Tabletop experiments for quantum gravity: a user's manual

The most basic prediction of the EFT picture is that massive objects will source gravitational fields tied to their center-of-mass positions. If we prepare a source mass in a non-classical state and bring in a test mass, gravity can entangle the two masses, and it is this effect that we could like to witness. We begin by explaining this idea using resonator systems, and then review the matter-wave variant proposed in [7, 8].

Throughout this section, we will focus on the entanglement generated between a pair of such masses. Our goal is to give a simple description of the main points of principle and to estimate the size of the gravitational signal. Thus our models will not involve any noise backgrounds, non-gravitational decoherence channels, mechanical dissipation, or even any non-gravitational interactions between the masses. In a practical implementation these of course all need to be understood and somehow subtracted out or otherwise accounted for.

Consider a pair of harmonic oscillators of equal mass m and frequency, as in figure 4. $\omega$. In the non-relativistic limit, graviton exchange leads to an effective Newtonian potential (3) between the masses. Thus the Hamiltonian reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle H = \sum_{i=1,2} \frac{\mb{p}_i^2}{2m} + \frac{1}{2} m \omega^2 \mb{x}_i^2 - \frac{G_{\rm N} m^2}{|\mb{d} -(\mb{x}_1 - \mb{x}_2)|}. \nonumber \end{align} \tag{ 4 }$$

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Here $\newcommand{\mb}{\mathbf} \mb{d}$ is the vector between the two equilibrium positions of the oscillators. Assuming that $\newcommand{\mb}{\mathbf} |\mb{d}| \gg |\mb{x}_1 - \mb{x}_2|$, we can expand out the denominator in a multipole expansion⁶. The zeroth order term is an overall constant and may be dropped, the first order term is proportional to $\newcommand{\mb}{\mathbf} \mb{x}_1 - \mb{x}_2$ and can thus be eliminated by re-definition (at order $G_{\rm N}$) of the equilibrium positions, and the second-order term contains both $\newcommand{\mb}{\mathbf} \mb{x}_1^2 + \mb{x}_2^2$ and $\newcommand{\mb}{\mathbf} \mb{x}_1 \cdot \mb{x}_2$. The squared terms can similarly be eliminated by re-definition (at order $G_{\rm N}$) of the oscillator frequencies. The order $G_{\rm N}$ corrections are negligible and we drop them. For further simplicity we will restrict to a single spatial dimension, by e.g. appropriately designing the oscillators or sharply trapping the particles in the transverse directions. Finally, we take the rotating wave approximation, which here is excellent due to the tiny coupling, allowing us to drop the interaction terms $\newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger}a_1 a_2 + a_1^{\dagger} a_2^{\dagger}$. In the end, we obtain the Hamiltonian

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle \label{2osc-H} H = \sum_{i=1,2} \hbar \omega a_i^{\dagger} a_i - \hbar \lambda_g (a_1 a_2^{\dagger} + a_1^{\dagger} a_2) \nonumber \end{align} \tag{ 5 }$$

where the coupling constant is

Here $x_{\rm ZPF} = \sqrt{\hbar/2 m \omega}$ is the ground-state uncertainty in the harmonic oscillator position. Note that the ratio m/d³ can be no larger than the material density of the resonators.

The Hamiltonian (5) manifestly generates entanglement between the oscillators. As a simple example, consider preparing one oscillator in its ground state and the other in its first excited Fock state, that is $\renewcommand{\ket}[1]{|#1\rangle} \ket{\psi} = \ket{10}$. Treating the gravitational interaction as a perturbation, this state will evolve to

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\Oi}{\mathcal{O}} \displaystyle \label{10example} \ket{10} \to \ket{10} - {\rm i} \lambda_g \Delta t \ket{01} + \Oi(\lambda_g^2) \nonumber \end{align} \tag{ 7 }$$

in a time $\Delta t$, up to an overall normalization. The state (7) is entangled, in the sense that it cannot be reduced to a product state of the two oscillators. The amount of entanglement is small, since it is proportional to the product $\lambda_g \Delta t$, but it could in principle be measured by a DLCZ-style scheme [59]. For example, if these oscillators are mirrors in an optical cavity, we can apply a red-detuned optical pulse, destroying the joint oscillator phonon state and mapping it onto a pair of optical photons, whose interference visibility can be measured using standard optics techniques. This technique has been successfully used to demonstrate entanglement in a pair of oscillators in the evenly-weighted state $\renewcommand{\ket}[1]{|#1\rangle} \ket{1\,0} + \ket{0\,1}$ generated using photon–phonon couplings [60].

