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We formulate a self-consistent model of the integer quantum Hall effect on an infinite strip, using boundary conditions to investigate the influence of finite-size effects on the Hall conductivity. By exploiting the translation symmetry along the strip, we determine both the general spectral properties of the system for a large class of boundary conditions respecting such symmetry, and the full spectrum for (fibered) Robin boundary conditions. In particular, we find that the latter introduce a new kind of states with no classical analogues, and add a finer structure to the quantization pattern of the Hall conductivity. Moreover, our model also predicts the breakdown of the quantum Hall effect at high values of the applied electric field., Comment: 41 pages, 12 figures

The conventional model of the gauge vector field is invariant under the local conformal symmetry only in the four-dimensional space ($4d$). Conformal generalization to an arbitrary dimension $d$ is impossible even for the free theory, differently from scalar and fermion fields. We discuss how to overcome this restriction and eventually construct four vector conformal actions. One of these models is the particular case of the previously known conformal theory of $n$-forms and others are new, up to our knowledge. In some of these models the gauge invariance is preserved, two of the new models are described by local actions with auxiliary compensating scalar fields, and the extended version of one of these models is on shell equivalent to the last, non-analytic, purely metric version., 15 pages. A few formulations made more precise, fits the version accepted in EPJP

The ultraviolet completion of gauge theories by higher derivative terms can dramatically change their behavior at high energies. The requirement of asymptotic freedom imposes very stringent constraints that are only satisfied by a small family of higher derivative theories. If the number of derivatives is large enough ($n > 4$) the theory is strongly interacting both at extreme infrared and ultraviolet regimes whereas it remains asymptotically free for a low number of extra derivatives ($n \leqslant 4$). In all cases the theory improves its ultraviolet behavior leading in some cases to ultraviolet finite theories with vanishing $��$-function. The usual consistency problems associated to the presence of extra ghosts in higher derivative theories may not harm asymptotically free theories because in that case the effective masses of such ghosts are running to infinity in the ultraviolet limit., 7 pages, 1 figure. Matches published version

The effect of supercritical charge impurities in graphene is very similar to the supercritical atomic collapses in QED for Z > 137, but with a much lower critical charge. In this sense graphene can be considered as a natural testing ground for the analysis of quantum field theory vacuum instabilities. We analyze the quantum transition from subcritical to supercritical charge regimes in gapped graphene in a common framework that preserves unitarity for any value of charge impurities. In the supercritical regime it is possible to introduce boundary conditions which control the singular behavior at the impurity. We show that for subcritical charges there are also non-trivial boundary conditions which are similar to those that appear in QED for nuclei in the intermediate regime 118, Comment: 24 pages, 7 figures

Novel bulk-edge dualities have recently emerged in topological materials from the observation of some phenomenological correspondences. The similarity of these dualities with string theory dualities is very appealing and has boosted a quite significant number of cross field studies. We analyze the bulk-edge dualities in the integer quantum Hall effect, where due to the simpler nature of planar systems the duality can be analysed by powerful analytic techniques. The results show that the correspondence is less robust than expected. In particular, it is highly dependent of the type of boundary conditions of the topological material. We give a formal proof of the equivalence of bulk and edge approaches to the quantization of Hall conductivity for metallic plates with local boundary conditions. However, the proof does not works for non-local boundary conditions, like the Atiyah-Patodi-Singer conditions, due to the appearance of gaps between the bulk and edge states.

One of the striking features of QED is that charged particles create a coherent cloud of photons. The resultant coherent state vectors of photons generate a nontrivial representation of the localized algebra of observables that do not support a representation of the Lorentz group: Lorentz symmetry is spontaneously broken. We show in particular that Lorentz boost generators diverge in this representation, a result shown also by Balachandran et al. [Eur. Phys. J. C 75, 89 (2015)EPCFFB1434-604410.1140/epjc/s10052-015-3305-0] (see also the work by Balachandran et al. [Mod. Phys. Lett. A 28, 1350028 (2013)MPLAEQ0217-732310.1142/S0217732313500284]. Localization of observables, for example in the Rindler wedge, uses Poincaré invariance in an essential way [Int. J. Geom. Methods Mod. Phys. 14, 1740008 (2017).0219-887810.1142/S0219887817400084]. Hence, in the presence of charged fields, the photon observables cannot be localized in the Rindler wedge. These observations may have a bearing on the black hole information loss paradox, as the physics in the exterior of the black hole has points of resemblance to that in the Rindler wedge.

