In this paper, we consider the convergence properties of the polarization tensor of graphene obtained in the framework of thermal quantum field theory in three-dimensional space-time. During the last years, this problem attracted much attention in connection with calculation of the Casimir force in graphene systems and investigation of the electrical conductivity and reflectance of graphene sheets. There are contradictory statements in the literature, especially on whether this tensor has an ultraviolet divergence in three dimensions. Here, we analyze this problem using the well known method of dimensional regularization. It is shown that the thermal correction to the polarization tensor is finite at any D, whereas its zero-temperature part behaves differently for D=3 and 4. For D=3, it is obtained by the analytic continuation with no subtracting infinitely large terms. As to the space-time of D=4, the finite result for the polarization tensor at zero temperature is found after subtracting the pole term. Our results are in agreement with previous calculations of the polarization tensor at both zero and nonzero temperature. This opens possibility for a wider application of the quantum field theoretical approach in investigations of graphene and other two-dimensional novel materials.

The anomalous magnetic moment of the electron, first calculated by Schwinger, lowers the ground state energy of the electron in a weak magnetic field. It is a function of the field and changes signs for large fields, ensuring the stability of the ground state. This has been shown in the past 50 years in numerous papers. The corresponding corrections to the mass of the electron have also been investigated in strong fields using semiclassical methods. We critically review these developments and point out that the calculation for low-lying excited states raises questions. Also, we calculate the contribution from the tadpole diagram, the relevance of which was observed only quite recently.

The chromomagnetic vacuum of SU(2) gluodynamics is considered in the background of a finite radius flux tube (center vortex) with a homogeneous field inside and a zero field outside. In this background, there are tachyonic modes. These modes cause an instability. It is assumed that the self-interaction of these modes stops the creation of gluons, and it is assumed that a condensate will be formed. For constant condensates, the minimum of the effective potential is found at the tree level. In the background of these condensates, all tachyonic modes acquire non-zero real masses, which will result in a real effective potential of this system. Considering only the tachyonic modes and adding the energy of the background field, the total energy is found to have a minimum at some value of the background field, which depends on the coupling of the initial SU(2) model. For small coupling, this dependence is polynomial in distinction from the Savvidy vacuum where it is exponentially suppressed. The minimum of this energy will deepen with a shrinking radius of the flux tube. It can be expected that this process can be stopped by adding quantum effects. Using the high-temperature expansion of the effective potential, it can be expected that the symmetry, which is broken by the condensate, will be restored at sufficiently high temperatures.

We review instabilities that appear from the coupling of spin-one fields to a magnetic background in a non-Abelian theory. Such coupling results, due to asymptotic freedom in a negative quantum, contribute to the effective potential. In QCD, the Savvidy vacuum results. However, due to the tachyonic mode, such a state is not stable, and the question about the true ground state of QCD is still open. In the electroweak model, the corresponding instability is postponed to very large background fields and may be of relevance in the early universe, at best. We start with an introduction to the topic and display the necessary formulas and methods. Then, we consider the one-particle spectra of the fields in a magnetic background and the related Euler–Heisenberg Lagrangians. In addition, we discuss the potential instability connected with the anomalous moment of the electron. The main part is on the quantum correction to the energy in non-Abelian fields, including massive ones. Here, the focus is on so-called electroweak magnetism and the search for a classical solution of the field equations and their approximations by a lattice of flux tubes. Finally, we review approaches with non-homogeneous background fields and the background of an A0-field.

I consider the chromomagnetic vacuum in SU(2). The effective Lagrangian in one loop approximation is known to have a minimum below zero which results in a spontaneously generated magnetic field. However, this minimum is not stable; the effective action has an imaginary part. Over the past decades, there were many attempts to handle this situation which all were at some point unsatisfactory. I propose an idea for a new solution by assuming that the tachyonic mode, at low temperature, acquires a condensate and, as a result, undergoes a phase transition like in the Higgs model. I consider the approximation where all gluon modes are dropped except for the tachyonic one. For this mode, we have a O(2)-model with quartic self-interaction in two dimensions. I apply the CJT (2PI) formalism in Hartree approximation. As a result, at zero and low temperatures, a minimum of the effective action at a certain value of the condensate and of the background fields is observed and there is no imaginary part. Raising the temperature, this minimum becomes shallower and at a critical temperature, the perturbative state becomes that with lower effective potential; the symmetry is restored. The physical interpretation says that the unstable mode creates tachyons until these come into equilibrium with their repulsive self-interaction and form a condensate. The relation to the Mermin-Wagner theorem is discussed.