A more dramatic example would be to prepare the source oscillator not in a low-lying Fock state like $\renewcommand{\ket}[1]{|#1\rangle} \ket{1}$ but a highly non-classical state like a 'cat code' $\renewcommand{\ket}[1]{|#1\rangle} \ket{\alpha} + \ket{-\alpha}$. [29] Here $\renewcommand{\ket}[1]{|#1\rangle} \ket{\alpha}$ is a coherent state, which has average spatial displacement $\newcommand{\mb}{\mathbf} \renewcommand{\bra}[1]{\langle#1|} \renewcommand{\braket}[1]{\langle#1\rangle} \braket{x} = {{\rm Re}}(\alpha)$ from the oscillator equilibrium. Consider preparing the first oscillator in this state and the second oscillator in its ground state $\renewcommand{\ket}[1]{|#1\rangle} \ket{0}$. We can easily find the exact time-evolution of the joint state without resorting to perturbation theory. Defining as usual the normal modes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle x_{\pm} = \frac{x_1 \pm x_2}{\sqrt{2}} \nonumber \end{align} \tag{ 8 }$$

the Hamiltonian can be written

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle \label{H-2osc-rwa} H = \omega_+ a_+^{\dagger} a_+ + \omega_- a_-^{\dagger} a_-, \ \ \ \omega_{\pm} = \omega \pm \lambda_g. \nonumber \end{align} \tag{ 9 }$$

Using this transformation, it is straightforward to show that a coherent state times the ground state evolves as

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \ket{\alpha}_1 \ket{0}_2 & \approx \ket{\alpha/\sqrt{2}}_+ \ket{-\alpha/\sqrt{2}}_- \nonumber \\ & \to \ket{{\rm e}^{-{\rm i} \omega_+ t} \alpha/\sqrt{2}}_+ \ket{-{\rm e}^{-{\rm i} \omega_- t} \alpha/\sqrt{2}}_- \nonumber \\ & \approx \ket{\alpha_t \cos \lambda_g t}_1 \ket{-{\rm i} \alpha_t \sin \lambda_g t}_2.\nonumber \end{align} \tag{ 10 }$$

Here we defined $\alpha_t = \alpha {\rm e}^{-{\rm i} \omega t}$, and approximated $\omega_{\pm} \approx \omega$ in the widths of the ground states of the $\pm$ modes; this creates errors in the final answers only at second order in $\lambda_g$. Extending this calculation by linearity, we see that the initial cat-vacuum state

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \ket{\psi} = \frac{\ket{\alpha} + \ket{-\alpha}}{\sqrt{2}} \otimes \ket{0} \nonumber \end{align} \tag{ 11 }$$

will evolve to

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \label{2-osc-state-final} \ket{\psi(t)} = \left(\ket{\alpha_t \cos \lambda_g t}\ket{-{\rm i} \alpha_t \sin \lambda_g t} + \ket{-\alpha_t \cos \lambda_g t} \ket{{\rm i} \alpha_t \sin \lambda_g t} \right)/\sqrt{2}. \nonumber \end{align} \tag{ 12 }$$

This state is a highly non-classical entangled state; for large $\newcommand{\mb}{\mathbf} {{\rm Re}}(\alpha)$ the two branches are essentially orthogonal. This also plainly demonstrates the transfer of quantum information from one oscillator into the other: for example, at $t = \pi/2\lambda_g$, the initial state of the first oscillator has been completely swapped (up to some phases) into the second oscillator.

The above examples show how the Newtonian force in the EFT picture of gravity acts as a 'quantum' interaction. What this means precisely is that the interaction is capable of transmitting quantum information: information about a superposition of one system can be mapped into another system, and so forth. This is in stark contrast to known models in which a classical gravitational field somehow couples to quantum matter, as we will see in section 4. More technically, this is the statement that the Newtonian interaction $V = Gm^2/|r_1-r_2|$ is an operator acting on the joint Hilbert space of two masses. Note that these results are not confined to the non-relativistic regime. Quite generally, as a consequence of Einsteins equations, the gravitational field is constrained to follow the matter field; and because of the equivalence principle, it does so in a way independent of the nature of the matter field. Thus the gravitational field must engage in any superpositions that the matter field happens to be engaged in⁷.

