The Coset Spin-4 Casimir Operator and Its Three-Point Functions with Scalars
Changhyun Ahn
备注：
65 pages; the ambiguity for the two coefficient functions is clarified and the abstract, the introduction, the subsection 3.4 and the conclusion are improved and to appear in JHEP
We find the GKO coset construction of the dimension 4 Casimir operator that contains the quartic WZW currents contracted with completely symmetric SU(N) invariant tensors of ranks 4, 3, and 2. The requirements, that the operator product expansion with the diagonal current is regular and it should be primary under the coset Virasoro generator of dimension 2, fix all the coefficients in spin-4 current, up to two unknown coefficients. The operator product expansion of coset primary spin-3 field with itself fixes them completely. We compute the three-point functions with scalars for all values of the 't Hooft coupling in the large N limit. At fixed 't Hooft coupling, these three-point functions are dual to that found by Chang and Yin recently in the undeformed AdS_3 bulk theory (higher spin gravity with matter).
We generalize the Goddard-Kent-Olive (GKO) coset construction to the dimension 5/2 operator for $ \hat{so} (5) $ and compute the fourth order Casimir invariant in the coset model $\hat{SO} (5)_{1} \times \hat{SO} (5)_{m} / \hat{SO} (5)_{1+m} $ with the generic unitary minimal $ c < 5/2 $ series that can be viewed as perturbations of the $ m \rightarrow \infty $ limit, which has been investigated previously in the realization of $ c= 5/2 $ free fermion model.
We compute the operator product expansion (OPE) between the spin-4 current and itself in the WD_4 coset minimal model with SO(8) current algebra. The right hand side of this OPE contains the spin-6 Casimir current which is also a generator of WD_4 coset minimal model. Based on this N=8 result, we generalize the above OPE for the general N(in the WD_{N/2} coset minimal model) by using two N-generalized coupling constants initiated by Hornfeck sometime ago: the simplest OPE for the lowest higher spin currents. We also analyze the similar OPE in the WB_3(and WB_{(N-1)/2}) coset minimal model with…
Starting from SO(N) current algebra, we construct two lowest primary higher spin-4 Casimir operators which are quartic in spin-1 fields. For N is odd, one of them corresponds to the current in the WB_{\frac{N-1}{2}} minimal model. For N is even, the other corresponds to the current in the WD_{\frac{N}{2}} minimal model. These primary higher spin currents, the generators of wedge subalgebra, are obtained from the operator product expansion of fermionic (or bosonic) primary spin-N/2 field with itself in each minimal model respectively. We obtain, indirectly, the three-point functions with two re…
By calculating the second-order pole in the operator product expansion (OPE) between the spin-3 Casimir operator and the spin-4 Casimir operator known previously, the spin-5 Casimir operator is obtained in the coset model based on (A_{N-1}^{(1)} \oplus A_{N-1}^{(1)}, A_{N-1}^{(1)}) at level (k,1). This spin-5 Casimir operator consisted of the quintic, quartic (with one derivative) and cubic (with two derivatives) WZW currents contracted with SU(N) invariant tensors. The three-point functions with two scalars for all values of 't Hooft coupling in the large N limit were obtained by analyzing th…
The general non-split scalar coset of supergravity theories is discussed.The symmetric space sigma model is studied in two equivalent formulations and for different coset parametrizations.The dualisation and the local first order formulation is performed for the non-split scalar coset G/K when the rigid symmetry group G is a real form of a non-compact semisimple Lie group (not necessarily split) and the local symmetry group K is G's maximal compact subgroup.A comparison with the scalar cosets arising in the T^{10-D}-compactification of the heterotic string theory in ten dimensions is also ment…
In the N=1 supersymmetric coset minimal model based on (B_N^{(1)} \oplus D_N^{(1)}, D_N^{(1)}) at level (k,1) studied recently, the standard N=1 super stress tensor of spins (3/2,2) is reviewed. By considering the stress tensor in the coset (B_N^{(1)}, D_N^{(1)}) at level k, the higher spin-2' Casimir current was obtained previously. By acting the above spin-3/2 current on the higher spin-2' Casimir current, its superpartner, the higher spin-5/2 current, can be generated and combined as the first higher spin supercurrent with spins (2', 5/2). By calculating the operator product expansions (OPE…
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The Casimir force on two-dimensional pistons for massive scalar fields with both Dirichlet and hybrid boundary conditions is computed. The physical result is obtained by making use of generalized $ζ$-function regularization technique. The influence of the mass and the position of the piston in the force is studied graphically. The Casimir force for massive scalar field is compared to that for massless scalar field.
