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Matthew J. Lake

Publications and source records attributed to Matthew J. Lake.

At least 19 recordsLinked to original sources
arXiv2026

How many degrees of freedom describe a quantum N-particle state?

In Newtonian spacetime, the canonical description of a classical $N$-particle system requires $3N$ degrees of freedom. Not all of these are physical, however, since the conservation of the net momentum implies that only $3(N-1)$ accelerations are independent. Hence, three constraints can be used to eliminate the unphysical centre-of-mass variables, at the level of the Lagrangian, leaving only the subset of observable displacements and momenta, which are relational. Imposing the constraints does not change the dynamics of these variables, at the classical level, and is analogous to a gauge-fixing procedure, which removes redundancy in the description of the system. In classical physics, therefore, the number of physical degrees of freedom equals the number of independent relational degrees of freedom. Here, we show that this is not the case in quantum mechanics. While an operator-analogue of the classical net momentum exists, it cannot be used to impose constraints that restrict the degrees of freedom in the theory, without a loss of physical information. This means that all $3N$ canonical degrees of freedom are physical, even though only $3(N-1)$ of them are relational. We explore the physical consequences of the non-relational variables and show that they give rise to generalised uncertainty relations (GURs), for the relational quantities that define the quantum reference frame (QRF). Hence, it is shown that the non-relational degrees of freedom refer to the frame itself and that the non-Heisenberg terms in the GURs define its Galilean-invariant spreads, in both real space and momentum space. The implications of this result for recent work on relational models, including the ``perspective neutral'' framework for QRFs, are discussed. Its implications for the wider relational program, and, in particular, the relevance of the latter to quantum gravity research, are also critically assessed

quant-ph↗
arXiv2026

Towards a Quantum Erlangen Program

The classical Erlangen Program sought to classify metric spaces entirely in terms of their symmetries. In physical spacetimes, these symmetries define transformations between classical reference frames, yielding a one-to-one correspondence between frame transformations and the underlying geometry. More recently, the classical notion of ideal frame has been extended to the quantum regime, by considering observers as embodied physical systems, subject to the laws of quantum mechanics. Here, we build on this approach, but outline an alternative definition of the term `quantum reference frame', which differs somewhat from the mainstream view. We then show how the new definition can be used to construct a simple model of Planck-scale spacetime, which makes contact with existing quantum gravity phenomenology. Finally, we show how classical spacetime symmetries can be ``mathematically preserved but operationally broken'', in the new model, suggesting that {\it quantum} spacetime may be classified, at least locally, in terms of transformations between quantised frames of reference.

physics.gen-ph↗
arXiv2025

Towards a group structure for superluminal velocity boosts

Canonical subluminal Lorentz boosts have a clear geometric interpretation. They can be neatly expressed as hyperbolic rotations, that leave both the family of $2$-sheet hyperboloids within the light cone, and the family $1$-sheet hyperboloids exterior to it, invariant. In this work, we construct a map between the two families of hypersurfaces and interpret the corresponding operators as superluminal velocity boosts. Though a physical observer cannot `jump' the light speed barrier, to pass from one regime to the other (at least not classically), the existence of superluminal motion does not, by itself, generate paradoxes. The implications of this construction for recent work on the `quantum principle of relativity', proposed by Dragan and Ekert, are discussed. The geometric picture reproduces their `superboost' operator in $(1+1)$ dimensions but generalises to $(1+3)$ dimensions in a very different way. This leaves open an important possibility, which appears to be closed to existing models, namely, the possibility of embedding the superluminal boosts within a group structure, without generating additional unwanted phenomenology, that contradicts existing experimental results. We prove that the set containing both subluminal and superluminal boosts forms a group, in $(1+1)$-dimensional spacetimes, and outline a program to extend these results to higher-dimensional geometries.

gr-qc↗
arXiv2025

The (1+3)-dimensional 'quantum principle of relativity' is Einstein's principle of relativity

We show that the $(1+3)$-dimensional `superboost' operators, proposed in Dragan and Ekert's most recent work on superluminal reference frames \cite{Dragan:2022txt}, are simply the canonical Lorentz boosts, expressed in nonstandard notation. Their $(1+3)$-dimensional `superflip', which is claimed to interchange time and space dimensions for a superluminal observer, travelling with infinite speed, is equivalent to applying the identity operator together with an arbitrary relabeling. Physically, it corresponds to staying put within the canonical rest frame, then renaming space as `time' and time as `space'. We conclude that their extension of the `quantum principle of relativity', proposed in earlier work on $(1+1)$-dimensional spacetimes \cite{Dragan:2019grn}, to ordinary Minkowski space \cite{Dragan:2022txt}, is simply Einstein's principle of relativity, proposed in 1905.

