given by the hypergeometric functions

*L*, consistent with Eq. (1) are given by,

It is easy to see that in the asymptotic past, \(t_R\rightarrow 0\) or \(t_L\rightarrow 0\), each of the sets Eqs. (4) and (7) splits into two positive frequency (representing particles) and two negative frequency (representing anti-particles) modes.

*R*or

*L*are orthonormal, i.e.,

Note also that since we are interested in constructing field theory in \(R\cup L\) and the region *C* is causally disconnected from them, we shall not be concerned about the modes in *C* in this work. We further refer our reader to [42, 53] on the normalisability of the mode functions for Dirac spinors and massive vectors in that region.

*R*and

*L*, we also require, apart from the above local modes, the notion of global modes which have support in \(R\cup L\). First noting that the functions in Eq. (6) have branch points at \(z=\pm 1\) and \(\infty \), using Eq. (1) and some identities of the hypergeometric function [52], such global modes are achieved by analytically continuing the local modes of

*R*to

*L*or of

*L*to

*R*along a complex path going through

*C*[42]. We review this procedure in Appendix A. The resulting global mode functions originating from the \(R\rightarrow L\) analytic continuation are given by,

*do not*form an orthogonal set under the global inner product, in contrast to the scalar field theory [8]. As we have discussed in Sect. 1, we cannot simply treat such non-orthogonal modes as our global basis modes in the field quantisation [48, 49]. Hence we first need to orthonormalise the modes of Eqs. (10), (12).

## 3 Constructing the global orthonormal modes

*not*orthogonal (although inter-pair orthogonality is satisfied) with respect to the global inner product, given by

*a priori*in our orthogonalisation scheme, we must treat both the solutions simultaneously in an

*equal*footing. This can be achieved by introducing a continuous parametrisation to obtain two orthogonal global modes,

Note that the quantisation of the Dirac field with the modes of Eq. (16) will effectively thus give de Sitter \(\alpha \)-vacua like structure (see e.g. [30, 31, 32, 54]). However, we emphasise that unlike the usual cases of such vacua, introducing the parametrisation \(\theta _{\mathrm{RL}}\) was *a priori necessary* in our current scenario, in order to maintain sufficient generality in the orthogonalisation procedure. This is the main result of this work. We shall see later that \(\theta _{\mathrm{RL}}=0, \pi /2\) values correspond to the usual thermal distribution of created massless particles in *R* or *L*.

## 4 Computation of the entanglement and the Rényi entropies

### 4.1 The field quantisation and the Bogoliubov coefficients

*all*other anti-commutators vanish. We define the local vacua \( |0\rangle _R, |0\rangle _L\),

*p*,

*j*,

*m*, for the sake of brevity. Similarly, by taking the inner products with the seven other modes \(\Psi _2,\,\Psi _3 \dots , \Psi _8\), we obtain

We once again emphasise here that had we not properly orthonormalised our global modes, we would not have retained the above canonical anti-commutation structure, essential to preserve the spin-statistics of the field theory.

*N*is the normalisation. Also, since the operators

*c*’s and

*d*’s anti-commute Eq. (19), we may further decompose the vacua defined in Eq. (20) as,

*R*and

*L*with respect to the global vacuum. Since the states belonging to

*R*and

*L*cannot be factored out, those pairs will be entangled. Also, since \(|0\rangle ^{(1)} \) depends upon \(\theta _{\mathrm{RL}}\) through \(\xi _1\) and \(\xi _2\), the particle creation and the entanglement will also depend upon it.

