General-relativistic spin system

The models of spin systems defined on Euclidean space provide powerful machinery for studying a broad range of condensed matter phenomena. While the non-relativistic effective description is sufficient for most of the applications, it is interesting to consider special and general relativistic extensions of such models. Here, we introduce a framework that allows us to construct theories of continuous spin variables on a curved spacetime. Our approach takes advantage of the results of the non-linear field space theory, which shows how to construct compact phase space models, in particular for the spherical phase space of spin. Following the methodology corresponding to a bosonization of spin systems into the spin wave representations, we postulate a representation having the form of the Klein-Gordon field. This representation is equivalent to the semi-classical version of the well-known Holstein-Primakoff transformation. The general-relativistic extension of the spin wave representation is then performed, leading to the general-relativistically motivated modifications of the Ising model coupled to a transversal magnetic field. The advantage of our approach is its off-shell construction, while the popular methods of coupling fermions to general relativity usually depend on the form of Einstein field equations with matter. Furthermore, we show equivalence between the considered spin system and the Dirac-Born-Infeld type scalar field theory with a specific potential, which is also an example of k-essence theory. Based on this, the cosmological consequences of the introduced spin field matter content are preliminarily investigated.


I. INTRODUCTION
The models of spin systems, for instance the Ising model, Heisenberg model, XY model [1], or Hubbard model [2], provide a theoretical description of such phenomena as magnetism (ferromagnetism and antiferromagnetism), superconductivity and topological phase transitions [3]. While these models are adequate in the context of the ground-based table-top experiments, where local Euclidean geometry is the precise approximation, their extensions to a general curved spacetime are almost unknown. Due to the weakness of gravity in the shortdistance interactions, it is mostly irrelevant in condensed matter physics 1 . In consequence, surprisingly little is known about the condensed matter phenomena in curved spacetime. Although rather no one would ask the question about what would happen with a ferromagnet during cosmic inflation, the lack of the definite answer reflects theoretical deficiencies in the lattice spin models. Analysis of their relativistic extensions may not only bring us to the better fundamental understanding of the inter-action between gravity and spins but also provide theoretical foundations for possible future condensed matter experiments on Earth's orbit 2 . More abstract thought experiments, like the analysis of a superfluid state in the vicinity of a black hole horizon or the measurement of gravitational waves near a merger of black holes propagating through a spin glass, might be pondered in light of our results as well.
In the case of spin models describing the ground-based laboratory experiments, spins are considered to be attached to given space points, i.e. forming a fixed lattice. Consequently, such models explicitly break general covariance by distinguishing a certain reference frame. A useful step towards a frame-independent formulation is the continuous limit of a spin system, in which a fieldtheoretic description of spins is obtained. The analogous continuous spin system approximations play a powerful role in theoretical condensed matter physics, e.g. in the theory of topological phase transitions [3].
The system in such a case becomes a spin field S(x) defined on some spatial hypersurface Σ. In the standard condensed matter considerations, the spatial manifold is chosen as Σ = R d , where its dimension d is 1, 2 or 3.
From this perspective, one could naively presume that the desired generalization of a spin system is provided by the relativistic field theory of the vector field S(x), with the constraint S := || S|| = S · S = const. The constraint is due to the fixed norm of the spin vector. Such a framework, an example of which is the famous non-linear σ−model [7,8], however, does not lead to the correct theory of a spin field.
In our analysis, we identify the notion of a spin at each point of space with its semiclassical description by a vector S and initially restrict to the spatial manifold Σ given by the 3-dimensional Euclidean space. We are going to construct the relativistic extension of such a continuous distribution of spins. A natural, but naive, proposal for a generalization of a spin field on Euclidean space to either the spatial sector of spacetime or the Cauchy hypersurface constructed through the ADM decomposition does not lead to a well-working model 3 . The reason is that the spin field is not a standard classical tensor field.
In quantum physics, the model of spin is a finitedimensional Hilbert space representation of the SU(2) group of symmetries, i.e. rotations, and the su(2) algebra of observables (whose generators are spin operatorsŜ x , S y andŜ z ). The su(2) algebra is noncommutative, which leads to the uncertainty relations between the three spin operators and does not allow to measure them simultaneously. One often considers the auxiliary object that is a vector S (called the spin vector), whose components are the expectation values of theŜ x ,Ŝ y andŜ z operators in a given state. At the (semi) classical level, this vector spans the phase space, which is the two-sphere S 2 . On the one hand, S could be (incorrectly) interpreted as an object containing the separate configurational and momentum variables. However, S 2 does not have the structure of a cotangent bundle and, consequently, the decomposition of the considered space into the product of configurational and momentum subspaces is possible only locally. From this perspective, S 2 should be called the configurational space, since canonical analysis tells us that there is no canonically conjugate momentum to S. On the other hand, a Hilbert space is the quantum version of a phase space. Consequently, it is convincing to interpret S 2 as the phase space. These arguments justify the practice to use (in the case of spin) the names phase space and configurational space interchangeably.
