# The points of space are not points: their wavefunctions remove the ultraviolet divergences of quantum field theory

Kevin Cahill

► To cite this version:

Kevin Cahill. The points of space are not points: their wavefunctions remove the ultraviolet divergences of quantum field theory. 2024. hal-04610320

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# The points of space are not points: their wavefunctions remove the ultraviolet divergences of quantum field theory

Kevin Cahill

► To cite this version:

Kevin Cahill. The points of space are not points: their wavefunctions remove the ultraviolet divergences of quantum field theory. 2024. hal-04610320

HAL Id: hal-04610320

https://hal.science/hal-04610320v1

Preprint submitted on 13 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# The points of space are not points: their wavefunctions remove the ultraviolet divergences of quantum field theory

Kevin Cahill*

Department of Physics and Astronomy

University of New Mexico

Albuquerque, New Mexico 87106

(Dated: June 11, 2024)

The points $p(x)$ of space are observables that obey quantum mechanics with average values $\langle p(x) \rangle$ that obey general relativity. In empty flat 3-space, a point $p(x)$ may be represented as the sum $p(x) = \langle p(x) \rangle + \epsilon(x)$ of its average value $\langle p(x) \rangle = \mathbf{x}$ and its quantum remainder $\epsilon(x)$. The ubiquitous Fourier exponential $\exp(ik \cdot p(x))$ is then $\exp(ik \cdot (\mathbf{x} + \epsilon))$. If the probability distribution of the quantum remainders $\epsilon(x)$ is normal and of width $\sigma$, then the average of the exponential $\langle \exp(ik(\mathbf{x} + \epsilon)) \rangle$ contains a gaussian factor $\exp(-\sigma^2 \mathbf{k}^2/2)$ which makes self-energies, Feynman diagrams and dark energy ultraviolet finite. This complex of ideas requires new bosons and suggests that supersymmetry is softly broken.

## INTRODUCTION

Einstein's equations $G_{ik} = 8\pi G T_{ik}$ relate the energy-momentum tensor $T_{ik}$ to the Einstein tensor $G_{ik}$ which is a sum of products of the metric $g_{ik}$, its inverse and its derivatives. The metric $g_{ik}(x)$ itself is quadratic in the derivatives

$$g_{ik}(x) = \frac{\partial p(x)}{\partial x^i} \cdot \frac{\partial p(x)}{\partial x^k} \equiv - \frac{\partial p^0(x)}{\partial x^i} \frac{\partial p^0(x)}{\partial x^k} + \sum_{\alpha=1}^{s} \frac{\partial p^\alpha(x)}{\partial x^i} \frac{\partial p^\alpha(x)}{\partial x^k} \quad (1)$$

of the point $p(x)$ which lies in a flat semi-euclidian embedding space of one dimension of time and $s \le 19$ dimensions of space [1–4]. Einstein’s equations therefore relate derivatives of the points $p(x)$ of our 4-dimensional curved space to the energy-momentum tensor $T_{ik}$ which is a sum of products of hermitian operators. It follows that the points $p(x)$ are themselves hermitian operators. Their average values $\langle p(x) \rangle$ obey general relativity, and their remainders $\epsilon(x) = p(x) - \langle p(x) \rangle$ obey quantum mechanics. Their coordinates $x$ are merely labels.
# The points of space are not points: their wavefunctions remove the ultraviolet divergences of quantum field theory

Kevin Cahill*

Department of Physics and Astronomy

University of New Mexico

Albuquerque, New Mexico 87106

(Dated: June 11, 2024)

The points $p(x)$ of space are observables that obey quantum mechanics with average values $\langle p(x) \rangle$ that obey general relativity. In empty flat 3-space, a point $p(x)$ may be represented as the sum $p(x) = \langle p(x) \rangle + \epsilon(x)$ of its average value $\langle p(x) \rangle = \mathbf{x}$ and its quantum remainder $\epsilon(x)$. The ubiquitous Fourier exponential $\exp(ik \cdot p(x))$ is then $\exp(ik \cdot (\mathbf{x} + \epsilon))$. If the probability distribution of the quantum remainders $\epsilon(x)$ is normal and of width $\sigma$, then the average of the exponential $\langle \exp(ik(\mathbf{x} + \epsilon)) \rangle$ contains a gaussian factor $\exp(-\sigma^2 k^2/2)$ which makes self-energies, Feynman diagrams and dark energy ultraviolet finite. This complex of ideas requires new bosons and suggests that supersymmetry is softly broken.

