Casimir effect in DFR space-time

Though it is the oldest known force, the nature of gravity is still unknown on microscopic scale. Understanding the nature of gravity at the quantum regime is one of the pressing issues in Physics. Many approaches have been developed to study quantum theory of gravity. Some of these approaches are string theory, loop gravity, emergent gravity, and non-commutative geometry[1, 2, 3, 4, 5, 6, 7]. All of these approaches introduce a fundamental length scale, below which quantum effect of gravity comes into play[3, 4]. Non-commutative geometry provides a path to incorporate such a fundamental length scale[1, 2, 3, 4]. Non-commutative geometry also appears in the low energy limit of string theory [2]. In low energy limit, quantum gravity model based on spin-form reduce to a field theory in a non-commutative space-time known as $\kappa$-space-time [8].

In recent times non-commutative space-times and models on such space-times are being investigated intensively. One of the initial motivations behind introducing non-commutative space-time was to remove divergence from QFT. In [9], non-commutative space-time was proposed, but it was shown that the divergences were not completely removed in this non-commutative theory[10]. Research activities in different areas, such as non-commutative geometry, string theory, fuzzy sphere, etc., rekindled the interest in non-commutative space-times and study of physics on such space-times[1, 2, 3, 4, 5, 6, 7, 11]. One of the well studied non-commutative space-time is the Moyal space-time, whose coordinates satisfy

$$[\hat{x}^{\mu},\hat{x}^{\nu}]=i\theta^{\mu\nu},$$

where $\theta^{\mu\nu}$ is antisymmetric, constant tensor with dimension of $(length)^{2}$. This constant $\theta^{\mu\nu}$ leads to the violation of Lorentz invariance in field theory [2] and breaking of rotational symmetry in non-relativistic theory. The violation of Lorentz symmetry makes definition of particles problematic and also it brings vacuum birefringence in theory[12].

To recover Lorentz symmetry in non-commutative theory, a transformation rule has been assigned to the non-commutative parameter $\theta^{\mu\nu}$ under Lorentz transformation[4] and further the components of the non-commutative parameter $\theta_{\mu\nu}$ have been elevated to coordinates of the space-time, thereby increasing the dimension of the underlying space-time[13]. This space-time is called DFR space-time [4] and associated symmetry algebra is known as DFR algebra. The coordinates of the phase space corresponding to DFR space-time are $x_{i}$, $p_{i}$, $\theta_{i}$ and $k_{i}$. To get complete Fourier decomposition of field and to maintain Lorentz invariance in DFR space-time, canonical conjugate momentum corresponding to non-commutative coordinate $\theta^{\mu\nu}$ have been introduced [14, 15, 16, 17]. The associated symmetry algebra is called DFRA algebra.

Since quantum theories of non-commutative space-time have an inherent feature of introducing length parameters (along with nonlocality and nonlinearity), it is of intrinsic interest to study the implications of this length scale in physical phenomena. One such phenomenon where length scale plays a significant role in the commutative space-time is the Casimir effect[18, 19]. It was shown that when two conducting plates are placed parallel to each other at a small separation, they experience an attractive force, and this force depends on the separation between the plate[19]. It was shown that the zero-point energy of the electromagnetic field changes when the plates are introduced and the attractive force arises due to the change in the zero-point energy. The boundary condition imposed on the electromagnetic field by the introduction of plates is different from that in their absence, and this changes the zero-point energy in two situations. This change in the zero-point energy was shown to result in an attractive force between parallel conductors in [18]. Subsequently, this effect was studied by calculating the change in the zero-point energy of various quantized fields such as a real scalar field and fermionic field, for conductors with different geometries [20, 19]. The electromagnetic field at either side of the plates consists of two modes (TE and TM), while the real scalar field has only one degree of freedom but obeys the same boundary conditions as the electromagnetic field. The Casimir energy and force calculated using the scalar field was shown to be half of the corresponding values obtained by considering the electromagnetic field, and this numerical factor difference is due to the differences in the number of modes associated with the scalar and the electromagnetic fields[19, 20, 21].

