Wave Function of the Universe, Path Integrals and Gauge Invariance

Abstract

The paper is devoted to some of the difficulties which the Wheeler -DeWitt quantum geometrodynamics encountered, in particular, a strong mathematical proof that this theory is gauge-invariant, the definition of the wave function of the Universe through a path integral and the illegality of asymptotic boundary conditions in quantum gravity, the derivation of the Wheeler -DeWitt equation from the path integral and the equivalence of the Dirac quantization scheme with other approaches, the problem of definition of physical states in quantum gravity, possible realizations of the Everett concept of "relative states". The problems are rarely discussed in the literature. They are related with the guiding idea that quantum theory of gravity must gauge invariant. It will lead to the question if it is possible to achieve this goal in a mathematically consistent way.Universe. But it contradicts to the fact that a solution to the Wheeler -DeWitt equation does not depend on time and, therefore, the probability distribution determined by the wave function must be true at every moment of the Universe existence. Thus, if the probability distribution predicted that geometries with small values of the scale factor are more feasible, the Universe would have had no chance to become macroscopic. Otherwise, if the probability distribution predicted that large values of the scale factor are more probable, it would mean just the statement of the fact that our Universe is macroscopic, but it would not tell us anything about the early stage of its existence. If so, the Wheeler -DeWitt theory can hardly add something to our understanding of the beginning of the Universe.But the analogy with quantum mechanics goes even further. In fact, some authors state that spacetime does not exist in quantum gravity. For instance, Kiefer [5] wrote:"In classical canonical gravity, a spacetime can be represented as a 'trajectory' in configuration space -the space of all three-metrics. . . Since no trajectories exist anymore in quantum theory, no spacetime exists at the most fundamental, and therefore also no time coordinates to parameterize any trajectory."The same idea one can find in the book by Rovelli [6]:". . . in quantum gravity the notion of spacetime disappears in the same manner in which the notion of trajectory disappears in the quantum theory of a particle." However, the structure of spacetime plays a significant role in general relativity. Do we really need to reject it when quantizing gravity? Rovelli [7] called the Wheeler -DeWitt equation "a strange equation, full of nasty features", but, in spite of all this, it is "a milestone in the development of general relativity". What gives us confidence that this equation is fundamental? How can we be so sure that it expresses gauge invariance of quantum gravity? The Wheeler -DeWitt equation is the result of application of the Dirac quantization scheme [8,9] to gravitational theory.Dirac just postulated that after quantization any constraint must become a condition on a state vector. But this post...

Is the Copenhagen Interpretation Inapplicable to Quantum Cosmology?

“…Accordingly, I consider the Dirac conjecture as a postulate, the validity of which is questionable. This point is important for the soundness of the extended phase space approach and discussed in details in [30,48].…”

On the Appearance of Time in the Classical Limit of Quantum Gravity

“…Then, we substitute (19) and (20) into Equations ( 13) and (17). The highest order, O(M 2 ), yields:…”

“…In the framework of the Wheeler-DeWitt quantum geometrodynamics, let us mention the well-known and often-cited review papers by Kuchař [17] and Isham [18]. Among others are the papers [19,20] and the recent work [21]. In the framework of supersymmetric quantum cosmology, we would like to refer to the paper [22,23] and especially to [24,25], where the semiclassical approximation is also discussed.…”

On A. D. Sakharov’s Hypothesis of Cosmological Transitions with Changes in the Signature of the Metric

“…In the U regions the path integral is a probability amplitude g 2 , φ 2 , S 2 | g 1 , φ 1 , S 1 to go from a state with a spacetime metric g 1 and matter fields φ 1 on a hypersurface S 1 to a state with a spacetime metric g 2 and matter fields φ 2 on a hypersurface S 2 . It is a sum over all field configurations g and φ [1,13,18]. If hypersurfaces S 1 , S 2 correspond to some points in time, t 1 and t 2 , one can formally consider the probability amplitude as a matrix element of the evolution operator:…”