## INTRODUCTION

Einstein's equations $G_{ik} = 8\pi G T_{ik}$ relate the energy-momentum tensor $T_{ik}$ to the Einstein tensor $G_{ik}$ which is a sum of products of the metric $g_{ik}$, its inverse and its derivatives. The metric $g_{ik}(x)$ itself is quadratic in the derivatives

$$g_{ik}(x) = \frac{\partial p(x)}{\partial x^i} \cdot \frac{\partial p(x)}{\partial x^k} \equiv - \frac{\partial p^0(x)}{\partial x^i} \frac{\partial p^0(x)}{\partial x^k} + \sum_{\alpha=1}^{s} \frac{\partial p^\alpha(x)}{\partial x^i} \frac{\partial p^\alpha(x)}{\partial x^k} \quad (1)$$

of the point $p(x)$ which lies in a flat semi-euclidian embedding space of one dimension of time and $s \le 19$ dimensions of space [1–4]. Einstein's equations therefore relate derivatives of the points $p(x)$ of our 4-dimensional curved space to the energy-momentum tensor $T_{ik}$ which is a sum of products of hermitian operators. It follows that the points $p(x)$ are themselves hermitian operators. Their average values $\langle p(x) \rangle$ obey general relativity, and their remainders $\epsilon(x) = p(x) - \langle p(x) \rangle$ obey quantum mechanics. Their coordinates $x$ are merely labels.
## EMPTY, FLAT SPACE

In empty flat space $\mathcal{E}$, the average $\langle \mathcal{E} | \boldsymbol{p}(x) | \mathcal{E} \rangle \equiv \langle \boldsymbol{p}(x) \rangle$ of the spatial components of a point $\boldsymbol{p}(\boldsymbol{x})$ is its position $\langle \boldsymbol{p}(x) \rangle = \boldsymbol{x}$ in the rest frame of the CMB. A point $\boldsymbol{p}(x)$ of space is the sum of its position $\boldsymbol{x}$ and its quantum-mechanical remainder $\boldsymbol{\epsilon}(x)$

$$ \boldsymbol{p}(x) = \langle \mathcal{E} | \boldsymbol{p}(x) | \mathcal{E} \rangle + \boldsymbol{\epsilon}(x) = \boldsymbol{x} + \boldsymbol{\epsilon}(x). \qquad (2) $$

In empty flat space $\mathcal{E}$ and in the rest frame of the CMB, the probability distribution $P(\boldsymbol{\epsilon}) = |\langle \boldsymbol{\epsilon} | \mathcal{E} \rangle|^2$ of the three components $\boldsymbol{\epsilon}(x)$ should depend only upon $\boldsymbol{\epsilon}^2$ and not upon $\boldsymbol{x}$ and be tightly concentrated about $\boldsymbol{\epsilon} = 0$. For simplicity I will assume that the components $\boldsymbol{\epsilon}(x)$ are normally distributed

$$ P(\boldsymbol{\epsilon}) = |\langle \boldsymbol{\epsilon} | \mathcal{E} \rangle|^2 = \frac{1}{(2\pi)^{3/2}\sigma^3} \exp\left(-\frac{\boldsymbol{\epsilon}^2}{2\sigma^2}\right) \qquad (3) $$

in which $\sigma$ is a length of the order of $M_{\text{Planck}}^{-1}$. The present formalism may entail violations of Lorentz invariance on scales of the order of the inverse of the Planck mass.

The average value of the exponential $\exp(\pm i\boldsymbol{k} \cdot (\boldsymbol{x} + \boldsymbol{\epsilon}(x)))$ in flat space $\mathcal{E}$ is then

$$ \begin{aligned} \langle \mathcal{E} | \exp(\pm i\boldsymbol{k} \cdot (\boldsymbol{x} + \boldsymbol{\epsilon}(x))) | \mathcal{E} \rangle &= \int d^3\boldsymbol{\epsilon}(\boldsymbol{x}) P(\boldsymbol{\epsilon}(x)) \exp(\pm i\boldsymbol{k} \cdot (\boldsymbol{x} + \boldsymbol{\epsilon}(x))) \\ &= \int d^3\boldsymbol{\epsilon}(\boldsymbol{x}) |\langle \boldsymbol{\epsilon}(\boldsymbol{x}) | \mathcal{E} \rangle|^2 \exp(\pm i\boldsymbol{k} \cdot (\boldsymbol{x} + \boldsymbol{\epsilon}(x))) \\ &= \int \frac{d^3\boldsymbol{\epsilon}}{\sigma^3(2\pi)^{3/2}} e^{\pm i\boldsymbol{k} \cdot (\boldsymbol{x}+\boldsymbol{\epsilon}) - \boldsymbol{\epsilon}^2/2\sigma^2} = \exp\left(\pm i\boldsymbol{k} \cdot \boldsymbol{x} - \sigma^2\boldsymbol{k}^2/2\right) \end{aligned} \qquad (4) $$

which is the original exponential $\exp(\pm i\boldsymbol{k} \cdot \boldsymbol{x})$ multiplied by the damping factor $\exp(-\sigma^2\boldsymbol{k}^2/2)$.

