Research Article | Open Access
Nonequilibrium Dynamics of the -Model Modes on the de Sitter Space
The two-dimensional -model with the de Sitter target space is locally canonic in the north pole diamond of the Penrose diagram in the cosmological gauge. The left and right moving modes on the cylindrical base space are entangled among themselves and interact with the de Sitter metric. Firstly, we show that the untangled oscillators can be obtained from the entangled operators by applying a set of Bogoliubov transformations constrained by the requirement that the partial evolution generator be diagonal. Secondly, we determine the nonequilibrium dynamics of the untangled modes in the nonequilibrium thermofield dynamics formalism. The thermal modes are represented as thermal doublet oscillators that satisfy partial evolution equations of Heisenberg-type. From these we compute the local free one-body propagator of an arbitrary mode between two times. Thirdly, we discuss the field representation of the thermal modes. We show that there is a set of thermal doublet fields that satisfy the equal time canonical commutation relations, are solutions to the -model equations of motion, and can be decomposed in terms of thermal doublet oscillators. Finally, we construct a local partial evolution functional of Hamilton-like form for the thermal doublet fields.
The two-dimensional nonlinear -model with the de Sitter target space describes a nontrivial class of field theories of which quantum dynamics is still largely unknown. Since the -models have proved themselves instrumental to the description of the string theory on several flat and curved space-time manifolds, it is likely that, by studying them on the de Sitter space, we will gain important insight into the structure of the string theory compactified to the four-dimensional expanding space-time and, more generally, on the time-dependent hyperbolic manifolds.
A considerable hindrance to the formulation of the quantum field theories on the de Sitter space has been the global structure of the latter that has made unclear the physical interpretation of different mathematical objects, for example, the generators of the de Sitter group, and the application of fundamental quantum concepts, for example, the -matrix. To some degree, it is possible to apply the Wightman two-point function formalism to noninteracting fields in the principal series representations of the group used to define the massive de Sitter fields and the Gupta-Bleuler method to the massless fields in the discrete series representations . However, there are still complications related to the causal horizons, the background temperature, and the instability of the system to the vacuum decay. Due to these particularities, it is extremely difficult to apply consistently even the noncovariant quantization methods to the field theories on the de Sitter space.
In contrast to other field theories, the two-dimensional -model is naturally localized on its base space which can be embedded inside a local causal region of the de Sitter space that is accessible to a single observer when the typical scale of the base space is much smaller than that of the target space. This localization usually implies that some local symmetries of the model have been fixed or linearized. In these conditions, the standard quantization methods can be employed (The obvious drawback of this approach is that the quantization is both local and noncovariant. Yet, in the absence of a covariant method, this is the only approach that has provided any information on the quantum properties of the σ-model on the de Sitter space so far.). Moreover, if the base space is considered to be a closed two-dimensional surface, then the corresponding -model contains an inner interaction of modes propagating in opposite directions on the base space. This interaction can be nontrivial in the de Sitter space even in a linearized theory.
There has been some work done recently along this line of investigation in the framework of the canonical quantization in time-dependent gauges that attempted to interpret the -models as tree level bosonic string theories (The de Sitter space-time is not a proper string background as it is not a solution of the string β-function equations.). The results obtained so far include solving the equation of state for the perturbations around the center of mass of different string configurations in the conformal gauge, the construction of the corresponding time-dependent Fock space, and the calculation of semiclassical quantities such as the mass of states and the maximum number of single string excitations [2–7]. The same quantities were derived for the little strings in an arbitrary time-dependent gauge in . The divergences of the Nambu-Goto model were discussed in the path integral formulation in  and the backreaction in the classical theory was analysed in .
In a very recent paper, the -model with a two-dimensional cylindrical base space has been quantized in a new time-dependent gauge, called the cosmological gauge, that is natural to the model in the sense that the equal time commutation relations of the fields and the canonical commutation relations of their oscillating modes are compatible with each other in this gauge . It was shown there that the left and right quantum excitations display a time-dependent entanglement generated by the partial evolution map that acts between the states at two instances of time and has the symmetry (States with symmetry in the early universe have been discussed recently in .). We note that this inner entanglement is different from the entanglement of field theories in the de Sitter space-time as each of the left and right moving modes lives on the full base space and in the same region of the de Sitter space-time . The partial evolution map can be expressed as unitary evolution operator generated locally by a Hermitian Hamiltonian-like operator in the north pole diamond of the Penrose diagram of the de Sitter space. But in contrast with the flat space-time, the oscillating modes are quenches of the generator of the partial evolution map and not eigenvectors of it. That raises the question whether there is a representation of the oscillating modes in which the generator is diagonal and the oscillators are untangled. Finding such representation is a necessary step to addressing the issue of the physical properties of the -model excitations.