Modifications to the Casimir effect due to different dimensions of space-time, due to the presence of medium between plates, due to thermal effect, etc., using real scalar field theory have been studied[20, 22]. Experimental investigations of Casimir force for separation of few micrometers between the plates have been reported [23, 24, 25]. Apart from the experimental study for parallel conducting plates, the Casimir effect has been investigated for other geometrical configurations [26, 27, 28]. Using these experimental results, constraints on the Yukawa type correction to Newtonian gravity are obtained [29].

In [30, 31, 32, 33, 34], Casimir effect has been studied in different non-commutative space-times such as Moyal space-time [2] and $\kappa$-deformed space-time [35], using scalar field theory. For even-dimensional Moyal space-time, using coherent state approach and smeared boundary condition, Casimir force is calculated between two parallel conducting plates [30, 31] using scalar field theory. In [32], Casimir energy in 2+1 dimensional non-commutative space-time with non-trivial topologies was derived and analyzed. In [36, 37], the Casimir effect on compact non-commutative space was studied. In [33, 34], the Casimir effect was investigated for $\kappa$-deformed scalar theory. Casimir effect was investigated using the Green’s function approach, and a bound on the non-commutative length scale was calculated by comparing it with experimental results[34]. In $\kappa$-space-time, space coordinates commute among themselves, but commutation relation between time coordinate and space coordinates vary with space coordinates, i.e., they satisfy

$$[x^{i},x^{j}]=0,~{}~{}~{}[x^{0},x^{i}]=ax^{i},~{}~{}~{}a=\frac{1}{\kappa}.$$

(1.1)

where $a$ is the deformation parameter. This indicates that as in Moyal space-time, in the $\kappa$ space-time also the Lorentz symmetry is broken. But this problem is avoided in DFR space-time with introduction of transformation for the non-commutative parameter $\theta^{\mu\nu}$ [4]. Latter, components of $\theta_{\mu\nu}$ have been promoted to coordinates, enlarging the dimension of the space-time[13]. In commutative space-time, Casimir force in presence of extra dimensions has been explored in recent times[20, 38, 39, 40, 41]. Thus it is of intrinsic interest to study the Casimir effect in DFR space-time which has extra dimensions.

In the commutative space-time, Casimir effect between two parallel plates has been evaluated by modelling the plates with two $\delta$-function potentials at $x=0$ and $x=L$ [42, 19], respectively. Casimir force on the plate at $x=L$ has been derived by taking difference of vacuum expectation values of Energy-Momentum tensor on either side of the plate. For massless scalar field theory, $xx$ component of Energy-Momentum tensor was found at a point, just left to the plate at $x=L$, i.e., $\hat{T}^{xx}\Big{|}_{x=L^{-}}$ and at another point, just right to the plate at $x=L$, i.e., $\hat{T}^{xx}\Big{|}_{x=L^{+}}$. Then force was calculated at $x=L$ by taking difference between the vacuum expectation value of the stress-tensor, in the strong interaction limit of $\delta$-function potential as,

$$\hat{F}_{\gamma,\gamma^{\prime}\rightarrow\infty}=<\hat{T^{xx}}>\Big{|}_{x=L^{-}}-<\hat{T^{xx}}>\Big{|}_{x=L^{+}}.$$

(1.2)

One way to derive vacuum expectation value of stress tensor was by rewriting Energy-Momentum tensor as an operator acting on the vacuum expectation value of the time ordered product of fields at nearby points, $x$ and $x^{\prime}$ as,

	$\displaystyle<\hat{T}^{\mu\nu}_{x,x^{\prime}}>$	$\displaystyle=$	$\displaystyle<\hat{O}^{\mu\nu}(\partial,\partial^{\prime})T(\phi\phi^{\prime})>$
		$\displaystyle=$	$\displaystyle\hat{O}^{\mu\nu}(\partial,\partial^{\prime})<T(\phi\phi^{\prime})>$

where $\hat{O}^{\mu\nu}(\partial,\partial^{\prime})$ is a derivative operator, $\phi=\phi(x)$ and $\phi^{\prime}=\phi(x^{\prime})$. Since

$$<T(\phi(x)\phi(x^{\prime}))>=iG(x,x^{\prime}),$$

(1.3)

where $G(x,x^{\prime})$ is the Green’s function. One can relate the vacuum expectation value of the Energy-Momentum tensor to the Green’s function. Thus to calculate RHS of Eqn.(1.2), Green’s functions in different regions are first derived by solving Euler-Lagrange equation of scalar field theory with Dirichlet boundary condition and using these Green’s functions, Casimir force is obtained. Casimir energy is calculated by integrating the Casimir force over the separation distance between two parallel plates, as

$$E_{\gamma,\gamma^{\prime}\rightarrow\infty}=-\int\hat{F}_{\gamma,\gamma^{\prime}\rightarrow\infty}dx$$

