A Consistency Check for the Free Scalar Field Theory Realization of the Doubly Spacial Relativity

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Mohsen Khodadi^,¹^,^???, Kourosh Nozari^,²^,³^,^???

1 Young Researchers and Elite Club, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran;

2 Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran;

3 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P. O. Box 55134-441, Maragha, Iran;

Corresponding authors: ? ? E-mail:knozari@umz.ac.ir? ? E-mail:mohsen.khodadi@gmail.com

Received:2019-01-12Online:2019-06-1

Fund supported:

Supported by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under Research Project.1/6025-73

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Mohsen Khodadi, Kourosh Nozari. A Consistency Check for the Free Scalar Field Theory Realization of the Doubly Spacial Relativity ^*. [J], 2019, 71(6): 677-684 doi:10.1088/0253-6102/71/6/677

1 Introduction

Although the ultimate nature of quantum gravity (QG) theory has not been yet revealed thoroughly, one feature is definitively accepted among experts in this field: there is a minimum threshold length scale or equivalently a natural minimum length cutoff of the order of Planck length, $\ell_p$.^§(§ Note that for the Planck length there are also two other important counterpart scales, the "Planck time" $t_p\varpropto l_p$ and "Planck energy" $E_p \varpropto \ell_p^{-1}$, that address the existence of a minimum time and a maximum energy cutoff in nature respectively). This minimum length can be considered as the spatial size of the universe at the beginning in a quantum spacetime picture. In around and beyond the mentioned threshold scale, transition towards quantum spactime happens naturally, so that the geometrical description via general relativity (GR) loses its validity. However, it is a reasonable expectation that QG should meet the special relativity (SR) for all experiments planned to explore the nature of spacetime even at length scales far from the $\ell_p$. Given that close to $\ell_p$ (or equivalently $E_p$) one expects emergence of new phenomena, so the question then arises that hin what reference frame the Planck scale $\ell_p$ (and also its other peers, Planck time and Planck energy) is the boundary for observation of new phenomena? From another perspective, SR also taught us that, due to the issue of h{length contraction}, the Planck length cannot be a unit boundary from viewpoint of all inertial observers. Overall, for the passage of this issue, two views have been raised in recent years. Firstly, by discarding the "Relativity Principle" (RP) as the heart of SR, one suggests the appearance of a preferred reference frame due to existence of an invariant length scale $\ell_p$. As a consequence, the local and global Lorentz invariance in the presence of $\ell_p$ are broken so that no longer this symmetry can be regarded as a fundamental symmetry of the nature.^[1-6] While some physicists insist on violation of Lorentz symmetry in scales close to the Planck scale^[7-9] and believe that in near future we will receive signals for this violation via, for instance, cosmic ray spectra^[10] and gamma ray bursts,^[1-4] yet there is no direct observational support for this issue. In this approach the correction due to existence of cutoff $\ell_p$ is considered just into the relativistic on-shell (energy-momentum) relation while other relations have no Planck scale corrections. Secondly, there is another proposal (that includes different versions) which by keeping RP, tries to fix the above mentioned problem via finding the modifications of the standard Lorentz transformations. To be more concrete, SR is developed to a framework called the "Doubly Special Relativity" (DSR) in which the standard Poincar$\acute{e}$ algebra is extended to a non-linear structure, see Refs. ^[11-12] and ^[13-14]. The troubles appeared within the first approach due to discarding the Lorentz symmetry have made the DSR models to gain more popularity in recent years. It seems that DSR(s) can be imagined as candidates for the role of a flat space-time limit of QG in the absence of gravitational interaction.^[15-17] In DSR theories, the relevant Lorentz transformations of the momentum space are modified by some non-linear terms so that the resulted transformations still protect the RP. However, due to the existence of non-linear modification terms, we are dealing with a more complicated non-linear invariant instead of quadratic invariant.^?(? The idea of having a non-linear invariant as a quantity involving the metric cannot be surprising in the sense that based on other prominent approaches to QG, such as the Loop quantum gravity (LQG), beyond the Planck invariant scales ($\ell_p,t_p,E_p$) the concept of a smooth metric is worthless.) This makes the dispersion relations (DR) to depart from the standard form $E^2- p^{2}=m^2$ at least up to the leading order of the Planck length.^[18-19] Overall, DSR theories absolutely respect the relativity of inertial frames so that all observers agree on the existence of a borderline given by the Planck invariant scale(s). It is interesting to mention that some arguments based on observational evidences raised in Refs. ^{[1-4, 10]} are justifiable by some DSR models, which highlights the phenomenological strength of these theories. Owing to the non-linear modification, rediscovering the position space of DSR (which was originally formulated in the momentum space as a consequence of the modified dispersion relations (MDR)), is non-trivial. In other words, physical interpretation of outcomes derived in momentum space formulation can be evaluated when the status of the connection between momentum space and its dual i.e. position space, is determined. As an example one can mention the troubles encountered when one defines the physical velocity within DSR models; see Refs. ^[20-21] for extensive reviews of the related issues. In Ref. ^[22], in order to solve the mentioned issue, by applying two possible routes the authors were able to display the position space within DSR. By concerning on the issue of internal consistency within the first framework used in Ref. ^[22], the authors have demanded that free field theories (in particular scalar field theory) should have plane wave solutions with four-momentum fulfilling the set of MDR relevant to a certain DSR model. It is worth noting that the most important outcome of embedding such a maximum energy into quantum field theory (QFT) is fixing the problem of renormalizability when interactions are regarded. To see various examples for the impacts of MDR on effective QFT, we refer to Ref. ^[23]. Apart from all these discussions, we know from standard QFT that there are two parallel routes to derive scalar field equation of motion known as the Klein-Gordon (KG) field equation.^[24-25] A well-known method starts by applying the first quantization scheme on the classical relativistic particle theory to get a relativistic quantum mechanics. Then by generalizing the states and commutation relations $[x_i,p_j]$ to fields $[\phi_r,\pi_s]$, the scalar QFT is generated. The other method starts by applying direct second quantization scheme on the classical relativistic field theory. However, as well as there is a third alternative method, the so called Heisenberg picture field's equation of motion, which in standard QFT is regarded as a reliable consistency check for the scalar field theory. In other words, deriving a KG field equation similar to the same thing that is acquired in common methods, expresses the fact that scalar QFT is a self-consistent theory. We note that there is another alternative way defined in pure FRW cosmology, e.g. making use of appropriate Lagrangians, different from the one of harmonic oscillator in effective field theory. We refer to Ref. ^[26] for a recent work in this direction. Also for related issue in the framework of entanglement in quantum cosmology see Refs. ^[27-28]. In this letter, by focusing on the free scalar QFT realization of DSR (in particular, the version constructed by Maguejo and Smolin (MS) in Refs. ^[13-14]), we are going to use the above mentioned alternative methods to provide a consistency check of the scalar field theory modified due to the presence of a natural Lorentz invariant energy cutoff. In Sec.~2, based on the MDR in MS model, we propose a relevant Lagrangian and subsequently we derive the modified, free KG field equation of motion via the Euler-Lagrange field equation. The main ingredient of the paper is reported in Sec.~3 where, to do a consistency check of the DSR modified scalar field theory at hand, we have derived the KG equation of motion now through an alternative path, that is, the Heisenberg picture equation of motion of the field (Hamiltonian formulation). It is done based on two postulates: firstly, the invariance of the linear contraction between position space and its dual which for the first time proposed in Ref. ^[22]. Secondly, preserving unitary time evolution which guarantees the conservation of the total probability. We observed that contrary to the standard case (in the absence of natural cut-offs), the KG field equation obtained in Secs.~2 and 3 are not identical. Rather, these two approaches result in the plane wave solutions that are corresponding to wave propagation in two mediums with different dispersion relations. This issue can be viewed from different perspectives: it may refer to a pathological feature of the extended QFT framework or it may be a signal that Lagrangian and Hamiltonian formulations are not necessarily equivalent at the Planck energy scale. This may be also a signal that pictures in quantum mechanics and QFT are not necessarily equivalent in quantum gravity regime.

