Quantum Gravity As a Metaphysical Problem;

Physics & Astronomy International Journal Review Article Open Access Quantum gravity as a metaphysical problem Abstract Volume 1 Issue 5 - 2017 The problem with quantum gravity is usually presented as if it would be difficult to construct even a single quantum theory of relativistic gravity. This is shown to be wrong. Ilja Schmelzer A straightforward approach using standard, well-studied methods allows to construct Independent Researcher with expertise in Theoretical Physics, mathematically well-defined quantum theories which give, in a certain classical limit, the Elementary Particle Physics, Germany Einstein equations of GR: GR may be transformed into a field theory on a fixed background Ilja Schmelzer, Independent Researcher with by breaking diffeomorphism symmetry using harmonic coordinates. The resulting field Correspondence: expertise in Theoretical Physics, Quantum Physics, Elementary theory may be regularized using standard lattice approximation techniques. The result is Particle Physics, Germany, Tel 49306316251, a well-defined canonical theory with a finite number of degrees of freedom, which can be Email quantized without problems in a canonical way. Received: September 01, 2017 | Published: November 29, Why such a straightforward way to quantize gravity is simply ignored? We identify missing 2017 explanation of relativistic symmetry as an important argument, and propose a solution. evaluate possible explanations why this simple possibility to construct a theory of quantum gravity is ignored. While a lot of different metaphysical and sociological reasons play a role, we identify as a main point a preference of the scientific community for the relational philosophy behind the spacetime interpretation of GR, in opposition to the Newtonian concept of absolute space and time (substantivalism). We conclude that the quantization of gravity is not a problem of physics, but a metaphysical problem. It is a problem of the relational philosophy of space and time in the tradition of Descartes and Leibniz, which is the base of the spacetime interpretation of GR, because this philosophy is incompatible with the known examples of theories of quantum gravity. Keywords: quantum gravity; relativistic gravity; metaphysical; faddeev-popov ghost Introduction How GR can be quantized in a simple way Quantum gravity is usually represented as being one of the central The first step is known as the field-theoretic approach to GR. As a open problems of modern fundamental physics. It is presented as being starting reference, one can use Feynman’s comments at the Chapel Hill “a dream, a theoretical need and a scientific goal. It is a theory which conference 1957.2 The key idea is to consider the gravitational field as still does not exist in complete form”, a “yet-to-be-found theory”.1 The a field on some affine background spacetime. For weak gravitational starting point of this paper is that such a non-existence claim is wrong: fields, one would say consider fluctuations of the gravitational field There exists a simple and straightforward way to construct a quantum gxµν () around the vacuum solution, the Minkowski metric ε µν . The theory of gravity-that means, a well-defined quantum theory, which in metric gµν is then, “just a shorthand for hxµν ()”, with hxµν () some well-defined limit gives the classical Einstein equations of GR. being the gravitational field. Feynman says: “Obviously one loses the Moreover, the technical way how to do this is well-known and well- beauty of geometry but this is not primary. What is primary is that understood in all of its parts. one had a new field and tried his very best to get a spin-two field as consistent as possible”. Feynman was aware of general covariance We start with the description of this straightforward construction as a symmetry of these equations, and comments: “From the other of a theory of quantum gravity. After this, we discuss the metaphysics viewpoint the geometry is important, but from this viewpoint gravity of quantum gravity. We find that missing explanation of relativistic is just another field. I am sure that an enormous amount of formulae symmetry would be the most important metaphysical objection to would be collected without having the generally covariant quantum straightforward QG. But this problem appears solvable. Another theory”. important argument proposed by Rovelli is instead, nothing but a metaphysical preference in the age-old philosophical dispute Let’s note that using GR as a field theory is clearly already between relational and absolute theories to space and time. And a modification of GR, because it already fixes the topology of the 4 given that quantum gravity on an absolute background exists, but a spacetime to be . But quantum gravity is not obliged to preserve relational, background-independent quantum gravity doesn’t, this is GR in all details. Given that we have not yet observed any GR ε + not an argument against quantum gravity with a background, but an solutions with non-trivial topology, a restriction of the theory to gxµν()= µν hx µν () argument against relationalism. is viable, and this is., all one needs to preserve the empirical success of GR is a theory which, in certain limit, gives the Einstein equations We also discuss, shortly, various historical, psychological of GR. and sociological influences which distort the actual metaphysical discussion. Submit Manuscript | http://medcraveonline.com Phys Astron Int J. 2017;1(5):163‒170. 