Why space must be quantised on a different scale to matter

“Why space must be quantized on a different scale to matter” (scipost_202009_00001v1) - Revised Manuscript

Dear Editor(s),
Please see attached the revised manuscript of scipost_202009_00001v1. I have amended the text in order to address the referee’s comments and a detailed list of all changes is given below. These mostly take the form of footnotes, in order to preserve the essay-style flow of the original draft. I thank the referee for taking the time to read through the manuscript carefully and giving feedback. I hope that these changes have satisfactorily addressed his / her remaining concerns.
Please excuse me for taking so long to resubmit the paper, especially since the required changes were relatively minor, but I was caught up in unfortunate administrative duties these past few months.
With best wishes,

Matt

Changes:
• A footnote has been added at the end of the last sentence on pg 1 clarifying the meaning of the term ‘quantisation scale’.
• A footnote has been added at the end of the sentence below Eq. (3.1). This is to clarify the way in which whole functions, $g(x’-x)$, may be associated with single points, $x$, as well as how the model defines `points’ in both the quantum and classical regimes.
• A footnote was added at the end of sentence following Eq. (3.3). This is to clarify the precise meaning of the term ‘quantum geometry wave’.
• The final sentence of the Conclusions has been changed from “These include the proposal that the observed vacuum energy is related to the quantisation scale of space itself \cite{Lake:2019oaz,Lake:2019nmn}.” to “These include the proposal that the observed vacuum energy, and the present-day accelerated expansion of the universe that it drives, are related to the quantum properties of space-time \cite{Lake:2019oaz,Lake:2019nmn}. In this model, a measurement of the dark energy density constitutes a de facto measurement of the geometry quantisation scale, $\beta$, fixing its value to $\beta \simeq \hbar \times 10^{-61}$.” Although this is clearly a postdiction rather than a prediction of the model, it provides a concrete link between observations and the smeared-space model parameters, in response to the referee’s final point.
• Two new references have been added, which appear as refs. [5] and [23] in the new draft.

Attachment:

Why_space_must_be_quantised_on_a_different_scale_SciPost_v2.pdf

Reply to referee report for “Why space must be quantized on a different scale to matter” (scipost_202009_00001v1)

For clarity, we reproduce the referee’s comments in full, before replying to them point-by-point. Excerpts from the referee report are written in italics and our replies are written in normal type.

In the manuscript the author investigates the possibility that space-time is quantized on a different scale as compared to matter. The manuscript may be publishable in SciPost Physics Proceedings if the author would fully consider the following points:

1. The meaning of the term "scale" is not clear. The quantization of matter takes place on the energy level; the quantization of gravity, if possible, would involve a geometrical scale. These two scales are not overlapping, and they are distinct, unless one introduces an energy scale for gravity. The meaning of "scale" in the present approach must be clarified.

The 'scale’ referred to here is an action scale. In other words, the fundamental quantum of action for geometry in this model is $\beta = \sqrt{\rho_{\Lambda}/\rho_{\rm Pl}} \simeq 10^{-61} \hbar$, as in Eq. (3.3).

We note that the canonical quantisation of matter is also based on the adoption of a characteristic action scale, namely $\hbar$. Wave-particle duality is consistent with Poincare invariance in the relativistic limit and, hence, with Galilean invariance in the non-relativisitc limit, if and only if $\vec{p} \propto \vec{k}$ and $E \propto \omega$, as noted by Weinberg in his seminal work on QFT. Ultimately, it is the constant of proportionality, $\hbar$, that determines the length and momentum (energy) scales at which quantum effects become important for material bodies.

Similarly, in the model of nonlocal geometry considered here, $\beta << \hbar$ sets the length and momentum (energy) scales at which quantum effects become important for the background geometry, in which the canonical quantum matter propagates.

These issues are treated in detail in refs. [15], [17], [23] and [24] of the text. However, due to the limited time afforded to the conference talk on which this submission is based, and to the limited space afforded to the conference proceedings, it was not possible to elaborate on them more fully in this context.

2. How does the author defines a point?

Classical points are defined, where necessary, as in standard differential geometry. However, the model considered here is not based on classical points or on the fixed manifolds that form the mathematical basis of classical spacetimes.