(1.4)

where $\gamma$ and $\gamma^{\prime}$ are coupling strengths of $\delta$-function potentials, modelling the plates and $\gamma,\gamma^{\prime}\rightarrow\infty$ denote the strong interaction limit. The Casimir force and energy calculated using the scalar field differs from the corresponding results obtained using electromagnetic field by a factor of $\frac{1}{2}$.

In this paper, we study the Casimir effect between two parallel plates in DFR space-time by analyzing DFRA scalar field theory [14, 15, 16, 17]. Since the non-commutative coordinates do not show up in the commutative limit, and our aim is to investigate the corrections to the Casimir force in the commutative space-time, we consider the situation where the plates are kept only in the x-directions (commutative directions). We start with the action of $10$-dimensional DFRA scalar field theory. For unitarity of this field theory one sets temporal part of non-commutative coordinate, $\theta^{\mu\nu}$ to be vanishing, i.e., $\theta^{0i}=0$ and further we take definitions, $\theta^{i}=\frac{1}{2}\epsilon^{ijl}\theta_{jl}$ and $k^{i}=\frac{1}{2}\epsilon^{ijl}k_{jl}$ [45, 43, 44]. After these, we get the action of $7$-dimensional DFRA scalar field theory. We start our study from the Lagrangian of DFRA scalar field theory with plates modelled through $\delta$-function potentials, as in the commutative space-time. Next, we vary the corresponding action and derive the general form of the Euler-Lagrange equation and components of the Energy-Momentum tensor. Without loss of generality, we restrict our attention to $4+1$ dimensions $(x^{\mu},\theta^{1})$, and study the Casimir effect in two ways. In first way, we treat extra $\theta$-dimension due to noncommutativity of the $4+1$-dimensional DFR space time, in the same footing as transverse directions, $y$ and $z$. Then following the standard calculational procedure given in [20, 19] we show that the Casimir force vary as $\frac{1}{L^{5}}$, where $L$ is the distance between the parallel plates. This force expression exactly coincides with result given in [19] for extra-dimensional commutative space-time. In second approach, we compactify the new extra dimension $\theta$ introduced by noncommutativity in DFR space-time. After compactifying the extra dimension, we solve the equation of motion using Dirichlet boundary condition and get Green’s functions in different regions. We also derive the $xx$ component of the Energy-Momentum tensor at either side of the plate at $x=L$. Then we obtain the Casimir force by taking the difference between vacuum expectation values of stress tensors at $x=L$ as shown in Eqn.(1.2) for the commutative case. For this, we use the relation between Green’s function and time-ordered product of fields in the nearby points in $4+1$-dimensional DFR space-time, viz;

$$<T(\phi(x,\eta)\phi(x^{\prime},\eta^{\prime}))>=iG(x,x^{\prime};\eta,\eta^{\prime}),$$

(1.5)

where $G(x,x^{\prime};\eta,\eta^{\prime})$ is the Green’s function obtained from Euler-Lagrange equations. Here $\eta=\frac{\theta}{R}$ is compactified direction and R is the size of the extra compactified dimension. Using this, we calculate Casimir force and derive Casimir energy. Further, we study finite temperature corrections to the Casimir effect for DFRA real scalar field theory and obtain Casimir force in the high temperature and low-temperature limits. We also show that our results reduce to commutative space-time results in the appropriate limit of the size of the extra compactified dimension. We show that the Casimir force gets two types of corrections due to non-commutativity of space-time, one is $\frac{L}{R}$ dependent modified Bessel function-$K_{1}$ whoes coefficient vary as $\frac{1}{L}$ and the other is $\frac{L}{R}$ dependent modified Bessel function-$K_{2}$ whose coefficient vary as $\frac{1}{L^{2}}$. In the high-temperature limit, we find the correction terms that vary as $\frac{1}{L}$ and $\frac{1}{L^{3}}$ while in the low-temperature limit Casimir force has correction terms that vary as $\frac{1}{L^{2}}$ and $\frac{1}{L^{4}}$. We also show that the modifications to the Casimir force in both zero temperature and the finite temperature, has dependency on $R$, size of the extra compactified dimension. To understand the generic effect of the non-commutativity on the Casimir force, we then consider most significant correction terms and analyzing the plots for the Casimir force with these correction terms, we obtain constraints on the size of the extra compactified dimension $R$.