gr-qc↗
arXiv2023

Quantum reference frames, revisited

The topic of quantum reference frames (QRFs) has attracted a great deal of attention in the recent literature. Potentially, the correct description of such frames is important for both the technological applications of quantum mechanics and for its foundations, including the search for a future theory of quantum gravity. In this letter, we point out potential inconsistencies in the mainstream approach to this subject and propose an alternative definition that avoids these problems. Crucially, we reject the notion that transformations between QRFs can be represented by unitary operators and explain the clear physical reasons for this. An experimental protocol, capable of empirically distinguishing between competing definitions of the term, is also proposed. The implications of the new model, for uncertainty relations, spacetime symmetries, gauge symmetries, the quantisation of gravity, and other foundational issues are discussed, and possible directions for future work in this field are considered.

gr-qc↗
arXiv2023

An Introduction to Noncommutative Physics

In recent years, many new developments in theoretical physics, and in practical applications rely on different techniques of noncommutative algebras. In this review, we introduce the basic concepts and techniques of noncommutative physics in a range of areas, including classical physics, condensed matter systems, statistical mechanics, and quantum mechanics, and we present some important examples of noncommutative algebras, including the classical Poisson brackets, the Heisenberg algebra, Lie and Clifford algebras, the Dirac algebra, and the Snyder and Nambu algebras. Potential applications of noncommutative structures in high-energy physics and gravitational theory are also discussed. In particular, we review the formalism of noncommutative quantum mechanics based on the Seiberg--Witten map and propose a parameterization scheme to associate the noncommutative parameters with the Planck length and the cosmological constant. We show that noncommutativity gives rise to an effective gauge field, in the Schrödinger and Pauli equations. This term breaks translation and rotational symmetries in the noncommutative phase space, generating intrinsic quantum fluctuations of the velocity and acceleration, even for free particles. This review is intended as an introduction to noncommutative phenomenology for physicists, as well as a basic introduction to the mathematical formalisms underlying these effects.

hep-th↗
arXiv2023

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

We present a simple gedanken experiment in which a compact object traverses a spacetime with three macroscopic spatial dimensions and $n$ compact dimensions. The compactification radius is allowed to vary, as a function of the object's position in the four-dimensional space, and we show that the conservation of gravitational self-energy implies the dimensional dependence of the mass-radius relation. In spacetimes with extra dimensions that are compactified at the Planck scale, no deviation from the four-dimensional result is found, but, in spacetimes with extra dimensions that are much larger than the Planck length, energy conservation implies a deviation from the normal Compton wavelength formula. The new relation restores the symmetry between the Compton wavelength and Schwarzschild radius lines on the mass-radius diagram and precludes the formation of black holes at TeV scales, even if large extra dimensions exist. We show how this follows, intuitively, as a direct consequence of the increased gravitational field strength at distances below the compactification scale. Combining these results with the heuristic identification between the Compton wavelength and the minimum value of the position uncertainty, due to the Heisenberg uncertainty principle, suggests the existence of generalised, higher-dimensional uncertainty relations. These relations may be expected to hold for self-gravitating quantum wave packets, in higher-dimensional spacetimes, with interesting implications for particle physics and cosmology in extra-dimensional scenarios.

gr-qc↗
arXiv2023

Generalised Uncertainty Relations from Finite-Accuracy Measurements

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 GUP-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 $σ_g > 0$. This length scale is the standard deviation of the envelope function, $g$, that defines the POVM elements. Similarly, an EUP-type relation emerges when the standard momentum operator is replaced by a POVM that localises the wave packet to within a region $\tildeσ_g > 0$ in momentum space. The usual GUP and EUP are recovered by setting $σ_g \simeq \sqrt{\hbar G/c^3}$, the Planck length, and $\tildeσ_g \simeq \hbar\sqrt{Λ/3}$, where $Λ$ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schr{\" o}dinger and Heisenberg equations, remain unchanged. This demonstrates that GUP and EUP 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.

gr-qc↗
arXiv2023

Problems with Modified Commutators

The purpose of this paper is to challenge the existing paradigm on which contemporary models of generalised uncertainty relations (GURs) are based, that is, the assumption of modified commutation relations. We review an array of theoretical problems that arise in modified commutator models, including those that have been discussed in depth and others that have received comparatively little attention, or have not been considered at all in the existing literature, with the aim of stimulating discussion on these topics. We then show how an apparently simple assumption can solve, or, more precisely, evade these issues, by generating GURs without modifying the basic form of the canonical Heisenberg algebra. This simplicity is deceptive, however, as the necessary assumption is found to have huge implications for the quantisation of space-time and, therefore, gravity. These include the view that quantum space-time should be considered as a quantum reference frame (QRF) and, crucially, that the action scale characterising the quantum effects of gravity, $β$, must be many orders of magnitude smaller than Planck's constant, $β\sim 10^{-61} \times \hbar$, in order to recover the present day dark energy density. We argue that these proposals should be taken seriously, as a potential solution to the pathologies that plague minimum length models based on modified commutators, and that their implications should be explored as thoroughly as those of the existing paradigm, which has dominated research in this area for almost three decades.