The stress tensor of a massless scalar field satisfying Robin boundary conditions on two one-dimensional wall in two-dimensional Schwarzschild background is calculated. We show that vacuum expectation value of stress tensor can be obtained explicitly by Casimir effect, trace anomaly and Hawking radiation.
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In this paper we calculate the Casimir energy for spherical shell with massless self-interacting scalar filed which satisfying Dirichlet boundary conditions on the shell. Using zeta function regularization and heat kernel coefficients we obtain the divergent contributions inside and outside of Casimir energy. The effect of self-interacting term is similar with existing of mass for filed. In this case some divergent part arises. Using the renormalization procedure of bag model we can cancel these divergent parts.
The zeta function regularization technique is used to study the Casimir effect for a scalar field of mass $m$ satisfying Dirichlet boundary conditions on a spherical surface of radius $a$. In the case of large scalar mass, $ma\gg1$, simple analytic expressions are obtained for the zeta function and Casimir energy of the scalar field when it is confined inside the spherical surface, and when it is confined outside the spherical surface. In both cases the Casimir energy is exact up to order $a^{-2}m^{-1}$ and contains the expected divergencies, which can be eliminated using the well established …
In this paper we use the scalar field constructed in phase space to analyze the analogous Stefan-Boltzman law and Casimir effect both of them at finite temperature. The temperature is introduced by Thermo Field Dynamics (TFD) and the quantities are analyzed once projected in the coordinates space.
The general structure of the scalar cosets of the Maxwell-Einstein supergravities is given. Following an introduction of the non-linear coset formalism of the supergravity theories a comparison of the coset algebras of the Maxwell-Einstein supergravities in various dimensions is discussed.
In this paper we consider a Lorentz-breaking extension of the theory for a real massive scalar quantum field in the region between two large parallel plates, with our manner to break the Lorentz symmetry is CPT-even, aether-like. For this system we calculated the Casimir energy considering different boundary conditions. It turns out to be that the Casimir energy strongly depends on the direction of the constant vector implementing the Lorentz symmetry breaking, as well as on the boundary conditions.
In this article, we study the finite temperature Casimir effect for scalar field with Robin boundary conditions on two parallel plates in a background spacetime that has a compact internal manifold with arbitrary geometry. The finite temperature Casimir force acting on the plates is derived using the piston approach, with which we show that the divergences of the Casimir forces from the region between the plates and the region outside the plates cancel each other. The nature and asymptotic behaviors of the Casimir force at different limits such as small and large plate separation, low and high…
We consider the vacuum energy for a scalar field subject to a frequency dependent boundary condition. The effect of a frequency cut-off is described in terms of an {\it incomplete} $ζ$-function. The use of the Debye asymptotic expansion for Bessel functions allows to determine the dominant (volume, area, >...) terms in the Casimir energy. The possible interest of this kind of models for dielectric media (and its application to sonoluminescence) is also discussed.
Following a field-theoretical approach, we study the scalar Casimir effect upon a perfectly conducting cylindrical shell in the presence of spontaneous Lorentz symmetry breaking. The scalar field is modeled by a Lorentz-breaking extension of the theory for a real scalar quantum field in the bulk regions. The corresponding Green's functions satisfying Dirichlet boundary conditions on the cylindrical shell are derived explicitly. We express the Casimir pressure (i.e. the vacuum expectation value of the normal-normal component of the stress-energy tensor) as a suitable second-order differential o…
We compute the first radiative correction to the Casimir energy of a massive scalar field with a quartic self-interaction in the presence of two parallel plates. Three kinds of boundary conditions are considered: Dirichlet-Dirichlet, Neumann-Neumann and Dirichlet-Neumann. We use dimensional and analytical regularizations to obtain our physical results.
The Casimir interaction between two concentric corrugated cylinders provides the mechanism for non-contact gears. To this end, we calculate the Casimir torque between two such cylinders, described by $δ$-potentials, which interact through a scalar field. We derive analytic expressions for the Casimir torque for the case when the corrugation amplitudes are small in comparison to the corrugation wavelengths. We derive explicit results for the Dirichlet case, and exact results for the weak coupling limit, in the leading order. The results for the corrugated cylinders approach the corresponding ex…
The finite temperature Casimir effect for a charged, massive scalar field confined between very large, perfectly conducting parallel plates is studied using the zeta function regularization technique. The scalar field satisfies Dirichlet boundary conditions at the plates and a magnetic field perpendicular to the plates is present. Four equivalent expressions for the zeta function are obtained, which are exact to all orders in the magnetic field strength, temperature, scalar field mass, and plate distance. The zeta function is used to calculate the Helmholtz free energy of the scalar field and …
We re-evaluate the zero point Casimir energy for the case of a massive scalar field in $\mathbf{R}^{1}\times\mathbf{S}^{3}$ space, allowing also for deviations from the standard conformal value $ξ=1/6$, by means of zero temperature zeta function techniques. We show that for the problem at hand this approach is equivalent to the high temperature regularisation of the vacuum energy.