2 Lagrangian Based Derivation of Klien-Gordon Equation with a Natural Lorentz Invariant Energy Cutoff

For compatibility of the invariant energy cutoff with other principles governing the SR theory, the standard mass-shell condition for the particles should be modified in the following general form

(1) $E^{2}f_{1}^{2}(E)-p^{2}f_{2}^{2}(E)=m^{2}\,,$

where the functional form of the energy dependent functions $f_{1}$ and $f_{2}$ are DSRs model-dependent. Due to the deformations appeared in the above dispersion relation, it can not remain invariant under the linear Lorenz transformations anymore. Indeed, Eq. (1) is consistent with the relativity principle of SR in the case where one adopts a nonlinear representation of the Lorentz group through the relation

(2) $K^{i}=U^{-1}L_{0}^{i}U~,$

with

(3) $U_{a}(E,p_i)=(U_{0},U_i)=(Ef_{1}(E),p_i f_{2}(E))\,,$

where $U:\cal{P}\rightarrow \cal{P}$ is a nonlinear mapping of the momentum space onto itself. The components of $L_{0}^{i}$ denote the standard Lorentz generators which act on the momentum. Note that the $U$ map defined in Eq. (3) is equivalent to the modified mass-shell condition (1) from the viewpoint that it can wholly address the one particle segment of any given DSR model formulated in the momentum space. Therefore, by choosing various cases for the $U$ map, one can find numerous nonlinear realizations of the action of the Lorentz group as well as modified dispersion relations which are showcase of new invariant quantities. With regard to Magueijo-Smolin (MS) model of DSR,^[13-14] a modified generator of boosts can be present as^Ⅱ(Ⅱ Note that $\ell_p$-dependent term in Eq. (4), the angular momentums $j^i$ and boosts $K^i$ still fulfill the standard Lorentz algebra i.e. $[J^i,K^j]= \epsilon^{ijk}K_k$ and $[K^i,K^j]=\epsilon^{ijk}J_k$. Also, the origin of the nonlinear action of $K^i$ on the momentum space goes back to the term $p_i$ in Eq. (4).)

