Dear Colleagues,

The notion of entropy, and particularly of statistical entropy, arises in many areas of physics, being related to the property of macroscopic irreversibility possibly arising in the finite/asymptotic behavior of dynamical systems, both classical and quantum. Although its role originates from thermodynamics and classical statistical mechanics (CSM) the concept of entropy has been greatly extended and now includes quantum gravity (QG).

In this Special Issue progress will be reported in the theory of quantum gravity, with particular reference to its covariant and manifestly-covariant realizations, which may actually help achieve insight into the possible role of quantum entropy. In this context, the latter is expected to be identified, in analogy to CSM, with a suitable Boltzmann-Shannon statistical entropy associated with the relevant quantum probability density function (PDF) which is characteristic of theory. Thus, generally it should correspond to the occurrence of non-stationary (in some suitable sense) quantum states. Specifically, to this end, consideration will be given to recent developments of QG that concern the adoption of either covariant or manifestly-covariant canonical approaches for the quantization of the space-time metric tensor. In both cases, the quantum PDF and the same quantum entropy can (or must) always be prescribed in terms of suitable $4-$scalars, with the second one being associated with an appropriate quantum expectation value of the same quantum PDF. For this purpose, review articles, as well as original research works, will be presented. Emphasis will be devoted in particular to a number of difficult/unsolved related issues, including:

• The explicit construction of the quantum entropy in vacuum together with its corresponding entropy production;
• The extension of the theory to include the treatment of event horizons as well either internal or external domains of black holes and the corresponding event horizons;
• The investigation of the asymptotic behavior of the quantum entropy and, correspondingly, the possible validity, in analogy to non-relativistic quantum mechanics, of an asymptotic and/or local constant H-theorem;
• The role of the cosmological constant in the determination of the quantum entropy and the related issue of the physical origin of the same cosmological constant.

Guest Editor

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• Quantum gravity
• Covariant and manifestly covariant theories
• Non-stationary quantum states
• Asymptotic behavior
• Quantum probability density function
• Quantum entropy
• Local/asymptotic H-theorems
• Event horizons
• Cosmological constant