In this paper I study the Casimir effect caused by a charged and massive scalar field that breaks Lorentz invariance in a CPT-even, aether-like manner. I investigate the case of a scalar field that satisfies Dirichlet or mixed (Dirichlet-Neumann) boundary conditions on a pair of very large plane parallel plates. The case of Neumann boundary conditions is straightforward and will not be examined in detail. I use the $ζ$-function regularization technique to study the effect of a constant magnetic field, orthogonal to the plates, on the Casimir energy and pressure. I investigate the cases of a ti…
We consider Casimir force acting on a three dimensional rectangular piston due to a massive scalar field subject to periodic, Dirichlet and Neumann boundary conditions. Exponential cut-off method is used to derive the Casimir energy in the interior region and the exterior region separated by the piston. It is shown that the divergent term of the Casimir force acting on the piston due to the interior region cancels with that due to the exterior region, thus render a finite well-defined Casimir force acting on the piston. Explicit expressions for the total Casimir force acting on the piston is d…
We re-evaluate the zero point Casimir energy for the case of a massive scalar field in $\mathbf{R}^{1}\times\mathbf{S}^{3}$ space, allowing also for deviations from the standard conformal value $ξ=1/6$, by means of zero temperature zeta function techniques. We show that for the problem at hand this approach is equivalent to the high temperature regularization of the vacuum energy, as conjectured in a previous publication. Two different, albeit equally valid, ways of doing the analytic continuation are described.
The Casimir stress on a cylinderical shell in background of conformally flat space-time for massless scalar field is investigated. In the general case of Robin (mixed) boundary condition formulae are derived for the vacuum expectation values of the energy-momentum tensor and vacuum forces acting on boundaries. The special case of the dS bulk is considered then different cosmological constants are assumed for the space inside and outside of the shell to have general results applicable to the case of cylindrical domain wall formations in the early universe.
In this work, we obtain the Casimir energy for the real scalar field and the Elko neutral spinor field in a field theory at a Lifshitz fixed point (LP). We analyze the massless and the massive case for both fields using dimensional regularization. We obtain the Casimir energy in terms of the dimensional parameter and the LP parameter. Particularizing our result, we can recover the usual results without LP parameter in (3+1) dimensions presented in the literature. Moreover, we compute the effects of the LP parameter in the thermal corrections for the massless scalar field.
A noncommutative complex scalar field, satisfying the deformed canonical commutation relations proposed by Carmona et al. [27]-[31], is constructed. Using these noncommutative deformed canonical commutation relations, a model describing the dynamics of the noncommutative complex scalar field is proposed. The noncommutative field equations are solved, and the vacuum energy is calculated to the second order in the parameter of noncommutativity. As an application to this model, the Casimir effect, due to the zero point fluctuations of the noncommutative complex scalar field, is considered. It tur…
Recent work in the literature had evaluated the energy-momentum tensor of a Casimir apparatus in a weak gravitational field, for an electromagnetic field subject to perfect conductor boundary conditions on parallel plates. The Casimir apparatus was then predicted to experience a tiny push in the upwards direction, and the regularized energy-momentum tensor was found to have a trace anomaly. The latter, unexpected property made it compelling to assess what happens in a simpler case. For this purpose, the present paper studies a free, real massless scalar field subject to homogeneous Dirichlet c…
We study the canonical realization of BMS symmetry for a massive scalar field introduced in reference \cite{LM}. We will construct an invariant scalar product for the generalized momenta. As a consequence we will introduce a quadratic Casimir with the supertranslations.
In this work we calculate the functional generator of the Green's functions of the Kalb-Ramond field in 3+1 dimensions. We also calculate the functional generator, and corresponding Casimir energy, of the same field when it is submitted to boundary conditions on two parallel planes. The boundary conditions we consider can be interpreted as a kind of conducting planes for the field in compearing with the Maxwell case. We compare our result with the standard ones for the scalar and Maxwell fields.