In SU(N) gluodynamics, above the de-confinement temperature, the effective potential has minima at non-zero $$A_0$$ -background fields in the two-loop approximation. Also, it has a minimum at non-zero chromomagnetic background field, known as ’Savvidy’-vacuum, which shows up on the one-loop level. In this paper, we join these two approaches. We formulate, at finite temperature, the effective action, or the free energy, in SU(2) gluodynamics on the two-loop level, with both, $$A_0$$ background and magnetic background present at the same time, which was not done so far. We provide the necessary representations for both, effective numerical calculation and high-temperature expansions. The results are represented as a 3D plot of the real part of the effective potential. Also, we reproduce for zero either, the $$A_0$$ -background or the magnetic background, the known minima and compare them. The imaginary part is, on the two-loop level, still present. We mention that, as is known from literature for the case without $$A_0$$ -background, the imaginary part is compensated by the ring (’daisy’) diagrams. However, in our two-loop approximation, the results reveal an unnatural, singular behavior of the real part of the effective potential in the region, where the imaginary part sets in. Our conclusion is that one has to go beyond the two-loop approximation and its ring improved version, in order to investigate the minimum of the effective action as a function of $$A_0$$ and chromomagnetic field, and its stability, at least in the approximation of super daisy diagrams, i.e., the Hartree approximation in the CJT formalism.

Exploiting methods of Quantum Field Theory we compute the bulk polarization tensor and bulk dielectric functions for Dirac materials in the presence of a mass gap, chemical potential, and finite temperature. Using these results (and neglecting eventual boundary effects), we study the Casimir interaction of Dirac materials. We describe in detail the characteristic features of the dielectric functions and their influence on the Casimir pressure.

In the present paper, we return to the problem of spontaneous generation of the A_0 -background field in QCD at finite temperature and a quark chemical potential, μ . On the lattice, this problem was studied by different approaches where an analytic continuation to the imaginary potential i μ has been used. Here we consider both, real and imaginary chemical potential, analytically within the two-loop gauge-fixing independent effective potential W_eff. . We realize the gauge independence in to ways: (1) on the base of Nielsen’s identity and (2) expressing the potential in terms of Polyakov’s loop. Firstly we reproduce the known expressions in terms of Bernoulli’s polynomials for the gluons and quarks. Then, we calculate the μ -dependence, either for small μ as expansion or numerically for finite μ , real and imaginary. One result is that the chemical potential only weakly changes the values of the condensate fields, but quite strongly deepens the minima of the effective potential. We investigate the dependence of Polyakov’s loop in the minimum of the effective potential, thermodynamic pressure and Debye’s mass on the chemical potential. Comparisons with other results are given.

In the present paper, we return to the problem of spontaneous generation of the $$A_0$$ -background field in QCD at finite temperature and a quark chemical potential, $$\mu $$ . On the lattice, this problem was studied by different approaches where an analytic continuation to the imaginary potential $$i \mu $$ has been used. Here we consider both, real and imaginary chemical potential, analytically within the two-loop gauge-fixing independent effective potential $$W_{eff.}$$ . We realize the gauge independence in to ways: (1) on the base of Nielsen’s identity and (2) expressing the potential in terms of Polyakov’s loop. Firstly we reproduce the known expressions in terms of Bernoulli’s polynomials for the gluons and quarks. Then, we calculate the $$\mu $$ -dependence, either for small $$\mu $$ as expansion or numerically for finite $$\mu $$ , real and imaginary. One result is that the chemical potential only weakly changes the values of the condensate fields, but quite strongly deepens the minima of the effective potential. We investigate the dependence of Polyakov’s loop in the minimum of the effective potential, thermodynamic pressure and Debye’s mass on the chemical potential. Comparisons with other results are given.

We calculate the vacuum (Casimir) energy for a scalar field with ϕ4 self-interaction in (1 + 1) dimensions non perturbatively, i.e., in all orders of the self-interaction. We consider massive and massless fields in a finite box with Dirichlet boundary conditions and on the whole axis as well. For strong coupling, the vacuum energy is negative indicating some instability.

We investigate Bose-Einstein condensation of a gas of non-interacting Bose particles moving in the background of a periodic lattice of delta functions. In the one-dimensional case, where one has no condensation in the free case, we showed that this property persist also in the presence of the lattice. In addition we formulated some conditions on the spectral functions which would allow for condensation.

