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

Abstract

Annalen der PhysikVolume 526, Issue 5-6 p. A47-A50 EXPERT OPINIONFree Access From Dirac theories in curved space-times to a variation of Dirac's large–number hypothesis Ulrich D. Jentschura, Ulrich D. JentschuraSearch for more papers by this author Ulrich D. Jentschura, Ulrich D. JentschuraSearch for more papers by this author First published: 14 July 2014 https://doi.org/10.1002/andp.201400808Citations: 3 Ulrich D. Jentschura Department of Physics, Missouri University of Science and Technology, Rolla, Missouri, MO65409-0640, USA E-mail: [email protected] AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Some physicists, even today, may think that it is impossible to connect relativistic quantum mechanics and general relativity 1-3. 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. 9-11). 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) therein. Speculations about conceivable effects, caused by the noncommutativity of space-time, in hydrogen and other high-precision experiments, have been recorded in the literature 12 (these rely on noncommuting coordinates which define a Moyal–Weyl plane). It is instructive to realize that relativistic quantum mechanics can easily be combined with general relativity if the latter is formulated classically, while the former relies on wave functions expressed in terms of the space-time coordinates. 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 space-time. 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 space-time; they have been investigated by Kostelecky et al. in a series of papers (see Refs. 13-15 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 and 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 the Dirac Hamiltonian based on the correspondence principle of classical and quantum physics 5, 6, 19-21. For illustration, the free Dirac Hamiltonian is , the Dirac–Coulomb Hamiltonian is , but the Dirac–Schwarzschild 22-25 Hamiltonian is not simply given as . Here, Z is the nuclear charge number, α is the fine-structure constant, G is Newton's gravitational constant, m and M are the masses of the two involved particles, and r is the distance from the gravitational center. We use the Dirac and β matrices in the standard representation 25. Natural units with are employed. Key to the calculation of the gravitational coupling of Dirac particles is the observation that the Dirac–Clifford algebra needs to be augmented to include the local character of the space-time metric, (1)and yet, the local Dirac matrices have to be in full consistency with the requirement that the covariant derivative of the metric vanishes (axiom of local Lorentz frames in general relativity). Curved and flat space-time matrices are denoted with and without the argument x, respectively. This leads to the following ansatz for the covariant derivative of a Dirac spinor, (2)and the spin-connection matrices are expressed in terms of the as follows 17, 26, (3)where is a local spin matrix and the a and b coefficients belong to the square root of the metric, , and , while the Christoffel symbols are . This formalism was later shown to be compatible with the exponentiation of the local generators 27, 19, 21 of the Lorentz group in the spinor representation, as explained for flat space-time in detail in Ref. 7. Contrary to wide-spread belief, it is indeed possible to combine relativistic quantum mechanics and general relativity, at least on the level of quantum mechanics and quantum fields, in curved space-times, which are defined by a nontrivial (but classical, non-quantized) metric . This formalism is free from ambiguities and has been used in order to evaluate perturbative corrections to the hydrogen spectrum in curved space-times with a nontrivial metric 28, 29. Recent developments in this field include a series of papers 30, 23, 24, 31-33, 26, 25, 34, 35, where the gravitationally coupled Dirac particle is formulated first in a fully relativistic setting, and then, a generalized Foldy—Wouthuysen transformation is applied in order to isolate the nonrelativistic limit, plus correction terms. For the photon emission in curved space-time, we refer to the result for the vector transition current in Ref. 25. A certain pitfall, which is hard to spot at first glance because a deceptively elegant Eriksen–Kolsrud 36 variant of the standard Foldy–Wouthuysen transformation exists, otherwise leads to the rather unfortunate appearance of spurious, parity-violating terms. In Ref. 34, the conclusion has been reached that, in a typical, nontrivial space-time geometry, the parity-breaking terms in the in the "chiral" modification 36 of the Foldy—Wouthuysen transformation change the interpretation of the wave functions and of the transformed operators to the extent that the transformed entities cannot be associated any more with their "usual" physical interpretation after the transformation, questioning the usefulness of the "chiral" transformation (for more details, see Sec. 4 of Ref. 34). We here take the opportunity to discuss the Foldy–Wouthuysen transformation of gravitationally coupled Dirac particles in some further detail. Of particular importance is the Dirac–Schwarzschild