(4) $K^i\equiv L_{0}^{i}+\ell_{p}p^iD~,$

in which $D=p_a({\partial}/{\partial p_a})$ is a dilatation generator and acts on the momentum space as $D\circ p_a =p_a$. Note that by choosing $U\equiv\exp (\ell_p ED)$ in Eq. (2) and keeping the leading order terms containing $\ell_p$, one recovers Eq. (4). Therefore, the suggested $U$ map can produce a nonlinear representation of the Lorentz group such that by acting on the momentum, one finds

(5)$U\circ p_a=\frac{p_a}{1-\ell_p E}~.$

One can check that $U$ is not unitary and also it diverges at $E =\ell_{p}^{-1}$ which refers to the appearance of a new invariant. Modified mass shell condition relevant to the $U$ presented in Eq. (5), reads as

(6) $\frac{\eta^{ab}p_ap_b}{(1-\ell_p E)^2}=m^2~.$

By imposing the postulate that "there should be plane wave solutions for free field theories", the authors in Ref. ^[22] were able to provide a dual position space for the momentum space counterpart (i.e. a position space version of the nonlinear relativity). It should be emphasized that $p_a$ in plane wave solutions as $\exp(- i p_ax^a)$ is restricted to satisfy the above MDR (or generally Eq. (1)) in one side and also retaining the linear contraction $p_ax^a$ in other side in order to meet the plane wave solution. In what follows, we proceed by deriving the KG equation of motion for a scalar field theory realized in MS model of DSR.^[13-14]

Let us firstly present the following standard Lagrangian density

(7) $\cal{L}=\kappa(\partial_{a} \phi^{\dagger}(x,t) \partial^{a}\phi(x,t)-\mu^2 \phi^\dagger (x,t) \phi(x,t))\,, \\ \Big(\mu^2=\frac{m^2}{\hbar^2} \quad {\rm and} \quad c=1\Big)\,,$

for a free scalar field $\phi(x,t)$ including an arbitrary numerical constant $\kappa$ and the mass $m$. By replacing

(8) $\partial_a\rightarrow\partial_a(1+\text{i}\hbar \ell_p\partial_0)~,$

that is applied on the MDR (6) as well as Eq. (5), one obtains a modified Lagrangian density as

(9)$ \cal{L}_{ MS}=\kappa[(1+2\text{i}\hbar \ell_p \partial_0)\partial_{a}\phi^{\dagger}(x,t) \partial^{a}\phi(x,t) \\ -\mu^{2}\phi^\dagger(x,t) \phi(x,t)+{O}(\ell_{p}^{2})]~, $(9)

for the relevant Lagrangian density of a free scalar field theory in DSR. Substituting the modified Lagrangian density into the Euler-Lagrange field equation

(10) $\frac{\text{d}}{\text{d}x^\nu}\Big(\frac{\partial {\cal{L}}}{\partial\phi^{r}_{,\nu}(x,t)} \Big)-\frac{\partial{\cal{L}}}{\partial\phi^{r}(x,t)}=0~,\quad \nu=0,1,2,3$

yields the KG equation for the scalar field and its conjugate, where the values $r=1,2$ signify respectively the field $\phi(x,t)$ and its complex conjugate $\phi^\dagger(x,t)$,

(11) $[(1+2\text{i}\hbar \ell_p \partial_0)\Box+\mu^2]\phi(x,t)=0\,, \\ [(1+2\text{i}\hbar \ell_p \partial_0)\Box+\mu^2]\phi^{\dagger}(x,t)=0~.$

Here, we choose $\kappa=1$ and restrict our calculations to the first order of the Planck length by neglecting all higher order terms. The existence of term $i\hbar\ell_p\partial_0$ in Eq. (11) addresses a scalar field theory supported by a natural Lorentz invariant energy cutoff since here $E$ is the relevant energy of plane wave employed to probe spacetime in such a way that its value cannot be greater than $E_p$. Finally, the above couple of equations can be rewritten in the following compact forms

(12) $(\tilde{\Box}+\mu^2)\phi(x,t)=0\,, \quad (\tilde{\Box}+\mu^2)\phi^{\dagger}(x,t)=0~,$

where here $\tilde{\Box}$ is a re-scaled d'Alembertian defined as

(13)$\tilde{\Box}\equiv \Box (1+2\text{i}\hbar\ell_p \partial_0)\,.$

Also, the modified equations (11) can be compactified in another equivalent form

(14) $(\tilde{\hbar}^2\Box+m^2)\phi(x,t)=0\,, \quad (\tilde{\hbar}^2\Box+m^2)\phi^{\dagger}(x,t)=0~,$

where $\tilde{\hbar}\equiv\hbar(1+\text{i}\hbar\ell_p \partial_0)$ refers to an effective energy (or length) dependent Planck constant, similar to what is expected from MS model of DSR.^[13-14] It can be easily proved that the general solution of the above modified KG field equations is a superposition of the plane waves similar to the standard case, but with a difference that here $p_a$ satisfies the relevant MDR (6), not its standard counterpart. Another important point is that the modified KG field equations (12) and (14) are directly extractable by replacing the $E\rightarrow \text{i}\hbar\partial_0$ and $p_a\rightarrow -\text{i}\hbar\partial_a$ in MDR (6). This indicates the validity of the proposed Lagrangian (9) for MS version of the DSR field theory. However, our principal motivation for introducing Lagrangian is investigation of the self-consistency of the underlying DSR-based free scalar field theory. This is done by extracting the KG field equations from the Hamiltonian formulation as a preferred way in the standard QFT.

