Toolkit for scalar fields in universes with finite-dimensional Hilbert space

Let us first recap the conventional, infinite-dimensional construction of a real scalar field on a curved spacetime, with the action

$$\begin{equation}S=\frac{1}{2}\int \sqrt{-g}\,{\mathrm{d}}^{4}x\left[{g}^{\alpha \beta }{\phi }_{,\alpha }{\phi }_{,\beta }-{m}^{2}{\phi }^{2}\right],\end{equation} \tag{ 2.1 }$$

where g is the determinant, and g^αβ are the components of the inverse of the metric tensor. In a flat Friedmann Universe, and using the conformal form of the metric

$$\begin{equation}\mathrm{d}{s}^{2}={a}^{2}\left(\mathrm{d}{\eta }^{2}-\mathrm{d}{\boldsymbol{x}}^{2}\right),\qquad {[\cdot ]}^{\prime }\equiv \frac{\partial }{\partial \eta }[\cdot ],\end{equation} \tag{ 2.2 }$$

this becomes

$$\begin{align}\hfill S& =\frac{1}{2}\int \mathrm{d}\eta \,{\mathrm{d}}^{3}x{a}^{2}\left[{({\phi }^{\prime })}^{2}-{(\nabla \phi )}^{2}-{m}^{2}{a}^{2}{\phi }^{2}\right]\hfill \\ \hfill & =\frac{1}{2}\int \frac{\mathrm{d}\eta \,{\mathrm{d}}^{3}k}{{(2\pi )}^{3}}{a}^{2}\left[\vert {\phi }_{\boldsymbol{k}}^{\prime }{\vert }^{2}-(\vert \boldsymbol{k}{\vert }^{2}+{m}^{2}{a}^{2})\vert {\phi }_{\boldsymbol{k}}{\vert }^{2}\right],\hfill \end{align} \tag{ 2.3 }$$

where in the second line we moved to Fourier space, with k labelling a co-moving Fourier mode. Expressing the Fourier transform of the field in terms of real and imaginary parts, ϕ _k = A _k + iB _k , we must have A _k = A_{− k} and B _k = −B_{− k} because ϕ is real. This allows us to define a new field

$$\begin{equation}{q}_{\boldsymbol{k}}=\sqrt{2}\left\{\begin{matrix}\hfill {A}_{\boldsymbol{k}}\quad \mathrm{f}\mathrm{o}\mathrm{r}\ {k}_{1}\leqslant 0\hfill \\ \hfill {B}_{\boldsymbol{k}}\quad \mathrm{f}\mathrm{o}\mathrm{r}\ {k}_{1} > 0\hfill \end{matrix}\right.\end{equation} \tag{ 2.4 }$$

such that the action can be re-written as [39]

$$\begin{equation}S=\int \frac{\mathrm{d}\eta \,{\mathrm{d}}^{3}k}{{(2\pi )}^{3}}\left[\frac{{a}^{2}}{2}{({q}_{\boldsymbol{k}}^{\prime })}^{2}-\frac{{a}^{2}(\vert \boldsymbol{k}{\vert }^{2}+{m}^{2}{a}^{2})}{2}{q}_{\boldsymbol{k}}^{2}\right].\end{equation} \tag{ 2.5 }$$

This can be interpreted as an action corresponding to a collection of harmonic oscillators with time dependent mass a² and time dependent frequency $\sqrt{\vert \boldsymbol{k}{\vert }^{2}+{m}^{2}{a}^{2}}$ [38, 39]. To make this analogy more explicit, let us restrict the field ϕ to a finite box of co-moving side length L_c, imposing periodic boundary conditions. This modifies the action to

$$\begin{equation}{S}_{\text{box}}=\int \mathrm{d}\eta \frac{1}{{L}_{\mathrm{c}}^{3}}\sum\limits _{\boldsymbol{k}}\left[\frac{{a}^{2}}{2}{({q}_{\boldsymbol{k}}^{\prime })}^{2}-\frac{{a}^{2}(\vert \boldsymbol{k}{\vert }^{2}+{m}^{2}{a}^{2})}{2}{q}_{\boldsymbol{k}}^{2}\right],\end{equation} \tag{ 2.6 }$$

where we have used the fact that ${\mathrm{d}}^{3}k\to {\Delta}{k}^{3}={(2\pi /{L}_{\mathrm{c}})}^{3}$ and the sum is over all k = (k₁, k₂, k₃) with ${k}_{i}\in \left\{2\pi n/{L}_{\mathrm{c}}\ \vert \ n\in \mathbb{Z}\right\}$. In order to extract the Hamiltonian from that action, let us re-write it in terms of physical time dt = a dη (a dot over a quantity represents its derivative with respect to physical time), i.e.

$$\begin{align}\hfill {S}_{\text{box}}& =\int \mathrm{d}t\frac{{a}^{3}}{{L}_{\mathrm{c}}^{3}}\sum\limits _{\boldsymbol{k}}\left[\frac{1}{2}{({\dot{q}}_{\boldsymbol{k}})}^{2}-\frac{(\vert \boldsymbol{k}{\vert }^{2}/{a}^{2}+{m}^{2})}{2}{q}_{\boldsymbol{k}}^{2}\right]\equiv \int \mathrm{d}t{L}_{\text{box}}\left(\left\{{q}_{\boldsymbol{k}}\right\},\left\{{\dot{q}}_{\boldsymbol{k}}\right\},t\right).\hfill \end{align} \tag{ 2.7 }$$

Here the last equality serves as a definition of the Lagrangian L_box of the discretized field. It is literally the Lagrangian of a set of harmonic oscillators with masses ${a}^{3}/{L}_{\mathrm{c}}^{3}$ and frequencies $\sqrt{\vert \boldsymbol{k}{\vert }^{2}/{a}^{2}+{m}^{2}}$. The corresponding Hamiltonian is given by

$$\begin{equation}{H}_{\text{box}}\left(\left\{{q}_{\boldsymbol{k}}\right\},\left\{{p}_{\boldsymbol{k}}\right\},t\right)=\sum\limits _{\boldsymbol{k}}\left[\frac{{L}_{\mathrm{c}}^{3}}{2{a}^{3}}{p}_{\boldsymbol{k}}^{2}+\frac{{a}^{3}}{{L}_{\mathrm{c}}^{3}}\frac{(\vert \boldsymbol{k}{\vert }^{2}/{a}^{2}+{m}^{2})}{2}{q}_{\boldsymbol{k}}^{2}\right],\end{equation} \tag{ 2.8 }$$

where we introduced the conjugate momenta ${p}_{\boldsymbol{k}}=\partial {L}_{\text{box}}/\partial {\dot{q}}_{\boldsymbol{k}}$. To obtain the quantum theory of this field one would usually promote q _k and p _k to conjugate Hermitian operators satisfying the Heisenberg commutation relations

$$\begin{equation}[{\hat{q}}_{\boldsymbol{k}},{\hat{p}}_{{\boldsymbol{k}}^{\prime }}]=\mathrm{i}{\delta }_{\boldsymbol{k},{\boldsymbol{k}}^{\prime }}\end{equation} \tag{ 2.9 }$$

such that the Hamiltonian operator of the field becomes

$$\begin{equation}\hat{H}(t)=\sum\limits _{\boldsymbol{k}}\left[\frac{{L}_{\mathrm{c}}^{3}}{2{a}^{3}}{\hat{p}}_{\boldsymbol{k}}^{2}+\frac{{a}^{3}}{{L}_{\mathrm{c}}^{3}}\frac{(\vert \boldsymbol{k}{\vert }^{2}/{a}^{2}+{m}^{2})}{2}{\hat{q}}_{\boldsymbol{k}}^{2}\right]\end{equation} \tag{ 2.10 }$$

which at any time t has the minimum eigenvalue

$$\begin{equation}{\lambda }_{\mathrm{min}}\left[\hat{H}(t)\right]=\frac{1}{a}\sum\limits _{\vert \boldsymbol{k}\vert < {k}_{\mathrm{max}}}\frac{\sqrt{\vert \boldsymbol{k}{\vert }^{2}+{m}^{2}{a}^{2}}}{2}.\end{equation} \tag{ 2.11 }$$

Here we have introduced a co-moving ultra-violet cut-off k_max in order to regularise this otherwise divergent expression. Such a sharp cut-off has been criticized because it breaks Lorentz symmetry [1, 36]. There are however reasons to believe that the breaking of Lorentz symmetry is physical [5, 28, 37], including the premise of this paper: finite-dimensionality of Hilbert space.