gr-qc↗
arXiv2023

Series solution of the time-dependent Schrödinger-Newton equations in the presence of dark energy via the Adomian Decomposition Method

The Schrödinger-Newton model is a nonlinear system obtained by coupling the linear Schrödinger equation of canonical quantum mechanics with the Poisson equation of Newtonian mechanics. In this paper we investigate the effects of dark energy on the time-dependent Schrödinger-Newton equations by including a new source term with energy density $ρ_Λ = Λc^2/(8πG)$, where $Λ$ is the cosmological constant, in addition to the particle-mass source term $ρ_m = m|ψ|^2$. The resulting Schrödinger-Newton-$Λ$ (S-N-$Λ$) system cannot be solved exactly, in closed form, and one must resort to either numerical or semianalytical (i.e., series) solution methods. We apply the Adomian Decomposition Method, a very powerful method for solving a large class of nonlinear ordinary and partial differential equations, to obtain accurate series solutions of the S-N-$Λ$ system, for the first time. The dark energy dominated regime is also investigated in detail. We then compare our results to existing numerical solutions and analytical estimates, and show that they are consistent with previous findings. Finally, we outline the advantages of using the Adomian Decomposition Method, which allows accurate solutions of the S-N-$Λ$ system to be obtained quickly, even with minimal computational resources.

gr-qc↗
arXiv2022

A New Approach to Generalised Uncertainty Relations

We outline a new model in which generalised uncertainty relations are obtained without modified commutation relations. While existing models introduce modified phase space volumes for the canonical degrees of freedom, we introduce new degrees of freedom for the background geometry. The phase space is therefore enlarged but remains Euclidean. The spatial background is treated as a genuinely quantum object, with an associated state vector, and the model naturally gives rise to the extended generalised uncertainty principle (EGUP). Importantly, this approach solves (or rather, evades) well known problems associated with modified commutators, including violation of the equivalence principle, the `soccer ball' problem for multiparticle states, and the velocity dependence of the minimum length. However, it implies two radical conclusions. The first is that space must be quantised on a different scale to matter and the second is that the fundamental quanta of geometry are fermions. We explain how, in the context of the model, these do not contradict established results including the no go theorems for multiple quantisation constants, which still hold for species of material particles, and the spin-$2$ nature of gravitons.

gr-qc↗
arXiv2022

Fractal properties of particle paths due to generalised uncertainty relations

We determine the Hausdorff dimension of a particle path, $D_{\rm H}$, in the recently proposed `smeared space' model of quantum geometry. The model introduces additional degrees of freedom to describe the quantum state of the background and gives rise to both the generalised uncertainty principle (GUP) and extended uncertainty principle (EUP) without introducing modified commutation relations. We compare our results to previous studies of the Hausdorff dimension in GUP models based on modified commutators and show that the minimum length enters the relevant formulae in a different way. We then determine the Hausdorff dimension of the particle path in smeared momentum space, $\tilde{D}_{\rm H}$, and show that the minimum momentum is dual to the minimum length. For sufficiently coarse grained paths, $D_{\rm H} = \tilde{D}_{\rm H} = 2$, as in canonical quantum mechanics. However, as the resolutions approach the minimum scales, the dimensions of the paths in each representation differ, in contrast to their counterparts in the canonical theory. The GUP-induced corrections increase $D_{\rm H}$ whereas the EUP-induced corrections decrease $\tilde{D}_{\rm H}$, relative to their canonical values, and the extremal case corresponds to $D_{\rm H} = 3$, $\tilde{D}_{\rm H} = 1$. These results show that the GUP and the EUP affect the fractal properties of the particle path in fundamentally different, yet complimentary, ways.

gr-qc↗
arXiv2022

Modelling Cosmic Springs with Finsler and Generalised Finsler Geometries

We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a $(3+1)$-dimensional perspective these can be viewed as dispersion relations for waves propagating in the string interior and are analogous to those for current-carrying topological defects. This allows us to construct a unified framework with which to study and interpret the internal structure of various field-theoretic and fundamental string species, in a simple physically intuitive coordinate system, without the need for dimensional reduction or approximate effective actions. This, in turn, allows us to identify the precise conditions under which higher-dimensional strings and current-carrying defects are observationally indistinguishable, for macroscopic observers. Our approach naturally incorporates the description of so-called `cosmic springs', whose dynamics are expressed in terms of an effective Finsler geometry, for circular loops, or generalised Finsler geometry, for non-circular configurations. This demonstrates the importance of these novel geometric structures and their utility in modelling complex physical phenomena in cosmology and astrophysics.