## FIELDS IN FLAT SPACE

The usual Fourier decomposition of a scalar field in flat space is

$$ \phi(x) = \int \frac{d^3 k}{\sqrt{(2\pi)^3 2\omega_k}} \left[ e^{ikx} a(k) + e^{-ikx} a^{\dagger}(k) \right] \qquad (5) $$

in which $kx = \boldsymbol{k} \cdot \boldsymbol{x} - \omega_k x^0$ and $\omega_k = \sqrt{\boldsymbol{k}^2 + m^2}$. Replacing the exponentials $e^{\pm ikx}$ by their average values (4) in empty flat space, we find for the averaged field

$$ \langle \mathcal{E} | \phi(x) | \mathcal{E} \rangle = \int \frac{d^3 k}{\sqrt{(2\pi)^3 2\omega_k}} \left[ e^{ikx} a(k) + e^{-ikx} a^{\dagger}(k) \right] e^{-\sigma^2 \boldsymbol{k}^2 / 2}. \qquad (6) $$

As $\sigma \to 0$, the averaged field $\langle \phi(x) \rangle$ becomes the usual field (5), but as long as $\sigma \neq 0$, the factor $e^{-\sigma^2 \boldsymbol{k}^2 / 2}$ makes Feynman diagrams finite.
## EMPTY, FLAT SPACE

In empty flat space $\mathcal{E}$, the average $\langle \mathcal{E} | \mathbf{p}(x) | \mathcal{E} \rangle \equiv \langle \mathbf{p}(x) \rangle$ of the spatial components of a point $\mathbf{p}(\mathbf{x})$ is its position $\langle \mathbf{p}(x) \rangle = \mathbf{x}$ in the rest frame of the CMB. A point $\mathbf{p}(x)$ of space is the sum of its position $\mathbf{x}$ and its quantum-mechanical remainder $\epsilon(x)$

$$ \mathbf{p}(x) = \langle \mathcal{E} | \mathbf{p}(x) | \mathcal{E} \rangle + \epsilon(x) = \mathbf{x} + \epsilon(x). \qquad (2) $$

In empty flat space $\mathcal{E}$ and in the rest frame of the CMB, the probability distribution $P(\epsilon) = |\langle \epsilon | \mathcal{E} \rangle|^2$ of the three components $\epsilon(x)$ should depend only upon $\epsilon^2$ and not upon $\mathbf{x}$ and be tightly concentrated about $\epsilon = 0$. For simplicity I will assume that the components $\epsilon(x)$ are normally distributed

$$ P(\epsilon) \equiv |\langle \epsilon | \mathcal{E} \rangle|^2 = \frac{1}{(2\pi)^{3/2}\sigma^3} \exp\left(-\frac{\epsilon^2}{2\sigma^2}\right) \qquad (3) $$

in which $\sigma$ is a length of the order of $M_{\text{Planck}}^{-1}$. The present formalism may entail violations of Lorentz invariance on scales of the order of the inverse of the Planck mass.

The average value of the exponential $\exp(\pm i\mathbf{k} \cdot (\mathbf{x} + \epsilon(x)))$ in flat space $\mathcal{E}$ is then

$$ \begin{aligned} \langle \mathcal{E} | \exp(\pm i\mathbf{k} \cdot (\mathbf{x} + \epsilon(x))) | \mathcal{E} \rangle &= \int d^3\epsilon(\mathbf{x}) P(\epsilon(x)) \exp(\pm i\mathbf{k} \cdot (\mathbf{x} + \epsilon(x))) \\ &= \int d^3\epsilon(\mathbf{x}) |\langle \epsilon(\mathbf{x}) | \mathcal{E} \rangle|^2 \exp(\pm i\mathbf{k} \cdot (\mathbf{x} + \epsilon(x))) \\ &= \int \frac{d^3\epsilon}{\sigma^3(2\pi)^{3/2}} e^{\pm i\mathbf{k} \cdot (\mathbf{x}+\epsilon) - \epsilon^2/2\sigma^2} = \exp\left(\pm i\mathbf{k} \cdot \mathbf{x} - \sigma^2\mathbf{k}^2/2\right) \end{aligned} \qquad (4) $$

which is the original exponential $\exp(\pm i\mathbf{k} \cdot \mathbf{x})$ multiplied by the damping factor $\exp(-\sigma^2\mathbf{k}^2/2)$.

## FIELDS IN FLAT SPACE

The usual Fourier decomposition of a scalar field in flat space is

$$ \phi(x) = \int \frac{d^3 k}{\sqrt{(2\pi)^3 2\omega_k}} \left[ e^{ikx} a(k) + e^{-ikx} a^{\dagger}(k) \right] \qquad (5) $$

in which $kx = \mathbf{k} \cdot \mathbf{x} - \omega_k x^0$ and $\omega_k = \sqrt{\mathbf{k}^2 + m^2}$. Replacing the exponentials $e^{\pm ikx}$ by their average values (4) in empty flat space, we find for the averaged field

$$ \langle \mathcal{E} | \phi(x) | \mathcal{E} \rangle = \int \frac{d^3 k}{\sqrt{(2\pi)^3 2\omega_k}} \left[ e^{ikx} a(k) + e^{-ikx} a^{\dagger}(k) \right] e^{-\sigma^2 \mathbf{k}^2 / 2}. \qquad (6) $$

As $\sigma \to 0$, the averaged field $\langle \phi(x) \rangle$ becomes the usual field (5), but as long as $\sigma \neq 0$, the factor $e^{-\sigma^2 \mathbf{k}^2 / 2}$ makes Feynman diagrams finite.
## BASIC COMMUTATION RELATION

The equal-time commutation relation for a scalar field with $x$ and $y$ replaced by $x + \epsilon(x)$ and $y + \epsilon(y)$ is

$$[\phi(t, x + \epsilon(x)), \pi(t, y + \epsilon(y))] = i \int \frac{d^3 k}{(2\pi)^3} \exp \left( i \boldsymbol{k} \cdot \left( \boldsymbol{x} + \epsilon(x) - \boldsymbol{y} - \epsilon(y) \right) \right). \quad (7)$$

