From Dirac theories in curved space‐times to a variation of Dirac's large–number hypothesis

Abstract

Some physicists, even today, may think that it is impossible to connect relativistic quantum mechanics and general relativity [1][2][3]. This impression, however, is false, as demonstrated, among many other recent works, in an article published not long ago in this journal [4]. In order to understand the subtlety, let us remember the meaning of first [5,6], second [7,8], and third quantization (see Refs. [9][10][11]). In first quantization, what was previously a well-defined particle trajectory now becomes smeared and defines a probability density of the quantum mechanical particle. In second quantization, what was previously a well-defined functional value of a physical field at a given space-time point now becomes a field operator, which is subject to quantum fluctuations. E.g., quantum fluctuations about the flat space-time metric can be expressed in terms of the graviton field operator, for which an explicit expression can be found in Eq. (5) of Ref. [8].In third quantization, what was previously a well-defined space-time point now becomes an operator, defining a conceivably noncommutative entity, which describes space-time quantization. E.g., the emergence of the noncommutative Moyal product in quantized string theory is discussed in a particularly clear exposition in Ref. [11], in the derivation leading to Eq. (32) We thus observe that space-time quantization is not necessary, a priori, in order to combine general relativity and quantum mechanics. Indeed, before we could ever conceive to observe effects due to space-time quantization, we should first consider the leading-order coupling of the Dirac particle to a curved spacetime. Space-time curvature is visible on the classical level, and can be treated on the classical level [3]. Deviations from perfect Lorentz symmetry caused by effects other than space-time curvature may result in conceivable anisotropies of spacetime; they have been investigated by Kostelecky et al. in a series of papers (see Refs. [13][14][15] and references therein) and would be visible in tiny deviations of the dispersion relations of the relativistic particles from the predictions of Dirac theory. Measurements on neutrinos [15] can be used in order to constrain the Lorentz-violating parameters. Other recent studies concern the modification of gravitational effects under slight global violations of Lorentz symmetry [16]. Brill and Wheeler [17] were among the first to study the gravitational interactions of Dirac particles, and they put a special emphasis on neutrinos. The motivation can easily be guessed: Neutrinos, at the time, were thought to be massless and transform according to the fundamental ( ) representations of the Lorentz group. Yet, their interactions have a profound impact on cosmology [18]. Being (almost) massless, their gravitational interactions can, in principle, only be formulated on a fully relativistic footing. This situation illustrates a pertinent dilemma: namely, the gravitational potential, unlike the the Coulomb potential, cannot simply be inserted into th...

Gravitational correction to vacuum polarization

“…[7]. Recently, the gravitationally coupled Dirac equation has been investigated [34][35][36][37], with particular emphasis on the Dirac-Schwarzschild problem, which is the equivalent of the Dirac-Coulomb problem for electrostatic interactions and describes a particle bound to a central gravitational field. From now on, for the remainder of this article, we revert to natural units with = c 0 = ǫ 0 = 1, because we no longer consider a conceivable "correction" of the form (2).…”

Attempts at a determination of the fine-structure constant from first principles: a brief historical overview

“…Work on this project has been supported by the National Science Foundation (Grants PHY-1068547 and PHY-1403973). Recently, the gravitational bound-state problem has been investigated in a relativistic quantum framework [84][85][86][87][88]. Gravity and QED are the only two long range interactions mediated by a massless gauge boson (photon and graviton), while gluons have never been observed as free particles due to their confinement into hadrons.…”

“…(A6) by the simple substitution α G → α QED . Based on an analysis of the Dirac-Schwarzschild central-field problem in general relativity [84,85,87], it has recently been verified that α G is the analogue of the QED coupling constant for gravitational phenomena, on the basis of a calculation of the relativistic corrections for the Dirac-Schwarzschild central-field problem [84,87].…”

Gravitational Interactions and Fine-Structure Constant

“…If we are ever to gain a better understanding of the relationship of gravitational interactions and electrodynamics in the quantum world, then a very practical approach is to try to solve a number of important example problems in gravitational theory, whose solution is known in electromagnetic theory, to try to generalize the approach to the gravitational analogue, and to compare. In order to proceed, it is not necessarily required to quantize space-time itself [1]. Indeed, the formulation of quantum mechanics on curved-space backgrounds in itself constitutes an interesting problem [2][3][4][5][6].…”

“…where the γ α are the flat-space Dirac matrices, which are preferentially used in the Dirac representation [1,[9][10][11][12]. The Christoffel symbols are Γ α νρ ≡ Γ α νρ (x).…”