While a natural ultraviolet cutoff, presumably at the Planck length, is widely assumed to exist in nature, it is nontrivial to implement a minimum length scale covariantly. This is because the presence of a fixed minimum length needs to be reconciled with the ability of Lorentz transformations to contract lengths. In this paper, we implement a fully covariant Planck scale cutoff by cutting off the spectrum of the d’Alembertian. In this scenario, consistent with Lorentz contractions, wavelengths that are arbitrarily smaller than the Planck length continue to exist. However, the dynamics of modes of wavelengths that are significantly smaller than the Planck length possess a very small bandwidth. This has the effect of freezing the dynamics of such modes. While both wavelengths and bandwidths are frame dependent, Lorentz contraction and time dilation conspire to make the freezing of modes of trans-Planckian wavelengths covariant. In particular, we show that this ultraviolet cutoff can be implemented covariantly also in curved spacetimes. We focus on Friedmann Robertson Walker spacetimes and their much-discussed trans-Planckian question: The physical wavelength of each comoving mode was smaller than the Planck scale at sufficiently early times. What was the mode's dynamics then? Here, we show that in the presence of the covariant UV cutoff, the dynamical bandwidth of a comoving mode is essentially zero up until its physical wavelength starts exceeding the Planck length. In particular, we show that under general assumptions, the number of dynamical degrees of freedom of each comoving mode all the way up to some arbitrary finite time is actually finite. Our results also open the way to calculating the impact of this natural UV cutoff on inflationary predictions for the cosmic microwave background.
Received 01 October 2012Accepted 17 January 2013Published online 13 February 2013
This work has been in part supported by the Discovery and Canada Research Chairs Programs of the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the National Research Foundation of South Africa. A.K. is grateful for the very kind hospitality during his sabbatical stay at the Centre for Quantum Computing Technology at the University of Queensland.
Article outline: I. INTRODUCTION II. REVIEW OF BASIC SAMPLING THEORY OF BANDLIMITED FUNCTIONS III. FLAT SPACETIME A. Aside: Fixed temporal modes of bandlimited fields on flat spacetime IV. EXPANDING FRW SPACETIMES A. Deficiency indices and self-adjoint extensions 1. The deficiency indices are generically (2, 2) B. Example: de Sitter spacetime 1. The zero mode V. THE NUMBER OF TEMPORAL DEGREES OF FREEDOM OF A FIXED COMOVING SPATIAL MODE A. The freezing of comoving modes at early times B. Example: de Sitter spacetime VI. THE IMPACT OF THE COVARIANT ULTRAVIOLET CUTOFF ON THE SPECTRUM OF QUANTUM FIELD FLUCTUATIONS A. The two-point function B. The two-point function in Minkowski space VII. CONCLUSIONS AND OUTLOOK
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