In this paper, we investigate a continuous distribution of spin vectors S = S(x) (a normalized vector field), which can be described as a system consisting of spherical symplectic manifolds attached at each point of space. Through a procedure of the boson mapping of operator algebras (bosonisation) 4 , the degrees of freedom associated to a spin distribution can be represented by spin waves (a comprehensive description of this issue is given in [15]). Our particular choice is the classical analogue of a spin wave reinterpreted as a bosonic field paired with its conjugate momentum. In this case, we will construct a scalar field representation, with the property that values of the field and its momentum are compactified to the sphere. Such a bosonic representation will allow us to construct the natural extension to a general-relativistic model. Translating it back to the spin vector's distribution is the main goal of our analysis, defining a method never (up to our knowledge) studied before. Our approach belongs to the recently introduced framework of Non-linear Field Space Theories (NFSTs) [16], which extends the standard field theories to the case where the phase space of a field has the non-trivial topology. This generalization is related to numerous ideas and theories, including: principle of finiteness [17], Born reciprocity [18], Relative Locality [19], Metastring Theory [20] and polymer quantization in Loop Quantum Gravity [21]. Some of the relations have been already discussed in detail in Refs. [16,[22][23][24]. It is worth noting that, as it has been suggested in the above mentioned literature, the compact phase space versions of NFSTs naturally implement the principle of finiteness of physical quantities, which underlies the Born-Infeld theory. However, the principle in the two theories is implemented in different manners and a direct relation between the actions of Born-Infeld and NFST has not been found so far. Here, we show that the similarity between these theories is not only conceptual and a scalar field with the specific relativistic spherical phase space reduces to Dirac-Born-Infeld (DBI)-type scalar field theory [25][26][27].
The idea to relate spin systems and scalar fields is enabled by the equal dimensions of the local phase spaces of a spin system (S 2 ) and a scalar field (R 2 ) 5 . Based on this observation, a new possibility of linking spin systems with field theories has been proposed in Ref. [22]. The exact matching of phase spaces (of a standard scalar field theory and a spin field model) is obtained by taking the large spin limit (S → ∞). Following this reasoning, it has been shown that in the large spin limit the continuous Heisenberg XXX model is dual to the non-relativistic scalar field theory with the quadratic dispersion relation [22]. The result has been thereafter generalized to the Heisenberg XXZ model with a dimensionless anisotropy parameter ∆. In this case, it has been demonstrated that taking both the large spin limit (S → ∞) and the isotropic limit ∆ → 0, we reduce the XXZ model to the relativistic Klein-Gordon field [23]. However, if the spin limit is not taken exactly, the next to the leading order terms violate relativistic symmetries [23]. Therefore, the question is whether the construction can be improved to preserve the special relativistic and, furthermore, general relativistic symmetries, also for the spherical phase space field theories with an arbitrary value of the spin vector norm S. The purpose of this paper is to address this issue, directly constructing and analyzing the spin field theory that obeys general relativistic symmetries. What we do differently in the current paper is the relation between a spin field and a scalar field, which is now imposed in way equivalent -at the level of excitationsto the bosonization into spin waves.
One might wonder whether our framework could give some hints what is the correct approach for the inclusion of intrinsic angular momentum (i.e. spin) in gravitation. Namely, is it the standard Einsteinian general relativity [30] or the metric-affine gauge theory [31], the simplest example of which is Einstein-Cartan(-Sciama-Kibble) theory [32]? In the latter cases, the intrinsic angular momentum turns out to be the source of the non-vanishing torsion. However, imposing a priori this relation assumes using the equations of motion, hence it is an on-shell method. This motivates us to look for the off-shell methods, restricting all the matter degrees of freedom to contribute only to the stress-energy tensor (the right-hand site of Einstein equations).
It should also be mentioned that all gauge theories of gravity are considered extensions of the standard general relativity that could be more accurate descriptions of classical gravity, deviating from the standard framework only at the very high mass densities. In particular, Einstein-Cartan theory is expected to become relevant when the density of spin squared is comparable to the density of mass, which (for nucleons) in the terms of mass density happens at ρ EC ≈ 10 54 g/cm 3 . This is still much smaller than the Planck density ρ Pl ≈ 10 93 g/cm 3 and from the perspective of early cosmology -a long way from the onset of inflation. Therefore, the role of Einstein-Cartan theory or metric-affine gravity in cosmology is definitely worth investigating.
Nevertheless, in the spin-field correspondence considered in this paper, a continuous distribution of spins and a massive field are treated as two faces of the same entity, which is why it makes no sense to compare the spin density with mass density. In particular, in the spinformulation we have no mass density, while in the fieldformulation of the same theory, fermionic degrees of freedom are not present. Moreover, we eventually translate the general-relativistic version of our field model back to the spin model, where the general-relativistic modifications of the former model contribute to the noninteracting Hamiltonian of the spin field. These modifications can be interpreted then as the source of energymomentum corrections, rather than the torsion ones.
The construction of the field-formulation is based on the bosonization procedure of the fermionic interactions in a solid spin system. This procedure, first introduced in [33], aims to effectively describe the particle-holelike excitations in the low-energetic regime [34]. The known phenomenological realization of this method is the Tomonaga-Luttinger liquid model [33,35], in which, under particular constraints, second-order interactions between electrons are represented by bosonic interactions. The model allows to derive the exact spectrum of the Hamiltonian operator, free energy of noninteracting fermions and dielectric constant [34] 6 .
