The fact that time and the radial coordinate coordinate do not commute means that it is possible to precisely localize a state in space, at the price of not knowing anything about its time localisation. And conversely, a sharp localisation in time implies complete delocalization as far the distance from the origin is concerned.
One of the tenets of Quantum Mechanics is that the observer is classical, usually macroscopic, and that therefore we “know” how to deal with them. In quantum gravity this may not be the case. While it is true that the smallness of the Planckian constants suggests this, there may be amplifying effects, and conceptual aspects to deals with.
Quantum Spacetime in turn requires quantum observers. This is of course true for quantum phase space as well. There we became (more or less) used to deal with the contradictions of the quantum/classical interaction. We learned how to deal with noncommuting observables for example. But a quantum spacetime will pose further challenges and other layers to our understanding, in this respect see DOUBLE QUANTIZATION
In order to produce a doubly quantized model, with both phase-space noncommutativity, governed by ℏ, and space-time noncommutativity, governed by λ, one cannot simply
apply the two twists, (14) and (17), one after the other, since they would not be acting on the correct algebra of functions. The solution is represented by the symplectic
embedding worked out in Sec. II, which allows for expressing the Poisson tensor of λ-deformed spacetime as a projection of a Poisson tensor on the full phase space,
according to the mapping (5)...This twist provides the double quantization.... the
separability between spacetime and phase-space quantizations is a feature in general not shared by other models.
noncommutative quantum wormholes?
Results suggest a granular structure of spacetime at the Planck scales.
In fact, the noncommutativity of space-time could be intrinsically connected with gravity [2, 6, 7],...
we consider the (backreaction) effects of tiny modifications of the Schwarzschild geometry induced noncommutative geometry, which in turn affect the Hawking temperature, to the deformation of the Heisenberg uncertainty principle, that is the GUP...
We have found that the -correction to the canonical commutation relations of Heisenberg algebra is negative, suggesting a discrete nature of spacetime at the Planck scales,...
These aspects appear particularly interesting in perspective of laboratory-scale imitation of the black hole horizon, with the subsequent possible emission of an analogue Hawking radiation [115, 116].
Entangled squeezed states in noncommutative spaces with minimal length uncertainty relations
Noncommutative systems are found to be more entangled than the usual quantum mechanical systems. The noncommutative parameter provides an additional degree of freedom in the construction by which one can raise the entanglement of the noncommutative systems to fairly higher values beyond the usual systems. Despite having classical-like behavior, coherent states in noncommutative space produce a small amount of entanglement and therefore they possess slight nonclassicality as well, which is not true for the coherent states of an ordinary harmonic oscillator.
the noncommutative space provides an extra degree of freedom by which one can increase the degree of entanglement beyond the ordinary systems.
Acoustic black hole noncommutative
The connection between black hole physics and the theory of supersonic acoustic flow is now well established and has been developed to investigate the Hawking radiation and other phenomena for understanding quantum gravity. Acoustic black holes were found to possess many of the fundamental properties of black holes in general relativity.
we wonder whether the noncommutativity of the spacetime affects the GUP itself. This is also motivated by the fact that in high energy physics both strong spacetime noncommutativity and quark gluon plasma (QGP) may take place together. Thus, it seems to be natural to look for acoustic black holes in a QGP fluid with spacetime noncommutativity in this regime.
It was found in [23] that the spacetime noncommutativity affects the rate of loss of mass of the black hole. Thus for suitable values of the spacetime noncommutativity parameter a wider or narrower spectrum of particle wave function can be scattered with increased amplitude by the acoustic black hole. This increases or decreases the superressonance phenomenon previously studied in [28], [29].