To obtain the vacuum energy density of the field we need to divide this eigenvalue by the physical volume of the box, i.e. by

$$\begin{equation}{V}_{\text{ph}}={L}_{\text{ph}}^{3}\equiv {(a{L}_{\mathrm{c}})}^{3},\end{equation} \tag{ 2.12 }$$

where L_ph = aL is the physical box size. The energy density of the vacuum is then given by

$$\begin{equation}{{\epsilon}}_{\text{vac}}=\frac{1}{{a}^{4}{L}_{\mathrm{c}}^{3}}\sum\limits _{\vert \boldsymbol{k}\vert < {k}_{\mathrm{max}}}\frac{\sqrt{\vert \boldsymbol{k}{\vert }^{2}+{m}^{2}{a}^{2}}}{2}.\end{equation} \tag{ 2.13 }$$

For a constant co-moving box size L_c this seems to indicate that _vac ∝ a⁻⁴, which is the behaviour of a relativistic fluid, and not that of a cosmological constant. However, it is usually assumed that the scale regularising a QFT is some fixed physical scale Λ_UV, which for the rest of this paper we will take to be equal to the Planck mass. In an expanding Universe we would then have k_max = aΛ_UV, such that equation (2.13) becomes

$$\begin{equation}{{\epsilon}}_{\text{vac}}=\frac{1}{{a}^{4}{L}_{\mathrm{c}}^{3}}\sum\limits _{\vert \boldsymbol{k}\vert < a{{\Lambda}}_{\text{UV}}}\frac{\sqrt{\vert \boldsymbol{k}{\vert }^{2}+{m}^{2}{a}^{2}}}{2}.\end{equation} \tag{ 2.14 }$$

For Λ_UV ≫ m the sum in this expression is proportional to a⁴, so that _vac is indeed constant. This is still somewhat curious, because a direct calculation of the vacuum pressure p_vac from the vacuum stress–energy tensor yields p_vac ≈ _vac/3 [1], which is again the behaviour of a relativistic fluid. Note however, that the number of modes k over which we sum in equation (2.14) is now itself a function of time, and that this compensates for the energy loss that a relativistic fluid would experience in an expanding Universe [37, 43]. This can be interpreted in terms of a modified continuity equation for the vacuum energy density [43].

We want to stress an important subtlety: since the Hamiltonian in equation (2.10) is time dependent, it is not possible for the field to remain in a state of minimum energy. Instead, each of the q _k behaves as a driven harmonic oscillator and the expansion of the Universe will inevitably lead to particle production [38, 39]. If the period of the oscillators around the cut-off Λ_UV is much smaller than the characteristic time scales over which a changes, then particle production will be negligible and the quantum state will undergo adiabatic evolution, i.e., stay in the instantaneous minimum energy eigenstate to a good approximation as time evolves. We will employ this adiabaticity assumption for the remainder of this paper. The assumption is well justified in late-time cosmology because the time scales relevant for the recent cosmic expansion history are much longer than the period of any cut-off scale that is relevant to well understood particle physics.

We now return to our premise that the dimension of the Hilbert space of the observable Universe should be finite. In this case, the dimensions of the Hilbert spaces corresponding to individual modes k also need to be finite. In the conventional infinite-dimensional setting, such as non-relativistic quantum mechanics of a single particle, classical conjugate variables q and p are promoted to Hermitian Hilbert space operators which obey the Heisenberg canonical commutation relation (CCR)

$$\begin{equation}[\hat{q},\hat{p}]=i,\end{equation} \tag{ 2.15 }$$

where we have set ℏ = 1. In a quantum field theory, the field and its conjugate momentum are operator-valued functions on spacetime which obey a continuous version of the CCR, labelled by the field modes, as done in the previous section. The Stone–von Neumann theorem guarantees that there is an irreducible representation of the Heisenberg CCR, but only on separable (i.e., that possesses a countable dense subset) and in particular infinite-dimensional Hilbert spaces [34]. The theorem also implies that this representation is unique and that the operators $\hat{q}$ and $\hat{p}$ must have an unbounded spectrum of eigenvalues, that is, the eigenvalues will either no upper and/or lower limit. By extension, the Stone–von Neumann theorem then implies that, on finite-dimensional spaces, there are no irreducible representations of equation (2.9), and one needs to consider a more general algebraic structure than the one imposed by Heisenberg's CCR.

Before we switch to a finite-dimensional construction, let us define convenient, dimensionless versions of our conjugate variables as

$$\begin{equation}{Q}_{\boldsymbol{k}}\equiv {q}_{\boldsymbol{k}}/{L}_{\mathrm{c}}^{2},\qquad {P}_{\boldsymbol{k}}\equiv {p}_{\boldsymbol{k}}{L}_{\mathrm{c}}^{2}.\end{equation} \tag{ 2.16 }$$

We would like to promote these to finite-dimensional, Hermitian operators ${\hat{Q}}_{\boldsymbol{k}},{\hat{P}}_{\boldsymbol{k}}$, which satisfy traditional properties of conjugate variables while also approaching the Heisenberg CCR in the large dimensional limit. In order to achieve this we follow the ansatz of [11, 42] and model ${\hat{Q}}_{\boldsymbol{k}},{\hat{P}}_{\boldsymbol{k}}$ in terms of generalized Pauli operators ${\hat{A}}_{\boldsymbol{k}},{\hat{B}}_{\boldsymbol{k}}$ GPOs as

$$\begin{equation}{\hat{A}}_{\boldsymbol{k}}=\mathrm{exp}\left(-\mathrm{i}{\alpha }_{\boldsymbol{k}}{\hat{P}}_{\boldsymbol{k}}\right),\qquad {\hat{B}}_{\boldsymbol{k}}=\mathrm{exp}\left(\mathrm{i}{\beta }_{\boldsymbol{k}}{\hat{Q}}_{\boldsymbol{k}}\right),\end{equation} \tag{ 2.17 }$$

which are defined on a Hilbert space of finite dimension d _k and satisfy the Weyl commutation relation [48]

$$\begin{equation}{\hat{A}}_{\boldsymbol{k}}{\hat{B}}_{\boldsymbol{k}}=\mathrm{exp}\left(\frac{-2\pi \mathrm{i}}{{d}_{\boldsymbol{k}}}\right){\hat{B}}_{\boldsymbol{k}}{\hat{A}}_{\boldsymbol{k}},\end{equation} \tag{ 2.18 }$$

and the closure property ${\hat{A}}_{\boldsymbol{k}}^{{d}_{\boldsymbol{k}}}={\mathbb{I}}_{{d}_{\boldsymbol{k}}}={\hat{B}}_{\boldsymbol{k}}^{{d}_{\boldsymbol{k}}}$, where ${\mathbb{I}}_{{d}_{\boldsymbol{k}}}$ is the identity operator on the Hilbert space of dimension d _k . Equation (2.18) above is an exponentiated form of Heisenberg's CCR in the sense that when the real parameters, α _k and β _k satisfy