The basic excitations of coupled spin systems with fixed, homogeneous distribution of spin vectors (but not their orientations) are called spin waves (see [15] for a detailed introduction). One can distinguish two different kinds of such excitations. The first is described by the wave of deflected dipolar magnetic moments produced by elementary spins shifted from their equilibrium positions that propagates through the solid system. This type of excitations is significant at very long wavelengths comparing to the spacing between individuals, and forms a macroscopic characteristic of the system. The second type of excitations relates to microscopic (quantum) properties of the spin lattice and is relevant for very short wavelengths, comparable with the lattice spacing. In both cases, in the simplest realization, only the nearestneighbour interactions are considered, forming a net of oscillators.
Since we are interested in the continuous limit, in which the lattice spacing tends to zero, the first type of excitations is relevant in our context. Furthermore, we are interested in the semiclassical description of the spin waves, understood as the linearly propagating vector's precession phase. When one considers closely spaced frequency components, they can be viewed as a wave packet that moves like a particle. This quasi-particle is called a magnon and it does not obey the Pauli exclusion principle. We are going to describe properties of this bosonic field in the semiclassical picture, in which the spin variables are represented by the field variables corresponding to the matrix elements of the bosonic operators. The latter ones are constructed in the (semi-)classical equivalent of the Holstein-Primakoff transformation [9] (see Appendix A for details), which is one of the most popular maps selected for the bosonization procedure.
The paper is organized as follows. In Sec. II a spinrelated spherical phase space and its scalar field-type parametrization are introduced. Then, in Sec. III a general strategy behind the defining of the Hamiltonian of the spin system is discussed, based on which, in Sec. IV the special relativistic spin system is introduced. The results are generalised to the curved spacetime case in Sec. V. In Sec. VI the obtained model is shown to be be an example of the DBI-type k-essence model. Due to the cosmological relevance of the k-essence models, consequences of the considered field theory in the dynamics of Universe are preliminarily investigated in Sec. VII. The results are summarized and additional discussion is given in Sec. VIII.

II. PHASE SPACE OF A SPIN
As we mentioned in the Introduction, we will mostly restrict to the semiclassical description of spin by the vector S = (S x , S y , S z ), whose components are the generators of proper rotations in the 3-dimensional Euclidean space (in other words, it is an element of the Lie algebra of rotations). Although the corresponding classical symmetry group is SO(3), we are more interested in another group, SU (2). The latter one is a double cover of the former, 2spheres are orbits of both of them and their Lie algebras are isomorphic. The double cover property, which leads to the appearance of half-integer spins in the quantum theory, is the reason for selecting it to construct models of spin, as we do in this paper.
Another interesting property of spin is its dimension, [M L 2 T −1 ]. It is the dimension of the classical angular momentum or the Planck constant, . It is worth noting that this is also the dimension of the physical action. This latter property already suggests that the geometrical interpretation of the spin is linked with the phase space. The classical interpretation of the spin's states are different directions in R 2 , equivalent to points on S 2 , therefore elements of the associated configurational space are naturally represented by two angles, (φ, θ). As long as there is no velocity related to spin, neither its conjugate momentum is defined, the configurational space of S can be simultaneously interpreted as its phase space. These observations suggest to describe it using the Kirillov orbit method [36]. The method tells that if a phase space is the coadjoint orbit of a symmetry group G (i.e. an orbit in g * , e.g. su(2) * ∼ = R 3 ) for which the considered mechanical system remains invariant, then the corresponding quantum system should be described by irreducible unitary representations of G. It is well known that 2-spheres are coadjoint orbits of G = SU(2), as evidenced by the fact that a coset SU(2)/U(1) ∼ = S 2 is the unit 2-sphere. In consequence, the quantum models of spin are given by irreducible unitary representations of the SU(2) group, labelled by half-integers s = n 2 , where n ∈ N ∪ {0}, as expected.
In order to treat S 2 as the phase space of spin, one has to equip it with a symplectic form (a closed 2-form), so that it becomes a symplectic manifold. The natural choice for such a form is the area form, ω = S sin θdφ∧dθ, where (φ, θ) are the usual spherical angles. This allows us to introduce the Poisson bracket via the standard definition (turning S 2 into a Poisson manifold): Here, f and g are some smooth functions on phase space and ω −1 is inverse of the symplectic form ω. Every Poisson bracket is a Lie bracket by definition. Calculating (1) for components of the spin vector S, we verify that they generate the su(2) Lie algebra {S i , S j } = ijk S k . Furthermore, integrating the symplectic form over the whole solid angle, we find that the volume of phase space is 4π ω = 4πS < ∞, which correctly has the dimension of the Planck constant. If we subsequently follow the topological quantization procedure, the finiteness of the phase space volume leads to a specific, discrete spectrum of each of the three operatorsŜ i . Let us now consider a continuous spin system, so that the phase space becomes a two-sphere at each point x of space. In consequence, components of a continuous spin variable, S(x) = (S x (x), S y (x), S z (x)), become functions of a position vector x and still satisfy the su(2) algebra, where i, j, k ∈ {x, y, z} are the internal indices. Since the Poisson algebra, spanned by the spin components, is 3dimensional, it is associated with one propagating degree of freedom and one Casimir function, S 2 := S· S = const.