Having explained the basic idea of gravitational entanglement generation using resonator systems, we now remark briefly on a recent, concrete proposal to study the same phenomenon in a matter-wave system [7, 8]. Consider a pair of matter-wave interferometers placed next to each other as in figure 5. The matter waves consist of massive objects which have a spin-1/2 degree of freedom which can be used to both control and read out the spatial state of the masses; in the proposal [7] the masses are nitrogen-vacancy diamonds (with the NV site providing the spin) and the 'beam splitting' action is accomplished by engineered magnetic fields coupling to the spin [35]. We denote the beam arms $\renewcommand{\ket}[1]{|#1\rangle} \ket{L} = \ket{\uparrow}, \ket{R} = \ket{\downarrow}$ in the notation of section 2. Neglecting all interactions except for the Newtonian interaction between the two matter waves, the total time evolution produces a state of the form

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \label{bose-result} \ket{LL} \to {\rm e}^{{\rm i} \phi_{LL}} \ket{LL} + {\rm e}^{{\rm i} \phi_{LR}} \ket{LR} + {\rm e}^{{\rm i} \phi_{RL}} \ket{RL} + {\rm e}^{{\rm i} \phi_{RR}} \ket{RR}, \nonumber \end{align} \tag{ 13 }$$

with the phases $\phi_{ij} = G_{\rm N} m^2\Delta t/\hbar d_{ij}$, d_ij the spatial distances between the various pairs of beam arms. For simplicity consider a geometry for the interferometers such that $\phi_{LL} = \phi_{RR} = \phi_{LR} \neq \phi_{RL}$ (for example, the geometry in figure 5), so that we may write this as

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \ket{LL} \to \ket{LL} + \ket{LR} + {\rm e}^{{\rm i} \Delta \phi} \ket{RL} + \ket{RR}, \nonumber \end{align} \tag{ 14 }$$

up to an overall phase, with the differential phase of order

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Here d is the separation between the two interferometers, $\Delta x$ the separation between the beam arms in a single interferometer, $\Delta t$ the time of flight, and we have taken the approximation $\Delta x/d \ll 1$ (in terms of the geometry in figure 5, we are approximating $d_S \approx d, d_L \approx d + \Delta x$). This is manifestly an entangled state of the two masses: it is not a product state⁸. The entanglement can be verified by measuring the spin states, for example by looking for violations of a Bell inequality: if the entanglement witness

$$\begin{align} \renewcommand{\braket}[1]{\langle#1\rangle} \renewcommand{\bra}[1]{\langle#1|} \newcommand{\e}{{\rm e}} \displaystyle W = | \braket{\sigma_x^1 \sigma_z^2 - \sigma_y^1 \sigma_z^2} | \leqslant 1 \nonumber \end{align} \tag{ 16 }$$

then the state is provably entangled.

One of the most striking features about perturbatively quantized general relativity is the existence of low-energy fluctuations in the field: gravitons. There are strong arguments that it will be impossible to ever directly detect a single graviton, even in principle-it would generally require a detector so large and massive that a black hole would form [62–64]. However, there is an alternative: perhaps one could infer the existence of the graviton, or more precisely the ability of systems to spontaneously emit gravitons, by observing decoherence consistent with information lost into graviton states [65–68].

Unfortunately, as we review in this section, the rates of graviton-induced decoherence are extraordinarily low compared to more pedestrian channels like electromagnetic radiative damping, at least for small masses. Our primary purpose here is to explain the nature of these effects 'in principle', and in particular to demonstrate that standard, quantized gravitons can lead to decoherence in precise analogy with photons. This decoherence must be distinguished from the ad hoc 'gravitational decoherence' mechanisms discussed in section 5, which generally speaking have enormously higher rates than that coming from gravitons.