Superresonance Acoustic Black Holes
a scalar sound wave is reflected from such a vortex with an amplification for a specific range of frequencies of the incident wave, a scalar wave (a solution of the massless Klein–Gordon equation in the black-hole spacetime), entering the ergosphere, is reflected back outside with an amplitude that exceeds the amplitude of the incident wave, for a certain range of frequencies bounded from above by the angular velocity of the black hole. The wave extracts energy from the rotational energy of the hole, leading to a ‘spin-down’ of the latter. This wave analogue of the Penrose process was discovered by Zeldovich [5] and investigated in detail by Starobinsky [6] and Misner (who dubbed the phenomenon ‘superradiance’) [7].
We may mention that the quantum version of superradiance has been likened to‘stimulated emission of radiation’ inasmuch as Hawking radiation is ‘spontaneous emission’ from a black hole. In this paper, we investigate the possibility of the acoustic analogue of superradiance (a phenomenon that we call ‘superresonance’), i.e., the amplification of a sound wave by reflection from the ‘ergo’-region of a rotating acoustic black hole...
Draining bathtub flow...the radial velocity component exceeds the local sound velocity everywhere, behaves as an outer trapped surface in this ‘acoustic’ spacetime, and is identified with the (future) event horizon of the black-hole analogue... the reflection coefficient has a magnitude larger than unity. This is precisely the amplification relation that emerges in superradiance from rotating black holes in general relativity [6, 7]....By conservation of energy, this flux must equal the rate of loss of mass (energy) from the black hole...
the phenomenon of sound amplification due to reflection from a medium at
rest, with a supersonically moving boundary, had already been known for about four decades [5]. It is not completely clear to us whether this is the same as superresonance. There may also be some connection of superresonance with stimulated vortex sound [11]. If indeed these classical acoustic phenomena are examples of superresonance, then one would be led to conclude that evidence already exists of acoustic black-hole analogues existing in nature. SuperResonance
The frequency range is determined by angular momentum of the incident wave and angular velocity of the horizon. The extra energy that reflected waves carry is taken from the rotational energy of the condensate. As a result vortex motion is slowed down and eventually stops when all the rotational energy of the condensate is extracted out of it
Phase-space noncommutative formulation of Ozawa’s uncertainty principle
From another perspective, NC deformations of the Heisenberg-Weyl (HW) algebra have been investigated in the context of quantum cosmology and are shown to have relevant implications on the thermodynamic stability of black holes and as a possible regularization of the black hole singularities [27]. In the context of QM, phase-space noncommutativity could induce violations of Robertson- Schrödinger uncertainty principle [28] and also work as a source of Gaussian entanglement [29].
noncommutativity introduces a “noise” into the interaction, and so the transformation is no longer noiseless....NCQM encompasses more states than the standard QM. Thus, experimentally, a tiny imprint of noncommutativity could be identified in quantum systems,
if an effective deviation from OUP were detected.
https://arxiv.org/pdf/2207.07451.pdf
Another [uncertainty] form was proven much more recently and states that a wave function and its Fourier transform cannot both have their support in sets of finite volume. In other words, they cannot both vanish outside a region of finite volume. This formulation is very similar to the general support uncertainty principles that we will consider below (Section 4.3) in the finite dimensional setting. They have attracted attention more recently, in the context of signal analysis [36, 37], in the theory of the Fourier analysis on finite groups [38, 34], and in the study of nonclassicality in quantum systems with a finite dimensional Hilbert space [19], which is our focus here
Investigations of possible violations of the Pauli exclusion principle [PEP] represent critical tests of the microscopic space-time structure and properties. Space-time noncommutativity provides a class of universality for several quantum gravity models.
https://arxiv.org/pdf/2209.00074.pdf
From the experimental point of view, a most intriguing prediction of this class of non-commutative models is a small but different from zero probability for electrons to perform PEP violating atomic transitions (δ2), which depends on the energy scale of the observed transition
Space does not actually have points - noncommutativity has missing points - vid lecture
So the "diagonalization" of Hilbert Space requires the numbers to be positive called "state physicality" (Nick Huggett). If one of them for wavelength is negative then you have a noncommutativity that goes against the density operator for amplitude squared probability. There are no points in noncommutative space. The probability becomes imaginary that makes no sense since it's non-physical but it is nonlocal.