To simplify the problem we introduce a new variable, S, being a constant in space and defined through the relation In the limit where S is very large compared with all other scales, we require that an ordinary scalar field theory is recovered.
As a next step, we are going to set the relation between the spin's and the ordinary scalar field's phase spaces. This can be done introducing the canonical parametrization of the sphere in the following way: where π ϕ ∈ [−R π , R π ] and ϕ ∈ (−πR ϕ , πR ϕ ]. The fields ϕ and π ϕ are 'linear-like' canonical variables, in terms of which we are able to recover the ordinary scalar field. The constants R ϕ and R π were introduced due to dimensional reasons. They play roles of parameters, which control the accuracy in recovering the standard form of the scalar field's Hamiltonian in the large S limit. Moreover, as mentioned earlier, we only expect one independent Casimir variable, because the rank of the su(2) algebra is two, and its dimension is three. Therefore, performing a canonical rescaling, one can eliminate R π and R ϕ in favour of S. It is worth noting that when we consider the general-relativistic perspective, the weights of the scalar ϕ and the scalar density π ϕ will become relevant. Consequently, the parameters R π and R ϕ would also become a scalar and a scalar density, respectively.
In order to determine how the constants R ϕ and R π depend on S, we use the canonical bracket in the fieldformulation. The Poisson bracket is defined as usual, where A := A[ϕ(x), π ϕ (x)] and B := B[ϕ(y), π ϕ (y)] are some general functionals. Computing the following relation, {S z , S x } ϕ,π = S RϕRπ S y , it is easy to see that to maintain consistency with the su(2) algebra, the parameters have to be related via simple equation, Notice that we introduced the object q, denoting the determinant of the spatial metric tensor q ab (which encodes invariance under spatial diffeomorphisms in the fieldformulation of the model) on the Cauchy hypersurfacefrom the general-relativistic perspective this would balance both sides of the identity. It would also modify the Levi-Civita symbol in (2), which would no longer be a tensor, but a totally antisymmetric tensor co-density (of weight −1), Here,˜ abc is a tensor, while the indices a, b, c, ..., which replaced i, j, k, ..., emphasize that the considered space may be curved 7 . The former set of indices labels the internal coordinates (in the spin-formulation) curved accordingly to the spatial coordinates (in the field-formulation) due to the dependence on the same metric, q ab . Provided the identity in (8) is satisfied, the algebra in (2) is consistent with the assumption that the pair ϕ and π ϕ satisfy the standard canonical bracket, Let us also mention that we are interested in the cosmological application of the discussed spin-field correspondence. The standard form of the Poisson bracket will simplify the description of the cosmological perturbations, allowing to perform a decomposition of the phase space into homogeneous and inhomogeneous parts in the standard way [37]. 7 The interesting property of the spin's system is the locality of the orientation of the spin vector's components, indicating directions in R 2 with respect to the same point. In the single point model, associated with a rotationally-invariant reference frame, the flat and curved coordinates are indistinguishable. Therefore, we do not need to modify the labelling of the internal coordinates in (2). The contravariant or covariant position of the indices, however, is relevant -it helps to control the proper weights of different objects. In consequence, this appears to specify the isotropic contribution to the minimal coupling between the gravitational field and the spin.
It is necessary to stress that the parametrization (4-6) has been chosen so that the ϕ and π ϕ fields satisfy the canonical bracket (10). This is different from the spherical parametrization considered in Refs. [22,23,38], which led to a modified form of the canonical relation between the field variables ϕ and π ϕ .
The canonical parametrization of su(2) (4-6) is technically advantageous due to the harmonic behavior of the variables in the vicinity of the classical minimum and the canonical relationship between the scalar field ϕ and its momentum. On the other hand, its form might appear contrived. In order to motivate the form of this parametrization we note that it is the semiclassical limit of the well known Holstein-Primakoff transformation [9,39]. This transformation expresses su(2) generators in terms of creation and annihilation operators, furnishing the crucial canonical structure. While it is not difficult to demonstrate the correspondence between the Holstein-Primakoff transformation and our canonical parametrization, the details of this semiclassical limit are too long to be included here and, therefore, were moved to Appendix A.

III. SPIN-FIELD CORRESPONDENCE
Let us consider the following Hamiltonian where γ is a certain constant. In condensed matter physics it would be interpreted as the interaction term of a continuous distribution of internal magnetic moments (spins) with an external homogeneous magnetic field oriented along the x axis. In the absence of other interactions, the spin at each point would precess around the ground state, S = (S, 0, 0). Meanwhile, the same system, described in terms of the ϕ and π ϕ variables, would correspond to a set of oscillators, which become harmonic when the 'precession angle' tends to zero, which is pictorially represented in Fig. 1.
