Quantum Spacetime and the Universe at the Big Bang, Vanishing Interactions and Fading Degrees of Freedom
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Quantum Spacetime and the Universe at the Big Bang, Vanishing Interactions and Fading Degrees of Freedom Sergio Doplicher1 · Gerardo Morsella2
· Nicola Pinamonti3,4
Received: 7 May 2020 / Accepted: 5 November 2020 / © The Author(s) 2020
Abstract As discussed in Bahns et al. (2015) fundamental physical principles suggests that, close to cosmological singularities, the effective Planck length diverges, hence a “quantum point” becomes infinitely extended. We argue that, as a consequence, at the origin of times spacetime might reduce effectively to a single point and interactions disappear. This conclusion is supported by converging evidences in two different approaches to interacting quantum fields on Quantum Spacetime: (1) as the Planck length diverges, the field operators evaluated at a “quantum point” converge to zero, and so do the lowest order expressions for interacting fields in the Yang Feldman approach; (2) in the same limit, we find convergence of the interacting vacuum to the free one at all perturbative orders. The latter result is obtained using the adaptation, performed in Doplicher et al. (2020), of the methods of perturbative Algebraic Quantum Field Theory to Quantum Spacetime, through a novel picture of the effective Lagrangian, which maintains the ultraviolet finiteness of the perturbation expansion and allows one to prove also the existence of the adiabatic limit. It remains an open question whether the S matrix itself converges to unity and whether the limit in which the effective Planck length diverges is a unique initial condition or an unreachable limit, and only different asymptotics matter. Keywords Quantum spacetime · Noncommutative field theories · Evolution of the Universe · Perturbation theory · Quantum cosmology Mathematics Subject Classification (2010) 81T75 · 83C65 · 81T15 · 81R15
Gerardo Morsella
[email protected]
Extended author information available on the last page of the article.
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Math Phys Anal Geom
(2020) 23:46
1 Introduction As explained in [7], the basic principles of general relativity and quantum mechanics suggests that, at short distances, the structure of spacetime needs to be drastically modified. In particular, there, the quantum nature of spacetime emerges in the form of the uncertainty relations satisfied by the coordinates used to localize quantum events. These uncertainty relations are given in terms of a new length scale which is of the order of the Planck length. In [2] and in [8] it is furthermore argued that in the case of homogeneous and isotropic cosmological spacetimes these new length scale does not depend on the cosmological time. This is in contrast with the spatial dimensions of the Universe which tend to grow as the scale factor in an expanding universe and vanish at the initial singularity. Hence, the Planck length measured with respect to the actual spatial dimensions, which we call effective Planck length, scales as the inverse of the scale factor and thus it tends to diverge close to the initial singularity. However,
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