Its average in empty flat space $\mathcal{E}$ is

$$\begin{aligned} \langle \mathcal{E} | [\phi(t, \boldsymbol{x}), \pi(t, \boldsymbol{y})] | \mathcal{E} \rangle &= i \int \frac{d^3 k \, d^3 \boldsymbol{\epsilon}(x) \, d^3 \boldsymbol{\epsilon}(y)}{(2\pi\sigma)^6} e^{i\boldsymbol{k}(\boldsymbol{x}+\boldsymbol{\epsilon}(x)-\boldsymbol{y}-\boldsymbol{\epsilon}(y)) - (\boldsymbol{\epsilon}^2(x)+\boldsymbol{\epsilon}^2(y))/2\sigma^2} \\ &= i \int \frac{d^3 k}{(2\pi)^3} e^{i\boldsymbol{k} \cdot (\boldsymbol{x}-\boldsymbol{y}) - \sigma^2 \boldsymbol{k}^2} = \frac{i}{(2\sqrt{\pi}\sigma)^3} e^{-(\boldsymbol{x}-\boldsymbol{y})^2/4\sigma^2} \end{aligned} \quad (8)$$

which approaches the usual expression as $\sigma \to 0$ but is finite for all $x$ and $y$.

## FEYNMAN'S PROPAGATOR

For scalar fields, the Feynman propagator is

$$\Delta_F(x-y) = \int \frac{d^4 q}{(2\pi)^4} \frac{e^{iq(x-y)}}{q^2 + m^2 - i\epsilon}. \quad (9)$$

With $x \to x + \epsilon(x)$ and $y \to y + \epsilon(y)$, it is

$$\Delta_F(x-y) = \int \frac{d^4 q}{(2\pi)^4} \frac{e^{iq(x+\epsilon(x)-y-\epsilon(y))}}{q^2 + m^2 - i\epsilon}. \quad (10)$$

This expression averaged over the normal distribution (3) is

$$\begin{aligned} \langle \mathcal{E} | \Delta_F (x-y) | \mathcal{E} \rangle &= \int \frac{d^4 q \, d^3 \boldsymbol{\epsilon}(x) \, d^3 \boldsymbol{\epsilon}(y)}{(2\pi)^4} \frac{e^{iq(x+\boldsymbol{\epsilon}(x)-y-\boldsymbol{\epsilon}(y))}}{q^2 + m^2 - i\epsilon} P(\boldsymbol{\epsilon}(x)) P(\boldsymbol{\epsilon}(y)) \\ &= \int \frac{d^4 q \, d^3 \boldsymbol{\epsilon}_x \, d^3 \boldsymbol{\epsilon}_y}{(2\pi)^6 \sigma^6} \frac{e^{iq(x+\boldsymbol{\epsilon}(x)-y-\boldsymbol{\epsilon}(y)) - (\boldsymbol{\epsilon}^2(x)+\boldsymbol{\epsilon}^2(y))/2\sigma^2}}{q^2 + m^2 - i\epsilon} \\ &= \int \frac{d^4 q}{(2\pi)^4} \frac{e^{iq(x-y)-\sigma^2 q^2}}{q^2 + m^2 - i\epsilon} \equiv \Delta_{F,\sigma}(x-y) \end{aligned} \quad (11)$$

which reduces to the usual propagator as $\sigma \to 0$ but is finite even at $x = y$.

## SELF-ENERGY

In second-order $\phi^3$ theory with action density

$$\mathcal{L} = -\frac{1}{2} \partial_i \phi \partial^i \phi - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{3!} \phi^3 \quad (12)$$
BASIC COMMUTATION RELATION

The equal-time commutation relation for a scalar field with $x$ and $y$ replaced by $x + \epsilon(x)$ and $y + \epsilon(y)$ is

$$[\phi(t, x + \epsilon(x)), \pi(t, y + \epsilon(y))] = i \int \frac{d^3 k}{(2\pi)^3} \exp \left( i \boldsymbol{k} \cdot (x + \epsilon(x) - y - \epsilon(y)) \right). \quad (7)$$

Its average in empty flat space $\mathcal{E}$ is

$$\begin{aligned} \langle \mathcal{E} | [\phi(t, \boldsymbol{x}), \pi(t, \boldsymbol{y})] | \mathcal{E} \rangle &= i \int \frac{d^3 k \, d^3 \boldsymbol{\epsilon}(x) \, d^3 \boldsymbol{\epsilon}(y)}{(2\pi\sigma)^6} e^{i\boldsymbol{k}(\boldsymbol{x}+\boldsymbol{\epsilon}(x)-\boldsymbol{y}-\boldsymbol{\epsilon}(y)) - (\boldsymbol{\epsilon}^2(x)+\boldsymbol{\epsilon}^2(y))/2\sigma^2} \\ &= i \int \frac{d^3 k}{(2\pi)^3} e^{i\boldsymbol{k} \cdot (\boldsymbol{x}-\boldsymbol{y}) - \sigma^2 \boldsymbol{k}^2} = \frac{i}{(2\sqrt{\pi}\sigma)^3} e^{-(\boldsymbol{x}-\boldsymbol{y})^2/4\sigma^2} \end{aligned} \quad (8)$$

which approaches the usual expression as $\sigma \to 0$ but is finite for all $x$ and $y$.