3 Heisenberg Picture Equation of Motion for the Scalar Field with a Natural Lorentz Invariant Energy Cutoff: Hamiltonian Formulation

Based on the standard quantum field theoretical considerations, in this section we derive a KG equation of motion for the free scalar field that is deformed by a maximum energy cutoff, but contrary to Eq. (11), this time in the Heisenberg picture. If we achieve in this fashion an equation exactly as Eq. (11), we can claim that the underlying DSR free scalar field theory is a self-consistent theory just as SR-based one.

The important character of the standard Heisenberg equation of motion is that it guarantees the unitary time evolution as a natural and required constraint for any real system of physics. Due to the energy dependent Planck constant suggested via MS deformed commutator relation $[x^i,p_j]=\text{i}\hbar\delta^{i}_{j}(1-\lambda E)$ in Ref. ^[14] (where $\lambda=E_p^{-1}$ or $\lambda=\ell_p$), Schr$\ddot{o}$dinger equation can be rewritten as

(15) $\lambda \hbar^2 \frac{\partial^2}{\partial t^2}\psi(t)+\text{i}\hbar\frac{\partial}{\partial t}\psi(t)=H_{\rm MS}\psi(t)~,$

where its solution takes the following form

(16) $\psi(t)=\exp\Big(-\frac{\text{i}t}{\hbar}\Big(\frac{1\pm\sqrt{1-4\lambda H_{\rm MS}}}{2\lambda}\Big)\Big)\psi(0)~.$

So, by concerning on the minus sign as an acceptable solution (note that the wave function disappears in the limit $\lambda\rightarrow0$ for the positive sign), then the average value of any operator ${\cal{O}}$ in the the Schr$\ddot{o}$dinger picture reads as

(17) $\langle O \rangle _{S}=\langle\psi(0)|\exp\Big[\frac{\text{i}t}{\hbar}\Big(\frac{1-\sqrt{1-4\lambda H_{\rm MS}}}{2\lambda}\Big)\Big] {\cal{O}} \\ \times\exp\Big[-\frac{\text{i}t}{\hbar}\Big(\frac{1-\sqrt{1-4\lambda H_{\rm MS}}}{2\lambda}\Big)\Big]|\psi(0)\rangle. $

Then based on the supposed equivalence of the Schr$\ddot{o}$-dinger and Heisenberg pictures, $\langle{\cal{O}}\rangle_{S}=\langle{\cal{O}}\rangle_{H}$, one arrives at

(18) ${\cal{O}}(t)=\exp\Big[\frac{\text{i}t}{\hbar}\Big(\frac{1-\sqrt{1-4\lambda H_{\rm MS}}}{2\lambda}\Big)\Big] {\cal{O}} \\ \hphantom{{\cal{O}}(t)=} \times\exp\Big[-\frac{\text{i}t}{\hbar}\Big(\frac{1-\sqrt{1-4\lambda H_{\rm MS}}}{2\lambda}\Big)\Big]\,.$

The time evolution of ${\cal{O}}(t)$ is derived as follows

(19) $\frac{\partial}{\partial t}{\cal{O}}=\frac{\text{i}}{\hbar}\Big[\Big(\frac{1-\sqrt{1-4\lambda H_{\rm MS}}}{2\lambda}\Big) {\cal{O}} \\ \hphantom{\frac{\partial}{\partial t}{\cal{O}}=}-{\cal{O}}\Big(\frac{1-\sqrt{1-4\lambda H_{\rm MS}}}{2\lambda}\Big)\Big]\,.$

Given the fact that $\lambda$ is small, by applying the expansion $\sqrt{1-4\lambda H_{\rm MS}}\approx1-2\lambda H_{\rm MS}$, one finally can show that within the context of MS model of DSR the Heisenberg equation of motion has the standard formulation as

(20) $\text{i}\hbar\frac{\partial}{\partial t}{\cal{O}}=[{\cal{O}}, H_{\rm MS}]~\,.$

This means that the relevant Heisenberg equations of motion in the level of DSR deformation satisfy the unitary condition like some other approaches to QG such as noncommutative geometry and GUP (see for example Refs. ^[29-32]). Now, by inspiring Eq. (20), one can by keeping the unitary condition suggest the following deformed Heisenberg equations of motion

