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A Fundamental Problem in Quantizing General Relativity

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Abstract

We point out a fundamental problem that hinders the quantization of general relativity: quantum mechanics is formulated in terms of systems, typically limited in space but infinitely extended in time, while general relativity is formulated in terms of events, limited both in space and in time. Many of the problems faced while connecting the two theories stem from the difficulty in shoe-horning one formulation into the other. A solution is not presented, but a list of desiderata for a quantum theory based on events is laid out.

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Fig. 1

Notes

  1. The equivalence between the Heisenberg and Schrödinger pictures assumes the existence of a Hamiltonian with some regularity conditions [13] and assumes that the Hamiltonian does not have nontrivial dependence on the system’s parameters [14].

  2. Despite appearances, explicit covariance can be attained in quantum field theory in general. In the Heisenberg picture, where the dynamical equations can be written in covariant form, the state implicitly refers to a specific reference frame [15]: a Heisenberg-picture state encodes the system preparation [16] which physically happens at a specific time in some reference frame. This requires a specific timeslice of spacetime to write the Heisenberg state of a field [15] (even though this state is time-independent). Nonetheless, if one considers the preparation procedure as something that can happen on a spacelike surface (a la Tomonaga-Schwinger), then a state transformed through a unitary representation of the Lorentz group encodes the preparation in another reference frame and, in this sense, relativistic covariance is maintained.

  3. “There is no way of defining a relativistic proper time for a quantum system which is spread all over space” [18].

  4. Indeed its interpretation becomes unclear if one uses, as a complete set of observables, the momenta of all the particles. Even the limitation to position observables will not help in the case of multiple systems: it would identify multiple points in spacetime, the positions of particles conditioned on what a clock shows.

  5. Of course, general relativity can also be given a non covariant formulation (geometrodynamics), and also a Hamiltonian formulation [20, 35].

  6. Multi-fingered time refers to the fact that different observers, following different world lines, experience different proper times, e.g. [40,41,42,43].

  7. Incidentally, it appears that post-selected teleportation and the path integral formulation are equivalent procedures [72].

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Acknowledgements

I acknowledge many fruitful discussions and a longstanding collaboration with V. Giovannetti and S. Lloyd. I acknowledge funding from Unipv “Blue sky” Project-Grant No. BSR1718573 and the FQXi foundation Grant FQXi-RFP-1513 “the physics of what happens”.

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  1. Dip. Fisica and INFN Sez. Pavia, University of Pavia, Via Bassi 6, 27100, Pavia, Italy

    Lorenzo Maccone

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Maccone, L. A Fundamental Problem in Quantizing General Relativity. Found Phys 49, 1394–1403 (2019). https://doi.org/10.1007/s10701-019-00311-w

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