FEYNMAN'S PROPAGATOR

For scalar fields, the Feynman propagator is

$$\Delta_F(x-y) = \int \frac{d^4 q}{(2\pi)^4} \frac{e^{iq(x-y)}}{q^2 + m^2 - i\epsilon}. \quad (9)$$

With $x \to x + \epsilon(x)$ and $y \to y + \epsilon(y)$, it is

$$\Delta_F(x-y) = \int \frac{d^4 q}{(2\pi)^4} \frac{e^{iq(x+\epsilon(x)-y-\epsilon(y))}}{q^2 + m^2 - i\epsilon}. \quad (10)$$

This expression averaged over the normal distribution (3) is

$$\begin{aligned} \langle \mathcal{E} | \Delta_F (x-y) | \mathcal{E} \rangle &= \int \frac{d^4 q \, d^3 \boldsymbol{\epsilon}(x) \, d^3 \boldsymbol{\epsilon}(y)}{(2\pi)^4} \frac{e^{iq(x+\boldsymbol{\epsilon}(x)-y-\boldsymbol{\epsilon}(y))}}{q^2 + m^2 - i\epsilon} P(\boldsymbol{\epsilon}(x)) P(\boldsymbol{\epsilon}(y)) \\ &= \int \frac{d^4 q \, d^3 \boldsymbol{\epsilon}_x \, d^3 \boldsymbol{\epsilon}_y}{(2\pi)^6 \sigma^6} \frac{e^{iq(x+\boldsymbol{\epsilon}(x)-y-\boldsymbol{\epsilon}(y)) - (\boldsymbol{\epsilon}^2(x)+\boldsymbol{\epsilon}^2(y))/2\sigma^2}}{q^2 + m^2 - i\epsilon} \\ &= \int \frac{d^4 q}{(2\pi)^4} \frac{e^{iq(x-y)-\sigma^2 q^2}}{q^2 + m^2 - i\epsilon} \equiv \Delta_{F,\sigma}(x-y) \end{aligned} \quad (11)$$

which reduces to the usual propagator as $\sigma \to 0$ but is finite even at $x = y$.

SELF-ENERGY

In second-order $\phi^3$ theory with action density

$$\mathcal{L} = -\frac{1}{2} \partial_i \phi \partial^i \phi - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{3!} \phi^3 \quad (12)$$
(i = 0 to 3), the self-energy amplitude

$$A(k, k') = \frac{\lambda^2 \delta(k - k')}{2(2\pi)^3 \sqrt{2k^0 2k'^0}} \int \frac{d^4 q}{(q^2 + m^2 - i\epsilon)((q-k)^2 + m^2 - i\epsilon)} \quad (13)$$

is quadratically divergent.

If the propagators (11) averaged over fluctuations $\Delta_{F,\sigma}(x-y)$ are used, then the amplitude $A(k, k')$ becomes a convergent integral

$$A_{\sigma}(k, k') = \frac{\lambda^2 \delta(k - k')}{2(2\pi)^3 \sqrt{2k^0 2k'^0}} \int d^4 q \frac{e^{-\sigma^2(q^2 + (q-k)^2)}}{(q^2 + m^2 - i\epsilon)((q-k)^2 + m^2 - i\epsilon)} \quad (14)$$

which at $\sigma = 0$ is the usual formula (13) and at $k = 0$ is

$$A_{\sigma}(0, k') = \frac{\lambda^2 \delta(k')}{4(2\pi)^3 m} \int d^4 q \frac{e^{-2\sigma^2 q^2}}{(q^2 + m^2 - i\epsilon)^2} = - \frac{im\lambda^2\delta(k')}{4\pi\sigma^2} e^{\sigma^2 m^2} K_1(\sigma^2 m^2). \quad (15)$$

## VACUUM ENERGY

The hamiltonian for a single scalar field is

$$H = \frac{1}{2} \int \left( \dot{\phi}^2(x) + (\nabla\phi(x))^2 + m^2\phi^2(x) \right) d^3x. \quad (16)$$

Its average value in the flat-space vacuum $|\mathcal{E}\rangle$ is

$$\langle \mathcal{E} | H | \mathcal{E} \rangle = \frac{1}{2} \int \langle \mathcal{E} | \dot{\phi}^2(x) + (\nabla\phi(x))^2 + m^2\phi^2(x) | \mathcal{E} \rangle d^3x \quad (17)$$

in which, for instance, $\phi^2(x)$ is the limit of $\phi(x)\phi(y)$ as $y \to x$. Using the averaged field (6), we find for the energy density

$$\rho_m = \int \frac{d^3 k}{2(2\pi)^3} \sqrt{m^2 + \mathbf{k}^2} e^{-\sigma^2 \mathbf{k}^2} = \frac{m^2}{16\pi^2 \sigma^2} e^{\sigma^2 m^2 / 2} K_1(\sigma^2 m^2 / 2). \quad (18)$$

The energy density due to a single fermionic mode of mass $m_f$ is $-\rho_{m_f}$.