We analyze the unitarity properties of higher derivative quantum field theories which are free of ghosts and ultraviolet singularities. We point out that in spite of the absence of ghosts most of these theories are not unitary. This result confirms the difficulties of finding a consistent quantum field theory of quantum gravity.

We argue that the ground state of a field theory, in the presence of charged particles, becomes an entangled state involving an infinity of soft photons. The quantum field vacuum is altered by the passage of a uniformly moving charge, leaving in its wake a different dressed ground state. In this sense a charged particle leaves its electromagnetic scent even after passing by. Unlike in classical electrodynamics the effect of the charge remains even at infinite time. The calculation is done in detail for the ground state of a spacetime wedge, although the results are more general. This agrees in spirit with recent results over the infrared aspects of field theory, although the technical details are different. These considerations open the possibility that the information carried by quantum fields, being nonlocal, does not disappear beyond the horizon of black holes., Comment: 10 pages. Minor corrections and added references

We show that the space of observables of test particles carries a natural Jacobi structure which is manifestly invariant under the action of the Poincar\'{e} group. Poisson algebras may be obtained by imposing further requirements. A generalization of Peierls procedure is used to extend this Jacobi bracket on the space of time-like geodesics on Minkowski space-time., Comment: 13 pages Submitted to MPLA

A geometrical approach to the covariant formulation of the dynamics of relativistic systems is introduced. A realization of Peierls brackets by means of a bivector field over the space of solutions of the Euler-Lagrange equations of a variational principle is presented. The method is illustrated with some relevant examples., 14 pages

The meaning of local observables is poorly understood in gauge theories, not to speak of quantum gravity. As a step towards a better understanding we study asymptotic (infrared) transformation in local quantum physics. Our observables are smeared by test functions, at first vanishing at infinity. In this context we show that the equations of motion can be seen as constraints, which generate a group, the group of space and time dependent gauge transformations.This is one of the main points of the paper. Infrared nontrivial effects are captured allowing test functions which do not vanish at infinity. These extended operators generate a larger group. The quotient of the two groups generate superselection sectors, which differentiate different infrared sectors. The BMS group changes the superselection sector, a result long known for its Lorentz subgroup. It is hence spontaneously broken. Ward identities implied by the gauge invariance of the S-matrix generalize the standard results and lead to charge conservation and low energy theorems. Their validity does not require Lorentz invariance., Comment: 21 pages, 1 reference added and typos corrected

We review the basic principles of Quantum Field Theory (QFT) in a brief but comprehensive introduction to the foundations of QFT. The principles of QFT are introduced in canonical and covariant formalisms. The problem of ultraviolet (UV) divergences and its renormalization is analyzed in the canonical formalism. As an application, we review the roots of Casimir effect. For simplicity, we focus on the scalar field theory but the generalization for fermion fields is straightforward. However, the quantization of gauge fields require extra techniques which are beyond the scope of this paper. The special cases of free field theories and conformal invariant theories in lower space-time dimensions illustrate the relevance of the foundations of the theory. Finally, a short introduction to functional integrals and perturbation theory in the Euclidean formalism is included in the last section.