To make the spin-field correspondence evident, let us substitute the expression for S x given by (4) into the Hamiltonian in (11), finding, Up to the constant factor −γS, the leading order terms describe the free homogeneous scalar field. From the mechanical perspective, the field is simply a continuous distribution of harmonic oscillators. Furthermore, the form of the Hamiltonian in (11) has been chosen in the way The specification of parameters is in general not unique. This one, however, entails the homogeneous Klein-Gordon formulation of the spin wave representation related to the Holstein-Primakoff transformation of the Hamiltonian in (11), which describes interactions in a continuous distribution of spins in the presence of an external homogeneous magnetic field oriented along the x axis. Let us emphasize that once the transformation in (4)(5)(6), associated with the parametrization in (13)(14)(15), is 8 Notice that in (12) we did not consider the generally relativistic formulation of the field theory. Later, however, we are going to construct an extension to this formulation, therefore any map linking the spin-and field-formulation, has to already include a proper scaling with respect to the metric tensor q ab . Luckily, all the equations contributing to the map, which we discuss, being the Holstein-Primakoff transformation with a particular parametrization, involve only scalars or scalar densities and, being one of these two objects, fixed vector coordinates. Therefore, the proper scaling is provided only by appropriate powers of the determinant of q ab . assumed, it fixes the meaning of the bosonic field's representation of the spin system. Consequently, it would also specify the quantum spin waves' kinematics and dynamics, whose solutions on the non-degenerate eigenstates of observables correspond to the excitations of a magnon quasiparticle. This allows to study bosonic representations of spin systems, which are usually simpler generalizable and coupleable with other fields. Results of any analogous modification in the field-formulation of a model are traceable then in the spin-formulation. The parametrization in (13)(14)(15) used to expression (12) leads to the following form of the Hamiltonian, where we imposed the flat space constraint, q ab = δ ab , simplifying the determinant of the metric tensor to q = 1. Notice that the first term in the second line diverges with the spacetime volume. This term, however, does not contribute to classical dynamics and for convenience can be regulated by performing an infinite subtraction (setting the vacuum energy to zero). The equations of motion can be calculated from the Hamiltonian in (16) in the usual way. In particular, one finds,φ Clearly, in the limit S → ∞, the standard resultφ = π ϕ is correctly recovered. The Eq. 17 can be rewritten in the form: The ± sign corresponds to ϕ ∈ F + or ϕ ∈ F − , respectively, where these two sectors of possible values of ϕ (associated with two hemispheres) are: The sign ambiguity in (18) has consequences when one performs the inverse Legendre transform to recover the Lagrangian. Namely, we have to define two variants of the Lagrangian, These two possible choices for the Lagrangian define dynamics related to two hemispheres of the spherical phase space. However, only L + allows us to recover the standard scalar field theory, corresponding to the large spin limit. This argument specifies the interval of the possible values, which the scalar field can take, ϕ m S ∈ [−π/2, π/2], as well as it makes the ϕ → 0 limit achievable. Consequently, in the expansion of L + around the small field's values and the related velocities, the standard form of the homogeneous scalar field's Lagrangian is recovered.

IV. SPECIAL RELATIVISTIC EXTENSION
In the previous section the spin-field correspondence was introduced, where the field-side of this relation was constructed in the Hamiltonian formalism. The spin's phase space was parametrized in the way leading to the spatially homogeneous Klein-Gordon-like form of the Hamiltonian in the large spin limit. It is, however, much easier to construct an invariant theory at the level of an action -one just has to make sure that the action is a scalar. Difficulties appear when the Poisson bracket related to a continuous spin theory is non-canonical, and therefore derivation of the Lagrangian is neither straightforward, nor trivial. Another argument to focus on the field-formalism is the well-known model of the Lorentzinvariant Klein-Gordon field, which is the standard formulation of this theory. Before we will begin generalizing the field-side of our framework, let us mention how one could proceed with the spin-side. The standard (special) relativistic generalization of spin is introduced as follows (see e.g. [40]). A given system has the angular momentum tensor J µν = L µν +S µν , where L µν = X µ P ν −X ν P µ is the 'orbital' angular momentum, while the intrinsic part (i.e. spin) S µν can be expressed as S µν = ε µναβ u α S β . Consequently, the spin four-vector (known as Pauli-Lubański vector) is orthogonal to the four-velocity, u α S α = 0, and in the rest frame it is simply (S µ ) = (0, S). M µν and S µν become mixed under the action of Lorentz transformations unless M µν = 0, as in our case. However, what we do in the current paper is quite different. We do not consider the relativistic generalization of a single spin but of a spin field, which is first represented by a scalar field and only then generalized.
