# Weekly Papers on Quantum Foundations (4)

Operational Theories in Phase Space: Toy Model for the Harmonic Oscillator. (arXiv:2101.08323v1 [quant-ph])

We construct a toy model for the harmonic oscillator that is neither classical nor quantum. The model features a discrete energy spectrum, a ground state with sharp position and momentum, an eigenstate with non-positive Wigner function as well as a state that has tunneling properties. The underlying formalism exploits that the Wigner-Weyl approach to quantum theory and the Hamilton formalism in classical theory can be formulated in the same operational language, which we then use to construct generalized theories with well-defined phase space. The toy model demonstrates that operational theories are a viable alternative to operator-based approaches for building physical theories.

Quantum Correlations in Space-Time: Foundations and Applications. (arXiv:2101.08693v1 [quant-ph])

The absolute/relative debate on the nature of space and time is ongoing for thousands of years. Here we attempt to investigate space and time from the information theoretic point of view to understand spatial and temporal correlations under the relative assumption. Correlations, as a measure of relationship between two quantities, do not distinguish space and time in classical probability theory; quantum correlations in space are well-studied but temporal correlations are not well understood. The thesis investigates quantum correlations in space-time, by treating temporal correlations equally in form as spatial correlations and unifying quantum correlations in space and time. In particular, we follow the pseudo-density matrix formalism in which quantum states in spacetime are properly defined by correlations from measurements. We first review classical correlations, quantum correlations in space and time, to motivate the pseudo-density matrix formalism in finite dimensions. Next we generalise the pseudo-density matrix formulation to the Gaussian case, general continuous variables via Wigner representations, and general measurement processes like weak measurements. Then we compare the pseudo-density matrix formalism with other spacetime formulations: indefinite causal structures, consistent histories, generalised non-local games, out-of-time-order correlation functions, and path integrals. We argue that in non-relativistic quantum mechanics, different spacetime formulations are closely related via quantum correlations, except path integrals. Finally, we apply the pseudo-density matrix formulation to time crystals. By defining time crystals as long-range order in time, we analyse continuous and discrete time translation symmetry as well as discuss the existence of time crystals from an algebraic point of view. Finally, we summarise our work and provide the outlook for future directions.

Indefinite global time. (arXiv:2101.08739v1 [quant-ph])

By studying the set of correlations that are theoretically possible between physical systems without allowing for signalling of information backwards in time, we here identify correlations that can only be achieved if the time ordering between the systems is fundamentally indefinite. These correlations, if they exist in nature, must result from non-classical, non-deterministic time, and so may have relevance for quantum (or post-quantum) gravity, where a definite global time might not exist.

Fr\”{o}hlich Condensate of Phonons in Optomechanical Systems. (arXiv:2101.07723v2 [quant-ph] UPDATED)

We propose that the optomechanical systems can be potential platforms to implement the Fr\”{o}hlich condensate of phonons. We consider a one-dimensional array of membranes coupled to the cavity field via a quadratic interaction, and the cavity is pumped by an external laser. Analytical and numerical results predict that the high phonon occupancy of the lowest or highest mechanical mode is achievable depending on the detuning of the driving laser, the optomechnical strength, and the temperature. The decoherence of the Fr\”{o}hlich condensate can be largely suppressed by the large number of membranes. Our results shed light on narrow-linewidth phonon laser, energy conversion/transfer, and efficient multimode cooling.

Qudits for Witnessing Quantum Gravity Induced Entanglement of Masses Under Decoherence. (arXiv:2101.08086v2 [quant-ph] UPDATED)

Recently a theoretical and an experimental protocol known as quantum gravity induced entanglement of masses (QGEM) has been proposed to test the quantum nature of gravity using two mesoscopic masses each placed in a superposition of two locations. If, after eliminating all non-gravitational interactions between them, the particles become entangled, one can conclude that the gravitational potential is induced via a quantum mediator, i.e. a virtual graviton. In this paper, we examine a range of different experimental set-ups, considering different geometries and the number of spatially superposed states taken, in order to determine which would generate entanglement faster. We conclude that without decoherence, and given a maximum distance $\Delta x$ between any two spatial states of a superposition, a set of two qubits placed in spatial superposition parallel to one another will outperform all other models given realistic experimental parameters. Furthermore, when a sufficiently high decoherence rate is introduced, multi-component superpositions can outperform the two-qubit set-up. This is further verified with an experimental simulation, showing that $O(10^3)$ measurements are required to reject the no entanglement hypothesis with a parallel qubits set-up without decoherence at a 99.9$\%$ confidence level. The number of measurements increases when decoherence is introduced. When the decoherence rate reaches $0.125$~Hz, 6-dimensional qudits are required as the two-qubit system entanglement cannot be witnessed anymore. However, in this case, $O(10^6)$ measurements will be required. One can group the witness operators to measure in order to reduce the number of measurements (up to ten-fold). However, this may be challenging to implement experimentally.

Observations and predictions from past lightcones. (arXiv:2101.08324v1 [gr-qc])

Authors: Martin Lesourd

In a general Lorentzian manifold M, the past lightcone of a point is a proper subset of M that does not carry enough information to determine the rest of M. That said, if M is a globally hyperbolic Cauchy development of vacuum initial data on a Cauchy surface S and there is a point whose past lightcone contains S, then the contents of such a lightcone determines all of M (up to isometry). We show some results that describe what properties of M guarantee that past lightcones do indeed determine all or at least significant portions of M. Null lines and observer horizons, which are well known features of the de-Sitter spacetime, play a prominent role.

