The aim of this work is to discuss some aspects of the reduction of order formalism in the context of the Fadeev-Jackiw symplectic formalism, both at the classical and the quantum level. We start by reviewing the symplectic analysis in a regular theory (a higher derivative massless scalar theory), both using the Ostrogradsky prescription and also by reducing the order of the Lagrangian with an auxiliary field, showing the equivalence of these two approaches. The interpretation of the degrees of freedom is discussed in some detail. Finally, we perform the similar analysis in a singular higher derivative gauge theory (the Podolsky electrodynamics), in the reduced order formalism: we claim that this approach have the advantage of clearly separating the symplectic structure of the model into a Maxwell and a Proca (ghost) sector, thus complementing the understanding of the degrees of freedom of the theory and simplifying calculations involving matrices.
We propose a type of non-anticommutative superspace, with the interesting property of relating to Lee-Wick type of higher derivatives theories, which are known for their interesting properties, and have lead to proposals of phenomenologicaly viable higher derivatives extensions of the Standard Model. The deformation of superspace we consider does not preserve supersymmetry or associativity in general; however, we show that a non-anticommutative version of the Wess-Zumino model can be properly defined. In fact, the definition of chiral and antichiral superfields turns out to be simpler in our case than in the well known N = 1/2 supersymmetric case. We show that, when the theory is truncated at the first nontrivial order in the deformation parameter, supersymmetry is restored, and we end up with a well known Lee-Wick type of higher derivative extension of the Wess-Zumino model. Thus we show how non-anticommutative could provide an alternative mechanism for generation of these kind of higher derivative theories.
Non-anticommutative deformations have been studied in the context of supersymmetry (SUSY) in three and four space-time dimensions, and the general picture is that highly nontrivial to deform supersymmetry in a way that still preserves some of its important properties, both at the formal algebraic level (e.g., preserving the associativity of the deformed theory) as well as at the physical level (e.g., maintaining renormalizability). The Hopf algebra formalism allows the definition of algebraically consistent deformations of SUSY, but this algebraic consistency does not guarantee that physical models build upon these structures will be consistent from the physical point of view. We will investigate a deformation induced by a Drinfel'd twist of the N = 1 SUSY algebra in three space-time dimensions. The use of the Hopf algebra formalism allows the construction of deformed N = 1 SUSY algebras that should still preserve a deformed version of supersymmetry. We will construct the simplest deformed version of the Wess-Zumino model in this context, but we will show that despite the consistent algebraic structure, the model in question is not invariant under SUSY transformation and is not renormalizable. We will comment on the relation of these results with previous ones discussed in the literature regarding similar four-dimensional constructions.
Este trabalho tem como objetivo discutir o comportamento de campos quânticos à temperatura finita. Primeiramente introduziremos os ingredientes relativísticos, quânticos e térmicos de maneira fenomenológica (PlanckEinstein) ao descrever o comportamento das partículas de radiação em equilíbrio com uma cavidade, o celebrado problema do corpo negro. Em seguida, mostraremos que a linguagem implícita do fenômeno físico em questão é uma linguagem de campos escalares não massivos de Klein-Gordon-Fock em (3 + 1) dimensões em equilíbrio térmico. Com este intuito, estudaremos o problema no formalismo de operadores e, a posteriori, no formalismo de integração funcional via matriz densidade de estados. Além disso, abordaremos o problema em diferentes representações (coordenadas/momento). A conexão entre uma teoria quântica no espaço de Minkowski e uma teoria quântica em equilíbrio no espaço Euclidiano por meio de uma rotação de Wick (tempo imaginário) será mostrada por meio das transformações de similaridade. Palavras-chave: Teoria Clássica de Campos; Teoria Quântica de Campos; Teoria Quântica de Campos TérmicaThe aim of this work is to discuss the behavior of quantum fields at finite temperature. Initially, we will introduce phenomenologically the basic relativistic, quantum and thermal elements (Planck-Einstein), by describing the behavior of radiation in equilibrium with a cavity, the celebrated black body problem. Then, we will show that a proper language for describing this physical phenomena is the language of non-massive scalar fields (Klein-GordonFock) in (3 + 1) dimensions, at thermal equilibrium. For this purpose, we will study the problem in the formalism of operators and a posteriori in the formalism of functional integration, by considering the density matrix of states. In addition, we will approach the problem in different representations (coordinates/momentum). The connection between a quantum theory in Minkowski space-time and a quantum theory in thermal equilibrium in Euclidean space by means of a Wick rotation (imaginary time) will be implemented by means of the appropriate similarity transformations. Keywords: Classical Field Theory; Quantum Field Theory; Thermal Quantum Fields Theory. Aspectos introdutóriosA evolução das idéias sobre o significado do calor ao longo da jornada humana em busca do conhecimento tem um papel importante na história da ciência. Como uma fagulha, ela se origina no primeiro hominídeo que dominou a técnica necessária para se fazer o fogo, observando a existência do fenômeno na natureza, e se prolonga até os tempos atuais em que físicos estudam e especulam sobre a natureza e propriedades microscópicas da radiação e sua interação com a matéria, ao mesmo tempo que exploram questões ainda não respondidas sobre a termodinâmica de buracos negros, por exemplo. A descrição microscó-pica da natureza remonta de frases poéticas gregas como as de Demócrito, na qual a matéria consiste em pequenas partículas indivisíveis chamadas "átomos" movendo-se pelo espaço vazio, uma visão conceitualmente ...
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