In this paper, we propose a simple solution to the problem of the untangled oscillator representation by constructing a set of linear time-dependent Bogoliubov transformations that diagonalize the local Hamiltonian-like generator of the partial evolution map. We show that the real time-dependent functions that parametrize the Bogoliubov transformations satisfy a set of constraint equations that represent the diagonalization condition. Also, we determine the differential Heisenberg-like equations that describe the partial evolution of the untangled oscillators. The thermal effects in a quantum system in a curved space-time can be considered among the most fundamental physical properties. The untangled oscillator representation allows one to define and analyse the thermalization of the excitations of the -model in the de Sitter space by applying canonical methods. The fact that the frequencies depend on time suggests that the thermal dynamics of the oscillators should take place in nonequilibrium. We examine the thermalization of the untangled oscillators in the nonequilibrium thermofield dynamics (NETFD) formalism in which the degrees of freedom of the thermal oscillators are most naturally represented by thermal doublets [14–16]. That allows us to determine the local nonequilibrium dynamics of the untangled modes generated by the partial evolution operator and calculate the propagator of an arbitrary thermal mode between two different instants of time. In curved space-time, the compatibility between the particle excitations and the field description is an important problem to be analysed. In the present case, it is important to know whether the thermal modes correspond to a thermal field theory of the -model. We answer this question in affirmative by constructing the thermal doublet fields associated with the -model we started with and their canonically conjugate momenta. These fields have the following fundamental properties: their oscillating modes are the thermal doublet oscillators, they are solutions to the -model equations of motion in the cosmological gauge, and the canonically conjugate fields satisfy the standard equal time commutation relations. The partial evolution of the thermal fields is characterized by differential equations that can be interpreted as canonical extended Heisenberg equations. This observation allows us to derive the classical partial evolution functional of Hamiltonian-type for the thermal doublet fields.
The paper is organized as follows. In Section 2, we review the results from  on the quantization of the -model in the cosmological gauge that are necessary to obtain the untangled oscillator representation. Then we define the Bogoliubov transformations between the entangled and untangled oscillators and derive the constraints satisfied by the Bogoliubov parameters. From the diagonal generator of the partial evolution, we deduce the partial evolution equations of the untangled oscillators. In Section 3 we describe the thermalization of the untangled oscillators in the NETFD approach. We formulate the thermal degrees of freedom as thermal doublet oscillators. Then we give a second set of Bogoliubov transformations that map the untangled oscillators into two different representations: the time-independent and the time-dependent pseudoparticle representations, respectively, which are needed to define the vacuum in which the physical quantities should be computed and to determine the dynamics of the thermal oscillators. These are used to calculate the propagator of an arbitrary thermal doublet mode between two different values of time. In Section 4 we obtain the thermal doublet fields associated with the thermal doublet oscillators. We show that these fields satisfy the -model equations of motion and the standard equal time commutation relations. Also, the partial evolution functional corresponding to this structure is derived. In the last section we discuss the results. In Appendix, we briefly review the NETFD relations that are used throughout this work.
2. Quantum -Model in the Cosmological Gauge
In this section, we review some results on the quantization of the two-dimensional -model with the de Sitter target space obtained in  that are relevant for the construction of the untangled oscillator representation. Then we define the Bogoliubov transformations between the entangled and untangled oscillator representations and determine the constraints that must be satisfied by the Bogoliubov parameters and the partial evolution of the untangled oscillators.