hep-th↗
arXiv2022

Generalised Uncertainty Relations and the Problem of Dark Energy

We outline a new model in which generalised uncertainty relations, that govern the behaviour of microscopic world, and dark energy, that determines the large-scale evolution of the Universe, are intrinsically linked via the quantum properties of space-time. In this approach the background is treated as a genuinely quantum object, with an associated state vector, and additional fluctuations of the geometry naturally give rise to the extended generalised uncertainty principle (EGUP). An effective dark energy density then emerges from the field that minimises the modified uncertainty relations. These results are obtained via modifications of the canonical quantum operators, but without modifications of the canonical Heisenberg algebra, allowing many well known problems associated with existing GUP models to be circumvented.

gr-qc↗
arXiv2021

Why space could be quantised on a different scale to matter

The scale of quantum mechanical effects in matter is set by Planck's constant, $\hbar$. This represents the quantisation scale for material objects. In this article, we present a simple argument why the quantisation scale for space, and hence for gravity, may not be equal to $\hbar$. Indeed, assuming a single quantisation scale for both matter and geometry leads to the `worst prediction in physics', namely, the huge difference between the observed and predicted vacuum energies. Conversely, assuming a different quantum of action for geometry, $β\ll \hbar$, allows us to recover the observed density of the Universe. Thus, by measuring its present-day expansion, we may in principle determine, empirically, the scale at which the geometric degrees of freedom should be quantised.

gr-qc↗
arXiv2021

How does the Planck scale affect qubits?

Gedanken experiments in quantum gravity motivate generalised uncertainty relations (GURs) implying deviations from the standard quantum statistics close to the Planck scale. These deviations have been extensively investigated for the non-spin part of the wave function but existing models tacitly assume that spin states remain unaffected by the quantisation of the background in which the quantum matter propagates. Here, we explore a new model of nonlocal geometry in which the Planck-scale smearing of classical points generates GURs for angular momentum. These, in turn, imply an analogous generalisation of the spin uncertainty relations. The new relations correspond to a novel representation of {\rm SU(2)} that acts nontrivially on both subspaces of the composite state describing matter-geometry interactions. For single particles each spin matrix has four independent eigenvectors, corresponding to two $2$-fold degenerate eigenvalues $\pm (\hbar + β)/2$, where $β$ is a small correction to the effective Planck's constant. These represent the spin states of a quantum particle immersed in a quantum background geometry and the correction by $β$ emerges as a direct result of the interaction terms. In addition to the canonical qubits states, $\ket{0} = \ket{\uparrow}$ and $\ket{1} = \ket{\downarrow}$, there exist two new eigenstates in which the spin of the particle becomes entangled with the spin sector of the fluctuating spacetime. We explore ways to empirically distinguish the resulting `geometric' qubits, $\ket{0'}$ and $\ket{1'}$, from their canonical counterparts.

quant-ph↗
arXiv2020

Generalised uncertainty relations for angular momentum and spin in quantum geometry

We derive generalised uncertainty relations (GURs) for angular momentum and spin in the smeared-space model of quantum geometry. The model implements a minimum length and a minimum linear momentum, and recovers both the generalised uncertainty principle (GUP) and the extended uncertainty principle (EUP) within a single formalism. In this paper, we investigate the consequences of these results for particles with extrinsic and intrinsic angular momentum, and obtain generalisations of the canonical ${\rm so(3)}$ and ${\rm su(2)}$ algebras. We find that, although ${\rm SO(3)}$ symmetry is preserved on three-dimensional slices of an enlarged phase space, individual subcomponents of the generalised generators obey nontrivial subalgebras. These give rise to GURs for angular momentum while leaving the canonical commutation relations intact except for a simple rescaling, $\hbar \rightarrow \hbar + β$. The value of the new parameter, $β\simeq \hbar \times 10^{-61}$, is determined by the ratio of the dark energy density to the Planck density. Here, we assume the former to be of the order of the Planck length and the latter to be of the order of the de Sitter momentum $\sim \hbar\sqrtΛ$, where $Λ$ is the cosmological constant. In the smeared-space model, $\hbar$ and $β$ are interpreted as the quantisation scales for matter and geometry, respectively, and a quantum state vector is associated with the spatial background. We show that this also gives rise to a rescaled Lie algebra for generalised spin operators, together with associated subalgebras that are analogous to those for orbital angular momentum. Remarkably, consistency of the algebraic structure requires the quantum state associated with a flat background to be fermionic, with spin eigenvalues $\pm β/2$. Finally, the modified spin algebra leads to GURs for spin measurements.

gr-qc↗