Abstract
The subject of this paper deals with the mathematical formulation of the Heisenberg Indeterminacy Principle in the framework of Quantum Gravity. The starting point is the establishment of the so-called time-conjugate momentum inequalities holding for non-relativistic and relativistic Quantum Mechanics. The validity of analogous Heisenberg inequalities in quantum gravity, which must be based on strictly physically observable quantities (i.e., necessarily either 4-scalar or 4-vector in nature), is shown to require the adoption of a manifestly covariant and unitary quantum theory of the gravitational field. Based on the prescription of a suitable notion of Hilbert space scalar product, the relevant Heisenberg inequalities are established. Besides the coordinate-conjugate momentum inequalities, these include a novel proper-time-conjugate extended momentum inequality. Physical implications and the connection with the deterministic limit recovering General Relativity are investigated.
Highlights
In this paper, a basic challenge in theoretical and mathematical physics is addressed which concerns the axiomatic approach to Quantum Gravity and has fundamental implications in relativistic astrophysics, quantum gravity theory, and cosmology
The conjecture which is advanced here is that these should be in some sense analogous to those which apply in the context of both non-relativistic and relativistic Quantum Mechanics (QM)
The second one concerns the logical connection with classical physics. It is about the relationship of the Heisenberg indeterminacy principle with the Deterministic Principle which applies in the context of General Relativity (GR)
Summary
A basic challenge in theoretical and mathematical physics is addressed which concerns the axiomatic approach to Quantum Gravity and has fundamental implications in relativistic astrophysics, quantum gravity theory, and cosmology. The first one concerns, the mandatory physical requirements to be set on the theory of quantum gravity itself as a consequence of HIP, with particular reference to the tensor properties which the relevant dynamical variables should exhibit Another question concerns the conditions of application of the indeterminacy principle with respect to the solution of the quantum-gravity wave function and the global validity of the Heisenberg inequalities for arbitrary quantum PDFs. it is not a priori obvious that HIP should hold for an arbitrary quantum wave function to be associated with the space-time quantum state, i.e., which is a physically realizable solution of the relevant quantum-wave equation. To better understand how technically this can be achieved in practice and to grasp its actual physical motivations, in the following the same problem is first posed in the context of both non-relativistic and relativistic quantum mechanics (NRQM and RQM), where a rigorous solution of the analogous problem is possible
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