## DARK ENERGY, DARK MATTER AND SUPERSYMMETRY

So if there are B bosonic modes each of mass $m_b$ and F fermionic modes each of mass $m_f$, then the density of dark energy is an alternating sum of the densities $\rho_{m_b}$ and $\rho_{m_f}$

$$\rho_{\text{de}}^{\text{th}} = \frac{1}{16\pi^2\sigma^2} \left( \sum_{b=1}^{B} m_b^2 e^{\sigma^2 m_b^2 / 2} K_1(\sigma^2 m_b^2 / 2) - \sum_{f=1}^{F} m_f^2 e^{\sigma^2 m_f^2 / 2} K_1(\sigma^2 m_f^2 / 2) \right). \quad (19)$$
(i = 0 to 3), the self-energy amplitude

$$A(k, k') = \frac{\lambda^2 \delta(k - k')}{2(2\pi)^3 \sqrt{2k^0 2k'^0}} \int \frac{d^4 q}{(q^2 + m^2 - i\epsilon)((q-k)^2 + m^2 - i\epsilon)} \quad (13)$$

is quadratically divergent.

If the propagators (11) averaged over fluctuations $\Delta_{F,\sigma}(x-y)$ are used, then the amplitude $A(k, k')$ becomes a convergent integral

$$A_{\sigma}(k, k') = \frac{\lambda^2 \delta(k - k')}{2(2\pi)^3 \sqrt{2k^0 2k'^0}} \int d^4 q \frac{e^{-\sigma^2(q^2 + (q-k)^2)}}{(q^2 + m^2 - i\epsilon)((q-k)^2 + m^2 - i\epsilon)} \quad (14)$$

which at $\sigma = 0$ is the usual formula (13) and at $k = 0$ is

$$A_{\sigma}(0, k') = \frac{\lambda^2 \delta(k')}{4(2\pi)^3 m} \int d^4 q \frac{e^{-2\sigma^2 q^2}}{(q^2 + m^2 - i\epsilon)^2} = - \frac{im\lambda^2\delta(k')}{4\pi\sigma^2} e^{\sigma^2 m^2} K_1(\sigma^2 m^2). \quad (15)$$

## VACUUM ENERGY

The hamiltonian for a single scalar field is

$$H = \frac{1}{2} \int \left( \dot{\phi}^2(x) + (\nabla\phi(x))^2 + m^2\phi^2(x) \right) d^3x. \quad (16)$$

Its average value in the flat-space vacuum $|\mathcal{E}\rangle$ is

$$\langle \mathcal{E} | H | \mathcal{E} \rangle = \frac{1}{2} \int \langle \mathcal{E} | \dot{\phi}^2(x) + (\nabla \phi(x))^2 + m^2 \phi^2(x) | \mathcal{E} \rangle d^3 x \quad (17)$$

in which, for instance, $\phi^2(x)$ is the limit of $\phi(x) \phi(y)$ as $y \to x$. Using the averaged field (6), we find for the energy density

$$\rho_m = \int \frac{d^3 k}{2(2\pi)^3} \sqrt{m^2 + \mathbf{k}^2} e^{-\sigma^2 \mathbf{k}^2} = \frac{m^2}{16\pi^2 \sigma^2} e^{\sigma^2 m^2 / 2} K_1(\sigma^2 m^2 / 2). \quad (18)$$

The energy density due to a single fermionic mode of mass $m_f$ is $-\rho_{m_f}$.

## DARK ENERGY, DARK MATTER AND SUPERSYMMETRY

So if there are B bosonic modes each of mass $m_b$ and F fermionic modes each of mass $m_f$, then the density of dark energy is an alternating sum of the densities $\rho_{m_b}$ and $\rho_{m_f}$

$$\rho_{\text{de}}^{\text{th}} = \frac{1}{16\pi^2\sigma^2} \left( \sum_{b=1}^{B} m_b^2 e^{\sigma^2 m_b^2 / 2} K_1(\sigma^2 m_b^2 / 2) - \sum_{f=1}^{F} m_f^2 e^{\sigma^2 m_f^2 / 2} K_1(\sigma^2 m_f^2 / 2) \right). \quad (19)$$
There are $B = 30$ known bosonic modes and $F = 96$ known fermionic modes if the three neutrinos are nearly massless Dirac neutrinos.

The fundamental length $\sigma$ is assumed to be very short, perhaps of the order of the Planck length, so in the formula (19) for the density of dark energy $\rho_{de}^{th}$, the first two terms in the expansion of the modified Bessel function

$$K_1(\frac{1}{2}\sigma^2 m^2) = \frac{2}{\sigma^2 m^2} + \frac{\sigma^2 m^2}{8} \left( 2\gamma - 1 + 2\log(\frac{1}{4}\sigma^2 m^2) \right) + \dots \quad (20)$$

are the most important. Here $\gamma = 0.5772156649$ is the Euler-Mascheroni constant.