To understand the relative size of photonic versus gravitonic effects, we can estimate the rates of spontaneous emission of either a photon or a graviton. The rate for spontaneous emission of photons should depend on the system's electric dipole made into a scalar $\newcommand{\mb}{\mathbf} \mb{d}^2$, the speed of light c, Planck's constant $\hbar$, and the system's frequency of oscillation $\omega$. Forming a rate from these quantities, on dimensional grounds, gives the estimate

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Gamma_{\rm EM} \sim \mu_0 \frac{d^2 \omega^3}{\hbar c} \nonumber \end{align} \tag{ 17 }$$

for the number of photons emitted per unit time, where $\mu_0$ is the vacuum permeability. For spontaneous emission of a graviton, since we are dealing with spin-2 emission instead of spin-1, we should have a dependence on the quadrupole moment Q², and furthermore we should expect a factor of $G_{\rm N}$ from the graviton coupling to matter. Again dimensional analysis gives

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Gamma_{\rm GR} \sim \frac{G_{\rm N} Q^2 \omega^5}{\hbar c^5}. \nonumber \end{align} \tag{ 18 }$$

To get a conservative and concrete estimate, we can approximate the dipole as consisting of N elementary dipoles d  =  NeL, with L the linear dimension of the object and e the electron charge. Similarly we approximate the mass quadrupole Q  =  mL². Then we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{em-gr-power} \frac{\Gamma_{\rm EM}}{\Gamma_{\rm GR}} \sim \frac{\mu_0 N^2 e^2 c^4}{G_{\rm N} m^2 L^2 \omega^2}. \nonumber \end{align} \tag{ 19 }$$

To get a feel for this, consider an object with ; this gives $\Gamma_{\rm EM}/\Gamma_{\rm GR} \sim N^2 \times 10^{23}$. In other words, even for a fairly aggressive choice of experimental parameters, the electromagnetic radiation absolutely swamps any graviton-based effects. However, it should be noted that for very large masses the gravitational rate can actually start to dominate [69]. Furthermore, a real experiment would not use a material with a large number of permanent electric dipoles but rather something like a neutral dielectric, which would significantly suppress the electric radiation. Nevertheless, a realistic experiment looking for spontaneous emission seems exceedingly difficult to imagine. Beyond this order-of-magnitude estimate, we refer the reader to detailed calculations of spontaneous emission in a neutron quantum bouncer [70], simple harmonic oscillator [68], and especially the wonderful review of the $3d \to 1s$ transition in hydrogen presented in [63, 64].

Another possibility would be to look for thermalization due to an ambient graviton background. Indeed, little is known about the precise nature of the ambient graviton background, and observing decoherence due to this background would be remarkable, to put it mildly. A simple estimate suggests that, if the early universe went through a period of inflation, a background of thermalized gravitons at should be universally present, if nothing else [54]. Stars, in particular the sun, also emit gravitons at reasonable rates; a simple estimate leads to a flux of about 10⁻⁴ gravitons per square centimeter per second on the Earth [62, 71]. To estimate the thermalization rate, we can again use dimensional analysis: the rate $\Gamma^{\rm dec} \sim c n_{\rm grav} \sigma$, where $n_{\rm grav}$ is the number density of the graviton background, and $\sigma$ is an effective graviton absorption cross-section. The cross section can be conservatively estimated by with $L_{\rm p}$ the Planck length [62], so for example solar gravitons lead to a decoherence rate , which is utterly negligible.

One might also consider decoherence caused by the bremsstrahlung of gravitons during the acceleration of a massive object. For example, imagine using a matter-wave beam of particles of mass m and sending these through a beam-splitter. We can think of the beam-splitter as the unitary which implements a simple 90-degree rotation of the incoming momentum vector:

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \ket{p \hat{x}} \to \ket{p \hat{x}} + \ket{p \hat{y}} \nonumber \end{align} \tag{ 20 }$$

where the kets are momentum eigenstates with momenta p aligned along either the x or y axis; see figure 2. The process $p \hat{x} \to p \hat{y}$ involves acceleration of the beam, and thus will generate bremsstrahlung in both gravitons and, if the beam is electrically charged, photons. Thus in actuality, the beamsplitter acts as

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \ket{p \hat{x} , 0} \to \ket{p \hat{x}, 0} + \sum_{\gamma,h} c_{\gamma,h} \ket{p \hat{y} , \gamma,h} \nonumber \end{align} \tag{ 21 }$$

where the 0 indicates the vacuum state of the radiation field, $\gamma,h$ represent some possible photon and graviton radiation, and the $c_{\gamma,h}$ are the amplitudes for each of these states.