2.1. Canonical -Model in the Cosmological Gauge
The -model of interest is defined by the embedding of the two-dimensional cylinder into the north pole diamond of the Penrose diagram of the four-dimensional de Sitter target space denoted by . The base space is endowed with the metric and the comoving frame is chosen in in which the metric takes the following form:The spatial and temporal coordinates on the base space are denoted by and , respectively. The embedding is characterized by the orientation , where is the parameter of the local time-like Killing vector in the north pole diamond of . The model is invariant under the two-dimensional Weyl symmetry and the reparametrization of the variables and . By using these symmetries, one can pick up the cosmological gauge which is defined as the following time-dependent gauge of the Weyl symmetry:Here, denotes the Hubble’s constant. In the cosmological gauge, the canonical commutation relations of the quantum oscillators and the equal time commutation relations of the corresponding canonical fields are compatible to each other. This statement is not true for an arbitrary time-dependent gauge . In this setting, the action of the -model takes the following form:where are the spatial direction transverse to the local time-like Killing vector in that span the transverse section with the symmetry. The variational principle applied to with vanishing boundary terms provides the following equations of motion:The general solution of (4) and its conjugate momenta have the following linear decomposition in terms of the Hankel functions  that reflect the symmetry of the embedded base space :where the conjugate momenta are defined byExponential time-variables are conveniently defined by the following relation:The expansion coefficients and do not depend on time. As usual, by applying the canonical quantization method, the coefficients and are promoted to operators. The canonically conjugate fields and the time-independent operators must satisfy simultaneously the equal time commutation relations and the oscillator commutators, respectively, that are given by the following relations:for all and all . These equations are satisfied in the cosmological gauge but not in an arbitrary time-dependent gauge. The above relations show that the two sets of operators and , respectively, should be interpreted as oscillating modes that move along the embedded circle in opposed directions. These modes have time-dependent frequencies and can be paired according to the values of the indices and as in the flat space-time. The quantum dynamics is constrained to obey the following equations:whereis the contribution of the zero modes to . The constraints (10) and (11) result from the vanishing of the energy-momentum tensor . In the above relations, all products of oscillator operators are normally ordered. Their coefficients from the right-hand side of (10) are given by the following time-dependent functions:where and is the Bessel function of the second kind. It was shown in  that the operator can be interpreted as the generator of the translations along the spatial direction of the embedded cylinder. On the other hand, the operator is the generator of the entanglement between and sectors, respectively, that occurs during their local time evolution and it is identical to the canonical Hamiltonian up to a smooth time-dependent function. The entanglement is caused by the interaction of the fields of the -model with the components of the metric on the and it depends on time. The mapping of the entangled oscillator states between two instants of time can be described by the partial evolution map that is generated by the operator and it is given by the following relation:The operator acts on the states at time and maps them to states at time , respectively. The time-dependent states belong to the one-parameter family of time-dependent Hilbert spaces that has the structure of a direct product:The space contains the time-independent eigenstates of the operators and . The time-independent oscillator states belong to the Hilbert space . Their linear combinations are formed with coefficients from that denotes smooth time-dependent functions defined in the north pole diamond of the de Sitter space. At any given instant of time, a state of the system is represented by a superposition of elements from with coefficients that belong to the set . Two operators at different values of time that belong to one-parameter family of operators are mapped into one another by the induced partial map:Equations (16) and (18) determine the partial evolution of the -model in the north pole diamond of the de Sitter space as seen by the comoving observer (As discussed in , this map does not describe the complete evolution of the system which must also take into account the evolution of the background metric and would ideally be derived from a theory of the Quantum Gravity.). The physical states belong to the physical subspace of . This is defined by the following relation:For the explicit calculation of we refer to .
2.2. The Untangled Modes
The normal ordered operator from (10) contains nondiagonal terms that generate the entanglement between the and sectors, respectively. Moreover, the time-independent eigenstates of the oscillators from these sectors are not eigenstates of .
In order to find a basis in which the operator is diagonal, we consider the general Bogoliubov transformations that act on the pair with definite values of indices according to the following equations:where the parameters , , , and are real functions on time. They depend only on the oscillating mode and have the same form for all spatial directions. The operators and satisfy the standard commutation relations:The operator can be expressed in the representation by applying the transformations (20) and (21) to the right-hand side of (10). After some algebraic manipulations, we arrive at the following diagonal terms:where the frequencies of the modes are given by the following real functions on time:The frequencies are positive for all values of if . If , the oscillator frequencies are negative. The requirement that the nondiagonal part of vanishes in the representation imposes the following constraints on the parameters of the Bogoliubov transformations:where we have introduced the notation and we have dropped the variable for simplicity. The first equation above simply expresses the parameter in terms of other parameters of the Bogoliubov transformations and the Hankel functions. The parameters and are left arbitrary by these conditions. The fact that the parameters of the Bogoliubov transformations are functions on time implies that the modes and depend on time, too. Therefore, the commutation relations (22) should be understood at equal times. The second constraint (26) is necessary to guarantee that the functions are real and it determines the parameters ’s as functions of ’s which can be calculated from the relation (14). After some manipulations of the nonlinear algebraic equation (26), one can show that the trigonometric functions of are real and of modulus smaller than one. This proves that the constraints are consistent with the properties of their variables.