Keeping only the first term in this expansion (20), we get for the density of dark energy

$$\rho_{de}^{th} \approx \frac{B-F}{8\pi^2\sigma^4} = -\frac{33}{4\pi^2\sigma^4} \quad (21)$$

which is negative and too great by many orders of magnitude. So if the theoretical density of dark energy $\rho_{de}^{th}$ is approximately equal to the observed density of dark energy $\rho_{de}^{obs}$, then we must have $B = F$ as in supersymmetry. Thus there must be at least 66 bosonic modes. These modes must be very heavy, dark or both.

If $B = F$, then the estimate (19) for $\rho_{de}^{th}$ and the approximation (20) give

$$\rho_{de}^{th} \approx \frac{1}{16\pi^2} \sum_{\alpha} (-1)^{2j\alpha} \left[ \frac{m_{\alpha}^2}{\sigma^2} + \left( \frac{m_{\alpha}^4}{8} + \frac{\sigma^2 m_{\alpha}^6}{16} \right) \left( 2\gamma - 1 + 2\log\left(\frac{1}{4}\sigma^2 m_{\alpha}^2\right) \right) \right]. \quad (22)$$

In models with softly broken supersymmetry, the alternating sum of squared masses vanishes $\sum_{\alpha} (-1)^{2j} m_{\alpha}^2 = 0$ at tree level leaving

$$\rho_{de}^{th} \approx \frac{1}{128\pi^2} \sum_{\alpha} (-1)^{2j\alpha} \left[ m_{\alpha}^4 \left( 2\gamma - 1 + 2\log\left(\frac{1}{4}\sigma^2 m_{\alpha}^2\right) \right) + \sigma^2 m_{\alpha}^6 \log\left(\frac{1}{4}\sigma^2 m_{\alpha}^2\right) \right]. \quad (23)$$

This estimate $\rho_{de}^{th}$ is generically orders of magnitude greater than the observed dark-energy density $\rho_{de}^{obs} = 2.74 \times 10^{-47} (\text{GeV})^4$ unless the alternating sum of quartic masses $\sum_{\alpha} (-1)^{2j} m_{\alpha}^4 = 0$ vanishes.

## CONCLUSION

Einstein's equation $G_{ik} = 8\pi G T_{ik}$ relates derivatives of points $p(x)$ to the energy-momentum tensor which is a quantum-mechanical operator. The points $p(x)$ of space therefore are quantum-mechanical operators. Their average values $\langle p(x) \rangle$ obey general relativity, and their remainders $\epsilon(x) = p(x) - \langle p(x) \rangle$ obey quantum mechanics.
There are $B = 30$ known bosonic modes and $F = 96$ known fermionic modes if the three neutrinos are nearly massless Dirac neutrinos.

The fundamental length $\sigma$ is assumed to be very short, perhaps of the order of the Planck length, so in the formula (19) for the density of dark energy $\rho_{\text{de}}^{\text{th}}$, the first two terms in the expansion of the modified Bessel function

$$K_1(\frac{1}{2}\sigma^2 m^2) = \frac{2}{\sigma^2 m^2} + \frac{\sigma^2 m^2}{8} \left(2\gamma - 1 + 2\log(\frac{1}{4}\sigma^2 m^2)\right) + \dots \quad (20)$$

are the most important. Here $\gamma = 0.5772156649$ is the Euler-Mascheroni constant.

Keeping only the first term in this expansion (20), we get for the density of dark energy

$$\rho_{\text{de}}^{\text{th}} \approx \frac{B-F}{8\pi^2\sigma^4} = -\frac{33}{4\pi^2\sigma^4} \quad (21)$$

which is negative and too great by many orders of magnitude. So if the theoretical density of dark energy $\rho_{\text{de}}^{\text{th}}$ is approximately equal to the observed density of dark energy $\rho_{\text{de}}^{\text{obs}}$, then we must have $B = F$ as in supersymmetry. Thus there must be at least 66 bosonic modes. These modes must be very heavy, dark or both.

If $B = F$, then the estimate (19) for $\rho_{\text{de}}^{\text{th}}$ and the approximation (20) give

$$\rho_{\text{de}}^{\text{th}} \approx \frac{1}{16\pi^2} \sum_{\alpha} (-1)^{2j\alpha} \left[ \frac{m_{\alpha}^2}{\sigma^2} + \left( \frac{m_{\alpha}^4}{8} + \frac{\sigma^2 m_{\alpha}^6}{16} \right) \left( 2\gamma - 1 + 2\log\left(\frac{1}{4}\sigma^2 m_{\alpha}^2\right) \right) \right]. \quad (22)$$

In models with softly broken supersymmetry, the alternating sum of squared masses vanishes $\sum_{\alpha} (-1)^{2j} m_{\alpha}^2 = 0$ at tree level leaving

$$\rho_{\text{de}}^{\text{th}} \approx \frac{1}{128\pi^2} \sum_{\alpha} (-1)^{2j\alpha} \left[ m_{\alpha}^4 \left( 2\gamma - 1 + 2\log\left(\frac{1}{4}\sigma^2 m_{\alpha}^2\right) \right) + \sigma^2 m_{\alpha}^6 \log\left(\frac{1}{4}\sigma^2 m_{\alpha}^2\right) \right]. \quad (23)$$

This estimate $\rho_{\text{de}}^{\text{th}}$ is generically orders of magnitude greater than the observed dark-energy density $\rho_{\text{de}}^{\text{obs}} = 2.74 \times 10^{-47} (\text{GeV})^4$ unless the alternating sum of quartic masses $\sum_{\alpha} (-1)^{2j} m_{\alpha}^4 = 0$ vanishes.