The radiation spectrum has a divergent infrared tail: an arbitrarily small amount of energy can be radiated out into an arbitrarily large number of very low-energy bosons [71–73]. These in turn cannot be measured by a finite-energy detector, and should be traced. This divergence is regulated by the temporal duration $\tau$ of the experiment, because a boson with wavelength $\lambda \gtrsim c \tau$ cannot be produced. One can perform the trace to all orders in the number of photons and gravitons [74], and one finds that the off-diagonal component of the density matrix decays like

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{LR}(\tau) \sim \rho_{LR}(0) \exp \left\{- \left(\frac{p}{P_{\rm p}} \right)^2 \ln \left(\frac{m c^2 \tau}{\hbar} \right) \right\}. \nonumber \end{align} \tag{ 22 }$$

Here is the Planck momentum, and we have ignored the photon contributions-this is the pure graviton decoherence rate. For a typical matter-wave experiment we have , and so this exponential is astronomically close to being equal to unity. In other words, the decoherence is entirely negligible. To get a non-negligible contribution, one basically needs $\newcommand{\Oi}{\mathcal{O}} p/P_{\rm pl} \sim \Oi(1)$, i.e. momenta of order , which seems rather difficult to achieve. One can compare this with the electromagnetic version, in which we replace $p/P_{\rm pl} \to N^2 \alpha$ with $\alpha$ the fine structure constant and Ne is the charge of the beam particles; in this case, it is easy to imagine an experiment with non-negligible decoherence from photon bremsstrahlung.

Clearly, observing decoherence caused by gravitons will be quite challenging. Still, there seems to be room for new ideas: although detecting a single graviton can be fairly conclusively deemed impossible on simple grounds, here we have only sketched some very rudimentary schemes for decoherence-based detection. One obvious avenue may be to engineer a system with a particularly large mass quadrupole and minimal electromagnetic couplings. Another could be to exploit a Dicke-superradiant state of some system to enhance the spontaneous graviton emission rate [75]. We look forward to the development of new ideas in this vein, but for the time being, now turn our attention to gravitational models beyond the effective quantum field theory picture of general relativity.

The first attempts to grapple with the challenges noted above were in early remarks by Feynman [9, 10], and in an analysis of uncertainty relations involving masses coupled to gravity by Karolyhazy [115]. However, no attempt was made in this early work to provide any kind of alternative theory. The first serious attempt at such a theory was made by Kibble et al in [78, 79]. Quite apart from the details, three key points were made by Kibble et al, which have just as much force today. These were:

• (i)  
  Any semiclassical theory of gravity (in which the matter fields are all quantized, but the metric is taken to be classical in a sense to be specified) will differ in its predictions from a fully quantized theory in which both the matter fields and the metric are quantized together.In the literature, a semiclassical theory is, as before, typically taken to mean one for which the relation between the Einstein tensor and the stress–energy tensor is given by

  $$\begin{align} \renewcommand{\braket}[1]{\langle#1\rangle} \renewcommand{\bra}[1]{\langle#1|} \newcommand{\e}{{\rm e}} \displaystyle \label{scg-eq1} G_{\mu\nu} = \frac{8 \pi G_{\rm N}}{c^4} \braket{T_{\mu\nu}} \nonumber \end{align} \tag{ 34 }$$

  in which $\renewcommand{\bra}[1]{\langle#1|} \renewcommand{\braket}[1]{\langle#1\rangle} \braket{T_{\mu\nu}}$ is the expectation value of the quantum-mechanical $T_{\mu\nu}$. A simple example of the predictions for such a model were given in the previous section; let us put this in more general terms. Consider a two-slit experiment, in which the two paths for a mass M are significantly separated in space. In a theory with a quantized metric, such a superposition would take the form

  $$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \ket{\Psi} = a_L \ket{\Phi_L; g^{(L)}_{\mu \nu}} + a_R \ket{\Phi_R; g^{(R)}_{\mu \nu}}, \label{GRsup} \nonumber \end{align} \tag{ 35 }$$

