Spacetime is described by a torsion-free, -dimensional Riemannian manifold associated with a Lorentzian metric of signature and a metric compatible Levi-Civita connection determining the covariant derivative . In order to formulate our statistical mechanics and information theory action for gravity, we need first to define the eigenvalues the logarithm of rank-2 tensors . To this end we first define the eigenvalues and eigenvectors of the covariant tensor of elements in a Lorentz invariant way. These satisfy the eigenvalue problem

We say that a rank-2 tensor is positively defined if all its eigenvalues are positive. We notice that this definition of the eigenvalue of a rank-2 tensor reduces to the definition of the eigenvalue of the matrix as Eq. (1) can be rewritten as

One striking consequence of this definition is that the eigenvalues of the metric are all identically equal to one.

Assuming that the tensor is positively defined, we define the logarithm of this tensor as

and the inverse of a tensor as

It follows that if the tensor ^𝐆 is invertible, the logarithm of the inverse of a positively defined tensor is given by

Finally we define the trace of a rank two tensor as the sum of its eigenvalues, i.e.,

Thus the trace of a rank-2 tensor can be also calculated as usual in tensor calculus, as the trace of the matrix ^𝐆⁡𝑔−1, i.e.,

In this first section we are interested exclusively on rank-two tensors that are metrics between vectors (and 1-forms). In the subsequent paragraphs and in Appendix B we will extended the notion of eigenvalues also to metric matrices between bivectors (and 2-forms) represented by rank-4 tensors. Such an approach will be shown in Appendix B to be general and applicable to metric tensors between two 𝑛-vectors (an 𝑛-forms) with 𝑛 of any order.

Before we develop our theory, let us consider an instructive warm-up scenario that will help justify the theory that we will present in the following. Having defined the logarithm of positively defined rank-2 tensors, we are now in the position to define their Lorentz invariant quantum entropy 𝐻 for metric associated to 1-forms. This is inspired by the expression of the von Neumann entropy albeit we do not require the tensor to have trace one at every point in spacetime. Thus strictly speaking we interpret the generic metric tensor as a quantum operator [33], which has the physical interpretation of a renormalizable effective density matrix [18]. We consider positive definite, invertible metric for 1-forms, represented by the covariant tensor ^𝐆 of rank two. We define the entropy 𝐻 of ^𝐆 as

Note that the problem of defining eigenvalues and entropy of tensors is a topic of intense research and similar definitions have been provided for instance in the theory of elasticity [50] and in applied tensor analysis [51] (although not defined in a Lorentz invariant way). In the present warm-up scenario we will define the entropy and the quantum relation entropy of different metrics taking the form of rank-2 tensors. As a reference for the notation used, the reader can refer to Table I. Recalling that the metric 𝑔 has all the eigenvalues equal to one, it follows that the entropy of the metric 𝑔 is null, i.e.,

The fundamental assumption of the present theory is that spacetime is endowed with two metrics: the metric 𝑔 fully determining the spacetime geometry and the metric induced by the matter fields 𝐆 that fully capture the interplay between matter fields and geometry. Leaving the detailed discussion about the metric induced by the matter fields to the next paragraph, we postulate that the action should explicitly express the relation between these two metrics and their reciprocal coupling. By embracing a statistical mechanics approach we consider the Lagrangian ℒ given by the quantum relative entropy between the metric 𝑔 and the metric 𝐆 induced by the matter field defined as

Using 𝐻 =0 we thus obtain

We observe that since 𝑔 has all the eigenvalues equal to the identity, and Eq. (2) holds for ^𝐆 =𝐆, the Lagrangian ℒ can be also be expressed as

where with Tr𝑀⁡ln⁡𝐆⁢𝑔−1 we indicate the usual trace of the logarithm of the matrix ln⁡𝐆⁢𝑔−1 and with 𝜆′ the eigenvalues of the 2-tensor 𝐆 as defined in Eq. (2).

An important point here is that the Lorentz invariant definition of the quantum relative entropy given by Eq. (12) requires that the matrix 𝐆 has to be invertible as well as the matrix 𝑔. Thus this is another difference with respect to the strictly speaking density matrices that can be semidefinite positive. We are now in the position to consider an action 𝒮 associated to the Lagrangian ℒ, as given by

where |−𝑔| indicates the absolute value of the determinant of 𝑔 and ℓ𝑃 indicates the Planck length.