The infrared behavior of quantum field theories confined in bounded domains is strongly dependent on the shape and structure of space boundaries. The most significant physical effect arises in the behaviour of the vacuum energy. The Casimir energy can be attractive or repulsive depending on the nature of the boundary. We calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary conditions depending on four parameters. The analysis provides a powerful method to identify which boundary conditions generate attractive or repulsive Casimir forces between the plates. In the interface between both regimes we find a very interesting family of boundary conditions which do not induce any type of Casimir force. We also show that the attractive regime holds far beyond identical boundary conditions for the two plates required by the Kenneth-Klich theorem and that the strongest attractive Casimir force appears for periodic boundary conditions whereas the strongest repulsive Casimir force corresponds to anti-periodic boundary conditions. Most of the analysed boundary conditions are new and some of them can be physically implemented with metamaterials., 21 pages, 11 figures

A new picture of quark confinement based on the instability of Coulomb phase at low energy was introduced by Volodya Gribov in the early nineties. In QCD the effective [Formula: see text] coupling constant can reach very large values in the infrared regime what generates Coulomb phase instabilities. In the Gribov picture the instability leads to a vacuum decay into light quarks for coupling constants [Formula: see text] larger than a critical value [Formula: see text], for SU(N) gauge theories. The instability of Coulomb phase can be derived from first principles in any non-Abelian gauge theory for [Formula: see text], a value which is larger than the Gribov critical value. In this paper we review the analytic derivation of the Gribov mechanism from first principles and analyze the effects of dynamical quarks in the instability of the Coulomb phase. The instabilities associated to light quarks turn out to appear at larger values of [Formula: see text] than the ones induced from pure gluon dynamics, unlike it is expected in the standard Gribov scenario. The analytic results confirm the consistency of the picture where quark confinement is mainly driven by gluonic fluctuations.

The analytic behaviour of θ-vacuum energy is related to the existence of phase transitions in QCD and ℂPN sigma models. The absence of singularities different from Lee-Yang zeros only permits ∧ cusp singularities in the vacuum energy density and never ∨ cusps. This fact, together with the Vafa-Witten diamagnetic inequality, provides a key missing link in the Vafa-Witten proof of parity symmetry conservation in vector-like gauge theories and ℂPN sigma models. However, this property does not exclude the existence of a first phase transition at θ = π or a second order phase transition at θ = 0, which might be very relevant for interpretation of the anomalous behaviour of the topological susceptibility in the ℂP1 sigma model.

We scrutinize the effects of non-ideal data acquisition on the homodyne tomograms of photon quantum states. The presence of a weight function, schematizing the effects of the finite thickness of the probing beam or equivalently noise, only affects the state reconstruction procedure by a normalization constant. The results are extended to a discrete mesh and show that quantum tomography is robust under incomplete and approximate knowledge of tomograms., Comment: 7 pages, 1 figure, published version

In this paper we analyse the structure of the BRST charge of nonlinear superalgebras. We consider quadratic non-linear superalgebras where a commutator (in terms of (super) Poisson brackets) of the generators is a quadratic polynomial of the generators. We find the explicit form of the BRST charge up to cubic order in Faddeev-Popov ghost fields for arbitrary quadratic nonlinear superalgebras. We point out the existence of constraints on structure constants of the superalgebra when the nilpotent BRST charge is quadratic in Faddeev-Popov ghost fields. The general results are illustrated by simple examples of superalgebras., 15 pages, Latex, references added, misprints corrected, comments added

The dynamics of symplectic gauge fields provides a consistent framework for fundamental interactions based on spin three gauge fields. One remarkable property is that symplectic gauge fields only have minimal couplings with gravitational fields and not with any other field of the Standard Model. Interactions with ordinary matter and radiation can only arise from radiative corrections. In spite of the gauge nature of symplectic fields they acquire a mass by the Coleman-Weinberg mechanism which generates Higgs-like mass terms where the gravitational field is playing the role of a Higgs field. Massive symplectic gauge fields weakly interacting with ordinary matter are natural candidates for the dark matter component of the Universe., Comment: 16 pages

A sample of some relevant developments that have taken place during the last twenty years in classical and quantum tomography are displayed. We will present a general conceptual framework that provides a simple unifying mathematical picture for all of them and, as an effective use of it, three subjects have been chosen that offer a wide panorama of the scope of classical and quantum tomography: tomography along lines and submanifolds, coherent state tomography and tomography in the abstract algebraic setting of quantum systems.