(21)$\text{i}\hbar\frac{\partial}{\partial t}\phi=[\phi,H_{\rm MS}]~,$

for any scalar field $\phi$ within the context of underlying DSR model. More technically, by preserving the unitary time evolution as the second pustulate in this letter in the above deformed Heisenberg equations of motion, the probabilistic interpretation of the system (i.e. the total probability equals unity) and also the conservation of information in the presence of a natural Planck energy cutoff remains unchanged. Thus, we need a deformed Hamiltonian density $\cal{H}_{\rm MS}$ to find $H_{\rm MS}=\int \cal{H}_{\rm MS} \text{d}^3x$. Using the Legendre transformations, we can readily use Eq. (9) to find the Hamiltonian density as

(22) ${\cal{H}}_{\rm MS}=\frac{\partial {\cal{L}}}{\partial \dot{\phi}}\dot{\phi} +\frac{\partial {\cal{L}}}{\partial \dot{\phi}^{\dagger}}\dot{\phi}^{\dagger} -{\cal{L}_{\rm MS}} =\pi^{\dagger}\pi+\nabla\phi^{\dagger}\nabla\phi \\ \hphantom{\cal{H}_{\rm MS}=} +\mu^{2}\phi^{\dagger}\phi+2\text{i}\hbar \ell_p (\nabla\pi\nabla\phi+\nabla\phi^{\dagger}\nabla\pi^\dagger)~,$

where $\pi={\partial {\cal{L}}}/{\partial \dot{\phi}}$ is the field conjugate momentum. Note that here $\pi$ equals the complex conjugate of the time derivative of the scalar field i.e. $\pi= \dot{\phi^{\dagger}}$ and $\pi^\dagger$ equals the time derivative of the scalar field i.e $\pi^\dagger=\dot{\phi}$. Now, using Eq. (22) for $\cal{H}_{\rm MS}$, we have

(23) $\text{i}\hbar\frac{\partial}{\partial t}\phi(x,t)=\Big\{\phi(x,t),\int \text{d}^3x' [\pi^{\dagger}(x',t)\pi(x',t) \\ + \nabla'\phi^{\dagger}(x',t)\nabla'\phi (x',t)+\mu^{2}\phi^{\dagger}(x',t)\phi(x',t) \\ + 2\text{i}\hbar\ell_p (\nabla'\pi(x',t)\nabla'\phi(x',t) \\ +\nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t))]\Big\}~,$

where the quantities inside the integral are all functions of $x'$ and $t$. Since $\phi(x',t)$ is a function of $x'$, we can evaluate the commutator inside the integral. Before that, let us postulate that $\phi$ and $\pi$, as well as their complex conjugate counterparts i.e. ($\phi^{\dagger}$ and $\pi^{\dagger}$), are operators obeying the following equal-time field commutation relations

(24) $[\hat{\phi}^r(x,t), \hat{\pi}_s(x',t)]=\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\delta^{r}_{s} \delta(x-x')\,, \\ [\hat{\phi}^r(x,t), \hat{\phi}^s(x',t)] =[\hat{\pi}_r(x,t), \hat{\pi}_s(x',t)] \\ \quad =[\hat{\phi}^r(x,t),\hat{\pi}^{\dagger}_s(x',t)]=0\,.$

These are the same as what we expected from the standard QFT, except the first commutator that now contains an energy dependent Planck constant. As a reminder, the commutator relations in the standard QFT have a counterpart in quantum mechanics. Therefore, once again it should be emphasized that the above deformed commutator relations are inspired from the MS deformed commutator relation $[x^i,p_j]=\text{i}\hbar\delta^{i}_{j}(1-\lambda E)$ in Ref. ^[14] in which $E=\text{i}\hbar\partial_0$ and $\lambda$ can be positive or negative. Interestingly, the underlying MS deformed commutator relation and also the first equal-time field commutation relation above, explicitly tell us that the Planck energy $E=E_p$ is not only an invariant but also it seems to be classical in the sense that it is free of uncertainty.

In what follows we use $\hat{\phi}^r=\phi$, $\hat{\pi}^r=\pi$, for simplicity. From the integral (23), one easily reads off that the second and third expressions commute with the scalar field $\phi(x,t)$ and thus can be dropped out from this expression. As a result, Eq. (23) can be rewritten as follows

(25) $\text{i}\hbar\frac{\partial}{\partial t}\phi(x,t)= \Big\{\int \text{d}^3x'[\phi(x,t)\,\,\textbf{,}\,\, \pi^{\dagger}(x',t)\pi(x',t)] \\ \quad + 2\text{i}\hbar\ell_p \int \text{d}^3x'[\phi(x,t)\,\,\textbf{,}\,\,\nabla'\pi(x',t) \nabla'\phi(x',t)] \\ \quad + 2\text{i}\hbar \ell_p \int \text{d}^3x'[\phi(x,t)\,\,\textbf{,} \,\,\nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t)]\Big\}~.$