We note that the quantum relative entropy [17,37,38] is a central quantity in quantum information [39], in the theory of local quantum operators [33] and the mathematical foundations of quantum gravity [34–36].

Although we are not aware of previous interpretations of the metrics as quantum operators, it is well known that the quantum relative entropy can also be defined among quantum operators that generalize density matrices. In particular, in the fundamental theory of quantum operators algebras [37,38], quantum operators generalize the notion of density matrices in an analogous way of the metric tensors adopted here. In particular in this theory, quantum operators, like our metric matrices, might admit a finite trace at each point of the manifold but this trace might be not unitary and might be a function of the considered point of the manifold. For these quantum operators the Araki quantum relative entropy [37,38] is defined. This entropy reduces to the von Neumann entropy when the quantum operators reduce to density matrices of unitary trace but is defined also among quantum operators of nonunitary trace.

While we leave the discussion of the relation of our entropic action to the Araki quantum relative entropy to Appendix C, a series of physical observations are here in place to justify our interpretation of the metrics matrices as quantum operators. As remarked previously we treat the metric tensors as quantum operators or effective density matrices. The main differences between the metric tensor and the density operators include the fact that we require that the metric matrices are invertible and we do not require that they have unitary trace at each point of the manifold. The requirement of treating exclusively invertible metrics 𝑔 and 𝐆 is dictated by our desire to define the entropy in a Lorentz invariant way. In fact here we desire to treat the metrics 𝑔 and 𝐆 on the same footing of their inverse 𝑔−1 and 𝐆−1. It might be argued that also our relaxation of the requirement to have density matrices of trace one has similar roots. In fact if we require 𝑔 and 𝐆 to have unitary trace at each point of the manifold their inverse as well as their dual (defined in Appendix C) in general will not have a unitary trace.

We now apply this warm-up scenario in presence of scalar matter fields. We consider the complex valued scalar matter field 𝜙⁡(𝐱) ∈ℂ with 𝐱 ∈𝒦 and we consider the 𝑑-dimensional manifold immersed in 𝒦 ⊗ℂ defined by the points (𝐱,𝜙⁡(𝐱)). The metric 𝐆 induced on 𝒦 by this matter fields is given by

where 𝛼 is a real positive parameter, and 𝐌 is the rank-2 tensor of elements

where here and in the following ¯· indicates complex conjugation. Since 𝐆 should be adimensional it is convenient to work in the units ℏ =𝑐 =1 and to put 𝛼 =𝛼′⁢ℓ𝑑𝑃 where ℓ𝑃 is the Planck length and 𝛼′ is adimensional.

We observe that in the limit in which the field 𝜙 is real, i.e., 𝜙⁡(𝐱)∈ℝ and the metric flat and Euclidean, i.e., 𝑔𝜇⁢𝜈 =𝛿𝜇⁢𝜈, the metric induced by the matter fields 𝐆 given by Eq. (14) reduces to the first fundamental form of Gauss for the 𝑑 manifold immersed in ℝ𝑑+1 defined by the set of points (𝐱,𝜙⁡(𝐱)). For the underlying mathematical treatment of the metric induced by real scalar fields we refer the interested reader to Ref. [52].

Let us define the scalar product |𝛁𝜙|2 as

We observe that the inverse 𝐆−1 of the induced metric 𝐆 has metric given by

Adopting this notation we observe that the logarithm of the induced metric ln⁡𝐆 and the logarithm of ln⁡𝐆−1 have elements

where

where 𝑓⁡(𝑤) →𝛼 for |𝛼⁢𝑤| ≪1. With this choice of the metric induced by the matter field we obtain that the Lagrangian (11) reads

We observe that in the limit 𝛼⁢|∇𝜙| →0 and then we have ℒ →−𝛼⁢|𝛁𝜙|2, i.e., we recover the Lagrangian corresponding to the massless Klein-Gordon equation.