Closed Timelike Curves, Singularities and Causality: A Survey from G\”odel to Chronological Protection. (arXiv:2101.08592v1 [gr-qc])

Authors: Jean-Pierre Luminet (Laboratoire d’Astrophysique de Marseille)

I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. G\”odel in his rotating universe model of 1949. I emphasize that journeying into the past is intimately linked to spacetime models devoid of timelike singularities. Since such singularities arise as an inevitable consequence of the equations of general relativity given physically reasonable assumptions, time travel in the past becomes possible only when one or another of these assumptions is violated. It is the case with wormhole-type solutions. S. Hawking and other authors have tried to “save” the paradoxical consequences of time travel in the past by advocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantum gravity in which causality plays a seminal role.

The 6th Post-Newtonian Potential Terms at $O(G_N^4)$. (arXiv:2101.08630v1 [gr-qc])

Authors: J. BlümleinA. MaierP. MarquardG. Schäfer

We calculate the potential contributions of the Hamiltonian in harmonic coordinates up 6PN for binary mass systems to $O(G_N^4)$ and perform comparisons to recent results in the literature \cite{Bern:2021dqo} and \cite{Bini:2020nsb}.

Modified general relativity. (arXiv:1904.10803v7 [gr-qc] UPDATED)

Authors: Gary Nash

In a Lorentzian spacetime there exists a smooth regular line element field $(\bm{X},-\bm{X})$ and a unit vector $\bm{u}$ collinear with one of the pair of vectors in the line element field. An orthogonal decomposition of symmetric tensors can be constructed in terms of the Lie derivative along $\bm{X}$ of the metric and a product of the unit vectors; and a linear sum of divergenceless symmetric tensors. A modified Einstein equation of general relativity is then obtained by using the principle of least action, the decomposition and a fundamental postulate of general relativity. The decomposition introduces a new symmetric tensor $\varPhi_{\alpha\beta}$ which describes the energy-momentum of the gravitational field. It completes Einstein’s equation and addresses the energy localization problem. Variation of the action with respect to $X^{\mu}$ restricts $u_{\mu}$ to a particular value, which defines the possible Lorentzian metrics. $\Phi$, the trace of $\varPhi_{\alpha\beta}$, describes dark energy. The cosmological constant is dynamically replaced by $\Phi$. A cyclic universe that developed after the Big Bang is described. The dark energy density provides a natural explanation of why the vacuum energy density is so small, and why it dominates the present epoch. Assuming dark matter does not exist, a solution to the modified Einstein equation introduces two additional terms into the Newtonian radial force equation, from which the baryonic Tully-Fisher relation is obtained.

Yet Again, Quantum Indeterminacy is not Worldly Indecision

Corti, Alberto (2021) Yet Again, Quantum Indeterminacy is not Worldly Indecision. [Preprint]

Scientific Realism and Dark Matter: Conflicts In Theory Confirmation

Allzén, Simon (2021) Scientific Realism and Dark Matter: Conflicts In Theory Confirmation. [Preprint]

Out of Nowhere: Introduction: The emergence of spacetime

Huggett, Nick and Wuthrich, Christian (2021) Out of Nowhere: Introduction: The emergence of spacetime. [Preprint]

The infinite and contradiction: A history of mathematical physics by dialectical approach

Ueki, Ichiro (2021) The infinite and contradiction: A history of mathematical physics by dialectical approach. [Preprint]

Decay and recurrence of non-Gaussian correlations in a quantum many-body system

Nature Physics, Published online: 18 January 2021; doi:10.1038/s41567-020-01139-2

Starting from a strongly correlated state, with highly non-Gaussian correlations, a Gaussian state can emerge dynamically over time. Experiments with ultracold atoms show how the mixing between phase and density fluctuations plays the crucial role.

Missing the point in noncommutative geometry

Abstract

Noncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

Does general relativity highlight necessary connections in nature?

Abstract

The dynamics of general relativity is encoded in a set of ten differential equations, the so-called Einstein field equations. It is usually believed that Einstein’s equations represent a physical law describing the coupling of spacetime with material fields. However, just six of these equations actually describe the coupling mechanism: the remaining four represent a set of differential relations known as Bianchi identities. The paper discusses the physical role that the Bianchi identities play in general relativity, and investigates whether these identities—qua part of a physical law—highlight some kind of a posteriori necessity in a Kripkean sense. The inquiry shows that general relativistic physics has an interesting bearing on the debate about the metaphysics of the laws of nature.

We construct a toy model for the harmonic oscillator that is neither classical nor quantum. The model features a discrete energy spectrum, a ground state with sharp position and momentum, an eigenstate with non-positive Wigner function as well as a state that has tunneling properties. The underlying formalism exploits that the Wigner-Weyl approach to quantum theory and the Hamilton formalism in classical theory can be formulated in the same operational language, which we then use to construct generalized theories with well-defined phase space. The toy model demonstrates that operational theories are a viable alternative to operator-based approaches for building physical theories.