Spontaneously broken symmetry restoration of quantum fields in the vicinity of neutral and electrically charged black holes

We consider the restoration of a spontaneously broken symmetry of an interacting quantum scalar field around neutral, i.e., Schwarzschild, and electrically charged, i.e., Reissner-Nordstr\"om, black holes in four dimensions. This is done through a semiclassical self-consistent procedure, by solving the system of non-linear coupled equations describing the dynamics of the background field and the vacuum polarization. The black hole at its own horizon generates an indefinitely high temperature which decreases to the Hawking temperature at infinity. Due to the high temperature in its vicinity, there forms a bubble around the black hole in which the scalar field can only assume a value equal to zero, a minimum of energy. Thus, in this region the symmetry of the energy and the field is preserved. At the bubble radius, there is a phase transition in the value of the scalar field due to a spontaneous symmetry breaking mechanism. Indeed, outside the bubble radius the temperature is low enough such that the scalar field settles with a nonzero value in a new energy minimum, indicating a breaking of the symmetry in this outer region. Conversely, there is symmetry restoration from the outer region to the inner bubble close to the horizon. Specific properties that emerge from different black hole electric charges are also noteworthy. It is found that colder black holes, i.e., more charged ones, have a smaller bubble length of restored symmetry. In the extremal case the bubble has zero length, i.e., there is no bubble. Additionally, for colder black holes, it becomes harder to excite the quantum field modes, so the vacuum polarization has smaller values. In the extremal case, the black hole temperature is zero and the vacuum polarization is never excited.Comment: 16 pages, 4 figure

Entropy of an extremal electrically charged thin shell and the extremal black hole

There is a debate as to what is the value of the the entropy $S$ of extremal black holes. There are approaches that yield zero entropy $S=0$, while there are others that yield the Bekenstein-Hawking entropy $S=A_+/4$, in Planck units. There are still other approaches that give that $S$ is proportional to $r_+$ or even that $S$ is a generic well-behaved function of $r_+$. Here $r_+$ is the black hole horizon radius and $A_+=4\pi r_+^2$ is its horizon area. Using a spherically symmetric thin matter shell with extremal electric charge, we find the entropy expression for the extremal thin shell spacetime. When the shell's radius approaches its own gravitational radius, and thus turns into an extremal black hole, we encounter that the entropy is $S=S(r_+)$, i.e., the entropy of an extremal black hole is a function of $r_+$ alone. We speculate that the range of values for an extremal black hole is $0\leq S(r_+) \leq A_+/4$.Comment: 11 pages, minor changes, added references, matches the published versio

Qubit Picture of Virtual Particles

We show that virtual particles, despite being unobservable, can be described by quantum operators which can be interpreted under certain conditions as valid qubit quantum states. For a single virtual fermion, we prove that such a state is a separable mixed 2-qubit state with a well-defined finite temperature. For spin-1 virtual bosons, we find them to be associated to 4-qubit operators which can be interpreted as quantum states for some gauges. We also study the creation of virtual pairs of fermions, where the pair is shown to be associated to an entangled 4-qubit operator, and show the corresponding quantum circuit. Finally, we prove that renormalization does not structurally affect these results. These findings represent new connections between quantum field theory, quantum information and quantum thermodynamics.Comment: Published version in PRA. The paper was restructured with new sections added. The spin 1 boson and fermionic pair creation cases were introduce

O impacte da orientação nas comunidades vegetais – O caso do Grande Prémio de Orientação de Terras do Bouro

Na sociedade atual os hábitos de desporto e lazer viram-se cada vez mais para a utilização do meio natural para a sua prática, como é o caso da Orientação. O impacte da orientação no ambiente tem duas facetas, por um lado a degradação da sua qualidade, através dos resíduos que são deixados inadvertidamente nas áreas onde se realizam as provas, por outro lado na destruição de vegetação e perturbação de espécies de animais. Ambas as categorias de impactes podem ser minimizadas desde que se cumpram um conjunto de regras de boas práticas ambientais. O estudo em causa teve como objectivos a avaliação do impacte do Grande Prémio de Orientação de Terras do Bouro sobre a flora e vegetação da serra de Santa Isabel, e da sua capacidade de regeneração subsequente ao evento. Este evento realizou-se nos dias 11 e 12 de Novembro de 2006, e consistiu numa prova da Taça de Portugal de Orientação Pedestre. A primeira etapa correspondeu a uma prova de distância média e a segunda etapa a uma prova de distância longa. Com este trabalho pretende-se contribuir para um melhor conhecimento dos impactes originados pela realização de provas de orientação sobre a vegetação, de forma a fornecer informação que possibilite uma melhor gestão ambiental deste tipo de eventos desportivos.FPO - Federação Portuguesa de Orientação. FCT – Fundação para a Ciência e a Tecnologia no âmbito do projecto PEst-OE/AGR/UI0681/2011