Looking for the Lorentz-invariant analog of the correspondence defined by the transformation and parametrization in (4-6) and (13)(14)(15), respectively, is the primary problem of this section. We are going to achieve our goal by modifying the Lagrangian obtained in (21). This object is a functional of Lorentz scalars and in the large spin limit it takes the form of the homogeneous scalar field Lagrangian. Its generalization is then straightforward,φ where the Minkowski metric, η µν = diag(−1, 1, 1, 1), was introduced. In consequence, Eq. 21 leads to (from now on, the abbreviation h.o.t. denotes higher order terms). For the + case, the leading order of the Lagrangian describes the standard Klein-Gordon scalar field. The momentum canonically conjugate to ϕ is given by the formula, (25) Performing the inverse Legendre transformation, we obtain the Hamiltonian: The interesting consequence of the form of this Hamiltonian is that gradient of the field ϕ is bounded, |∇ϕ| ≤ √ mS. Thinking of the gradient as the momentum, this relation allows us to put an upper bound on the energy carried by the scalar field waves in a finite volume.
To get rid of the sign factor, one can take cos ϕ m S out of the square root, which is possible because different signs correspond to two sectors F ± of the ϕ values and, in consequence, to different signs of the cos ϕ m S function. However, this is done at the cost of moving cos 2 ϕ m S into the denominator, which diverges at ϕ m S = ± π 2 . As a result, the Hamiltonian becomes: Using the relations, and we can then express the relativistic Hamiltonian in (27) in terms of the spin variables: (30) In the large mass m limit and in a vicinity of the ground state (for which S x ≈ S and S y ≈ 0), the Hamiltonian can be approximated by the following expression In the context of condensed matter physics, this would be interpreted as a continuous Ising model coupled to a constant, transversal external magnetic field [41]. Interestingly, the model is not only of theoretical interest and finds an important application in adiabatic quantum computing [42]. Therefore, quantized Hamiltonian (30) may allow to study special relativistic generalisations of the quantum annealing process. Furthermore, in the following section we will extend the Hamiltonian (30) to the general relativistic case.

V. GENERAL RELATIVISTIC EXTENSION
Having established the canonical formulation of the bosonic wave representation of the spin system, characterized by the Lorentz symmetry, the most natural way to proceed is to look for an extension of the symmetry to the full generally relativistic case. This construction can be done in much the same way as in the special relativistic case, hence except instead of using the Minkowski metric, as in (24), we are going to use the general one, g µν . Consequently, the general relativistic extension of (21) involves the following replacing, Our goal is to formulate this scalar field's system in the canonical setting and therefore it is most convenient to decompose the metric tensor into ADM variables, Here, N denotes the lapse function, N a the shift vector and q ab = g ab the spatial metric, where a, b, ... = 1, 2, 3 label spatial coordinates. Consequently, the volume element takes the form √ −g = N √ q, where g = det(g µν ) and q = det(q ab ). The inverse of (33) is given by the matrix Using this object, we can express the kinetic term as: This form of the kinetic term in the field-formulation of the action allows to begin the canonical analysis and, in particular, to derive the momentum of the field ϕ.
As a next step, we postulate the fully relativistic generalization of the action based on the Lagrangian in (24), The result, as expected, took an explicitly invariant form. The transformation of the model back to the spinformulation can be done analogously to the procedure in the previous section. Dynamical analysis of the general-relativistic extension of a field theory have to involve gravity. This is needed, because the extension is realized via the minimal coupling of the field with the metric tensor. In order to make the spacetime geometry dynamical, we define the minimal coupling of the theories in the standard way -at the level of their actions, constructing the following one, The formulation of our model allows us also to easily add other elements to the theory such as a cosmological constant or other types of bosonic fields.
To complete the construction of the generally relativistic spin-field correspondence theory, we now need to perform the Legendre transformation of (38). The direct calculation gives the momentum canonically conjugate to ϕ: In the large spin limit (S → ∞) of the + case, we correctly recover π ϕ = √ q N a ∂ a ϕ). The matter Hamiltonian can now be written as where we introduced the field's diffeomorphism constraint: It took the same form as in the case of the ordinary Klein-Gordon field. Finally, the matter Hamiltonian constraint reads, Taking the limit where the field excitations are small compared with the scale set by S, we correctly recover the Hamiltonian constraint of the self-interacting scalar field minimally coupled to Einsteinian gravity.

A. Hypersurface deformation algebra
Even though we constructed the Hamiltonian in (40) using the general relativistic approach, it is useful to per-form an independent check that this Hamiltonian is indeed covariant. This can be done verifying that our spinfield contributions lead to the unmodified constraint algebra. In other words, our constraints should satisfy the hypersurface deformation algebra. Taking a different point of view, we can think of (42) and (41) as an ansatz for the contributions from the matter component to the gravitational constraint and then verify that they close the algebra.
The total constraints are sums of the gravity and mat-ter contributions, We will first check whether the following identity holds: To this end, we simplify the problem, collecting together the Hamiltonian contributions from different fields, The cross terms cancelled out because the minimal coupling of the matter field to gravity in the ADM formalism is given by coupling only with the spatial metric (and not with the gravitational momenta). Consequently, no integration by parts has to be performed when comput-ing the cross terms and their sum is proportional to N M − M N = 0. The first term in the second line of (45) is standard because we did not consider any modification of the gravitational constraint. Therefore, we only need to compute the second term, which by the direct calculation is found to be, This is the result that we expected -the contribution to the diffeomorphism constraint from the matter field takes the standard form for the Klein-Gordon field despite our modifications.