Instead, we associate each point in the classical background, labeled by $\vec{x}$, with a vector in a Hilbert space, $\langle g_{\vec{x}}\rangle$. The associated wave function, $\langle\vec{x}|g_{\vec{x}}\rangle = g(\vec{x}’-\vec{x})$, may be regarded as a Gaussian of width $\sigma_g \simeq l_{\rm Pl}$. This represents the quantum state of a delocalized `point’ in the quantum geometry, but this term is used here in an imprecise sense, only for illustration (hence the inverted commas).

3. Usually one can associate to a point the value of a function or of an operator, and not the function itself, so that in the point x_0 we may consider the couple (x_0,f(x_0)), where f is an arbitrary function or operator, but not the couple (x_0,f(x)), where x takes all real (or complex) values, let's say. The procedures of associating entire functions (operators), be they delta or of other type, to points, must be clearly explained.

In this model, a function is associated to each point by doubling the degrees of freedom in the classical phase space. Hence, the classical point labeled by $\vec{x}$ is associated with the quantum probability amplitude, $g(\vec{x}’-\vec{x})$. This is the mathematical representation of a `delocalized point’ in the nonlocal geometry, as discussed above.

For each $\vec{x}$, the additional variable $\vec{x}’$ may take any value in $\mathbb{R}^3$. Together, $\vec{x}$ and $\vec{x}’$ cover $\mathbb{R}^6$, which is interpreted as a superposition of 3D Euclidean spaces.

However, the smearing process is easiest to visualize in the case of a toy one-dimensional universe. In this case, the original classical geometry is the x-axis and the $(x,x’)$ plane on which the function $g(x’-x)$ is defined represents the smeared superposition of geometries.

Again, these issues are considered in detail in the refs. [15], [17], [23] and [24], but are not discussed at length in the present article for want of space. (In particular, see Fig. 1 of ref. [15].)

4. The author also mention "geometry wave"(s), to which the rules of standard quantum mechanics are applied, without much justification. Using Eq. (2) for the "geometry wave"(s) is at least problematic. One must make clear what is the meaning of geometry wave in the present context, and why should they obey exactly the rules of quantum mechanics for matter.

Eq. (2.1) follows directly from the standard properties of the Fourier transform, which is applied to the function $g(\vec{x}’-\vec{x})$ at the scale $\beta$, just as one applies the Fourier transform to the function $\psi(\vec{x})$ at the scale $\hbar$ in canonical QM, in order to obtain the momentum space representation of the theory.

In this sense, delocalized spatial 'points' in our model are analogous to delocalized point-particles (i.e., wave functions) in canonical QM. However, the former are not assumed to follow the rules of standard quantum theory per se. There are a number of subtle differences between the quantum treatments of matter and geometry, which is what allows the model to evade the standard no go theorems regarding multiple quantisation constants [16].

The meaning of 'geometry wave’, here, is none other than this: that the 'wave function’ of a delocalized spatial point, $g(\vec{x}’-\vec{x})$, can be expanded in terms of both Dirac deltas and plane waves. The latter are analogous to the plane waves used to expand the wave function $\psi(\vec{x})$ in canonical QM, but with the canonical quantisation scale $\hbar$ replaced by $\beta$.

5. The "quantum of action for geometry", given by Eq. (6), is an extremely small quantity, which looks very difficult to be measured directly, or even indirectly. Still the authors may suggest some other observational effects that could provide, at least in principle, some (realistic) observational signatures for β.

If this model is correct, then the presence of a vacuum energy density of order $\rho_{\Lambda} = \Lambda c^2/(8\pi G)$ is already an observational signature of the geometry quantisation scale $\beta$. This is discussed briefly in Sec. 4, but is clearly of post-diction, not a prediction, of the model.

However, that is not all. As also discussed in Sec. 4, the model suggests that the true dark energy field optimizes the generalized uncertainty relations (3.2), which leads to granularity (i.e., small-scale fluctuations in the effective strength of the gravitational field) on scales of order 0.1 mm. At present, there is tentative (2$\sigma$) evidence that such fluctuations have already been observed [21,22], but more data is needed to either confirm or rule out this possibility.