$$\begin{equation}{\alpha }_{\boldsymbol{k}}{\beta }_{\boldsymbol{k}}=\frac{2\pi }{{d}_{\boldsymbol{k}}},\end{equation} \tag{ 2.19 }$$

then equation (2.18) is equivalent to equation (2.9) in the limit d _k → ∞. The operators ${\hat{Q}}_{\boldsymbol{k}}$ and ${\hat{P}}_{\boldsymbol{k}}$ defined through equations (2.17) and (2.18) do indeed admit a unitary representation on a Hilbert space with finite dimension d _k . Moreover, the representation is still unique up to unitary equivalence via the Stone–von Neumann theorem, since a finite-dimensional Hilbert space is separable. For example, let the dimension of Hilbert space be d _k = 2l _k + 1 for some non-negative integer l _k . Then the GPOs have the following matrix representation (up to unitary equivalence)

$$\begin{align}\hfill & {\hat{A}}_{\boldsymbol{k}}={\left(\begin{matrix}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \dots & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\vdots}\hfill & \hfill {\vdots}\hfill & \hfill {\vdots}\hfill & \hfill \ddots \hfill & \hfill {\vdots}\hfill & \hfill {\vdots}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots & \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right)}_{{d}_{\boldsymbol{k}}\times {d}_{\boldsymbol{k}}}\qquad \hfill \\ \hfill & {\hat{B}}_{\boldsymbol{k}}={\left(\begin{matrix}\hfill \mathrm{exp}\left(\frac{2\pi \mathrm{i}}{{d}_{\boldsymbol{k}}}{l}_{\boldsymbol{k}}\right)\hfill & \hfill 0\hfill & \hfill \dots & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathrm{exp}\left(\frac{2\pi \mathrm{i}}{{d}_{\boldsymbol{k}}}({l}_{\boldsymbol{k}}-1)\right)\hfill & \hfill \dots & \hfill 0\hfill \\ \hfill {\vdots}\hfill & \hfill {\vdots}\hfill & \hfill \ddots \hfill & \hfill {\vdots}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots & \hfill \mathrm{exp}\left(\frac{-2\pi \mathrm{i}}{{d}_{\boldsymbol{k}}}{l}_{\boldsymbol{k}}\right)\hfill \end{matrix}\right)}_{{d}_{\boldsymbol{k}}\times {d}_{\boldsymbol{k}}},\hfill \end{align} \tag{ 2.20 }$$

which indeed satisfy equation (2.18). The construction works for even dimensions as well, but we focus on odd values to streamline the notation. It can then be shown [42] that the operators ${\hat{Q}}_{\boldsymbol{k}}$ and ${\hat{P}}_{\boldsymbol{k}}$ (defined from ${\hat{A}}_{\boldsymbol{k}}$ and ${\hat{B}}_{\boldsymbol{k}}$ via equation (2.17)) have bounded, discrete and linearly-spaced spectra which are given by

$$\begin{equation}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}({\hat{Q}}_{\boldsymbol{k}})=\left\{-{\ell }_{\boldsymbol{k}}{\alpha }_{\boldsymbol{k}},\dots ,{\ell }_{\boldsymbol{k}}{\alpha }_{\boldsymbol{k}}\right\};\qquad \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}({\hat{P}}_{\boldsymbol{k}})=\left\{-{\ell }_{\boldsymbol{k}}{\beta }_{\boldsymbol{k}},\dots ,{\ell }_{\boldsymbol{k}}{\beta }_{\boldsymbol{k}}\right\},\end{equation} \tag{ 2.21 }$$

i.e. the eigenvalue spacings of the two operators are given by α _k and β _k respectively. It can be shown that in the limit d _k → ∞ the commutator of ${\hat{Q}}_{\boldsymbol{k}}$ and ${\hat{P}}_{\boldsymbol{k}}$ indeed approaches the infinite-dimensional CCR [42].

We would now like to quantize the Hamiltonian of equation (2.8) in terms of these finite-dimensional operators. Taking into account the re-scaling from equation (2.16), the Hamiltonian operator becomes

$$\begin{align}\hfill \hat{H}& =\sum\limits _{\vert \boldsymbol{k}\vert < a{{\Lambda}}_{\text{UV}}}\left[\frac{{\hat{P}}_{\boldsymbol{k}}^{2}}{2{a}^{3}{L}_{\mathrm{c}}}+\frac{{a}^{3}{L}_{\mathrm{c}}(\vert \boldsymbol{k}{\vert }^{2}/{a}^{2}+{m}^{2})}{2}{\hat{Q}}_{\boldsymbol{k}}^{2}\right]\equiv \sum\limits _{\vert \boldsymbol{k}\vert < a{{\Lambda}}_{\text{UV}}}\left[\frac{{\hat{P}}_{\boldsymbol{k}}^{2}}{2M}+\frac{M{{\Omega}}_{\boldsymbol{k}}^{2}}{2}{\hat{Q}}_{\boldsymbol{k}}^{2}\right],\hfill \end{align} \tag{ 2.22 }$$

where we have defined M = a³ L_c and ${{\Omega}}_{\boldsymbol{k}}=\sqrt{\vert \boldsymbol{k}{\vert }^{2}/{a}^{2}+{m}^{2}}$ to make each individual mode formally resemble a standard quantum harmonic oscillator with time dependent mass M and time dependent frequency Ω _k . To determine the energy spectrum of this finite-dimensional constructions, we need to fix two ingredients: the dimension d _k of the Hilbert space of each mode k , and the spacing α _k of the eigenvalues of ${\hat{Q}}_{\boldsymbol{k}}$ (which via equation (2.19) also fixes the eigenvalue spacing of ${\hat{P}}_{\boldsymbol{k}}$).

As a proof of concept [42], have considered the situation where ${\alpha }_{\boldsymbol{k}}={\beta }_{\boldsymbol{k}}={(2\pi /{d}_{\boldsymbol{k}})}^{1/2}$. We investigate the impact of that choice on the vacuum energy of our scalar field in section 4.2, but for our fiducial construction, we opt for a different approach to selecting α _k and β _k . Recall that because of equation (2.19) any choice of α _k already fixes β _k . Hence, for any given values of d _k , Ω _k and M, the minimum energy E_{min, k} of the mode k only depends on α _k . Now to fix a choice of α _k , consider the time t _k when the mode k enters the sum of equation (2.22), that is, when | k | = a(t _k )Λ_UV. This is the time when the mode k is initialized, and we are going to choose α _k such that it maximises E_{min, k} at that time,

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}{\alpha }_{\boldsymbol{k}}}{E}_{\mathrm{min},\boldsymbol{k}}({t}_{\boldsymbol{k}})=0.\end{equation} \tag{ 2.23 }$$

As we show in appendix A, the eigenvalue spacings that satisfy this criterion are exactly given by

$$\begin{equation}{\alpha }_{\boldsymbol{k}}=\sqrt{\frac{2\pi }{{d}_{\boldsymbol{k}}M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}},\qquad {\beta }_{\boldsymbol{k}}=\sqrt{\frac{2\pi M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}{{d}_{\boldsymbol{k}}}}.\end{equation} \tag{ 2.24 }$$

It can be shown that finite-dimensionality can only decrease E_{min, k} compared to its infinite-dimensional value (cf [42] or our appendix B). Hence, the above choice for α _k and β _k minimizes finite-dimensional effects on the low energy spectrum of the Hamiltonian at the time t _k when the mode k is initialised. Since both M and Ω _k are functions of time, the Hamiltonian of each mode will eventually move away from that sweet spot. But as long as the vacuum energy of our field is dominated by UV modes, for which t _k ≈ t_today, our fiducial construction can be considered as conservative w.r.t. finite-dimensional effects.