  with $L,R$ denoting the two paths. In this superposition, those gravitational degrees of freedom that are tied to matter in the gravitational part $\renewcommand{\ket}[1]{|#1\rangle} \ket{g_{\mu \nu}}$ of the state vector are then completely entangled with the matter state $\renewcommand{\ket}[1]{|#1\rangle} \ket{\Phi}$. If M is small enough to treat these metrics as perturbations around flat spacetime, then this description is just what was given in the EFT picture above.Suppose we try to measure which path the mass M has followed using, for example, another mass m to measure the metric perturbation associated with the mass M. It is immediately obvious that in a fully quantized theory, the secondary 'apparatus' mass m will be deflected differently depending on which path the mass M follows: we will end up with a superposition

  $$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \label{GRsup2} \ket{\Psi_f} = a_L \ket{\Phi_L; g_{(L)} ; \chi_{(L)}} + a_R \ket{\Phi_R; g_{(R)} ;\chi_{(R)}} \nonumber \end{align} \tag{ 36 }$$

  where the apparatus mass states $\renewcommand{\ket}[1]{|#1\rangle} \ket{\chi_{(L)}}$, $\renewcommand{\ket}[1]{|#1\rangle} \ket{\chi_{(R)}}$ have the mass m deflected in different directions by the fields g_(L), g_(R). A standard projective measurement done jointly on the two masses would then see the apparatus deflected one way or the other in correlation with the source, with probabilities $|a_L|^2$ and $|a_R|^2$ respectively.On the other hand in a semiclassical theory where (34) is satisfied, we get a quite different result. The expectation value of the source mass's stress tensor is a classical sum of two locations. For example, suppose that $a_L = a_R = 1/\sqrt{2}$ in (35) and the apparatus is placed directly between the two source paths $L,R$. Then the apparatus mass will be pulled equally toward each path, and will remain undeflected. In particular, absolutely no entanglement is generated between the source and apparatus, as discussed above.Other writers have commented on this point of Kibble's [11, 116], and it was even claimed [11] that measurements have already decided in favour of the superposition (35). This claim is controversial for several reasons [117–119]; from our point of view the chief problem is that no sources of environmental decoherence sensitive to the path of the mass M are considered in any of the above, and such decoherence (which can come from many sources, including photons, phonons, and gas atoms) is likely to be large in any experiment where M is large enough to have appreciable gravitational effects.
• (ii)  
  Any semiclassical theory of gravity must necessarily involve non-linear time evolution in the matter fields, and hence break the superposition principle. This means, for example, that in the non-relativistic limit we will be dealing with a generalized, non-linear Schrödinger equation of form

  $$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle (\hat{{\mathcal H}} - {\rm i}\hbar \partial_t) \psi({\bf r},t) = f(\psi({\bf r},t), \psi^{\dagger}({\bf r},t)) \label{NLSE} \nonumber \end{align} \tag{ 37 }$$

  where $\newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger}f(\psi({\bf r},t), \psi^{\dagger}({\bf r},t))$ is some arbitrary function of the matter wave-function and its conjugate, whose origin is supposed to be gravitational (and so negligible for microscopic masses). Equations like this can be readily generalized to relativistic fields (see [78, 79] for details), where the non-linear term now depends on the expectation value of the field.However, irrespective of the theory involved, equations like these run into severe difficulties. It was noted early on [120] that the rescaling of the wave-function during measurements required a specific logarithmic form for the potential, and it is hard to make theories of this kind consistent if one also assumes conventional ideas about quantum measurements. Further investigations of non-linear Schrödinger equations by Weinberg [121, 122] uncovered similar problems. The work of Weinberg, although it had nothing to do with gravitation, had two great virtues: it encapsulated many rather general features that such non-linear generalizations of Schrödinger's equation should possess, and at the same time produced a specific theory for non-linearities at the microscopic scale which was experimentally testable (and indeed it was falsified within a year [123–125]).Further difficulties afflicting any non-linear quantum theory were also found by Polchinski and others [126–129]; these included non-violation of Bell inequalities, and superluminal signal propagation. In particular, the Schrödinger–Newton model admits an explicit protocol for superluminal signalling [84]. We note, however, that such effects would be quite invisible in experiments done so far, if the source of the non-linearity is assumed to be gravitational.
• (iii)  
  It seems very difficult to derive a consistent theory including gravity, whether it be semiclassical or fully quantized, if one also tries to use the traditional quantum-mechanical framework of measurements, operators, observables, and states in Hilbert space. As noted already, this problem arises clearly in the non-linear Schrödinger treatments, where normalization problems occur when measurement operations occur, and the usual Born rule is violated. It is certainly generic to any semiclassical treatment, which is inconsistent with the conventional wave function collapse—our example above showed this. Note that this problem is not obviated by assuming an 'epistemic' interpretation for the wave-function, according to which $\psi$ only represents an observer's information about the world, rather than any 'real' state it may have. This is because the non-linearity of the time evolution of $\psi$ is sourced by $\renewcommand{\bra}[1]{\langle#1|} \renewcommand{\braket}[1]{\langle#1\rangle} \braket{T_{\mu\nu}}$, which cannot be observer dependent or undergo sudden 'wave function collapse'; for more discussion of this point, see remarks by Unruh [116] and Kibble [130].