The Fock space of the -model at any given value of time is spanned by linear combinations of multiexcitation states of the following form:where we have used the standard multi-index notationHere, and denote the eigenvalues of the corresponding number operators and , respectively. The coefficients are smooth complex functions on the local time parameter. The vectors from the Fock space are obtained in the usual fashion by applying repeatedly the creation operators on the ground state of the oscillator system defined by the following relations:for all and all . As expected, the Fock space at a given time has the structure of the direct product: Thus, the full Hilbert space of the -model in the untangled representation is a one-parameter family of Hilbert spaces:where is the Hilbert space of the zero mode operators and , respectively. The physical subspace is defined asBy applying the Bogoliubov transformations (20) and (21) to the right-hand side of (11), one can see that the operator takes the following form:This shows that the space contains just states that satisfy the level matching condition at every value of time. On the other hand, there is a relationship among eigenvalues of different operators calculated in states that belong to the kernel , namely,where is the eigenvalue of . Equation (34) is an energy-momentum-like relation of which outcomes could be observed locally in the comoving frame in the north pole of the de Sitter space in the cosmological gauge. Note that the last term in the right-hand side of the above equation is the result of the interaction between the oscillators and the de Sitter background metric. In the ground state, (34) reduces toThe lowest excited oscillator state corresponds to and has the following form:For these states, (34) takes the following form:Since the quantities from the right-hand side of (35) and (37) are fixed, these equations impose restrictions on the values of and . Similar relations can be derived for states with a higher number of excitations.
We end this section with a brief comment on the time variation of the untangled oscillators. In this representation, the partial evolution operator from (16) is written in terms of the diagonal operator from the relation (23). It is easy to see that the action of the operator on the operators and given by (18) is compatible with the following Heisenberg-like equations:for all and all . The above equations together with the partial map of statesdetermine the partial evolution of the untangled excitations of the -model.
3. Nonequilibrium Dynamics of the Untangled Modes
In the previous section, we have shown that the left and right moving modes on the embedded base space can be untangled from each other. However, each of the untangled modes still interacts with the components of the de Sitter metric. The most important consequence of this fact is that the number of quanta of a given mode can vary with time as observed in the local frame associated with the embedded cylinder . Moreover, the frequencies of all modes are time-dependent. This shows that the untangled modes are subject to nonequilibrium dynamical equations that describe their evolution under the continuous exchange of energy with the classical time-dependent metric fields. In this section, we are going to determine these equations. Since the problem is formulated in the canonical formalism, we will apply the NETFD method that can be found in [14–16, 18].
3.1. The Thermal Doublet Modes
We begin by constructing the thermal doublets corresponding to the untangled modes of the -model. To this end, we associate with each oscillator an identical copy of it, denoted by a tilde. Then the Hilbert space of the total system is the direct sum of the Hilbert spaces of all untangled oscillators and their copies:The above relation shows that, for every pair of indices , there are two pairs of independent oscillators and , respectively, that act on the corresponding total Fock space which can be decomposed asThe oscillator operators satisfy the standard canonical commutation relations:Since the oscillators are time-dependent, the above relations should be understood as holding at equal times. The states from are constructed as usual by acting with the creation operators on the total vacuum: which is annihilated by all annihilation operators from the tilde as well as the nontilde sectors, respectively,for all and . In what follows, we are going to use the notation in which the time variable is explicit instead of . We recall that the relationship between the two variables is given by (8) above.
The time-dependent thermal doublets are two-dimensional vectors with operator entries defined by the following relations:where are two-dimensional vector indices. The row vectors can be obtained from the column vectors by applying the following bar-conjugation operation:where stands for the corresponding Pauli matrix. The action of the thermal doublets on the vacuum is defined by the action of their components as given by (44) above.