So how does space emerge? See Basil J. Hiley use of Jordan product for the answer.
nonlinear noncommutative waves have infinite propagation speed, i.e., if the initial conditions at time 0 have a compact support then for any positive time the support of the solution can be arbitrarily large.
https://arxiv.org/pdf/hep-th/0408190.pdf
By comparing the operator formalism of 2-D NC space with that of quantum harmonic oscillator problem, we have deduced that the operator corresponding to the square of length is analogous to the Hamiltonian of the oscillator and hence the length and area are quantized in 2-D NC space. This result is in conformity with and a special case of the already established
result in the context of spectral manifolds in NC geometry [25]. We have also succeeded in showing that the length is quantized in a 2-D NC subspace of a 3-D NC space. Since the length quantization is more fundamental, the quantization of area and volume [25]can be inferred from it for the special cases in which they directly depend on the length along the 2-D subspace. But our method does not work for the cases of 1+1 and 2+1 spacetime dimensions if the time is taken to noncommute with the spatial coordinates.
This is because the eigenvalue equation Eq.(10) which is the result of the commutator Eq.(4) gives imaginary eigenvalues for λ — the quantum of ˆL2. This essentially implies that the length is increased or decreased in steps of imaginary values which is unphysical.
Bekenstein Bound and Non-Commutative Canonical Variables
Due to its equivalence with HUP, we see that in the expression of the Bekenstein bound a pair of non-commuting canonical variables can be identified, namely, , which for weak-gravity or “light” systems coincide with the usual non-commuting canonical pair . The above sequence (10) of inequalities suggests that, as the energy of the system contained in the region R grows (namely, grows), then the analytic expression of the Bekenstein bound naturally tends to become the analytic expression of the holographic bound. Therefore, we can reasonably wonder if the pair of non-commuting canonical variables naturally goes to identify another pair of non-commuting canonical variables, i.e., or , this time more properly useful for the description of highly massive and gravitating systems. In other words, the expression of the holographic bound, , may suggest a different pair of dynamical non-commuting canonical variables , to be used in strong-gravity or quantum-gravity situations (e.g., black holes), where holography becomes crucial.
by Born’s principle of reciprocity [12], the coordinate operators do not commute on a curved momentum space. This prompted the formulation of Quantum Mechanics and Quantum Field Theory in Noncommutative spacetimes (also called Noncommutative QFT), and which might cast some light in the formulation of Quantum Gravity by encoding both key aspects of a curved and a noncommuting spacetime (a curved noncommuting spacetime).
quantum matter curves noncommuting spacetime, and vice versa, non-
commuting spacetime curves quantum matter (quantum momentum space) as
a result of the back-reaction of quantum spacetime on quantum matter.
We found that QM in noncommutative spaces leads to very different solutions,
eigenvalues, and uncertainty relations than ordinary QM in commutative spaces.
https://www.sciencedirect.com/science/article/abs/pii/S0039368122000334
Quantum reality is then just a noncommutative version of the classical reality.
Noncommutative-geometry wormholes based on the Casimir effect
The Casimir effect does not, however, guarantee that the small-scale violation is sufficient for supporting a macroscopic wormhole. The purpose of this paper is to connect the Casimir effect to noncommutative geometry, which also aims to accommodate small-scale effects, the difference being that these can now be viewed as intrinsic properties of spacetime. As a result, the noncommutative effects can be implemented by modifying only the energy momentum tensor in the Einstein field equations, while leaving the Einstein tensor unchanged. The wormhole can therefore be macroscopic in spite of the small Casimir effect.
Here it is emphasized in Ref. [7] that noncommutativity
is an intrinsic property of spacetime rather than a superimposed geometric structure. So it naturally affects the mass-energy and momentum distributions, which, in turn, determines the spacetime curvature, thereby explaining why the Einstein tensor can be left unchanged. As a consequence, the length scales can be macroscopic.
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