163 ©2017 Schmelzer. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: Quantum gravity as a metaphysical problem ©2017 Schmelzer 164 How to handle diffeomorphism symmetry preferred coordinates with a coordinate condition. In GR, there are not many pretenders for a coordinate condition which make sense Symmetries are in general beautiful, but they can cause serious as a physical equations too. Essentially there is only the harmonic problems for quantization. This is the case for the quantization of condition. This condition has two important forms – as a condition for GR, because GR has a very large symmetry group – the group of the metric field, if it is written in harmonic coordinates, which has the diffeomorphisms. In the field-theoretic approach to gravity this µν form of a conservation law ∂−µ (gg )=0 , and as an equation for diffeomorphism symmetry of GR has to be broken. It is essentially the preferred coordinates (which will be denoted with X µ ), namely µ broken already by the formula gxµν()=ε µν+ hx µν (), which the harmonic equation Xx( )=0 . Given that this is the equation for µν introduces a difference between the vacuum solution gµν () x dx dx µν a scalar field too, one can obtain this equation simply by adding the and another, non-trivial solution gµν () x dx dx . Make a coordinate usual term for a scalar field for each of the preferred coordinates. So transformation x′′= xx (). This does not change the metric–it remains we obtain the modified Lagrangian: the same metric, so, the same solution from point of view of GR. µν α α ′ µν′′ L= L+ L + cg −∂ g X ∂ X (1) But gµν′′( x ( x )) dx dx will be some completely different field GR matter α µν ′ µν gµν () x dx dx , thus, define some completely different function which depends on the preferred coordinates, thus, is no longer hxµν (). So, “the same” solution of GR has in the field-theoretic covariant. With covariance gone away, we have a classical field approach many different realizations. theory with the usual translational symmetry Xµµ() x→+ X () x const µ This situation is well-known from gauge theory, where we (the Lagrangian depends only on derivatives of the X ). And the have also a quite large group of continuous symmetries, gauge Noether theorem gives, as usual for such field theories, local energy transformations. And there have been developed methods to handle and momentum conservation laws. So, with such a radical breaking this, so that one can, nonetheless, use standard field theory. One of diffeomorphism symmetry we have also solved the GR problems standard approach proposed by Faddeev and Popov uses a gauge with local energy and momentum conservation, without any energy- condition to choose one field configuration out of each equivalence momentum pseudo-tensor. class, does all the computations using the field theory with this gauge Breaking diffeomorphism invariance leads to additional degrees of condition as an additional equation, and then corrects for the error freedom for the gravitational field: different choices of the coordinates which is nonetheless made: The abstract space of solutions one would would now correspond to different physical states, which are obtain by factorizing is nonetheless different than its embedding, with distinguished by these additional degrees of freedom. Their character a gauge condition, into the field theory. The correction is defined by can be easily seen from the Lagrangian: Small modifications of the a so-called “Faddeev-Popov ghost”. According to Donoghue, this coordinates would behave like four massless scalar fields, which do works nicely for GR as an effective field theory too: not interact in any way with any matter fields. The Einstein action is in fact readily quantized, using gauge- To summarize: Explicitly breaking diffeomorphism symmetry fixing and ghost fields ala Feynman, DeWitt, Faddeev, Popov. The solves all conceptual problems caused by this symmetry, in particular background field method used by Hooft and Veltman is most beautiful the problems with the definition of local energy and momentum in this context because it allows one to retain the symmetries of densities for the gravitational field–something we need if we want to general relativity in the background field, while still gauge-fixing the use canonical quantization of a field theory. It is not problematic for quantum fluctuations.

Load more