## CONCLUSION

Einstein's equation $G_{ik} = 8\pi G T_{ik}$ relates derivatives of points $p(x)$ to the energy-momentum tensor which is a quantum-mechanical operator. The points $p(x)$ of space therefore are quantum-mechanical operators. Their average values $\langle p(x) \rangle$ obey general relativity, and their remainders $\epsilon(x) = p(x) - \langle p(x) \rangle$ obey quantum mechanics.
In empty flat 3-space, a point $p(x)$ may be represented as the sum $\boldsymbol{p}(x) = \langle \boldsymbol{p}(x) \rangle + \boldsymbol{\epsilon}(x)$ of its average value $\langle \boldsymbol{p}(x) \rangle = \boldsymbol{x}$ and its quantum remainder $\boldsymbol{\epsilon}(x)$. The ubiquitous Fourier exponential $\exp(ik \cdot \boldsymbol{p}(x))$ is then $\exp(ik \cdot (\boldsymbol{x} + \boldsymbol{\epsilon}))$. If the probability distribution of the quantum remainders $\boldsymbol{\epsilon}(x)$ is normal and of width $\sigma$, then the average of the exponential $\langle \exp(ik(\boldsymbol{x}+\boldsymbol{\epsilon})) \rangle$ contains a gaussian factor $\exp(-\sigma^2\boldsymbol{k}^2/2)$ which makes self-energies, Feynman diagrams and dark energy ultraviolet finite. This complex of ideas requires dark matter and suggests that supersymmetry is softly broken.

ACKNOWLEDGMENTS

I am grateful to Rouzbeh Allahverdi, Michael Grady, David Kaiser and Malcolm Perry for conversations and email.

<footnote>* cahill@unm.edu</footnote>

<footnote>[1] R. E. Greene, Isometric embeddings of Riemannian and pseudo-Riemannian manifolds (American Mathematical Society, Providence, R.I., 1970) pp. iii+63, memoirs of the American Mathematical Society, No. 97.</footnote>

<footnote>[2] C. J. S. Clarke, On the global isometric embedding of pseudo-Riemannian manifolds, Proc. Roy. Soc. London Ser. A **314**, 417 (1970).</footnote>

<footnote>[3] O. Müller and M. Sánchez, Lorentzian manifolds isometrically embeddable in $\mathbb{L}^N$, Trans. Amer. Math. Soc. **363**, 5367 (2011).</footnote>

<footnote>[4] L. Aké, J. L. Flores, and M. Sánchez, Structure of globally hyperbolic spacetimes with timelike boundary, Rev. Matem. Iberoamericana, 45 (2020), arXiv:1808.04412 [gr-qc].</footnote>
In empty flat 3-space, a point $p(x)$ may be represented as the sum $\boldsymbol{p}(x) = \langle \boldsymbol{p}(x) \rangle + \boldsymbol{\epsilon}(x)$ of its average value $\langle \boldsymbol{p}(x) \rangle = \boldsymbol{x}$ and its quantum remainder $\boldsymbol{\epsilon}(x)$. The ubiquitous Fourier exponential $\exp(ik \cdot \boldsymbol{p}(x))$ is then $\exp(ik \cdot (\boldsymbol{x} + \boldsymbol{\epsilon}))$. If the probability distribution of the quantum remainders $\boldsymbol{\epsilon}(x)$ is normal and of width $\sigma$, then the average of the exponential $\langle \exp(ik(\boldsymbol{x}+\boldsymbol{\epsilon})) \rangle$ contains a gaussian factor $\exp(-\sigma^2\boldsymbol{k}^2/2)$ which makes self-energies, Feynman diagrams and dark energy ultraviolet finite. This complex of ideas requires dark matter and suggests that supersymmetry is softly broken.

ACKNOWLEDGMENTS

I am grateful to Rouzbeh Allahverdi, Michael Grady, David Kaiser and Malcolm Perry for conversations and email.

<footnote>* cahill@unm.edu</footnote>

<footnote>[1] R. E. Greene, Isometric embeddings of Riemannian and pseudo-Riemannian manifolds (American Mathematical Society, Providence, R.I., 1970) pp. iii+63, memoirs of the American Mathematical Society, No. 97.</footnote>

<footnote>[2] C. J. S. Clarke, On the global isometric embedding of pseudo-Riemannian manifolds, Proc. Roy. Soc. London Ser. A **314**, 417 (1970).</footnote>

<footnote>[3] O. Müller and M. Sánchez, Lorentzian manifolds isometrically embeddable in $\mathbb{L}^N$, Trans. Amer. Math. Soc. **363**, 5367 (2011).</footnote>

<footnote>[4] L. Aké, J. L. Flores, and M. Sánchez, Structure of globally hyperbolic spacetimes with timelike boundary, Rev. Matem. Iberoamericana, 45 (2020), arXiv:1808.04412 [gr-qc].</footnote>