We also show that the massless DFRA scalar theory in 7-dimensions gives a massive scalar theory in 4-dimensions under Kaluza-Klein reduction. Also the modification of the Casimir effect using this model is calculated.

This paper is organized as follows. In the next section, we briefly discuss DFR space-time and underlying DFRA algebra [14, 15, 16, 17]. In Sec.3., we derive the general form of the Euler-Lagrange equation, current density, and the components of Energy-Momentum tensor corresponding to the scalar theory on the $7$-dimensional DFR space-time. Our main results are obtained in Sec.4. Here we construct the Lagrangian describing massless DFRA scalar field theory in the presence of two parallel plates. We impose the Dirichlet boundary condition and calculate the Casimir force and the energy for massless scalar field with parallel plates in $4+1$-dimensional DFR space-time. The $\theta^{\mu\nu}$ dependent terms in the Lagrangian captures the effects of non-commutativity of space-time. First, we study the Casimir effect by treating the new extra dimension-$\theta$ introduced by non-commutativity in $4+1$ dimensional DFR space-time, in same footing with other two transverse commutative space dimensions- $y$ and $z$. Then by following the standard procedure, we show that the Casimir force expression is exactly same as the result obtained in case of commutative extra dimensional space-time. Then in Subsection.4.1, we study the Casimir effect by treating the additional dimension-$\theta$ as a compactified dimension in $4+1$ dimensional DFR space-time. This is carried out by first compactifying the extra dimension $\theta$ of DFR space-time. Then we derive Green’s function for different regions of interest by solving the Euler-Lagrangian equation, describing the interaction of scalar field with parallel plates in presence of extra compactified dimension in $4+1$-dimensional DFR space-time. Next we obtain the energy-momentum tensor corresponding to the massless DFRA real scalar theory from the general form of the stress-tensor constructed in Sec.3. Then we derive the vacuum expectation value of the energy-momentum tensor in terms of the obtained Green’s function. Using the reduced Green’s function obtained in different regions, we calculate the Casimir force and Casimir energy in presence of extra compactified dimension-$\theta$ for $4+1$ dimensional, massless DFRA scalar field theory. Further, we investigate finite temperature modifications to the Casimir effect in this model in Sec.5. We also study corrections to Casimir force for both the low-temperature limit and the high-temperature limit. Our concluding remarks are given in Sec.6. In the appendix, we discuss the study of the Casimir effect for massive scalar theory in $4$-dimensional Minkowski space-time, obtained using Kaluza-Klein dimensional reduction from a $7$-dimensional, massless, DFRA complex scalar theory. We use the metric, $\eta_{AB}=(\eta_{\mu\nu},-1)=diag(+1,-1,-1,-1,-1)$.

In this section, we present a brief summary of the DFR space-time, a non-commutative space-time that respects Lorentz invariance and also the symmetry algebra associated to this space-time[4]. The Moyal-space-time coordinates satisfy

$$[\hat{x}_{\mu},\hat{x}_{\nu}]=i{\theta}_{\mu\nu}.$$

(2.1)

Since $\theta_{\mu\nu}$ is a constant antisymmetric tensor, it results in the violation of Lorentz symmetry [2]. Violation of Lorentz invariance brings issues with the interpretation of particle-antiparticle states. Assigning an appropriate transformation to $\theta_{\mu\nu}$ under Lorentz transformation reinstates the Lorentz invariance of theory [4] and thus avoids the issue of defining particle-antiparticle states. In [13], the components of the $\theta_{\mu\nu}$ were promoted to coordinate operators $\hat{\theta}_{\mu\nu}$. Thus the non-commutative space-time gets six more dimensions and the resulting 10-dimensional space-time is known as DFR space-time. The corresponding algebra is given by