By minimizing the action 𝒮 with respect to the field 𝜙 and to the metric 𝑔 we obtain the Euler-Lagrange equations of motion,

where ℎ⁡(𝑤) is given by

In the limit 𝛼⁢𝑤 →0 we have ℎ⁡(𝑤) →𝛼 and the equation for the scalar field reduces to the simple massless Klein-Gordon equation. By putting equal to zero the variation of the action 𝛿⁢𝒮 =0 with respect to the metric 𝑔, we get

In empty spacetime 𝐌 =0 and ℒ =0, thus this equation is automatically satisfied independently on the value of 𝑔. Thus in this limit, the metric 𝑔 is not determined by the action.

Let us now make some remarks about this warm-up derivation. We proposed a statistical mechanics framework that is very inspiring as we get the massless Klein-Gordon equation as the outcome of the minimization of a quantum entropy action for low coupling, i.e., 0<𝛼⁢|∇𝜙|⁢2|≪1. However this approach has two important limitations. The first limitation is that the Klein-Gordon equation does not contain the mass term. The second limitation is that in absence of matter fields the metric is not determined. In the following we provide a more comprehensive framework to address these two limitations. This framework can be related to gravity as it gives rise to modified Einstein equations that reduce to the Einstein equations for low coupling.

In order to derive gravity from our entropic action we need to consider the topological bosonic matter field. The topological bosonic matter field is a type of Dirac-Kähler [40,41] boson given by the direct sum of a 0-form, a 1-form and a 2-form. Topological bosonic fields are receiving increasing attention in discrete theories developed in network [44] and lattice gauge theories [42,43]. Taking into consideration topological bosonic fields will allow us to introduce in the metric induced by the matter fields terms depending on the mass of the bosonic field. Thus, in this way, we address the first limitation of the warm-up scenario that we have presented above.

In order to address the second limitation of the warm-up scenario discussed previously, we include in the expression of the metric induced by the matter field also terms depending directly on the curvature of the manifold. These terms will be expressed in terms of the Ricci scalar 𝑅, the Ricci (covariant) tensor 𝐑 of elements 𝑅𝜇⁢𝜈 and the Riemann tensor of elements 𝑅𝜇⁢𝜈⁢𝜌⁢𝜎.

We define the topological fields |Φ⟩ as the direct sum between a 0- (complex valued) form 𝜙 and a (complex valued) 1-form 𝜔𝜇⁢𝑑⁢𝑥𝜇, and a (complex valued) 2-form 𝜁𝜇⁢𝜈⁢𝑑⁢𝑥𝜇 ∧𝑑⁢𝑥𝜈, with 𝜁𝜇⁢𝜈 =−𝜁𝜈⁢𝜇 i.e.,

and its conjugate topological field ⟨Φ| as

The considered covariant metric ˜𝑔 is defined as the direct sum of the metric among scalars (the identity), the metric 𝑔 among vectors introduced previously, and the metric 𝑔(2) among bivectors given by

or, exploiting the antisymmetry of the 2-forms,

with

It then follows that ˜𝑔 is given by

The local scalar product among topological fields is defined as

where 𝜔𝜇⁢𝜔𝜇 =𝜔𝜈⁢𝑔𝜈⁢𝜇⁢𝜔𝜇, while the local outer product is given by

Indicating here with 𝑑 the differential operator and with 𝛿 the codifferential operator, we define the Dirac operator 𝐷 as 𝐷 =𝛿 +𝑑, which is restricted to the space of topological bosons. Thus we define the action of 𝐷 over |Φ⟩ as

The metric ˜𝐆 induced by the topological matter field will have structure similar to ˜𝑔 that we can write in full generality as

Proceeding as in the warm-up scenario we might wish to define the metric induced by the topological field as

Thus in this general scenario we consider the two covariant topological metrics ˜𝑔 and ˜𝐆 each given by the direct sum of metrics among scalars, vectors and bivectors. As a reference on our notation for this general scenario we refer the reader to the Table II and the Table III.