The eigenvalue spacings of equation (2.24) can also be motivated from a different, but related point of view. The operators ${\hat{Q}}_{\boldsymbol{k}}$ and ${\hat{P}}_{\boldsymbol{k}}$ start to contribute to our scalar field and its conjugate momentum field at t _k , i.e. at the time when the mode k starts to enter the sum in equation (2.22). Let us assume that at this moment the sub-system corresponding to mode k is initialised in its instantaneous ground state which we denote by |0( k , t _k )⟩. We would like our construction to resemble the infinite-dimensional limit as closely as possible at that time of initialization. To achieve this, we employ a 'resolution criterion:' we demand that the system at t _k should have the same resolution in 'position'- and 'momentum'-space, i.e.

$$\begin{equation}\frac{\langle 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\vert {\hat{Q}}_{\boldsymbol{k}}^{2}\vert 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\rangle }{{\alpha }_{\boldsymbol{k}}^{2}}=\frac{\langle 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\vert {\hat{P}}_{\boldsymbol{k}}^{2}\vert 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\rangle }{{\beta }_{\boldsymbol{k}}^{2}}\end{equation} \tag{ 2.25 }$$

$$\begin{align}\hfill {\Rightarrow}\frac{\langle 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\vert {\hat{Q}}_{\boldsymbol{k}}^{2}\vert 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\rangle }{{\alpha }_{\boldsymbol{k}}^{4}}& =\frac{\langle 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\vert {\hat{P}}_{\boldsymbol{k}}^{2}\vert 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\rangle }{{(2\pi )}^{2}}{d}_{\boldsymbol{k}}^{2}\hfill \\ \hfill {\Rightarrow}{\alpha }_{\boldsymbol{k}}^{4}& =\frac{\langle 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\vert {\hat{Q}}_{\boldsymbol{k}}^{2}\vert 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\rangle }{\langle 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\vert {\hat{P}}_{\boldsymbol{k}}^{2}\vert 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\rangle }\frac{{(2\pi )}^{2}}{{d}_{\boldsymbol{k}}^{2}}.\hfill \end{align} \tag{ 2.26 }$$

This criterion ensures that the spread of |0( k , t _k )⟩ in the eigenbasis of ${\hat{Q}}_{\boldsymbol{k}}$ is equally well resolved by the eigenvalue spacing of ${\hat{Q}}_{\boldsymbol{k}}$ as the spread of |0( k , t _k )⟩ in the eigenbasis of ${\hat{P}}_{\boldsymbol{k}}$ by the eigenvalue spacing of ${\hat{P}}_{\boldsymbol{k}}$. If this was not case, then even a seemingly high resolution in ${\hat{Q}}_{\boldsymbol{k}}$-space could easily be identified as deviating from infinite-dimensional behaviour in ${\hat{P}}_{\boldsymbol{k}}$-space. This is also in line with the infinite-dimensional case where both conjugate variables are equally well resolved by construction, albeit infinitely well resolved.

To the best of our knowledge, in the finite-dimensional case, no closed form expressions for the quadratic expectation values of ${\hat{Q}}_{\boldsymbol{k}}$ and ${\hat{P}}_{\boldsymbol{k}}$ are available. We can however attempt to approximate them by the corresponding expectation values of an infinite-dimensional oscillator, which when combined with equation (2.25), results in

$$\begin{align}\hfill \langle 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\vert {\hat{Q}}_{\boldsymbol{k}}^{2}\vert 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\rangle & \approx \frac{1}{2M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})},\hfill \\ \hfill \langle 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\vert {\hat{P}}_{\boldsymbol{k}}^{2}\vert 0(\boldsymbol{k},{t}_{\boldsymbol{k}})\rangle & \approx \frac{M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}{2}\hfill \\ \hfill {\Rightarrow}{\alpha }_{\boldsymbol{k}}& \approx \sqrt{\frac{2\pi }{{d}_{\boldsymbol{k}}M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}},\qquad {\beta }_{\boldsymbol{k}}\approx \sqrt{\frac{2\pi M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}{{d}_{\boldsymbol{k}}}}.\hfill \end{align} \tag{ 2.27 }$$

This is indeed equivalent to equation (2.24). We consider this as further demonstration that our construction is conservative and minimizes finite-dimensional effects.

We show in appendix B that with the above choice for α _k , β _k , the minimum energy eigenvalue of the finite-dimensional harmonic oscillators at any time t becomes

$$\begin{equation}{E}_{\mathrm{min},\boldsymbol{k}}(t)\approx \frac{{{\Omega}}_{\boldsymbol{k}}(t)}{2}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}{12M(t){{\Omega}}_{\boldsymbol{k}}(t)}{d}_{\boldsymbol{k}}\right),\end{equation} \tag{ 2.28 }$$

which approximates the exact results of [42], making them more amenable for numerical implementation at high dimensions d _k . For M(t)Ω _k (t) ≫ M(t _k )Ω _k (t _k ) the right-hand side of equation (2.28) can significantly deviate from the infinite-dimensional result E_{min, k} = Ω _k (t)/2. The minimum eigenvalue of the total Hamiltonian is then

$$\begin{equation}{E}_{\mathrm{min}}(t)\approx \sum\limits _{\vert \boldsymbol{k}\vert < a(t){{\Lambda}}_{\text{UV}}}\frac{{{\Omega}}_{\boldsymbol{k}}(t)}{2}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}{12M(t){{\Omega}}_{\boldsymbol{k}}(t)}{d}_{\boldsymbol{k}}\right)\end{equation} \tag{ 2.29 }$$

and the corresponding vacuum energy density is

$$\begin{align}\hfill {{\epsilon}}_{\text{vac}}(t)& \approx \frac{1}{a{(t)}^{3}{L}_{\mathrm{c}}^{3}}\sum\limits _{\vert \boldsymbol{k}\vert < a(t){{\Lambda}}_{\text{UV}}}\frac{{{\Omega}}_{\boldsymbol{k}}(t)}{2}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}{12M(t){{\Omega}}_{\boldsymbol{k}}(t)}{d}_{\boldsymbol{k}}\right)\hfill \\ \hfill & \approx \ \frac{1}{a{(t)}^{4}}\underset{\vert \boldsymbol{k}\vert < a(t){{\Lambda}}_{\text{UV}}}{\int }\frac{{\mathrm{d}}^{3}\boldsymbol{k}}{{(2\pi )}^{3}}\ \frac{\sqrt{\vert \boldsymbol{k}{\vert }^{2}+{m}^{2}a{(t)}^{2}}}{2}\hfill \\ \hfill & \quad \times \mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}M({t}_{\boldsymbol{k}}){{\Omega}}_{\boldsymbol{k}}({t}_{\boldsymbol{k}})}{12M(t){{\Omega}}_{\boldsymbol{k}}(t)}{d}_{\boldsymbol{k}}\right).\hfill \end{align} \tag{ 2.30 }$$

If this integral is dominated by high k ≡ | k | ≈ aΛ_UV, and if the mass m of the field is much smaller than Λ_UV (as must be the case for all standard model particles), then the frequency of the modes that dominate vacuum energy is given by Ω _k ≈ k/a and we can further approximate _vac by

$$\begin{align}\hfill {{\epsilon}}_{\text{vac}}(t)\approx & \frac{1}{a{(t)}^{4}}\underset{k< a(t){{\Lambda}}_{\text{UV}}}{\int }\frac{\mathrm{d}k{k}^{3}}{{(2\pi )}^{2}}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}M({t}_{\boldsymbol{k}})a(t)}{12M(t)a({t}_{\boldsymbol{k}})}{d}_{\boldsymbol{k}}\right)\hfill \\ \hfill & =\underset{{k}_{\text{ph}}< {{\Lambda}}_{\text{UV}}}{\int }\frac{\mathrm{d}{k}_{\text{ph}}{k}_{\text{ph}}^{3}}{{(2\pi )}^{2}}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}M({t}_{a(t){k}_{\text{ph}}})a(t)}{12M(t)a({t}_{a(t){k}_{\text{ph}}})}{d}_{a(t){k}_{\text{ph}}}\right),\hfill \end{align} \tag{ 2.31 }$$

where k_ph is now physical (as opposed to co-moving) wave number. Without the error function in the integrand of this expression, this would be the standard result for vacuum energy density of a scalar field with a hard UV cut-off (cf equation (2.14)). In section 4 we will see that this modification can significantly suppress _vac(t) in a time-dependent way.