Abstract The basic thermodynamic quantities for a non-interacting scalar field in a periodic potential composed of either a one-dimensional chain of Dirac $$\delta -\delta ^\prime $$ δ-δ′ functions or a specific potential with extended compact support are calculated. First, we consider the representation in terms of real frequencies (or one-particle energies). Then we turn the axis of frequency integration towards the imaginary axis by a finite angle, which allows for easy numerical evaluation, and finally turn completely to the imaginary frequencies and derive the corresponding Matsubara representation, which this way appears also for systems with band structure. In the limit case $$T \rightarrow 0$$ T→0 we confirm earlier results on the vacuum energy. We calculate for the mentioned examples the free energy and the entropy and generalize earlier results on negative entropy.

We obtain new expressions for the Casimir energy between plates that are mimicked by the most general possible boundary conditions allowed by the principles of quantum field theory. This result enables to provide the quantum vacuum energy for scalar fields propagating under the influence of a one-dimensional crystal represented by a periodic potential formed by an infinite array of identical potentials with compact support.

We reconsider the composite string model introduced {30 years ago} to study the vacuum energy. The model consists of a scalar field, describing the transversal vibrations of a string consisting of piecewise constant sections with different tensions and mass densities, keeping the speed of light constant across the junctions. We consider the spectrum using transfer matrices and Chebyshev polynomials to get a closed formula for the eigenfrequencies. We calculate vacuum and free energy as well as the entropy of this system in two approaches, one using contour integration and another one using a Hurwitz zeta function. The latter results in a representation in terms of finite sums over polynomials. Several limiting cases are considered as well, for instance, the high-temperature expansion, which is expressed in terms of the heat kernel coefficients. The vacuum energy has no ultraviolet divergences, and the corresponding heat kernel coefficient $a_1$ is zero due to the constancy of the speed of light. This is in parallel to a similar situation in macroscopic electrodynamics with isorefractive boundary conditions.

We consider a Bose gas in a one-dimensional periodic background formed by generalized delta function potentials with one and two impurities. We investigate the scattering off these impurities and their bound state levels. Besides expected features, we observe a kind of long-range correlation between the bound state levels of two impurities. Further, we define and calculate the vacuum energy of the impurity. It causes a force acting on the impurity relative to the background. We define the vacuum energy as a mode sum. In order to get a discrete spectrum we start from a finite lattice and use Chebychev polynomials to get a general expression. These allow also for quite easy investigation of impurities in finite lattices.

We investigate Bose–Einstein condensation of a noninteracting gas of Bose particles moving in the background of a periodic lattice of delta functions. In the one-dimensional case, where one has no condensation in the free case, this property persists also in the presence of the lattice for all examples which are considered in the present paper and we could only formulate some conditions which are necessary for condensation. We also considered the three-dimensional case and showed that the lattice does not destroy condensation. We calculated, for small coupling, the change in the critical temperature, which is lowered by the lattice. Finally, we took another, more general view on the problem using heat kernel expansion, and discuss BEC for Casimir effect related configurations.

The quantum vacuum energy for a hybrid comb of Dirac $δ$-$δ'$ potentials is computed using the energy of the single $δ$-$δ'$ potential over the real line that makes up the comb. The zeta function of a comb periodic potential is the continuous sum of zeta functions over the dual primitive cell of Bloch quasi-momenta. The result obtained for the quantum vacuum energy is non-perturbative in the sense that the energy function is not analytical for small couplings

We calculate the photon dispersion relations generated by the quark loop in a quark-gluon plasma with the color $A_0$ background condensate $A_0^c = A_0^{c3} + A_0^{c8}$ = const. It is found that both transversal and longitudinal modes are exited. They have a gap at low momenta and are stable in high temperature approximation. The background fields act as imaginary chemical potentials and decrease the photon frequencies compared to the case of zero background. The comparison with QED plasma with chemical potential is discussed.

We consider the Casimir effect for a scalar field interacting with another scalar field that is confined to two half spaces. This model is aimed to mimic the interaction of the photon field with matter in two slabs. We use Dirichlet boundary conditions on the interfaces for the fields in the half spaces and calculate their one-loop contribution to the wave equation for the other field. We perform the ultraviolet renormalization and develop a convenient formalism for the calculation of the vacuum energy in this configuration.

Рассматривается простая $(1+1)$-модель взаимодействия Казимира-Полдера, которая включает в себя два осциллятора, связанных скалярным полем. Рассматривается диссипация в рамках подхода, основанного на первых принципах, при допущении, что осцилляторы взаимодействуют с термостатами. Для данной системы выведено уравнение для свободной энергии в терминах вещественных частот. На основе данного представления получено представление Мацубары для случая диссипации. Рассмотрен случай нулевых собственных частот осцилляторов. Показано, что в данном случае вклад от нулевой частоты Мацубары становится модифицированным и не возникает проблем, связанных с законами термодинамики.