The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space $\mathcal{M}$ of such extensions is shown to have a canonical group composition law structure. The obtained results are compared with von Neumann's Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. The geometry of the submanifold of elliptic self-adjoint extensions $\mathcal{M}_\mathrm{ellip}$ is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian $\mathbf{Gr}$. The topology of $\mathcal{M}_\mathrm{ellip}$ is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change. Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results. A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved. The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grasmmannian is obtained. Finally, an interpretation of spontaneous symmetry breaking is offered from the point of view of the theory of self-adjoint extensions., 77 pages

We elaborate on the notion of generalized tomograms, both in the classical and quantum domains. We construct a scheme of star-products of thick tomographic symbols and obtain in explicit form the kernels of classical and quantum generalized tomograms. Some of the new tomograms may have interesting applications in quantum optical tomography., 5 pages, 0 figures

The dynamics of quantum field theories on bounded domains requires the introduction of boundary conditions on the quantum fields. We address the problem from a very general perspective by using charge conservation as a fundamental principle for scalar and fermionic quantum field theories. Unitarity arises as a consequence of the choice of charge preserving boundary conditions. This provides a powerful framework for the analysis of global geometrical and topological properties of the space of physical boundary conditions. Boundary conditions which allow the existence of edge states can only arise in theories with a mass gap which is also a physical requirement for topological insulators., 11 pages

When spatial boundaries are inserted, SUSY can be broken. We show that in an $\mathcal{N}=2$ supersymmetric theory, all the boundary conditions allowed by self-adjointness of the Hamiltonian break $\mathcal{N}=2$ SUSY while only a few of these boundary conditions preserve $\mathcal{N}=1$ SUSY. We also show that for a subset of the boundary conditions compatible with $\mathcal{N}=1$ SUSY, there exist fermionic ground states which are localized near the boundary., Comment: Section 2 rewritten; Equations corrected; Results unchanged; 15 pages

We analyse the effects of Robin boundary conditions on quantum field theories of spin 0, 1 and 1/2. In particular, we show that these conditions always lead to the appearance of edge states that play a significant role in quantum Hall effect and topological insulators. We prove in a rigorous way the existence of spectral lower bounds on the kinetic term of the Hamiltonian, which guarantees the stability and consistency of massive field theories when the mass is larger than the lower bound of the kinetic term. We also find an upper bound for the deepest edge state. The explicit dependence of both bounds on the boundary conditions and the size of the system is derived under very general conditions. For fermionic systems we analyse the case of Atiyah-Patodi-Singer and chiral bag boundary conditions. We point out the existence of edge states also in these cases and show that they disappear for small enough systems. Stability is granted in this case., Comment: 34 pages with 4 pdf figures. Updated version

The complex projection of any n-dimensional quaternionic unitary dynamics defines a one-parameter positive semigroup dynamics. We show that the converse is also true, i.e. that any one-parameter positive semigroup dynamics of complex density matrices with maximal rank can be obtained as the complex projection of suitable quaternionic unitary dynamics.

The highly non-trivial structure of the $\theta$--vacuum encodes many of the fundamental properties of gauge theories. In particular, the response of the vacuum to the $\theta$--term perturbation is sensitive to the existence of confinement, chiral symmetry breaking, etc. We analyze the dependence of the vacuum energy density on theta around two special values, $\theta=0$ and $\theta=\pi$. The existence or not of singular behaviors associated to spontaneous breaking of CP symmetry in these vacua has been a controversial matter for years. We clarify this important problem by means of continuum non-perturbative techniques. The results show the absence of first order cusp singularities on the vacuum energy density at $\theta=0$ and $\theta=\pi$ for some gauge theories. This smooth dependence of the energy on $\theta$ might have implications for long standing cosmological problems like the baryonic asymmetry and the cosmological constant problem., Comment: 8 pages, 1 figure. Invited talk at "Renormalization Group and Anomalies in Gravity and Cosmology", Ouro Preto,Brazil, March 17-23, 2003.v2: Added references