The final ingredient we are missing in order to evaluate the above result for _vac(t) is the dimension d _k of the mode Hilbert spaces. It was motivated by [11] that d _k ≲ 1/πk² should be an upper bound for this dimension, based on the requirement that the maximum energy in each mode should be smaller than the Schwarzschild energy of the Universe. They however argue that this is a rather loose bound, and furthermore, their derivation was carried out with a choice for the eigenvalue spacings α _k , β _k that differs from our construction. In the following, we will use an agnostic parametrisation of the form

$$\begin{equation}{d}_{\boldsymbol{k}}=D{(k/{{\Lambda}}_{\text{UV}})}^{{n}_{D}}+{d}_{\mathrm{min}},\end{equation} \tag{ 2.32 }$$

where we take D > 0 and d_min = 2 to ensure that every mode is initialised with at least the Hilbert space of a qubit. Division by the physical cut-off Λ_UV has been introduced to keep D a dimensionless parameter, but note that k is still co-moving wave number. Note also that equation (2.32) should be interpreted as an approximation to what should actually be a function with discrete, integer values. In sections 3 and 4 we investigate how different values of D and n_D impact the behaviour of our finite-dimensional field and how demanding consistency with low energy physics can constrain the D − n_D space.

In the previous subsections we have quantized our field in a box of finite physical side length L_ph = aL_c. In the following we want to interpret this length as approximating the size of the observable Universe. This is a strong simplification since we would expect any meaningful boundary of the Universe to be spherical. To account for such a spherical geometry we would in principle have to change the way we discretized the field. Instead of a decomposition in terms of Fourier modes we would e.g. need to expand the field in terms of 3D Zernicke polynomials [32]. We do not expect such a procedure to qualitatively change the results of the remainder of this paper, and we leave more rigorous investigation of field discretization to follow-up work. For now, let us simply identify L_ph with the radius of a spherically bounded Universe.

What should we choose this radius to be at any given time? As we argued in section 1, our classical notion of spacetime is likely to emerge from an underlying quantum theory [9, 10, 13, 14, 18, 46], and the correct choice of L_ph should be informed by that emergence map. To work out this map is far beyond the scope of this work, but we can at least guess a number of candidate scales. We could e.g. choose the particle horizon

$$\begin{equation}{L}_{\mathrm{p}}(t)=a(t){\int }_{{t}_{i}}^{t}\frac{\mathrm{d}{t}^{\prime }}{a({t}^{\prime })}\equiv a(t)[\eta (t)-{\eta }_{i}],\end{equation} \tag{ 2.33 }$$

which defines the volume about which an observer can have information at time t. Other natural choices would be the curvature scale

$$\begin{equation}{L}_{R}=\frac{1}{\sqrt{R}},\end{equation} \tag{ 2.34 }$$

where R is the Ricci scalar, or the Hubble scale

$$\begin{equation}{L}_{H}=\frac{1}{H}.\end{equation} \tag{ 2.35 }$$

We could also assume that the Universe has some constant co-moving size L_c and that its physical size grows proportional to the scale factor, i.e.

$$\begin{equation}{L}_{a}=a{L}_{\mathrm{c}}.\end{equation} \tag{ 2.36 }$$

As a proof of concept, and following [4, 11, 43], we will indeed consider that last choice, with the constant co-moving IR scale L_c given by the asymptotic, co-moving particle horizon (i.e. the co-moving particle horizon in the infinite future). In our dark energy dominated FLRW Universe, this (co-moving) horizon is indeed finite and about 4.5 times as large as today's Hubble radius.

Choosing the co-moving IR scale to be constant brings with it a number of technical conveniences. For example, it results in a constant spacing 2π/L_c of the grid we used to discretize the field, such that the sum that e.g. appears in equation (2.22) is over sub-sets of the same set of modes k at any time. This especially means that our decomposition of the total Hilbert space into individual mode Hilbert spaces is well defined and constant in time. Another advantage of a constant co-moving scale L_c is that it allows us to simplify our expression for vacuum energy density (cf equation (2.31)) to

$$\begin{equation}{{\epsilon}}_{\text{vac}}(t)\approx \underset{{k}_{\text{ph}}< {{\Lambda}}_{\text{UV}}}{\int }\frac{\mathrm{d}{k}_{\text{ph}}{k}_{\text{ph}}^{3}}{{(2\pi )}^{2}}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}{a}^{2}({t}_{a(t){k}_{\text{ph}}})}{12{a}^{2}(t)}{d}_{a(t){k}_{\text{ph}}}\right)\end{equation} \tag{ 2.37 }$$

(where we have again assumed that the mass m of the field is negligible). These are however just technical conveniences, and some aspects of our derivation will need to be revisited if L_c changes in time.

We had argued previously, that there are several candidate scales which could act as the physical size L_ph of the Universe, and that the correct choice among those scales will depend on the mapping through which the effective notion of space emerges from an underlying purely quantum theory. Working out the details of this mapping is far beyond the scope of this work, and as a proof of concept we simply assume that the Universe has a constant co-moving size L_c and that L_ph(t) = L_c a(t), i.e. that the physical size of the Universe is proportional to the scale factor. As explained in section 2.3, we will furthermore choose the constant co-moving IR scale L_c to be the asymptotic, co-moving particle horizon, which in our dark energy dominated FLRW Universe is about 4.5 times as large as today's Hubble radius.

With a constant co-moving IR scale we can simplify our expression for _vac in equation (2.37) even further, because the scale factor a(t _k ) at the time of mode entry t _k can be calculated as

$$\begin{align}\hfill {k}_{\text{ph}}a(t)& =k={{\Lambda}}_{\text{UV}}a({t}_{\boldsymbol{k}})\hfill \\ \hfill \\ \hfill {\Rightarrow}\frac{{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}& =\frac{a({t}_{\boldsymbol{k}})}{a(t)}.\hfill \end{align} \tag{ 4.1 }$$

Together with our parametrization of d _k from equation (2.32) this gives

$$\begin{equation}{{\epsilon}}_{\text{vac}}(t)=\underset{{k}_{\text{ph}}\leqslant {{\Lambda}}_{\text{UV}}}{\int }\frac{\mathrm{d}{k}_{\text{ph}}{k}_{\text{ph}}^{3}}{{(2\pi )}^{2}}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}}{12}{\left[\frac{{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right]}^{2}\left\{D{\left(\frac{a(t){k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}+{d}_{\mathrm{min}}\right\}\right),\end{equation} \tag{ 4.2 }$$

where, as mentioned before, we take D > 0 and d_min = 2, such that every mode is initialised with at least the Hilbert space of a qubit. We can define an equation of state parameter w_vac = w_vac(a) for this energy density through

$$\begin{align}\hfill {{\epsilon}}_{\text{vac}}\propto & \mathrm{exp}\left(-3{\int }^{a}\frac{\mathrm{d}{a}^{\prime }}{{a}^{\prime }}(1+w)\right)\hfill \\ \hfill {\Rightarrow}{w}_{\text{vac}}& =-\left(1+\frac{1}{3}\frac{\mathrm{d}\,\mathrm{ln}\,{{\epsilon}}_{\text{vac}}}{\mathrm{d}\,\mathrm{ln}\,a}\right).\hfill \end{align} \tag{ 4.3 }$$

We display w_vac for a = 1 as a function of D and n_D in figure 2. Note that the range of D and n_D over which we plot w_vac in that figure is the same as that of figure 1. So the region of parameter space where w_vac deviates most strongly from −1 seems to roughly coincide with the region in which the maximum entropy attainable with our construction deviates most from a volume-scaling.