$$\begin{split}[\hat{x}_{\mu},\hat{x}_{\nu}]&=i\hat{\theta}_{\mu\nu},~{}~{}[\hat{x}_{\mu},\hat{\theta}_{\nu\rho}]=0,\\ [\hat{x}_{\mu},\hat{p}_{\nu}]&=i\eta_{\mu\nu},~{}~{}[\hat{\theta}_{\mu\nu},\hat{\theta}_{\rho\lambda}]=0\\ [\hat{p}_{\mu},\hat{\theta}_{\nu\lambda}]&=0,~{}~{}~{}~{}~{}~{}[\hat{p}_{\mu},\hat{p}_{\nu}]=0\end{split}$$

(2.2)

For Fourier decomposition of field, and also for the Lorentz invariance of the field theory defined on this non-commutative space-time, one needs to introduce conjugate momentum $\hat{k}_{\mu\nu}$ - corresponding to $\hat{\theta}_{\mu\nu}$, apart from $\hat{x}^{\mu}$, $\hat{p}_{\mu}$ and $\hat{\theta}_{\mu\nu}$. In DFR space-time [14, 15, 16], non-commutative coordinates $\hat{\theta}_{\mu\nu}$ and their canonical conjugate momenta $\hat{k}_{\mu\nu}$ are considered as operators alongside with the commutative coordinates, $\hat{x}_{\mu}$ and their canonical conjugate momenta $\hat{p}_{\mu}$ . Thus algebra is enlarged by

$$\begin{split}[\hat{x}_{\mu},\hat{k}_{\nu\lambda}]&=-\frac{i}{2}(\eta_{\mu\nu}\eta_{\rho\lambda}-\eta_{\mu\lambda}\eta_{\nu\rho})\hat{p}^{\rho},~{}~{}[\hat{p}_{\mu},\hat{k}_{\nu\lambda}]=0,\\ [\hat{\theta}_{\mu\nu},\hat{k}_{\rho\lambda}]&=i(\eta_{\mu\rho}\eta_{\nu\lambda}-\eta_{\mu\lambda}\eta_{\nu\rho}),~{}~{}[\hat{k}_{\mu\nu},\hat{k}_{\rho\lambda}]=0,\end{split}$$

(2.3)

in addition to the ones given in Eqn.(2.2). The relations in Eqn.(2.2) and Eqn.(2.3) together is a closed algebra-DFRA algebra[14, 15, 16, 17]. The corresponding Lorentz generator is defined as [14, 17]

$$M_{\mu\nu}=\hat{X}_{\mu}\hat{p}_{\nu}-\hat{X}_{\nu}\hat{p}_{\mu}-\hat{\theta}_{\mu\lambda}{\hat{k}}_{\nu}~{}^{{\lambda}}+\hat{\theta}_{\nu\lambda}{\hat{k}}_{\mu}~{}^{\lambda}$$

(2.4)

where,

$$\hat{X}_{\mu}=\hat{x}_{\mu}+\frac{1}{2}\hat{\theta}_{\mu\nu}\hat{p}^{\nu}.$$

(2.5)

The shifted coordinates in the above equation satisfy commutation relations

$$[\hat{X}_{\mu},\hat{X}_{\nu}]=0,~{}~{}~{}[\hat{X}_{\mu},\hat{p}_{\nu}]=i\eta_{\mu\nu}.$$

(2.6)

With the relations given in Eqn.(2.2), Eqn.(2.3) and Eqn.(2.4), one sees that the generators of DFRA algebra are closed, i.e.,

$$\begin{split}[M_{\mu\nu},\hat{p}_{\lambda}]&=i(\eta_{\mu\lambda}\hat{p}_{\nu}-\eta_{\nu\lambda}\hat{p}_{\mu}),\\ [M_{\mu\nu},\hat{k}_{\alpha\beta}]&=i(\eta_{\mu\beta}\hat{k}_{\alpha\nu}-\eta_{\mu\alpha}\hat{k}_{\nu\beta}+\eta_{\nu\alpha}\hat{k}_{\beta\mu}-\eta_{\nu\beta}\hat{k}_{\alpha\mu}),\\ [M_{\mu\nu},M_{\lambda\rho}]&=i(\eta_{\mu\rho}M_{\nu\lambda}-\eta_{\nu\rho}M_{\lambda\mu}-\eta_{\mu\lambda}M_{\rho\nu}+\eta_{\nu\lambda}M_{\rho\mu}),\end{split}$$

