Quantum-induced Stress-Energy Tensor in the Relativistic Regime

The generalized noncommutative Heisenberg algebra based on the generalized uncertainty principle imposes a minimal length uncertainty to quantum mechanics (QM). On the other hand, quantum-induced spacetime is suggested as an additional curvature on the relativistic eight-dimensional tangent bundle (phase-space), with a complimentary term combining reconciling principles of QM with General Relativity (GR) and comprising the minimal length discretization and the first-order derivatives of tangent covectors, the quantum-induced torsion-free metric tensor could be constructed. Accordingly, quantum-induced corrections imposed on the symmetric stress-energy tensor, the source of spacetime curvature, and the energy density associated with the electromagnetic and scalar Lagrangian are also suggested. Besides the classical version of the stress-energy tensor, the proposed quantization introduces additional Lagrangian densities and potentials together with coefficients depending on the metric tensor, tangent covector derivatives, and physical constants including the gravitational constant, Planck constant, speed of light, and Planck length. The vanishing covariant derivative of the quantum-induced stress-energy tensor confirms Einstein’s GR and suggests that the corresponding continuity equation implies that the gravitational fields do work on the classical and quantum matter and vice versa. For vanishing tangent covector’s first derivative and/or vanishing minimal length uncertainty, the classical GR and the undeformed (orthodox) QM are fully retained. Accordingly, the Einstein stress-energy tensor is also retrieved. Thus, we conclude that the suggested quantum-induced stress-energy tensor is, in principle, suitable for both quantum and classical field equations.