In spite of all these problems, the semiclassical model can serve as a useful heuristic picture, and one can imagine attempting to test it in some limited sense, in which we simply accept the interpretation of $\renewcommand{\ket}[1]{|#1\rangle} \ket{\psi}$ as a wavefunction and blindly apply the Born rule. A basic prediction of this type is the lack of entanglement production in the matter wave experiment as discussed in section 4. Another early proposal, described by Carlip and Salzmann in 2006, was to look for a loss of coherence in matter-wave Talbot–Lau interferometry [81, 82, 131], and more recently there have been proposals to look for distortions in the energy spectrum of optically trapped nanoparticles [132–134].

The initial formulation of quantum mechanics involved a fundamental split between the classical and quantum worlds, with the associated 'Copenhagen interpretation' providing the glue between the two. In an attempt to make sense of this, von Neumann in 1932 proposed the idea of a succession of entanglements between ever larger objects (the 'von Neumann chain') truncated by a probabilistic 'wave function collapse' into one particular state [135]. The state in (36) is a typical von Neumann state, and one supposes that one has the transition

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \displaystyle \ket{\Psi_{\rm in}} = \ket{\Phi_o} \sum_k c_k \ket{\phi_k} \rightarrow \sum_k c_k \ket{\Phi_k; \phi_k^{\prime}} \label{vNmmt} \nonumber \end{align} \tag{ 38 }$$

where the initial 'system' state $\renewcommand{\ket}[1]{|#1\rangle} \sum\nolimits_k \ket{\phi_k}$ couples to the initial 'apparatus' state $\renewcommand{\ket}[1]{|#1\rangle} \ket{\Phi_o}$ in such a way that these two systems entangle with a perfect correlation between the initial system states $\renewcommand{\ket}[1]{|#1\rangle} \ket{\phi_k}$ and the apparatus states $\renewcommand{\ket}[1]{|#1\rangle} \ket{\Phi_k}$, where we allow the final system states $\renewcommand{\ket}[1]{|#1\rangle} \ket{\phi_k^{\prime}}$ to be different from the initial ones. This perfect correlation is the defining property of a perfect measurement operation (in the basis frame of the $\{\phi_k \}$). However, to then engineer a definite final state with probability $P_k = |c_k|^2$, we require a 'collapse' of the superposition into just one of its branches, the kth component. The lack of any theory whatsoever for how this collapse occurs is the famous measurement problem.

In what follows we only discuss ideas that have attempted to get around this problem using a gravitational mechanism. There are basically two of these, stochastic collapse models invoking a gravitational noise source, and Penrose's gravitational collapse model. In their simplest forms, these models also lead to a kind of semiclassical model of gravitationally-induced wavefunction collapse [136, 137]. Because this leads to non-linearity in the Schrödinger equation, it is then clear that this idea will have problems of the kind just discussed [84]. Thus any theory of this kind has to address problems of internal consistency.