We determine the effect of Lorentz invariance violation in the vacuum energy and stress between two parallel plates separated by a distance $L$, in the presence of a massive real scalar field. We parametrize the Lorentz-violation in terms of a symmetric tensor $h^{\,μν}$ that represents a constant background. Through the Green's function method, we obtain the global Casimir energy, the Casimir force between the plates and the energy density in a closed analytical form without resorting to perturbative methods. With regards to the pressure, we find that $\mathcal{F}_c(L)=\mathcal{F}_0(\tilde{L}…
We study the field theoretical model of a real scalar field in presence of spacial inhomogeneity in form of a finite width mirror (material layer). The interaction of the scalar field with the defect is described with position-dependent mass term. We calculate the propagator of the theory, the Casimir energy and the pressure on the boundaries of the layer. We discuss the renormalization procedure for the model in dimensional regularization.
The Casimir energy of a massless scalar field is semiclassically given by contributions due to classical periodic rays. The required subtractions in the spectral density are determined explicitly. The so defined semiclassical Casimir energy coincides with that obtained using zeta function regularization in the cases studied. Poles in the analytic continuation of zeta function regularization are related to non-universal subtractions in the spectral density. The sign of the Casimir energy of a scalar field on a smooth manifold is estimated by the sign of the contribution due to the shortest peri…
A Lorentz symmetry violation aether-type theoretical model is considered to investigate the Casimir effect and the generation of topological mass associated with a self-interacting massive scalar fields obeying Dirichlet, Newman and mixed boundary conditions on two large and parallel plates. By adopting the path integral approach we found the effective potential at one- and two-loop corrections which provides both the energy density and topological mass when taken in the ground state of the scalar field. We then analyse how these quantities are affected by the Lorentz symmetry violation and co…
A global approach with cut-off exponential functions previously proposed is used to obtain the Casimir energy of a massless scalar field in the presence of a spherical shell. The proposed method makes the use of two regulators, one of them to makes finite the sum of the orders of Bessel functions and the other, to regularizes the integral involving the zeros of Bessel function. This procedure ensures a consistent mathematical handling in the calculations of the Casimir energy for a scalar field and allows to show all types of divergences. We consider separately the contributions of the inner a…
We compute the $O(λ)$ correction to the Casimir energy for the massive $λϕ^4$ model confined between a pair of parallel plates. The calculations are made with Dirichlet and Neumann boundary conditions. The correction is shown to be sensitive to the boundary conditions, except in the zero mass limit, in which case our results agree with those found in the literature.
In this paper, the first order radiative correction to the Casimir energy for a massive scalar field in the $ϕ^4$ theory on a spherical surface with $S^2$ topology was calculated. In common methods for calculating the radiative correction to the Casimir energy, the counter-terms related to free theory are used. However, in this study, by using a systematic perturbation expansion, the obtained counter-terms in renormalization program were automatically position-dependent. We maintained that this dependency was permitted, reflecting the effects of the boundary conditions imposed or background sp…
It is well known that the toroidal dimensional reduction of supergravities gives rise in three dimensions to theories whose bosonic sectors are described purely in terms of scalar degrees of freedom, which parameterise sigma-model coset spaces. For example, the reduction of eleven-dimensional supergravity gives rise to an E_8/SO(16) coset Lagrangian. In this paper, we dispense with the restrictions of supersymmetry, and study all the three-dimensional scalar sigma models G/H where G is a maximally-non-compact simple group, with H its maximal compact subgroup, and find the highest dimensions fr…
In this paper we examine Casimir effect in the case of tachyonic field, which is connected with particles with negative four-momentum square. We consider here only the case of one dimensional, scalar field. In order to describe tachyonic field, we use the absolute synchronization scheme preserving Lorentz invariance. The renormalized vacuum energy is calculated by means of Abel-Plana formula. Finaly, the Casimir energy and Casimir force as the functions of distance are obtained. In order to compare the resulting formula with the standard one, we calculate the Casimir energy and Casimir force f…
Violation of the Lorentz symmetry has important effects on physical quantities including field propagators. Therefore, in addition to the leading order, the sub-leading order of quantities may be modified. In this paper, we calculate the next to leading (NLO) radiative corrections to the Casimir energy in the presence of two perfectly conducting parallel plates for $ϕ^4$ theory with a Lorentz-breaking extension. We do the renormalization and investigate these NLO corrections for three distinct directions of the Lorentz violation; temporal direction, parallel and perpendicular to the plates.