The behaviour of 0–vacuum and π–vacuum under CP symmetry is analyzed in two different families of quantum field theories: gauge theories in 4D and [Formula: see text] sigma models in 2D. In particular, the possibility of spontaneous CP symmetry breaking is analyzed by new continuum non-perturbative techniques. In the case of θ = 0–vacuum we find that the dependence of the vacuum energy density on θ around θ = 0 has not a first order cusp singularity, a missing requirement of the proof of the Vafa-Witten theorem. The result is based on the absence of Lee-Yang singularities for pure imaginary values of the θ–parameter. In the case of θ = π the result follows from the analysis of the nodal structure of the vacuum functional in the Schrödinger representation. The dynamics of both theories favors the localization of vacuum nodes at sphalerons instead of classical vacua as corresponds to parity even states. The existence of a dynamical level repulsion between parity even and parity odd states at low energies is the mechanism behind the absence of CP symmetry breaking in π–vacuum.

In the setting of a multisymplectic formalism for Hamiltonian theories on manifolds with boundary a class of admissible boundary conditions based on the principle of preservation of the gauge symmetries of the theory is presented. Such admissible conditions are characterized as those boundary conditions determined by Lagrangian submanifolds on the space of fields at the boundary lying in the zero level set of the moment map of the group of gauge transformations at the boundary. The examples of gauge covariant fields and pure Yang–Mills theories are analyzed.

The Gribov picture to quark confinement is based on the Coulomb phase instability due to the very large values that the effective αs coupling constant can reach in the infrared regime. The Gribov instability is driven by a vacuum decay into light quarks beyond a critical value of the coupling constant αs3π(1-2/3)/4 (for SU(3) gauge group). From first principles it has been shown the existence of an instability of the Coulomb phase in pure gauge theories for α≥2, much beyond the Gribov critical value. In this paper we analyze the effect of dynamical quarks in the instability of the Coulomb phase. We find a critical value of the coupling α = 3 where a quark-antiquark pair creation mechanism leads to vacuum instability. However, the new critical value turns out to be larger than the pure gauge critical value α = 2, unlike it is expected in the standard Gribov scenario. The result is analytically derived from first principles and provides further consistency to the picture where quark confinement is mainly dr...

We show that parity symmetry is not spontaneously broken in the CP^N sigma model for any value of N when the coefficient of the $\theta$--term becomes $\theta=\pi$ (mod $2\pi$). The result follows from a non-perturbative analysis of the nodal structure of the vacuum functional $\psi_0(z)$. The dynamical role of sphalerons turns out to be very important for the argument. The result introduces severe constraints on the possible critical behavior of the models at $\theta=\pi$ (mod $2\pi$).

We analyze the perturbative implications of the most general high derivative approach to quantum gravity based on a diffeomorphism invariant local action. In particular, we consider the super-renormalizable case with a large number of metric derivatives in the action. The structure of ultraviolet divergences is analyzed in some detail. We show that they are independent of the gauge fixing condition and the choice of field reparametrization. The cosmological counterterm is shown to vanish under certain parameter conditions. We elaborate on the unitarity problem of high derivative approaches and the distribution of masses of unphysical ghosts. We also discuss the properties of the low energy regime and explore the possibility of having a multi-scale gravity with different scaling regimes compatible with Einstein gravity at low energies. Finally, we show that the ultraviolet scaling of matter theories is not affected by the quantum corrections of high derivative gravity. As a consequence, asymptotic freedom is stable under those quantum gravity corrections., Comment: LaTeX, 25 pages + 3 ps-figures, Minor typos corrected

We analyse some non-perturbative properties of the Yang-Mills vacuum in twodimensional spaces in the presence of Chern-Simons interactions. We show that the vacuum functional vanishes for some gauge field configurations. We have identified some of those nodal configurations which are characterized by the property of carrying a non-trivial magnetic charge. In abelian gauge theories this fact explains why magnetic monopoles are suppressed by ChernSimons interactions. In non-abelian theories it suggests a relevant role for nodal gauge field configurations in the confinement mechanism of Yang-Mills theories. In topological Chern-Simons theories nodal configurations belong to Atiyah-Bott strata with non-null codimension in the space of gauge field configurations. In the presence of external static quarks some nodes of the vacuum functional with non-trivial magnetic charge are removed and they are responsible for the increase of vacuum energy.