The NETFD prescription  requires the introduction of a time-independent pseudoparticle doublet for each thermal doublet that is defined by the following relations:where the Bogoliubov matrices are given by the following relations (see Appendix):The time-independent operators are necessary to define the time-independent pseudoparticle vacuum in which all physical quantities should be computed . For example, the number of untangled excitations from (52) and (53), respectively, is defined by the vacuum expectation value of the thermal doublet operators in the pseudoparticle vacuum:In order to determine the evolution of the thermal doublets under the action of the partial evolution map , one needs to construct a second set of time-dependent pseudoparticle operators and that differ from and , respectively, by the time-dependent phases:Two pseudoparticle doublets and , respectively, are associated with the same untangled doublet if . In this case, the Bogoliubov matrix defined by (52) maps to as can be easily verified. After some algebra, one can show that the operators and satisfy the following partial evolution equations:where we have introduced the following notation:The equations that describe the partial evolution of the operators and can be obtained from (57) and (58), respectively, by replacing to quantities that refer to the -modes by the ones corresponding to the -modes. Equations (57) and (58) should be compared with the action of the partial evolution map from (16) that characterizes the dynamics of the -model in the de Sitter background . By using (18), one can easily see that the normal ordered operator generates the first two terms from each of (57) and (58) if . These terms define the evolution of the pseudoparticle operators and , respectively. The last term is determined by the operator that contains the contribution of the Bogoliubov matrix to and which is also present in the interaction theory where it generates a counterterm in the interacting Hamiltonian . The same considerations can be made for the excitations from the -sector and we reach at similar conclusions.
This analysis shows that the nonequilibrium dynamics of the untangled modes has the standard NETFD form for time-dependent system of oscillators. The dynamics are determined in terms of the operators and that generate together the following nonequilibrium evolution map of the total system:whereHere, operator has the same form as operator but with replacing in (59) above.
By extending in a simple manner the NETFD formalism, one can associate thermal doublets with the degrees of freedom and that are necessary to describe a full copy of the original -model . To this end, we firstly define the following variables: Next, we introduce the zero mode oscillator operators and that satisfy the canonical commutation relations . Then it is easy to show that (62) can be rewritten as follows:One can verify that the operators and obey the same canonical commutation relations in both representations. In order to construct the thermal degrees of freedom, we associate identical copies with the zero modes oscillators denoted by and . These operators can be obtained from (63) by applying the tilde-conjugation rules from Appendix. Then the thermal doublet of the zero modes is given by (45). The result can be written as follows:where . Equations (57), (58), and (65) determine the partial evolution of the thermal doublets of all untangled modes of the -model including the center of the circle . They can be used to calculate physical quantities associated with local observations in a neighbourhood of the embedded base space as we are going to see in the next subsection.
3.2. Free One-Body Propagator
One important quantity to be calculated in nonequilibrium is the free one-body propagator of an oscillating mode between two different values of time and . Since the and -sectors do not interact with each other, there is just one propagator for each of these sectors and no cross-sector propagation. These are defined by the following vacuum expectation values:where is the vacuum state of the time-independent pseudoparticle excitations. The propagators can be calculated by using the relationship between the two sets of operators and , respectively, as given by (48) and (49) above. A short computation produces the following results:whereThe explicit form of the components of the propagator can be obtained after a simple matrix algebra. The results in the -sector take the following form:As before, the calculations and the results in the -sector duplicate the ones in the -sector. The above equations show that the propagator of an untangled mode between and is completely determined by its frequency and the number of excitations . The partial evolution equation of the latter can be obtained from (38) and (54). It is easy to verify that the results arewhere the frequencies of the untangled modes are given by (24). If interactions among the modes are introduced in either or sectors, respectively, the propagators and receive corrections as the interactions change the number of excitations being created at each instant of time .
4. The Thermal Doublet Fields
We now turn to the question whether the nonequilibrium dynamics of the untangled excitations can be described in terms of fields that satisfy the equations of motion of the -model. These fields should have a thermal doublet structure since their modes should be the thermal doublet oscillators constructed in the previous section.
In order to construct the thermal fields of the -model, we start by expressing the solutions given by (5) and (6), respectively, in the representation. The result takes the following form:where we have introduced the following notations:It is a straightforward exercise to show that the fields and are canonically conjugate to each other in the untangled oscillator representation:The identical copy of the system can be constructed by applying the tilde-conjugation given in Appendix to the fields and , respectively. We write down the result for completeness:Then by combining (72) and (78) we obtain the following thermal doublet fields:The physical interpretation of the fields and is the following: they represent the degrees of freedom associated with the two-dimensional -model that lie on the embedded cylinder in terms of which the nonequilibrium dynamics of the untangled modes are described in the cosmological gauge. From the point of view of , the theory is local in a neighbourhood of as the observation of the nonequilibrium dynamics is accessible only to the observer defined in the Section 2.
The consistency of the present formulation of the nonequilibrium dynamics with the canonical quantization formalism requires the identification of the canonically conjugate field variables. They can be recognized from the set where the fields and can be obtained from and