Generalised uncertainty relations from finite-accuracy measurements

Abstract

In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A generalised uncertainty principle-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial region σg > 0. This length scale is the standard deviation of the envelope function, g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a region σ̃g>0 in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by setting σg≃ℏG/c3, the Planck length, and σ̃g≃ℏΛ/3, where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the “minimum” length, and the so-called soccer ball problem for multi-particle states.

Dimensionally-dependent uncertainty relations, or why we (probably) won’t see micro-black holes at the LHC, even if large extra dimensions exist

“…As extensions of the current analysis, we should consider relativistic corrections, as well as the incorporation of modified uncertainty principles, obtained from the quantum gravity literature, such as the generalised uncertainty principle (GUP) (Adler et al, 2001;Maziashvili, 2006;Xiang and Wen, 2009;Lake et al, 2019;Sakalli and Kanzi, 2022;Lake et al, 2023), extended uncertainty principle (EUP), and extended generalised uncertainty principle (EGUP) (Bolen and Cavaglia, 2005;Bambi and Urban, 2008;Park, 2008). Furthermore, in order to consistently incorporate the latter, we must also consider the conditions for the formation of gravitational bound states, in higher dimensions, in the presence of a positive cosmological constant (Burikham et al, 2015;Burikham et al, 2016).…”

“…still holds, where we have rewritten ℏ = R Pl M Pl c. Finally, before concluding this section, we note that, although Eq. 16 represents a form of generalised uncertainty principle, which is valid for self-gravitating objects in higher-dimensional spacetimes, this is not the same as the 'generalised uncertainty principle' (GUP), commonly referred to in the quantum gravity literature (see, for example (Adler et al, 2001;Maziashvili, 2006;Xiang and Wen, 2009;Lake et al, 2019;Sakalli and Kanzi, 2022;Lake et al, 2023), and references therein). In fact, the derivation of Eq.…”

Problems with modified commutators

“…We believe that this, accumulated evidence, should strongly motivate the GUR research community to seek alternative mathematical structures which are capable of generating the same phenomenology, without the inconsistencies, ambiguities, and headaches associated with modified commutation relations. To this end we outlined one such formalsim, originally proposed in a series of works coauthored by one of us (Lake, 2019;Lake et al, 2019;Lake, 2020;Lake et al, 2020;Lake, 2021a;Lake, 2021b;Lake, 2022a;Lake, 2022b;Lake et al, 2023), in the second part of this paper. Whether, ultimately, this model has anything to do with physical reality, or not, is perhaps less important than what is demonstrates: that GUP, EUP and EGUP phenomenology can be obtained without assuming modified commutation relations of a non-Heisenberg type.…”

“…Unfortunately, this apparently reasonable assumption has been shown to give rise to a variety of pathologies (Hossenfelder, 2013;Hossenfelder, 2014;Tawfik and Diab, 2014). These strongly suggest that modified commutator models are not mathematically self-consistent (Lake, 2020;Lake et al, 2023).…”