By concerning on the first integral in this equation, we arrive at

(26) $\int \text{d}^3x'[\phi(x,t), \pi^{\dagger}(x',t)\pi(x',t)] \\ \quad =\!\int \!\text{d}^3x'[\pi^{\dagger}(x',t)\phi(x,t)\pi(x',t) -\pi^{\dagger}(x',t)\phi(x,t)\pi(x',t) \\ \qquad +\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\pi^{\dagger}(x',t)\delta(x-x')] \\ \quad =\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\pi^{\dagger}(x,t)\,.$

For the second and third integrals in Eq. (25), we obtain (see Appendix A)

(27) $2\text{i}\hbar \ell_p \int \text{d}^3x'[\phi(x,t), \nabla'\pi(x',t) \nabla'\phi(x',t)] \\ \quad =2\hbar^2 \ell_p\nabla^2\phi(x,t)+{O}(\ell_{p}^{2})~,$

(28) $2\text{i}\hbar\ell_p \int \text{d}^3x'[\phi(x,t), \nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t)]=0~,$

respectively. By inserting the above expressions into Eq. (25) and applying ${\partial}/{\partial t}$ from the left side, we find

(29) $\text{i}\hbar\frac{\partial^2}{\partial t^2}\phi(x,t) =\text{i}\hbar\frac{\partial}{\partial t}\pi^{\dagger}(x,t)+ \lambda\hbar^2\frac{\partial^2}{\partial t^2}\pi^{\dagger}(x,t) \\ +2\hbar^2 \ell_p \frac{\partial} {\partial t}\nabla^2\phi(x,t)+{O}(\ell_{p}^{2})\,.$

Next, by using Eq. (21) when the operator is the complex conjugate of the canonical momentum we have

(30) $\text{i}\hbar\frac{\partial}{\partial t}\pi^{\dagger}(x,t)= \pi^{\dagger}(x,t), \int \text{d}^3x' [\pi^{\dagger}(x',t)\pi(x',t)+ \nabla'\phi^{\dagger}(x',t)\nabla'\phi (x',t) \\ +\mu^{2}\phi^{\dagger}(x',t)\phi(x',t) + 2\text{i}\hbar\ell_p (\nabla'\pi(x',t)\nabla'\phi(x',t)+\nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t))]\Big\}~.$

It is obvious that the first term inside the integral of Eq. (29) commutes with $\pi^{\dagger}(x,t)$. Therefore we find

(31) $\text{i}\hbar\frac{\partial}{\partial t}\pi^{\dagger}(x,t)=\Big\{\int \text{d}^3x'[\pi^{\dagger}(x,t) , \nabla'\phi^{\dagger}(x',t)\nabla'\phi(x',t)]+ \mu^2\int \text{d}^3x'[\pi^ {\dagger}(x,t), \phi^{\dagger}(x',t)\phi(x',t)] \\ \hphantom{\text{i}\hbar\frac{\partial}{\partial t}\pi^{\dagger}(x,t)=} + 2\text{i}\hbar\ell_p\int \text{d}^3x'[\pi^{\dagger}(x,t), \nabla'\pi (x',t)\nabla'\phi(x',t)]+ 2\text{i}\hbar\ell_p\int \text{d}^3x'[\pi^{\dagger}(x,t), \nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t)]\Big\}\,.$

Finally by substituting the results of Eq. (31) (see Appendix~B) into Eq. (29) as well as using the fact that $[\nabla^2,\partial_t]=0$, we get

(32) $\Big(\frac{\partial^2}{\partial t^2}-(1-2\text{i}\hbar\ell_p\partial_0)\nabla^2+\mu^2+ {O}(\ell_{p}^{2})\Big)\phi(x,t)=0\,, \\ (\Box+\mu^2)\phi(x,t)=-2\text{i}\hbar\ell_p\partial_0\nabla^2\phi(x,t)+{O}(\ell_{p}^{2})\,.$

These results obviously indicate that the free field KG equation of motion obtained in the Heisenberg picture (Hamiltonian formalism) is not necessarily equivalent with its counterpart obtained in the previous section. By comparing the right hand side of Eqs. (12) and (32), one explores that just in the case of $\nabla^2\phi=\Box\phi$, these two equations could be equal. However, in the non-relativistic limit these are equal since both recover the same modified Schr$\ddot{o}$dinger equation. Besides, it is not hard to prove that the above free field KG equation of motion has arisen from a DSR model generated by

(33) $U\circ (E,P)=\Big(E,\frac{p}{(1+\ell_p E)^2}\Big)~,$

not from the non-linear representation of the Lorentz group suggested in Eq. (5). The MDR relevant to the above representation can be written as

(34) $E^2-p^2(1-\ell_p E)^2=m^2\,,$

which is not in agrement with the MDR suggested in MS model (6).