### 4.2 The entanglement entropy

*L*region, inaccessible to an observer in

*R*. We thus obtain the reduced density operator in

*R*,

*R*states will give identical results. We find a matrix representation of \(\rho _R\) in terms of the

*R*-state vectors,

*p*and \(\theta _{\mathrm{RL}}\) value) is given by

*R*and

*L*is obtained by integrating over all

*p*modes. The final expression of the entanglement entropy is obtained by further integrating the result over the purely spatial section of Eq. (2). Since \(S(p, m; \theta _{\mathrm{RL}})\) has no spatial dependence, the integration, being done over a non-compact space, would diverge. Accordingly, one needs to put a cut off at some ‘large’ radial distance. The resultant regularised volume integral equals \(2\pi \), see [7] for details. The regularised entanglement entropy then equals

(a) In Fig. 2, the curves corresponding to \(\theta _{\mathrm{RL}}=0,\, \pi /2\), are exactly coincident and they show maximal \(R-L\) entanglement for all values of \(\nu ^2\). The coincidence corresponds to the fact that the coefficients \(\xi _{1,2}(\theta _{\mathrm{RL}}=0)=\pm \xi _{1,2}(\theta _{\mathrm{RL}}=\pi /2)\), in Eq. (28). (b) While most of the curves in Fig. 2 are monotonic, the curve corresponding to \(\theta _{\mathrm{RL}}=\pi /3\) shows extrema. (c) For any given value of \(\theta _{\mathrm{RL}}\), the entanglement entropy has its maximum value in the massless limit, \(\nu ^2=9/4\). This might correspond to the fact that in this limit the creation of particle–antiparticle pairs is energetically most favourable. Since such pairs are entangled, Eq. (30), it is perhaps reasonable to expect that the entanglement entropy also gets its maximum value in the massless limit.

### 4.3 The Rényi entropy

*q*values remains the same as that of the entanglement entropy. In particular, a) the values \(\theta _{\mathrm{RL}}=0, \pi /2\) gives maximum Rényi entropy for all values of \(\nu ^2\) and b) the extrema for \(\theta =\pi /3\) is still present.

## 5 Discussions

In this paper we have investigated the entanglement and the Rényi entropies between the *R* and *L* states of the Dirac field in the hyperbolic dS spacetime, Eqs. (2), (3), as a measure of the long range non-local quantum correlations between these two regions.

The chief results of these paper could be summarised as follows. First, the *natural emergence* of the continuous, one parameter family of global modes (cf. Sect. 3) and vacua, Eq. (30). Such vacua have structures similar to the de Sitter \(\alpha \)-vacua, though they have originated here from the mere necessity of an *a priori* general orthonormalisation scheme for the global modes. Such orthonormalisation of the modes is necessary to preserve the canonical anti-commutation structure of the field theory [48, 49]. Second, we have seen in Sects. 4.2, 4.3 that a) \(\theta _{\mathrm{RL}}=0, \pi /2\) reproduces the thermal spectrum for the created massless particles. The dependence of the entanglement and the Rényi entropies on the parametrisation \(\theta _{\mathrm{RL}}\) was depicted in Figs. 2, 3, 4a, b.

We recall that instead of taking \(\alpha \) as a usual independent parameter, one can also take it to be momentum dependent, see [30] for a discussion on the scalar field theory. Note that even though we have not taken any momentum dependence in \(\theta _{\mathrm{RL}}\) here, the coefficients of the linear combinations in the global modes, Eq. (16), are indeed momentum dependent. This effectively makes our construction qualitatively similar to that of [30]. Nevertheless, it will be interesting on top of this to further allow explicit momentum dependence in \(\theta _{\mathrm{RL}}\). In this case we need to make a suitable ansatz for it, such that the mode by mode normalisability of the states is achieved and also the various momentum integrals we encounter converge.

It seems interesting to investigate the effects of \(\theta _{\mathrm{RL}}\) into the other measures of quantum correlations e.g., the entanglement negativity, the violation of Bell’s inequality and the quantum discord etc, in order to quantify it further. We hope to address these issues in our future work.

## Notes

### Acknowledgements

We would like to acknowledge R. Basu, S. Chakraborty, R. Gupta, A. Lahiri, S. Mukherjee, B. Sathiapalan and I. S. Tyagi for fruitful discussions. SB’s research is partially supported by the ISIRD grant 9-289/2017/IITRPR/704. SC is partially supported by ISIRD grant 9-252/2016/IITRPR/708.

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