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The space of possible values for D and n_D should be constrained by the fact that finite-dimensional effects have not been observed on standard model scales. We have implemented two qualitative versions of this constraint, which are displayed as grey regions in figure 2. The first region results from demanding that at energies k_LHC ≈ 10 TeV, the dimension d _k of the individual mode Hilbert spaces be larger than some number N_min. This would require that

$$\begin{align}\hfill {N}_{\mathrm{min}}& < D{({k}_{\text{LHC}}/{{\Lambda}}_{\text{UV}})}^{{n}_{D}}+{d}_{\mathrm{min}}\hfill \\ \hfill {\Rightarrow}\mathrm{ln}\,D& > \mathrm{ln}({N}_{\mathrm{min}}-{d}_{\mathrm{min}})-{n}_{D}\,\mathrm{ln}({k}_{\text{LHC}}/{{\Lambda}}_{\text{UV}}).\hfill \end{align} \tag{ 4.4 }$$

It is beyond the scope of our work to determine which N_min would be sufficient to stay consistent with current experimental data (e.g. regarding the dispersion relation of the Higgs, which so far is the only scalar field of the standard model). But for the purpose of building intuition, we implement the above bound with N_min = 10⁶ as the grey region to the very left of figure 2. A second criterion we consider is the vacuum energy of modes with k_ph ≈ k_LHC. Even for large dimensions d _k , that energy can significantly deviate from the infinite-dimensional expectation k_ph/2, since it also depends on the eigenvalue spacings α _k , β _k as well as on the parameters M and Ω _k appearing in the Hamiltonian in equation (2.22). In order to ensure that finite-dimensional effects on the vacuum energy of IR scales are small we demand that

$$\begin{equation}\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}}{12}{\left[\frac{{k}_{\text{LHC}}}{{{\Lambda}}_{\text{UV}}}\right]}^{2}\left[D{\left(\frac{a(t){k}_{\text{LHC}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}+{d}_{\mathrm{min}}\right]\right) > x,\end{equation} \tag{ 4.5 }$$

where x is some number close to 1. It is again beyond the scope of this work to determine realistic values for x. But as an illustration we implement the above criterion with x = 0.99 as the second grey region to the left of figure 2. Note that both inequality (4.4) and inequality (4.5) could be tightened further, because also many scales below k_LHC should be close to infinite-dimensional. E.g., for n_D > 0, inequality (4.4) would always be violated if k_LHC was replaced by an arbitrarily small scale. Similarly, for n_D > −2 inequality (4.5) could always be violated if k_LHC was replaced by an arbitrarily small scale. It is unclear, down to which energy scales the standard model should be expected to stay accurate [8, 19, 31], so we do not attempt to strengthen our bounds in that way.

Equation (4.2) also allows us to consider the time dependence of vacuum energy density and we follow the evolution _vac(a) for different values of D and n_D in the bottom panel of figure 3. For any of the configurations (D, n_D ) shown there, the vacuum energy decays between two epochs of constant energy density. Correspondingly, deviations from w = −1 will peak at a certain time and then fall off again. This can be understood as follows: around the upper integration limit k_ph ∼ Λ_UV the integrand in equation (4.2) becomes time independent both for a → 0 and a → ∞. So if the integral is dominated by that upper limit, then _vac will become time independent in both the asymptotic past and future. We discuss this further at the end of section 4.2 where we also argue that equation (4.2) is indeed dominated by k_ph ∼ Λ_UV.

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So far we had chosen the eigenvalue spacings of our finite-dimensional conjugate operators as in equation (2.27). As we have argued around that equation (see also appendix A) this choice is minimizing finite-dimensional effects on the low-energy spectrum of Hamiltonian ${\hat{H}}_{\boldsymbol{k}}$ of each mode k at the time t _k when the mode emerges (cf section 3 for an interpretation of the emergence process). At the same time, vacuum energy density is dominated by the UV modes for which t _k is close to the present time. So our construction can be considered a conservative estimate of the impact of finite-dimensionality on _vac.

We now want to complement this estimate by an alternative choice of α _k and β _k that leads to more drastic consequences. We can do this because we take the finite-dimensional construction as more fundamental than its infinite-dimensional limit and demand only that the former approaches the latter when actually d _k → ∞. For our alternative scenario we follow [42] in choosing

$$\begin{equation}{\alpha }_{\boldsymbol{k}}=\sqrt{\frac{2\pi }{{d}_{\boldsymbol{k}}}}={\beta }_{\boldsymbol{k}}.\end{equation} \tag{ 4.6 }$$

These values for the eigenvalue spacings are treating both conjugate operators ${\hat{Q}}_{\boldsymbol{k}}$ and ${\hat{P}}_{\boldsymbol{k}}$ equal in an algebraic sense (though we note that ${\hat{Q}}_{\boldsymbol{k}}$ and ${\hat{P}}_{\boldsymbol{k}}$ are not uniquely determined by the infinite-dimensional limit, which was our original motivation for the resolution criterion of equation (2.25)).

As we show in appendix B, the vacuum energy at late time of each mode k with the new eigenvalue spacings is given by

$$\begin{equation}{E}_{\mathrm{min},\boldsymbol{k}}(t)\approx \frac{{{\Omega}}_{\boldsymbol{k}}(t)}{2}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}}{12M(t){{\Omega}}_{\boldsymbol{k}}(t)}{d}_{\boldsymbol{k}}\right),\end{equation} \tag{ 4.7 }$$

where as before M = a³ L_c and ${{\Omega}}_{\boldsymbol{k}}=\sqrt{\vert \boldsymbol{k}{\vert }^{2}/{a}^{2}+{m}^{2}}\approx \vert \boldsymbol{k}\vert /a$. Hence, the vacuum energy density now becomes

$$\begin{equation}{{\epsilon}}_{\text{vac}}(t)=\underset{{k}_{\text{ph}}\leqslant {{\Lambda}}_{\text{UV}}}{\int }\frac{\mathrm{d}{k}_{\text{ph}}{k}_{\text{ph}}^{3}}{{(2\pi )}^{2}}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}}{12a{(t)}^{3}{k}_{\text{ph}}{L}_{\mathrm{c}}}\left[D{\left(\frac{a(t){k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}+{d}_{\mathrm{min}}\right]\right).\end{equation} \tag{ 4.8 }$$

The top panel of figure 3 displays _vac as a function of the scale factor a for the same values of D and n_D as we had previously considered for our fiducial construction. And figure 4 shows the equation of state parameter corresponding to the alternative expression for _vac as a function of D and n_D . The behaviour of vacuum energy is now radically altered compared to our fiducial construction (cf figure 2). Now, in most parts of parameter space, _vac does not act as a dark energy at all and instead rapidly decays with w_vac ⩾ 0.