At this point, in order to conduct a consistency check which in essence was the main objective of this paper, we focus on the obtained results i.e. Eqs. (12) and (32). From a general perspective, the aforementioned equations both display a free scalar field wave equation in a homogeneous medium, which admits plan wave solutions. Focusing on the fact that any real physical medium allows only such waves to propagate for those combinations of $E$ and $p$ that satisfy the dispersion relation of the medium, so the plane wave solutions obtained in these two approaches are not equivalent. Because, the plane wave arisen from Eqs. (12) and (32) includes a four momentum $p^a$ which satisfies the MDRs (6) and (34) for these two approaches, respectively. As a consequence, taking a maximum energy (or a minimum length) cutoff into the free scalar field Lagrangian, causes (unlike the standard QFT) the equation of motion of $\phi(x,t)$ in Heisenberg picture not to be exactly equivalent to its counterpart that is extracted from the Lagrangian approach directly. This means that the scalar field theory arisen from MS version of DSR, unlike its standard counterpart, is not self-consistent. As it is seen clearly, by discarding the assumption of a Lorentz invariant natural cutoff (i.e. by setting $\ell_p\rightarrow0$ or $E_p\rightarrow\infty$), the above inconsistency fades, as expected.

4 Conclusion

In this paper, inspiring from the standard quantum field theory, we have applied two usually supposed equivalent ways: the Euler-Lagrange approach and the Heisenberg picture approach to the free scalar field equation of motion in order to provide a consistency check of the underlying QFT in a DSR framework. For this purpose we have focused on the derivation of the Klein-Gordon field equation arising from a free scalar field theory realization of DSR, in particular the Magueijo-Smolin model. This model is one of the known non-linear realization of the action of the Lorentz group on momentum space which addresses an invariant energy cutoff. This consistency check has been done based on two postulates. Firstly, the invariance of the linear contraction between position space and its dual which requires plan waves solutions for free scalar field theory. Secondly, preserving unitary time evolution which guarantees the conservation of the total probability and information at scales close to the Planck scale.

While incorporation of a Lorentz invariant maximum energy (or a minimum length) scale in the nature is able essentially to control the ultra-violate divergencies in interacting QFT, it can also cause some inconsistent results as we have shown via inconsistencies in Eqs. (12) and (32) for a free scalar field theory. However, in two cases the mentioned inconsistency in these equations fades away: in non-relativistic limit and also by relaxing the relevant natural cutoff. Generally, this incompatibility can bring two possible interpretations: Firstly, it may reflect the issue that the non-linear generalization of the scalar field theory at hand is not a self-consistent extended framework. Secondly, it may indicate that Lagrangian and Hamiltonian formulations are not necessarily equivalent in quantum gravity regime. We stress that by adopting other options of $U(p_0)$ as reported in Eq. (2), there is possibility of the numerous non-linear realization of the action of the Lorentz group which will subsequently result in different MDR(s). As an open question in this framework, it might be interesting to ask for a consistency check of the free scalar field theories arisen from some more general options of $U(p_0)$ (see Ref. ^[33] to find some other non-linear representations of the Lorentz group). In this way, by demanding equivalence between Lagrangian and Hamiltonian formulations, a self-consistent non-linear relativistic scalar field theory can be constructed. This issue is currently on progress.

Appendix A: Derivation of Eqs. (27) and (28)

Derivation of Eq. (27)

(A1) $2\text{i}\ell_p \int \text{d}^3x'[\phi(x,t), \nabla'\pi(x',t) \nabla'\phi(x',t)]=2\text{i}\ell_p \int \text{d}^3x'[\phi(x,t)\nabla'\pi(x',t) \nabla'\phi(x',t)-\nabla'\pi(x',t)\nabla'\phi(x',t)\phi(x,t)] \\ \quad =2\text{i}\ell_p \int \text{d}^3x'[\nabla'(\phi(x,t)\pi(x',t))\cdot \nabla'\phi(x',t)-\nabla'(\pi(x',t)\phi(x,t))\cdot\nabla'\phi(x',t)] \\ \quad =2\text{i}\ell_p \int \text{d}^3x'[\nabla'(\pi(x',t)\phi(x,t))\cdot \nabla'\phi(x',t)+\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\nabla'\delta(x-x')\cdot\nabla'\phi(x',t) \\ \qquad - \nabla'(\pi(x',t)\phi(x,t))\cdot\nabla'\phi(x',t)] =2\hbar^2(1-\text{i}\hbar\lambda\partial_0)\ell_p\nabla^2\phi(x,t)~, $

where $\int \text{d}^3x'\nabla'\delta(x-x')\cdot\nabla'\phi(x',t) =-\nabla\cdot\nabla\phi(x,t)$.

Derivation of Eq. (28)

(A2)$2\text{i}\hbar\ell_p \int \text{d}^3x'[\phi(x,t), \nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t)] \\ \quad =2\text{i}\ell_p \int \text{d}^3x'[\phi(x,t)\nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t)-\nabla'\phi^{\dagger}(x',t)\nabla'\pi^\dagger(x',t)\phi(x,t)] \\ \quad =2\text{i}\hbar\ell_p \int \text{d}^3x'[\nabla'\phi^{\dagger}(x',t)\cdot\nabla'(\phi(x,t)\pi^{\dagger}(x',t)) - \nabla'\phi^{\dagger}(x',t) \cdot \nabla'(\pi^{\dagger}(x',t)\phi(x,t))\pi^{\dagger}(x',t)] =0~, $

where $[\phi(x,t),\pi^{\dagger}(x',t)]=0=[\phi(x,t),\nabla'\phi^{\dagger}(x',t)]$ and $\nabla'\phi(x,t)=0$.