(2.7)

The above relations of DFRA algebra are consistent with the Jacobi identities (see [14, 17]). The Casimir operator associated with the DFRA algebra is given as [14]

$$\hat{P}^{2}=\hat{p}_{\mu}\hat{p}^{\mu}+\frac{\lambda^{2}}{2}\hat{k}_{\mu\nu}\hat{k}^{\mu\nu},$$

(2.8)

where $\lambda$ is the non-commutative parameter with the length dimension. In [14, 15], the construction of infinitesimal transformations of $x_{\mu},~{}\theta_{\mu\nu},~{}p_{\mu},~{}k_{\mu\nu}$ and $M_{\mu\nu}$ and their relevance for the DFR space-time are discussed in detail.

One way to evaluate the Casimir force experienced by a plate is to obtain the vacuum expectation value of the Energy-Momentum tensor on either side of this plate and find their difference. Since the Energy-Momentum tensor can be written as a differential operator acting on the product of fields, this vacuum expectation value of the Energy-Momentum tensor is related to the vacuum expectation value of the time-order product of fields at two space-time points acted upon by this operator. This allows one to express vacuum expectation value of Energy-Momentum tensor as the operator acting on Green’s function, i.e.,

	$\displaystyle<T_{x,x^{\prime}}^{\mu\nu}>$	$\displaystyle=$	$\displaystyle\hat{O}<T(\phi(x)\phi(x^{\prime}))>$		(3.1)
		$\displaystyle=$	$\displaystyle\hat{O}G(x,x^{\prime})$

The Green’s function on either side of the plate are calculated by solving the Euler-Lagrange equation for these different regions. In this section, we summarize the construction of action for the scalar field theory in DFR space-time using the definition of the Moyal star product [2]. By varying the action, we then obtain the equations of motion corresponding to the scalar field defined in the DFR space-time. Here we also discuss the implications of the $\theta$-dependent weight function introduced in the definition of action.

In this section, we derive the Casimir force felt by one of the parallel plates by analyzing the vacuum fluctuation of the DFRA scalar field. For this, we first calculate the vacuum expectation value of the Energy-Momentum tensor corresponding to the DFRA scalar field theory, using the relation between the expectation value of the time-ordered product of fields and Green’s function. Here the Green’s function is obtained by solving Euler-Lagrange equations. Using this vacuum expectation value of Energy-Momentum tensor, we obtain corrections to Casimir force and Casimir energy. We investigate Casimir effect in $4+1$ dimensions $(x^{\mu},\theta)$ and the result we obtain is generalized to $4+3$-dimensional DFR space-time. Here the metric used is $diag(+------)$.

We start with the DFRA massless real scalar field Lagrangian [14, 15, 16, 17],

$$\mathcal{L}_{0}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{\lambda^{2}}{2}\partial_{\theta^{i}}\phi\partial^{\theta^{i}}\phi$$

(4.1)

where in the limit $\lambda\rightarrow 0$, we get back Lagrangian of the real scalar field theory in commutative space-time.

To study Casimir effect between two parallel plates, we introduces these plates perpendicular to $x$-direction and model these plates, using delta function potentials at $x=0$ and $x=L$. Thus we get total Lagrangian as [19],

$$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{\lambda^{2}}{2}\partial_{\theta^{i}}\phi\partial^{\theta^{i}}\phi-{\frac{\gamma}{2L}\phi^{2}\delta(x)}-{\frac{\gamma^{\prime}}{2L}}\phi^{2}\delta(x-L),$$

(4.2)

where $\gamma,\gamma^{\prime}$ are coupling constant, which are dimension free quantities. Here we consider the Lagrangian in $4+1$ dimension given by

$$\mathcal{L}=\frac{1}{2}{\partial_{0}\phi\partial_{0}\phi}-\frac{1}{2}{\partial_{m}\phi\partial_{m}\phi}+\frac{\lambda^{2}}{2}\partial_{\theta^{1}}\phi\partial^{\theta^{1}}\phi-{\frac{\gamma}{2L}\phi^{2}\delta(x)}-{\frac{\gamma^{\prime}}{2L}}\phi^{2}\delta(x-L);~{}~{}m=1,2,3.$$

(4.3)

Note that the interaction part, mimicking the plates, is similar to that in the commutative space-time.