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To understand this strongly different behaviour, let us compare the error functions appearing in the integrands of equations (4.2) and (4.8). As functions of physical wave number k_ph they encode the amount by which the vacuum energy of the co-moving mode k = ak_ph is reduced with respect to the infinite-dimensional expectation of E_min ≈ k_ph/2 in our two scenarios. In our alternative scenario (i.e. the one of equation (4.8)), the error function can be approximated in terms of a piece wise power law as

$$\begin{equation}\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}\left[D{\left(\frac{a{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}+{d}_{\mathrm{min}}\right]}{12{a}^{3}{k}_{\text{ph}}{L}_{\mathrm{c}}}\right)\propto \left\{\begin{array}{ll}1\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\ D{\left(\frac{a{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}\gg {k}_{\text{ph}}{L}_{\mathrm{c}}\hfill \\ {({k}_{\text{ph}})}^{{n}_{D}-1}\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\ {k}_{\text{ph}}{L}_{\mathrm{c}}\gg D{\left(\frac{a{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}\gg {d}_{\mathrm{min}}\hfill \\ {({k}_{\text{ph}})}^{-1}\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\ {d}_{\mathrm{min}}\gg D{\left(\frac{a{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}\hfill \end{array}\right..\end{equation} \tag{ 4.9 }$$

For n_D = −3.5, y = −4.0 (a point in parameter space which results in a particularly high w in our alternative construction) and at a scale factor of a = 1 we demonstrate this behaviour in appendix C (cf figure 7 there). We also display the above error function for a number of different scale factors (a = 0.5, 0.8, 1.0) and on a reduced k_ph-range in the upper panel of figure 5. For negative values of n_D these scalings imply that deviations from the infinite-dimensional vacuum energy are monotonically increasing with k_ph (i.e. the error function is monotonically decreasing, cf figure 5) when choosing our alternative spacings for the eigenvalues of the conjugate operators. The wave number above which the error function starts to noticeably deviate from 1 is approximately given by

$$\begin{equation}{k}_{\text{ph}}^{\ast }\approx {\left(\frac{D{a}^{{n}_{D}}}{{({{\Lambda}}_{\text{UV}})}^{{n}_{D}}{L}_{\mathrm{c}}}\right)}^{\frac{1}{1-{n}_{D}}}.\end{equation} \tag{ 4.10 }$$

This can be seen as a threshold above which modes start to contribute significantly less to the vacuum energy density of the field than they would in the infinite-dimensional case. For negative n_D that threshold is decreasing as a(t) increases, which explains the strongly decaying behaviour of vacuum energy density for the alternative eigenvalue spacings of equation (4.6).

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The error function of our fiducial construction behaves quite differently. Because of an additional factor of ${k}_{\text{ph}}^{3}$ in its argument, it follows a piece wise scaling of

$$\begin{equation}\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{\frac{3}{2}}{k}_{\text{ph}}^{2}\left[D{\left(\frac{a{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}+{d}_{\mathrm{min}}\right]}{12{{\Lambda}}_{\text{UV}}^{2}}\right)\propto \left\{\begin{array}{ll}1\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\ {\left(\frac{{{\Lambda}}_{\text{UV}}}{{k}_{\text{ph}}}\right)}^{2-{n}_{D}}\gg D{a}^{{n}_{D}}\hfill \\ {({k}_{\text{ph}})}^{{n}_{D}+2}\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\ {\left(\frac{{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{2}\gg D{\left(\frac{a{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}\gg {d}_{\mathrm{min}}\hfill \\ \mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{\frac{3}{2}}{d}_{\mathrm{min}}}{12{{\Lambda}}_{\text{UV}}^{2}}{k}_{\text{ph}}^{2}\right)\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\ {d}_{\mathrm{min}}\gg D{\left(\frac{a{k}_{\text{ph}}}{{{\Lambda}}_{\text{UV}}}\right)}^{{n}_{D}}\hfill \end{array}\right..\end{equation} \tag{ 4.11 }$$

At least for n_D < −2 this implies a non-monotonic behaviour: deviations from the infinite-dimensional limit first increase with k_ph and then decrease again (i.e. the error function decreases and then increases again, cf lower panel of figure 5). And for any n_D the error function at k_ph ∼ Λ_UV in equation (4.2) will behave as

$$\begin{equation*}\sim \mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{\frac{3}{2}}\left[D{a}^{{n}_{D}}+{d}_{\mathrm{min}}\right]}{12}\right),\end{equation*}$$

which is always $\sim \mathcal{O}(1)$. As a consequence, the integral in equation (4.2) is always dominated by the UV. The resulting vacuum energy is then transitioning between two dark-energy-like regimes, as discussed at the end of section 4.1: a regime when the error function equals 1 over almost the entire integration range (which happens for a → 0 as long as n_D is negative) and a regime where the UV part of the integrand has approached the limit $\sim \mathrm{e}\mathrm{r}\mathrm{f}({\pi }^{\frac{3}{2}}{d}_{\mathrm{min}}/12)$ (which happens for a → ∞ as long as n_D is negative).

The stark difference between the behaviour of vacuum energy in our fiducial and in our alternative construction highlights, that effects of finite-dimensionality are highly sensitive to the details of how quantum field theory emerges from a finite-dimensional quantum theory. Neither the fiducial construction (which minimizes the impact of finite-dimensionality) nor the alternative construction (which attempts to treat both conjugate variables as algebraically equal) are likely to fully capture this emergence. In the following section we summarize the assumptions and limitations behind both scenarios and give an outlook on potential extensions and improvements.

Consider again the Hamiltonian of a finite-dimensional harmonic oscillator,

$$\begin{equation}\hat{H}=\frac{{\hat{P}}^{2}}{2M}+\frac{M{{\Omega}}^{2}}{2}{\hat{Q}}^{2}.\end{equation} \tag{ B.1 }$$

Our goal in this appendix is to derive an approximation to the minimum eigenvalue of this operator as a function of M, Ω, d and α (cf appendix A for notation) that is numerically feasible even for large d. We had seen in appendix A, that $\hat{H}$ can be re-written as

$$\begin{equation}\hat{H}=\frac{{(2\pi )}^{2}}{2M{d}^{2}{\alpha }^{2}}\left\{\hat{S}{\hat{L}}^{2}{\hat{S}}^{-1}+\frac{{\alpha }^{4}{M}^{2}{{\Omega}}^{2}{d}^{2}}{{(2\pi )}^{2}}{\hat{L}}^{2}\right\}\equiv \frac{{(2\pi )}^{2}}{2M{d}^{2}{\alpha }^{2}}\left\{\hat{S}{\hat{L}}^{2}{\hat{S}}^{-1}+X{\hat{L}}^{2}\right\}.\end{equation} \tag{ B.2 }$$

So to calculate the lowest energy level of $\hat{H}$ we need to know the minimum eigenvalues of operators of the form ${\hat{O}}_{X}\equiv \hat{S}{\hat{L}}^{2}{\hat{S}}^{-1}+X{\hat{L}}^{2}$. At the fix point α_* we have derived in appendix A the Hamiltonian becomes

$$\begin{equation}\hat{H}=\frac{{\Omega}}{2}\frac{2\pi }{d}\left\{\hat{S}{\hat{L}}^{2}{\hat{S}}^{-1}+{\hat{L}}^{2}\right\}.\end{equation} \tag{ B.3 }$$

At the same time, we had argued there that the fix point minimizes effects of finite-dimensionality, such that vacuum energy comes close to its infinite-dimensional limit Ω/2 (numerical calculation confirms that this is true to high accuracy even for d as low as 5). From that we can conclude that

$$\begin{equation}{\lambda }_{\mathrm{min}}({\hat{O}}_{X\approx 1})\approx \frac{d}{2\pi }.\end{equation} \tag{ B.4 }$$