Appendix~B: Derivation of Eq. (31)

The first integral of Eq. (31)

(A3) $\int \text{d}^3x'[\pi^{\dagger}(x,t), \nabla'\phi^{\dagger}(x',t)\nabla'\phi(x',t)] \\ \quad =\int \text{d}^3x'[\pi^{\dagger}(x,t)\nabla'\phi^{\dagger}(x',t)\nabla'\phi(x',t)- \nabla'\phi^{\dagger}(x',t)\nabla'\phi(x',t)\pi^{\dagger}(x,t)] \\ =\int \text{d}^3x'\{\nabla'(\pi^{\dagger}(x,t)\phi^{\dagger}(x',t))\cdot\nabla'\phi(x',t) -\nabla'\phi(x',t)\cdot\nabla'(\phi^{\dagger}(x',t)\pi^{\dagger}(x,t))\} \\ \quad =\int \text{d}^3x'[\nabla'(\phi^{\dagger}(x',t)\pi^{\dagger}(x,t))\cdot\nabla'\phi(x',t) -\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\nabla'\delta(x-x')\cdot\nabla'\phi(x',t) \\ \qquad - \nabla'\phi(x',t)\cdot\nabla'(\phi^{\dagger}(x',t)\pi^{\dagger}(x,t))] =\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\nabla^2\phi(x,t)\,, $

The second integral of Eq. (31)

(A4) $\mu^2\int \text{d}^3x'[\pi^{\dagger}(x,t), \phi^{\dagger}(x',t)\phi(x',t)] \\ =\mu^2\int \text{d}^3x'[\pi^{\dagger}(x,t) \phi^{\dagger}(x',t)\phi(x',t)-\phi^{\dagger}(x',t)\phi(x',t)\pi^{\dagger}(x,t)] \\ \quad =\mu^2\int \text{d}^3x'[\phi^{\dagger}(x',t)\pi^{\dagger}(x,t)\phi^{\dagger}(x,t) -\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\delta(x-x')\phi(x',t) \\ -\phi^{\dagger}(x',t)\pi^{\dagger}(x,t)\phi^{\dagger}(x,t)] =-\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\mu^2\phi(x,t)\,, $

where $[\pi^{\dagger}(x,t), \phi^{\dagger}(x',t)]=-\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\delta(x-x')$.

The third integral of Eq. (31)

(A5)$2\text{i}\hbar\ell_p\int \text{d}^3x'[\pi^{\dagger}(x,t) , \nabla'\pi (x',t)\nabla'\phi(x',t)] \\ =2\text{i}\hbar\ell_p\int {\rm d}^3x'[\pi^{\dagger}(x,t)\nabla'\pi (x',t)\nabla'\phi(x',t)-\nabla'\pi(x',t)\nabla'\phi(x',t)\phi^{\dagger}(x,t)] \\ =\text{i}\hbar\ell_p\int \text{d}^3x'[\nabla'\pi(x',t)\cdot\nabla'(\pi^{\dagger}(x,t)\phi(x',t)) -\nabla'\pi(x',t)\cdot\nabla'(\phi(x',t)\pi^{\dagger}(x,t))] =0\,, $

where $\nabla'\pi^{\dagger}(x,t)=0$.

The forth integral of Eq. (31)

(A6) $2\text{i}\hbar\ell_p\int \text{d}^3x'[\pi^{\dagger}(x,t), \nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t)] \\ =2\text{i}\hbar\ell_p\int \text{d}^3x'[\pi^{\dagger}(x,t)\nabla'\phi^{\dagger}(x',t) \nabla'\pi^\dagger(x',t)-\nabla'\phi^{\dagger}(x',t)\nabla'\pi^\dagger(x',t)\pi^{\dagger}(x,t)] \\ \quad =2\text{i}\hbar\ell_p\int \text{d}^3x'[\nabla'(\pi^{\dagger}(x,t)\phi^{\dagger}(x',t))\cdot\nabla'\pi^{\dagger}(x',t) -\nabla'(\phi^{\dagger}(x',t)\pi^{\dagger}(x,t)) \cdot \nabla'\pi^{\dagger}(x',t)] \\ \quad =2\text{i}\hbar\ell_p\int \text{d}^3x'[\nabla'(\phi^{\dagger}(x',t)\pi^{\dagger}(x,t))\cdot\nabla'\pi^{\dagger}(x',t) -\text{i}\hbar(1-\text{i}\hbar\lambda\partial_0)\nabla'\delta(x-x') \\ \times \nabla'\pi^{\dagger}(x',t) - \nabla'(\phi^{\dagger}(x',t)\pi^{\dagger}(x,t))\cdot\nabla'\pi^{\dagger}(x',t) ] =-2\hbar^2(1-\text{i}\hbar\lambda\partial_0)l_p\nabla^2\pi^{\dagger}(x,t)\,. $

The authors have declared that no competing interests exist.