Using Eqn.(3.13), we obtain Euler-Lagrange equation from the above Lagrangian as

$$\Big{(}\partial_{0}^{2}-\partial_{m}^{2}-\lambda^{2}\partial_{\theta^{1}}^{2}+\frac{\theta^{1}}{2\lambda^{2}}\partial_{\theta^{1}}+\frac{\gamma}{L}\delta(x)+\frac{\gamma^{\prime}}{L}\delta(x-L)\Big{)}\phi({\bf x},\frac{\theta^{1}}{\lambda})=0,$$

(4.4)

where ${\bf x}$ denote all the three space co-ordinates $x,y,z$ and $\theta^{1}$ is the non-commutative of space coordinate. Note that if weight function $W(\theta)$ is taken to be a constant, the third term, $\frac{\theta^{1}}{2\lambda^{2}}\partial_{\theta^{1}}$ will be absent. We re-express Eqn.(4.4) as,

$$\Big{(}\partial_{0}^{2}-\partial_{x}^{2}-\partial_{y}^{2}-\partial_{z}^{2}-\lambda^{2}\partial_{\theta^{1}}^{2}+\frac{\theta^{1}}{2\lambda^{2}}\partial_{\theta^{1}}+\frac{\gamma}{L}\delta(x)+\frac{\gamma^{\prime}}{L}\delta(x-L)\Big{)}\phi({\bf x},\frac{\theta^{1}}{\lambda})=0.$$

(4.5)

Here in the limit, $\lambda\rightarrow 0$ and $\theta=0$, we get back equation of motion in the commutative space-time [19]. As we are considering only one non-commutative direction $\theta^{1}$, we set $\theta^{1}=\theta$.

Casimir force between two parallel plates is obtained by taking the difference between vacuum expectation value of $xx$ component of Energy-Momentum tensor at either side of the plate. Since the vacuum expectation values of stress tensor is related to the Green’s function as in Eqn.(3.1), first we calculate the Green’s functions corresponding to Eqn.(4.5). We start with,

$$\Big{(}\partial_{0}^{2}-\partial_{x}^{2}-\partial_{y}^{2}-\partial_{z}^{2}-\lambda^{2}\partial_{\theta}^{2}+\frac{\theta}{2\lambda^{2}}\partial_{\theta}+\frac{\gamma}{L}\delta(x)+\frac{\gamma^{\prime}}{L}\delta(x-L)\Big{)}G({\bf x,x^{\prime}};\frac{\theta}{\lambda},\frac{\theta^{\prime}}{\lambda})=\delta^{3}({\bf x-x^{\prime}})\delta(t-t^{\prime})\delta(\frac{\theta}{\lambda}-\frac{\theta^{\prime}}{\lambda}).$$

(4.6)

Note that in the commutative limit, i.e., $\lambda\rightarrow 0$, $\theta=0$ above equation reduce to well known commutative result [19]. In the commutative space-time one solves for the Green’s function by using Fourier-transform for $t$, $y$ and $z$ dependence of the Green function. We can now treat the additional $\theta$-direction introduced by in the DFR space-time in the same manner as $y$ and $z$ directions. Then following the standard calculation we would get the Casimir force to be $F=-\frac{3.12}{32\pi^{2}L^{5}}$ (for weight function $W(\theta)=1$) which exactly match with the result obtained in [19] for 5-dimensional commutative space-time. This shows that the newly introduced $\theta$-dimension act just like extra dimensions in commutative space-time. Note that this expression for the Casimir force is independent of $\theta$ and/or $\lambda$ and the Casimir force in DFR space-time does not have a commuatative limit; thus this Casimir effect does not distinguise between 5-dimensional DFR space-time and 5-dimensional commutative space-time. Since non-commutativity is expected to bring in modifications which should smoothly vanish in the commutative limit, we adopt a different calculation scheme in the following. In [20, 38, 39, 40, 41] extra dimensions were treated as compact directions and the Casimir effect was studied. We next study the Casimir effect in DFR space-time by treating additional dimensions introduced by non-commutativity as compact dimensions.