To understand the situation for more general values of X, let us consider the matrix elements of ${\hat{O}}_{X}$ in the eigenbasis of $\hat{Q}$. They are given by [42]

$$\begin{equation}{[{\hat{O}}_{X}]}_{ij}=\frac{1}{4}\sum\limits _{n\ne i,j}\frac{1}{\mathrm{sin}\left(\frac{2\pi \ell }{2\ell +1}(n-i)\right)\mathrm{sin}\left(\frac{2\pi \ell }{2\ell +1}(n-j)\right)}+X{j}^{2}{\delta }_{ij},\end{equation} \tag{ B.5 }$$

where all integers run from −ℓ to ℓ (and d = 2ℓ + 1 as in appendix A). In the limit X → ∞ the second term in the above bracket dominates such that the eigenvector v of ${\hat{O}}_{X}$ with the lowest eigenvalue becomes

$$\begin{equation}{[\boldsymbol{v}]}_{i}\propto \left\{\begin{matrix}\hfill 1\hfill & \hfill \quad \mathrm{f}\mathrm{o}\mathrm{r}\ i=0\hfill \\ \hfill 0\hfill & \hfill \mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\hfill \end{matrix}\right.\end{equation} \tag{ B.6 }$$

with the corresponding eigenvalue

$$\begin{equation}{\lambda }_{\mathrm{min}}({\hat{O}}_{X})\approx {[{\hat{O}}_{X}]}_{00}=\frac{1}{2}\sum\limits _{n=1}^{\ell }\frac{1}{{\mathrm{sin}}^{2}\left(\frac{2\pi \ell }{2\ell +1}n\right)}=\frac{1}{2}\sum\limits _{n=1}^{\ell }\frac{1}{{\mathrm{sin}}^{2}\left(\pi n(1-1/d)\right)}.\end{equation} \tag{ B.7 }$$

For d ≫ 1 we can expand this in the small parameter x ≡ 1/d as

$$\begin{equation}{[{\hat{O}}_{X}]}_{00}\approx \frac{1}{2{\pi }^{2}{x}^{2}}\sum\limits _{n=1}^{\ell }\frac{1}{{n}^{2}}\approx \frac{{d}^{2}}{12}.\end{equation} \tag{ B.8 }$$

This is an approximation for ${\lambda }_{\mathrm{min}}({\hat{O}}_{X})$ as X approaches infinity. By the symmetry arguments we had employed in appendix A one can then show that ${\lambda }_{\mathrm{min}}({\hat{O}}_{X})\approx X{d}^{2}/12$ as X approaches zero. In summary, we obtain

$$\begin{equation}{\lambda }_{\mathrm{min}}({\hat{O}}_{X})\approx \left\{\begin{matrix}\hfill {d}^{2}/12\hfill & \hfill \mathrm{a}\mathrm{s}\ X\to \infty \hfill \\ \hfill d/(2\pi )\hfill & \hfill \!\!\mathrm{f}\mathrm{o}\mathrm{r}\ X\approx 1\hfill \\ \hfill X{d}^{2}/12\hfill & \hfill \mathrm{a}\mathrm{s}\ X\to 0\hfill \end{matrix}\right..\end{equation} \tag{ B.9 }$$

These asymptotics are matched exactly by the formula

$$\begin{equation}{\lambda }_{\mathrm{min}}({\hat{O}}_{X})\approx \frac{F}{{\left(1+{X}^{-3}+G{X}^{-3/2}\right)}^{1/3}}\end{equation} \tag{ B.10 }$$

with

$$\begin{equation}F=\frac{{d}^{2}}{12},\qquad G={\left(\frac{\pi d}{6}\right)}^{3}-2.\end{equation} \tag{ B.11 }$$

We compare that formula to the exact calculation of ${\lambda }_{\mathrm{min}}({\hat{O}}_{X})$ in the upper panel of figure 6—exact and approximated result agree to within $\sim 4\%$ accuracy over the entire range of X for d ⩾ 7 (and much better for most values of X). That accuracy reduces to $\sim 6\%$ for d = 5 and to $\sim 15\%$ for d = 3.

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We can obtain a more concise formula that directly approximates the minimum eigenvalue of $\hat{H}$ for the late-time cosmological situation of section 4. For both constructions we considered there, the eigenvalues spacing of $\hat{Q}$ was of the form

$$\begin{equation}\alpha =\sqrt{\frac{2\pi }{Ad}},\end{equation} \tag{ B.12 }$$

with A = M(t _k )Ω _k (t _k ) in our fiducial construction and A = 1 in section 4.2. The parameter X then becomes X = M²Ω²/A², which in our fiducial construction is given as a function of k and t by

$$\begin{equation}{X}_{\boldsymbol{k}}^{\text{fid}}(t)=\frac{M{(t)}^{2}{\Omega}{(t)}^{2}}{M{({t}_{\boldsymbol{k}})}^{2}{\Omega}{({t}_{\boldsymbol{k}})}^{2}}\approx {\left(\frac{a(t)}{a({t}_{\boldsymbol{k}})}\right)}^{4}.\end{equation} \tag{ B.13 }$$

In an expanding Universe this is clearly always larger than 1. On the other hand, in our alternative construction X becomes

$$\begin{equation}{X}_{\boldsymbol{k}}^{\text{alt}}(t)={L}_{\mathrm{c}}^{2}{a}^{4}{k}^{2}={({L}_{\mathrm{c}}{a}^{3}{k}_{\text{ph}})}^{2}.\end{equation} \tag{ B.14 }$$

In the late-time Universe, a ∼ 1, this is greater than 1 as long as k ≳ 1/L_c ≈ 3.3 × 10⁻³⁴ eV, i.e. on all scales relevant to the vacuum energy density of our scalar field. So for the late-time expansion we have considered in this work, we are indeed fine to consider only the two asymptotics of equation (B.9) with X ⩾ 1. In that case, and taking into account the relation between $\hat{H}$ and ${\hat{O}}_{X}$, the minimum energy eigenvalue will asymptotically behave as

$$\begin{align}\hfill {\lambda }_{\mathrm{min}}(\hat{H})\approx & \frac{A}{2M}\cdot \left\{\begin{matrix}\hfill \pi d/6\hfill & \hfill \mathrm{a}\mathrm{s}\ M{\Omega}/A\to \infty \hfill \\ \hfill 1\hfill & \hfill \mathrm{f}\mathrm{o}\mathrm{r}\ M{\Omega}/A\approx 1\hfill \end{matrix}\right.\hfill \\ \hfill \approx & \frac{{\Omega}}{2}\cdot \left\{\begin{matrix}\hfill \frac{\pi Ad}{6M{\Omega}}\hfill & \hfill \mathrm{a}\mathrm{s}\ M{\Omega}/A\to \infty \hfill \\ \hfill 1\hfill & \hfill \mathrm{f}\mathrm{o}\mathrm{r}\ M{\Omega}/A\approx 1\hfill \end{matrix}\right..\hfill \end{align} \tag{ B.15 }$$

This behaviour can be approximately matched by the ansatz

$$\begin{equation}{\lambda }_{\mathrm{min}}(\hat{H})\approx \frac{{\Omega}}{2}\,\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{{\pi }^{3/2}Ad}{12M{\Omega}}\right).\end{equation} \tag{ B.16 }$$

This is the approximation we used in order to derive the results of section 4. We compare it to the exact calculation of ${\lambda }_{\mathrm{min}}(\hat{H})$ over a limited range of d in the lower panel of figure 6. That figure only includes results for one set of values for Ω and M and for two different values for A. But we find that equation (B.16) agrees with the exact result to within $\sim 2\%$ accuracy for a wide range of values for (Ω, M, A) as long as d ⩾ 7. For d = 5 this reduces to $\sim 4\%$ accuracy and for d = 3 to $\sim 12\%$ accuracy.

Finally, we want to note that in the early Universe the third regime of equation (B.9) can indeed become relevant—at least in our alternative construction. There one would have X < 1 even at k_ph ≈ Λ_UV = 1 as long as a ≲ 3.0 × 10⁻²¹. So the approximation of equation (B.16) would e.g. be not appropriate to calculate the behaviour of our finite-dimensional field during inflation, and one would have to use equation (B.10) instead. Both equations (B.16) and (B.10) and the corresponding exact calculations are all implemented within our publicly available code package GPUniverse.