The calculation of the bracket between the diffeomorphism constraint and the Hamiltonian constraint is a little more subtle. Let the object H = H[q ab , π ab , ϕ, π ϕ ] denote the Hamiltonian density, then the following relation holds: We can use this intermediate result to compute the bracket between the Hamiltonian constraint and the diffeomorphism one, obtaining It is worth mentioning that we derived this result almost without any effort due to the unmodified matter contribution to the diffeomorphism constraint. Collecting all the resulting Poisson brackets together, we obtain the following list, We can then conclude this section with a remark that our spin-field matter contribution indeed lead to a generallyrelativistic invariant theory when coupled to a dynamical, possibly curved background.

B. General relativistic spin-field correspondence
Similarly to what we did in the special-relativistic case, we begin with absorbing the sign factor by moving the function cos 2 ϕ m S in (42) out of the square root, which leads to the following Hamiltonian, As a next step, we re-express this Hamiltonian in terms of spin variables, obtaining Notice that this result puts an upper bound on the magnitude of the gradient of arcsinh S y S x . It is also worth mentioning that deriving the expression above, we took the advantage of the correct construction of the map in (cf. (4)(5)(6)), in which the weights of π ϕ are balanced with the ones of R ϕ in the parametrization (13)(14)(15). To emphasize the importance of the correctness of the transformation's construction, let us recall the su(2) algebra from (2), expressing it in the explicitly covariant notation, Let us also recall the implicit coupling to the metric tensor of two objects in the formula above, S a = q ab S b and abc =˜ abc / √ q, where S b is the spin vector in the general coordinates, while˜ abc is the Levi-Civita tensor.
Finally, the spin-formulation of the diffeomorphism constraint in the generally-relativistic framework (given in (41)) reads, We re-expressed the matter contributions to the constraints in terms of spin variables but, although we have changed coordinates, the calculations done in the previous section with the hypersurface deformation algebra still hold. This is due to the independence of the overall Poisson structure of the choice of coordinates.

VI. DIRAC-BORN-INFELD THEORY PERSPECTIVE
We will now demonstrate that our model can be mapped to a Dirac-Born-Infeld (DBI) [25][26][27] model via the ap-propriate field redefinition. Let us begin moving the cosine square term out of the square root the action in (37), so that the sign factor is absorbed -analogously as in Section V), this time, however, at the level of the action, The resulting expression has similar form to the free Dirac-Born-Infeld (DBI) action for a scalar field ξ, Here, f (ξ) is a functional which in the case of the D3-brane inspired origin of the DBI action is a warp factor of the AdS-like throat [27]. To find the relation between (56) and (57), let us redefine the field variable in the former expression, setting ϕ = G(ξ). The form of the functional G(ξ) depends on whether the sector ϕ ∈ F + or ϕ ∈ F − (see (19)) is considered. The first sector corresponds to the non-negative sign of cos ϕ m S , while in the second one to the negative sign. To distinguish these situations, we introduce the additional labeling of G(ξ), i.e. the functional related to the first sector is going to be denoted by G + (ξ), while to the second one, G − (ξ).
Applying the change of variables, ϕ = G(ξ), to the action in (56), we obtain, This suggest to impose the following constraints on G ± (ξ), This leads the DBI-like form of the action, where the functional f (ξ) is given by the expression, The functional f is positive on the ϕ ∈ F + branch, negative on the ϕ ∈ F − one, and diverges when ϕ m S = ±π/2. The action in (60) matches with the DBI one in (57) only in the F + sector. The absolute value |f | ensures that the argument |f (ξ)| g µν ∂ µ ξ∂ ν ξ is non-negative always, and the square root is real-valued. This observation will have essential consequences when applying this model to cosmology, in particular during the inflation stage, what will be discussed later, in section VII.
Let us first consider the positive branch. Choosing G + (0) = 0 and G + (0) = 1, so that in a vicinity of ξ = 0 we have ϕ ≈ ξ, we find that the solution to the equation (59) takes the form: where am(u|n) is the Jacobi amplitude function and F (x|n) := Despite the presence of imaginary factors i, the above functions are real-valued. The minimal and maximal values of ξ are, The ϕ = G(ξ) function in the full range of variability of its argument (covering the G + (ξ) and G − (ξ) branches) is plotted in Fig. VI.
In the low energy limit, in which gradients are small, the DBI-like action (60) takes the approximate form where the ∓ sign refer to the F + or F − sector, respectively. In such a case, the functional f (ξ) plays the role of the effective potential, Then the mass of the field ξ is indeed represented by the scalar m. The quartic self interaction term is of the order O(1/S) as the neglected higher order kinetic term.
Finally, performing the large S expansion of the action in (57) with the functional f (ξ) given by the expression in (61), we obtain, This action gives the following equations of motion for ξ, corresponding to the sectors, F + and F − , respectively. To unify the notation, the identity, N √ q = √ −g, was used. The first two terms contribute to the standard wave equation for the Klein-Gordon on curved spacetime, while the terms of order O(S −1 ) specify the nonlinear corrections.

