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We introduce in this paper a tomographic analysis of the properties of a Friedmann–Lemaitre–Robertson–Walker (FLRW) universe with a perfect fluid. We first review previous works where the Schutz's parametrization in terms of Clebsch potentials was adopted to describe the perfect fluid. This approach allows to introduce a fiducial time in the Wheeler–De Witt equation. We revisit the properties of the perfect fluid and the introduced Clebsch potentials. In particular, we see that thermasy plays an extremely important role in the definition of fiducial time. The definition of a time and the condition a ≥ 0 for the expansion factor imply that the Hamiltonian operator must be self-adjoint which implies a restriction on the initial conditions for the wave packet. We show that these allow to obtain well-defined tomograms. Tomograms are marginal functions which incorporate all the information contained in the wave function of the universe, but have the properties of classical probability functions. They can be defined for classical distributions on the phase space as well, enabling us to describe quantum and classical states with the same family of functions. The aim of this paper is to compare the difference between classical tomograms where the Hawking and Penrose theorems imply the inevitability of an initial singular state and the well-defined initial quantum states found in literature. Finally the introduction of a time in the Wheeler–DeWitt allows us to consider the evolution of the classical and quantum initial states of the universe which can be accomplished by introducing a transition probability function for tomograms. [ABSTRACT FROM AUTHOR]

Quantum and classical physical states are represented in a unified way when they aredescribed by symplectic tomography. Therefore this representation allows us to study directly the necessary conditions for a classical universe to emerge from a quantum state. In a previous work on the de Sitter universe this was done by comparing the classical limit of the quantum tomograms with those resulting from the classical cosmological equations. In this paper, we first review these results and extend them to all the de Sitter models. We show further that these tomograms can be obtained directly from transposing the Wheeler–De Witt equation to the tomographic variables. Subsequently, because the classic limits of the quantum tomograms are identified with their asymptotic expressions, we find the necessary conditions to extend the previous results by taking the tomograms of the WKB solutions of the Wheeler–DeWitt equation with an any potential. Furthermore, in the previous works, we found that the de Sitter models undergo the quantum-to-classical transition when the cosmological constant decays to its present value, we discuss at the end how far we can extend this result to more general models. In the conclusions, after discussing any improvements and developments of the results of this work, we sketch a phenomenological approach from which to extract information about the initial states of the universe. [ABSTRACT FROM AUTHOR]

In this work, we show the importance of introducing the quantum tomography formalism to analyze the properties of wave functions in quantum cosmology. In particular, we examine the initial conditions of the universe proposed by various authors in the context of de Sitter's cosmology studying their classical limit and comparing it with the classical tomogram obtained from the Hamiltonian constraint in General Relativity. This comparison gives us the opportunity to find under which conditions there is a transition from the quantum universe to the classical one. A relevant result is that in these models the decay of the cosmological constant is a sufficient condition for this transition. [ABSTRACT FROM AUTHOR]

The importance of the tomographic approach is that either in quantum mechanics as in classical mechanics the state of a physical system is expressed with marginal probability functions called tomograms. The extension of this procedure to quantum cosmology is straightforward. But in this paper, instead of using the tomographic representation, we use tomograms to analyze the properties of the quantum and classical universes, starting from the wave functions in quantum cosmology and from the phase space distributions in classical cosmology. In this, there is a part where we resume the properties of the tomograms. Then, we apply them to study and discuss the properties of the initial conditions introduced by Hartle and Hawking, Vilenkin and Linde and finally we argue about their classical transition. According to the results of this paper it follows that the decay of the cosmological constant from the Planck scale to the present one could be responsible for the transition of the quantum universe to the classical one. [ABSTRACT FROM AUTHOR]

In the case of gauge theories, which are ruled by an infinite-dimensional invariance group, various choices of antisymmetric bilinear maps on field functionals are indeed available. This paper proves first that, within this broad framework, the Peierls map (not yet the bracket) is a member of a larger family. At that stage, restriction to gauge-invariant functionals of the fields, with the associated Ward identities and geometric structure of the space of histories, make it possible to prove that the new map is indeed a Poisson bracket in the simple but relevant case of Maxwell theory. The building blocks are available for gauge theories only: vector fields that leave the action functional invariant; the invertible gauge-field operator; and the Green function of the ghost operator. [ABSTRACT FROM AUTHOR]

In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang–Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functions giving rise to a Peierls bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on the space of all group-invariant functionals in two cases only: either the gauge-fixing is arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a linear functional of gauge fields, which are generic points of the space of histories. In both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal bracket is defined that reduces to iħ times the Peierls bracket to lowest order in ħ. [ABSTRACT FROM AUTHOR]

The quantum theory of a free particle in two dimensions with nonlocal boundary conditions on a circle is known to lead to surface and bulk states. Such a scheme is here generalized to the quantized Maxwell field, subject to mixed boundary conditions. If the Robin sector is modified by the addition of a pseudo-differential boundary operator, gauge-invariant boundary conditions are obtained at the price of dealing with gaugefield and ghost operators which become pseudo-differential. A good elliptic theory is then obtained if the kernel occurring in the boundary operator obeys certain summability conditions, and it leads to a peculiar form of the asymptotic expansion of the symbol. The cases of ghost operator of negative and positive order are studied within this framework. [ABSTRACT FROM AUTHOR]

We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism. [ABSTRACT FROM AUTHOR]

A set of brackets for classical dissipative systems, subject to external random forces, are derived. The method is inspired by the old procedure found by Peierls, for deriving the canonical brackets of conservative systems, starting from an action principle. It is found that an adaptation of Peierls'' method is applicable also to dissipative systems, when the friction term can be described by a linear functional of the coordinates, as is the case in the classical Langevin equation, with an arbitrary memory function. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism. [Copyright &y& Elsevier]