The Heisenberg Indeterminacy Principle in the Context of Covariant Quantum Gravity

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.

[1]  Claudio Cremaschini,et al.  Generalized Lagrangian Path Approach to Manifestly-Covariant Quantum Gravity Theory , 2018, Entropy.

[2]  Claudio Cremaschini,et al.  Quantum-Gravity Screening Effect of the Cosmological Constant in the DeSitter Space-Time , 2020, Symmetry.

[3]  W. Heisenberg.  Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , 1927 .

[4]  S. Hojman,et al.  Class of Exact Solutions for a Cosmological Model of Unified Gravitational and Quintessence Fields , 2015, 1509.09233.

[5]  H. P. Robertson.  The Uncertainty Principle , 1929 .

[6]  P. Nicolini,et al.  Physics on the smallest scales: an introduction to minimal length phenomenology , 2012, 1202.1500.

[7]  Quantum groups, gravity, and the generalized uncertainty principle. , 1993, Physical review. D, Particles and fields.

[8]  B. Dewitt.  Quantum Theory of Gravity. I. The Canonical Theory , 1967 .

[9]  C. Cremaschini,et al.  Generalized Lagrangian-Path Representation of Non-Relativistic Quantum Mechanics , 2016 .

[10]  Claudio Cremaschini,et al.  Space-Time Second-Quantization Effects and the Quantum Origin of Cosmological Constant in Covariant Quantum Gravity , 2018, Symmetry.

[11]  Kobe,et al.  Derivation of the energy-time uncertainty relation. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[12]  C. Cremaschini,et al.  Hamiltonian approach to GR – Part 1: covariant theory of classical gravity , 2016, 1609.04426.

[13]  Claudio Cremaschini,et al.  Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity , 2017, Entropy.

[14]  Claudio Cremaschini,et al.  Quantum-Gravity Stochastic Effects on the de Sitter Event Horizon , 2020, Entropy.

[15]  C. Cremaschini,et al.  Quantum theory of extended particle dynamics in the presence of EM radiation-reaction , 2015 .

[16]  Inverse kinetic theory for quantum hydrodynamic equations , 2006, quant-ph/0606091.

[17]  S. Liberati,et al.  Cosmological singularity resolution from quantum gravity: the emergent-bouncing universe , 2016, 1612.07116.

[18]  D. Pranzetti,et al.  Black hole quantum atmosphere for freely falling observers , 2019, Physics Letters B.

[19]  M. Nowakowski,et al.  Comparing two approaches to Hawking radiation of Schwarzschild–de Sitter black holes , 2008, 0810.5156.

[20]  James B. Hartle,et al.  Wave Function of the Universe , 1983 .

[21]  C. Cremaschini,et al.  Synchronous Lagrangian variational principles in General Relativity , 2015, 1609.04418.

[22]  F. Scardigli,et al.  Generalized Uncertainty Principle, Classical Mechanics, and General Relativity , 2020, Physics Letters B.

[23]  The Minimum and Maximum Temperature of Black Body Radiation , 2009, 0905.3762.

[24]  Claudio Cremaschini,et al.  Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription , 2020, Symmetry.

[25]  G. Veneziano.  A Stringy Nature Needs Just Two Constants , 1986 .

[26]  R. Brustein,et al.  Quantum hair of black holes out of equilibrium , 2017, 1709.03566.

[27]  C. Cremaschini,et al.  Hamiltonian approach to GR – Part 2: covariant theory of quantum gravity , 2016, 1609.04428.

[28]  Federico Re.  Distortions of Robertson–Walker metric in perturbative cosmology and interpretation as dark matter and cosmological constant , 2019, The European Physical Journal C.

[29]  C. Cremaschini,et al.  Manifest Covariant Hamiltonian Theory of General Relativity , 2016, 1609.04422.

[30]  Claudio Cremaschini,et al.  Role of Quantum Entropy and Establishment of H-Theorems in the Presence of Graviton Sinks for Manifestly-Covariant Quantum Gravity , 2019, Entropy.

[31]  Zbigniew Haba,et al.  Unification of dark energy and dark matter from diffusive cosmology , 2019, Physical Review D.

[32]  Basil J. Hiley,et al.  An ontological basis for the quantum theory , 1987 .