VII. COSMOLOGICAL IMPLICATIONS
What would a continuous spin system embedded into spacetime imply on its dynamics? In the previous section, a parallel between the general relativistic spin action and the DBI-type action has been established, which permits to make some predictions towards this direction, being of particular relevance in the cosmological context. This is because, since a few decades, works on string theory led to a renewed interest concerning the DBI ac-tion due to its link with the D-brane models [25]. Imposing the DBI action (57) to be real requires the square root 1 − f (ξ) g µν ∂ µ ξ∂ ν ξ to be real-valued. Under this condition, and considering negative f function, an upper bound on the scalar field velocity |ξ| is predicted. This upper bound is responsible for the so-called D-cceleration mechanism, and can be shown to introduce naturally a slow-roll inflation [26], with promising predictions on the non-Gausiannity of the power spectrum of the CMB [27]. This sections aims to investigate briefly if such cosmological features are also implied by our model, while we keep a more detailed analysis for forthcoming publication. While (81) has similar functional form as in the usual DBI cosmology [26], here we have α ≤ 1 always. This ensures that w ≤ −1, and in consequence the expansion of the universe is continually (super)-accelerated. This behaviour is similar to the one know from the case of phantom cosmologies [43]. Furthermore, assuming now that f is negative (ϕ ∈ F − ) such that the energy density is positive definite, since w < −1/3, the more the universe expands, the more ρ increases. In consequence, the more ρ increases, the faster the universe expands, thus engendering internal inflation as soon asȧ > 0. The D-cceleration leading to slow-roll inflation is, therefore, excluded by our model 9 .
One could, however, argue that the flatness and horizon problems can still be solved without the need for an inflationary phase. The other common solution requires a recollapse phase and refers to an ekpyrotic or cyclic universe [44]. If such a scenario would be a priori conceivable, it would be inadequate for producing scaleinvariant perturbations as long as f < 0 [45].
Alternatively, the both problems are solved through the emission of tachyacoustic perturbations which also happen to be scale-invariant [46]. One can indeed observe that the speed of adiabatic waves for our model is larger than the speed of light in vacuum [27,47]: It is important to notice that these tachyacoustic perturbations for k-essence models [48] do not violate causality [49] (see [46] for an application of the proof to the DBI case). Closing the discussion on the inflationary phase, the obtained DBI-like action may imply broader phenomenology and be considered as a candidate for dark energy [50]. Furthermore, worth stressing is also the relation to the Chaplygin gas models and Tachyon condensate. Another important property is that the DBItype k-essence models, to which our model belongs, have unique property from the viewpoint of the problem of introducing time in gravity, by virtue of the Brown-Kuchar mechanism [51].

VIII. SUMMARY
In this article, the general-relativistic model of a continuous spin system has been constructed. It has been derived as a generalization of a system of spins (magnetic moments) precessing in a constant magnetic field. An essential step was the application of the semi-classical version of the Holstein-Primakoff transformation, which allowed us to relate phase space of a spin field with phase space of a scalar field, so that the Poisson bracket on the latter remains canonical. The transformation is an example of a general procedure introduced recently in the context of Non-linear Field Space Theories (NFSTs).
The construction discussed in this article is not unique and the method can be used to obtain other theories of spin fields on curved spacetimes. However, the considered case is special because of a few reasons. Firstly, the investigated model reduces to the relativistic massive scalar field theory (Klein-Gordon field) in the large spin limit. Secondly, in the large mass limit, and in vicinity of a ground state, the model reduces to the continuous Ising model coupled to an external constant transversal magnetic field. Thirdly, the model is equivalent to a concrete realisation of the DBI-type k-essence scalar field theory, with the form of the f (ξ) function predicted by the model.
The third point is especially interesting since it indicates that there is a certain relation between spin fields and the Dirac-Born-Infeld theory. The spin field is in turn an example of NFST with the compact phase space. Actually, one of the motivations behind proposing the NFST programme was to impose constraints on the field values in the spirit of the original Born-Infeld theory. However, in the case of NFST this is done by introducing non-linearity to the field phase space. The results of our investigations confirm that some compact phase space realisations of NFST may be equivalent to the DBI-type scalar field theories. This also shows a possible connection between compact phase spaces and string theory, in the context of which the DBI models have been considered in the recent literature. Furthermore, reduction to the DBI-type k-essence opens a possibility of making phenomenological predictions, especially in cosmology [27], which has been preliminarily explored here.
The approach introduced in this paper provides a consistent method of coupling a spin field to gravity, which may be of relevance not only at the classical but also quantum level. While one possibility given by our framework is coupling a spin field to quantum gravity, not less interesting is analysis of quantum spin systems on curved backgrounds. Therefore, the considered model and the whole framework may find application in the domain of quantum many-body systems on curved spacetimes and the relativistic quantum information theory [52]. This may be relevant e.g. in theoretical description of such astrophysical objects as neutron stars, quark stars or white dwarfs, where both quantum and gravitational effects (but not quantum-gravitational) are relevant. For this purpose, quantization of the spin field considered in the paper has to be performed, which is a challenge that interested readers are encouraged to take.