On first attempts to reconcile quantum principles with gravity

Abstract

In his 1916's first paper on gravitational waves Einstein began to speculate on interactions between the principles of the old quantum theory and his theory of gravitation. With this contribution Einstein has stimulated a lot of similar speculations, during the dawn and the development of Quantum Mechanics. These speculations have culminated with the first attempt to quantize the gravitational field, that was provided by Rosenfeld in 1930. In this paper we briefly explain why this period (1916)(1917)(1918)(1919)(1920)(1921)(1922)(1923)(1924)(1925)(1926)(1927)(1928)(1929)(1930) should be inserted into the history of Quantum Gravity and then we focus on Klein's approach to the problem of reconciling Wave Mechanics with gravity, during the two-years period [1926][1927]. His attempt should be looked as the prehistory of Quantum Field Theory in a curved background. 1 Recent results [2] give approximately (4.354 ± 0.012) × 10 17 s. 2 See [4] for a commented English translation.3 Rosenfeld refers to this formalism as "the one-body theory proposed by Weyl [7]". 4 In those years physicists were used to talk about c-numbers (classical) and q-numbers (quantum). 7 We refer to Oskar Klein, not to be confused with Felix Klein. 8 In the old quantum theory there were two main approaches [25]: the particle approach and the wave approach. They led to the birth of Matrix Mechanics in 1925, developed by Heisenberg, Born and Jordan, and to the birth of Wave Mechanics at the beginning of 1926, developed by Schrödinger, respectively. 9 Klein himself admitted that his program failed because "as Dirac may well say, my main trouble came from trying to solve too many problems at a time!" [11].

Making a Quantum Universe: Symmetry and Gravity

“…Introduction of quantum mechanical concepts to general relativity was first mentioned by Einstein himself in his famous 1916 paper. The first detailed work on the topic was by Léon Rosenfeld in 1930 [41], in which the action of Einstein-Hilbert model with matter is quantized by replacing classical variables with hermitian operators, see e.g., [42] for the history of early approaches to quantum gravity. This canonical approach and its modern variants based on the quantization of 3 + 1 dimensional Hamiltonian description of dynamics, notably Wheeler-DeWitt (WD) formalism [43,44] and quantum geometrodynamics [45] lead to nonrenormalizable models.…”

Quantum Cultures during the Prehistory of Quantum Gravity: Léon Rosenfeld's Early Contributions to Quantum Gravity

“…(48) The expression in the squared brackets resembles the Lagrangian density of a complex scalar field in the presence of an electromagnetic and a gravitational field, but neither ψ nor J would have the correct dimensionality to be interpreted as a scalar field and a density Lagrangian respectively. Unlike Klein's functional, De Donder's functional (48) would have the correct sign in order to be interpreted as a Lagrangian density [Rocci 2013].…”

“…Indeed, Rosenfeld claimed that L should be the generalization of the Lagrangian considered by Gordon in [Gordon 1927], where Gordon himself suggested to consider the wave function and his complex conjugated as independent variables with vanishing variations at the boundary. Unlike Klein's functional, Rosenfeld's L functional had the correct sign to be interpreted as a Lagrangian density [Rocci 2013]. This follows from the fact that Rosenfeld was influenced by De Donder's approach presented above.…”

Towards energy discretization in quantum cosmology

“…Even though the pillars of modern Physics, General Relativity (GR) and quantum physics, have had their paths developed over the years separately, now a crucial challenge is to develop a quantum theory of gravitation. As early as 1930 Rosenfeld was the first to verify that there were problems quantizing gravity [3] , [4] , [5] and others attempts have been made until 1959 when Arnowitt, Deser and Misner proposed the Hamiltonian formulation of GR [6] . Thus the quantization of gravitation could have been achieved for instance with the help of Dirac method [7] but the concept of gravitational energy is controversy which imposes severe difficulties when to use the Hamiltonian formulation to achieve such a goal.…”