We use a functional approach to evaluate the Casimir free energy for a self-interacting scalar field in $d+1$ dimensions, satisfying Dirichlet boundary conditions on two parallel planes. When the interaction is turned off, exact results for the free energy in some particular cases may be found, as well as low and high temperature expansions based on a duality relation that involves the inverse temperature $β$ and the distance between the mirrors, $a$. For the interacting theory, we derive and implement two different approaches. The first one is a perturbative expansion built with a thermal pro…
We consider a Casimir apparatus consisting of two perfectly conducting parallel plates, subject to the weak gravitational field of the Earth. The aim of this paper is the calculation of the energy-momentum tensor of this system for a free, real massless scalar field satisfying Neumann boundary conditions on the plates. The small gravity acceleration (here considered as not varying between the two plates) allows us to perform all calculations to first order in this parameter. Some interesting results are found: a correction, depending on the gravity acceleration, to the well-known Casimir energ…
We compute the finite temperature Casimir energy for massive scalar field with general curvature coupling subject to Dirichlet or Neumann boundary conditions on the walls of a closed cylinder with arbitrary cross section, located in a background spacetime of the form $M^{d_1+1}\times \mathcal{N}^n$, where $M^{d_1+1}$ is the $(d_1+1)$-dimensional Minkowski spacetime and $\mathcal{N}^n$ is an $n$-dimensional internal manifold. The Casimir energy is regularized using the criteria that it should vanish in the infinite mass limit. The Casimir force acting on a piston moving freely inside the closed…
The Casimir effect for {mass dimension one fermion fields (sometimes called Elko)} in $3+1$ dimension is obtained using Dirichlet boundary conditions. It is shown the existence of a repulsive force four times greater than the case of the scalar field. The precise reason for such differences are highlighted and interpreted, as well as the right parallel of the Casimir effect due to scalar and fermionic fields.
The zeta function regularization technique is used to study the finite temperature Casimir effect for a charged and massless scalar field confined between parallel plates and satisfying Dirichlet boundary conditions at the plates. A magnetic field perpendicular to the plates is included. Three equivalent expressions for the zeta function are obtained, which are exact to all orders in the magnetic field strength, temperature and plate distance. These expressions of the zeta function are used to calculate the Helmholtz free energy of the scalar field and the pressure on the plates, in the case o…
In this study, the Casimir energy for massive scalar field with periodic boundary condition was calculated on spherical surfaces with $S^1$, $S^2$ and $S^3$ topologies. To obtain the Casimir energy on spherical surface, the contribution of the vacuum energy of Minkowski space is usually subtracted from that of the original system. In large mass limit for surface $S^2$; however, some divergences would eventually remain in the obtained result. To remove these remaining divergences, a secondary renormalization program was manually performed. In the present work, a direct approach for calculation …
In this work we show how to define the action of a scalar field in a such a way that Robin boundary condition is implemented dynamically, i.e., as a consequence of the stationary action principle. We discuss the quantization of that system via functional integration. Using this formalism, we derive an expression for the Casimir energy of a massless scalar field under Robin boundary conditions on a pair of parallel plates, characterized by constants $c_1$ and $c_2$. Some special cases are discussed; in particular, we show that for some values of $c_1$ and $c_2$ the Casimir energy as a function …
The Casimir stress on two concentric spherical shell in de Sitter background for massless scalar field is calculated. The scalar field satisfies Dirichlet boundary conditions on the spheres. The metric is written in conformally flat form to make maximum use of Minkowski space calculations. Then the Casimir stress is calculated for inside and outside of the shell with different backgrounds. This model may be used to study the effect of the Casimir stress on the dynamics of the domain wall formation in inflationary models of early universe.
In the present study, the first-order radiative correction to the Casimir energy for massive and massless scalar fields confined with mixed boundary conditions (Dirichlet-Neumann) between two points in ϕ^4 theory was computed. Two issues in performing the calculations in this work are essential: to renormalize the bare parameters of the problem, a systematic method was employed, allowing all influences from the boundary conditions to be imported in all elements of the renormalization program. This idea yields our counterterms appeared in the renormalization program to be position-dependent. Us…
We construct the lowest higher spin-2 current in terms of the spin-1 and the spin-1/2 currents living in the orthogonal SO(N+4)/[SO(N) x SO(4)] Wolf space coset theory for general N. The remaining fifteen higher spin currents are determined. We obtain the three-point functions of bosonic (higher) spin currents with two scalars for finite N and k (the level of the spin-1 current). By multiplying SU(2) x U(1) into the above Wolf space coset theory, the other fifteen higher spin currents together with the above lowest higher spin-2 current are realized in the extension of the large N=4 linear sup…