Hamiltonian 23-25, which describes the most basic central-field problem in gravitation and probably replaces the Dirac–Coulomb Hamiltonian 37-39 as the paradigmatic bound-state problem in gravitation. Inherently, the Foldy–Wouthuysen program is perturbative; for nontrivial geometries, one tries to disentangle the particles and the antiparticle degrees of freedom which are otherwise simultaneously described by the Dirac theory 26. A Hamiltonian, which preserves the full parity symmetry of the Schwarzschild geometry, and which allows for a clear identification of the relativistic corrections terms, has recently been presented in Ref. 25. It reads (4)Here, the Schwarzschild radius is given as . This Hamiltonian is equivalent to the one presented in Eq. (26) of the recent paper published in this journal 4, provided the V and W functions are expanded to first order in the gravitational constant G (first order in the Schwarzschild radius ). The subject is not without pitfalls: We note that Donoghue and Holstein 22, two authors otherwise known for their extremely careful analysis of a number of physical problems, obtained spurious parity-violating terms after the Foldy–Wouthuysen transformation [see the text surrounding Eq. (46) of Ref. 22]; the community seems to have converged on the result that these terms actually are absent. The manifestly and fully parity-conserving Hermitian form 4 allows for an immediate answer to a pressing question: If the electrodynamic interaction were absent, would the quantum mechanical Schrödinger problem of proton and electron still have a solution and feature bound states? The answer is yes. The leading nonrelativistic terms from Eq. 4 exactly have the structure expected from a naive insertion of the gravitational potential into the Schrödinger Hamiltonian, namely (5)where the gravitational fine-structure constant is and . The relativistic corrections are described by the expectation value of the remaining terms in Eq. 4 on Schrödinger–Pauli wave functions and lead to the gravitational fine-structure formula 35 (6)where n is the principal quantum number, ϰ is the Dirac angular quantum number, which summarizes both the orbital angular momentum ℓ as well as the total angular momentum j of the bound particle into a single integer, according to the formula . Initial steps toward the calculation of the fine-structure formula had been taken in Refs. 40, 41; the formula 6 is in agreement with the form of the gravitational Hamiltonian given in Eq. (26) of Ref. 4 and with Refs. 25, 34. The energy levels 6 lift the degeneracy known from the Dirac–Coulomb problem. For electron-proton interactions, the gravitational fine-structure constant is a lot smaller than the electromagnetic fine-structure constant , but can be a lot larger for particles bound to black holes even if the "tiny" black holes are lighter than the Earth. A "scatter plot" of a gravitational bound state (circular Rydberg state) which illustrates the quantum-classical correspondence is given in Fig. 1. Figure 1Open in figure viewerPowerPoint Probability density of a neutron in a quantum Rydberg state with and maximum orbital angular momentum quantum number , gravitationally bound to a (tiny) black hole located at the origin. The scaled coordinates is defined in the text immediately following Eq. 5. For a neutron bound to a black hole of mass equal to 10−15 times the mass of the Earth, the gravitational fine-structure constant has the numerical . The points are distributed with a probability density proportional to , i.e., proportional to the probability of finding the electron at a particular coordinate in the immediate vicinity of the plotted coordinate. The circular Rydberg state approximates a classical circular trajectory with . Dirac's large-number hypothesis 42 is based on the observation that the ratio (7) is approximately equal to the ratio of the age of the Universe, T, to the time it takes light to travel a distance equal to the classical electron radius, , where is the classical electron radius, . Eddington 43 observed that the ratio of αQED to the gravitational fine-structure constant for two gravitating electrons is approximately equal to the square root of the number N of charged particles in the Universe, . One may also curiously observe, as a variation of the other observations recorded in the literature, that (8)With the most recent CODATA value 44 for G plugged into the left-hand side of Eq. 8, the left-hand side and right-hand sides of Eq. 8 are in agreement to better than half a permille. It is unclear at present if the emergence of the famous factor "137" from an equation whose only adjustable prefactor is the number one [see Eq. 8] constitutes a pure accident, or if there is a deeper physical interpretation available. Variants of unified electromagnetic and gravitational theories, with Kaluza–Klein compactification, may lead to massless, or mass-protected spinors coupled to the gauge fields 45. Possible connections of gravitational and electromagnetic interactions have intrigued physicists ever since Gertsenshtein 46, as well as Zeldovich and Novikov 47, discovered the possibility of graviton-photon conversion. 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