In our interpretation of the metric as a quantum density matrix, this would correspond to density matrix corresponding to a pure state 𝐷|Φ⟩. However, if the induced metric ˜𝐆 is interpreted as a density matrix, it is natural to add further terms in the metric ˜𝐆. These terms will depend on the topological field and the geometry of the space which will allow us to describe mixed states. First, we introduce in ˜𝐆 a term proportional to |Φ⟩⟨Φ|. Specifically we introduce a term  (𝑚2 +𝜉⁢𝑅)|Φ⟩⟨Φ| where 𝑅 is the Ricci scalar and 𝜉 might include the case of conformal coupling 𝜉 =(𝑑 −2)/(4⁢(𝑑 −1)). This term can be also interpreted as a projector. Secondly, we introduce a term depending explicitly on the curvature of the manifold. Since ˜𝐆 involves the metric for scalars, vectors and bivectors on equal footing, it is natural to consider a further term involving ˜𝓡 given by the direct sum of the Ricci scalar 𝑅, the Ricci tensor of elements 𝑅𝜇⁢𝜈 and the Riemann tensor of elements 𝑅𝜇⁢𝜈⁢𝜌⁢𝜎, i.e.,

Including ˜𝓡 into the metric ˜𝐆 will allow us to describe more general metric matrices that are not decomposable into the sum of projectors.

From these considerations, it follows that in the induced metric ˜𝐆, we will substitute the term 𝐷|Φ⟩⟨Φ|𝐷 with ˜𝐌 defined as

As we will see in the following paragraph this choice will allow us to effectively overcome the first limitation of the warm-up scenario and to recover the Klein-Gordon equation in curved spacetime in full. Furthermore we consider also the additional term proportional to ˜𝓡 and we postulate that the metric ˜𝐆 induced by the geometry and the matter fields is given by

As we will see in the following the addition of the term proportional to ˜𝓡 will allow also us to overcome the second limitation of the warm-up scenario. Note that here 𝛼, 𝛽 are positive constants. In particular, since we require ˜𝐆 to be adimensional, we need to consider 𝛼 =𝛼′⁢ℓ𝑑𝑃 and 𝛽 =𝛽′⁢ℓ2𝑃 where ℓ𝑃 is the Planck length and 𝛼′,𝛽′ are adimensional in the units ℏ =𝑐 =1.

In the main body of this paper we will investigate only (bosonic) Dirac-Kähler matter fields. However gauge fields and fermionic Dirac fields can be included as well. For a discussion of the inclusion of Abelian gauge field see Appendix A.

Possibly this approach could be extended to include also higher-forms. However, for simplicity, we consider here only topological matter fields formed by the direct sum between a 0-form, a 1-form and a 2-form as this the minimal choice that will allow us to include in the action the Ricci scalar, the Ricci and the Riemann tensor explicitly.

We propose a statistical mechanics action formulated in terms of the quantum relative entropy between the metric ˜𝑔 and the metric ˜𝐆 induced by the topological matter fields. In order to define our action we need to extend the notion of eigenvalues to metric tensors between bivectors. This will allow us to define the entropy and the quantum relative entropy in this novel framework. For a detailed discussion of this mathematical background see Appendix B that constitutes also the reference for our notation. First of all we observe that the entropy associated to the metric ˜𝑔 remains zero. Indeed we define ˜𝐻 as

where we have used Eq. (9) and the identity derived in the Appendix B [in Eq. (B39)],

This equation is a direct consequence of the fact that not only 𝑔 but also 𝑔(2) has all the eigenvalues equal to one.

We are now in the position to consider the Lagrangian given by the quantum relative entropy between ˜𝑔 and ˜𝐆,

(see Fig. 1 for an illustration of the physical model beyond the choice of this Lagrangian). Since we have ˜𝐻 =0 we obtain

By treating separately the contributions of 𝐺(0), 𝐆(1) and 𝐆(2) we get the explicit expression for the Lagrangian ℒ given by

Since the metric matrices 𝑔(𝑛) have all their eigenvalues equal to one, using the notation developed in Appendix B we can express this Lagrangian as well as

The resulting statistical mechanics action 𝒮 associated to the Lagrangian ℒ is strictly related to the Araki quantum relative entropy [17,37,38] and can be formally derived from a mathematical theory of quantum operators (see Appendix C). Specifically, the considered action is given by

This action defines a modified theory of gravity. In the linearised limit 𝛼′ ≪1, 𝛽′ ≪1 this action reduces to the Einstein-Hilbert action with zero cosmological constant [46,47] coupled with the scalar topological field. Indeed in this limit ℒ reduces to

where