5.2.1. Stochastic collapse models.

Stochastic models of quantum mechanics have been advanced for a variety of reasons over a long period of time. In the non-relativistic limit these models have an equation of motion of general form

$$\begin{align} \renewcommand{\braket}[1]{\langle#1\rangle} \renewcommand{\bra}[1]{\langle#1|} \newcommand{\e}{{\rm e}} \displaystyle (\hat{{\mathcal H}} - {\rm i}\hbar \partial_t) \psi({\bf r},t) = \xi(\braket{\psi | F(\hat{O}_j(t)) | \psi}, t) \label{Sch-var} \nonumber \end{align} \tag{ 39 }$$

where F is some arbitrary function, and the argument of the 'noise' term $\xi(\langle F \rangle)$ involves a set of operators $\hat{O}_j(t)$ acting in the Hilbert space of the system. Work of this kind [138], motivated by the quantum measurement problem, almost always deals with non-relativistic QM, and assumes the usual QM structure of state vectors, Hilbert space, and projective quantum measurements. The noise term may depend on $|\psi \rangle$, making (39) non-linear. Note that in contrast to any uncertainty principle arguments [115], this approach definitely leads to decoherence, and 'wave-function collapse' caused precisely by the stochastic noise field.

Diosi [91], Ghirardi et al [139] and Pearle [140] have discussed the possibility of a gravitational origin for this noise (here we will follow the treatment of Diosi). In their models, the state of the matter system is taken to evolve according to a Schrödinger equation with a 'noise' potential, superficially similar to the starting point of the Schrödinger–Newton equation (27). One has

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle {\rm i} \hbar \partial_t \ket{\psi} = \left(H_{\rm mat} + \int {\rm d}^3\mb{x} \ M(\mb{x}) \Phi(\mb{x}) \right) \ket{\psi}, \nonumber \end{align} \tag{ 40 }$$

where $\newcommand{\mb}{\mathbf} M(\mb{x})$ is a mass density operator for the system, similarly to the mass density operator in section 4. In the SN model, the potential $\Phi$ is a semiclassical quantity sourced by the state $\renewcommand{\ket}[1]{|#1\rangle} \ket{\psi}$ itself. In the noise model, on the other hand, this potential is a stochastic quantity independent of $\renewcommand{\ket}[1]{|#1\rangle} \ket{\psi}$, whose statistical correlations involve the gravitational propagator:

$$\begin{align} \renewcommand{\braket}[1]{\langle#1\rangle} \renewcommand{\bra}[1]{\langle#1|} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle \braket{\Phi(\mb{x},t)} = 0, \ \ \ \braket{\Phi(t_1,\mb{x}) \Phi(t_2,\mb{y})}= G_{\rm N} \frac{\delta(t_1-t_2)}{|\mb{x}-\mb{y}|}, \nonumber \end{align} \tag{ 41 }$$

where $\renewcommand{\bra}[1]{\langle#1|} \renewcommand{\braket}[1]{\langle#1\rangle} \braket{\cdots}$ represents an average over realizations of the noise. Performing this average, we obtain an evolution equation for the system density matrix:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle \partial_t \rho = -{{\rm i} \over \hbar} [H_{\rm mat},\rho] - \frac{G_{\rm N}}{2} \int {\rm d}^3\mb{x} {\rm d}^3\mb{y} \frac{[M(\mb{x}),[M(\mb{y}),\rho]]}{|\mb{x}-\mb{y}|}. \label{diosiD} \nonumber \end{align} \tag{ 42 }$$

This equation leads to decoherence in position space, i.e. decay of the off-diagonal density matrix elements. Our definition of the mass density operator in section 4 involves pointlike mass distributions. This causes the integral in (42) to diverge! To remove this divergence, the massive objects were taken to have a uniform density. For the case of a spherically symmetric, uniform sphere of radius R₀, one gets a 'smeared' mass density operator

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle M(\mb{x}) = \frac{m}{4 \pi R_0^3/3} \int {\rm d}^3\mb{x}' \theta(R_0 - |\mb{x} - \mb{x}'|) \delta(\mb{x} - \hat{\mb{r}}), \label{massO} \nonumber \end{align} \tag{ 43 }$$

where $\newcommand{\mb}{\mathbf} \hat{\mb{r}}$ means the usual single-particle position operator, and $\theta$ is a step function. Then a position eigenstate $\newcommand{\mb}{\mathbf} \renewcommand{\ket}[1]{|#1\rangle} \ket{\mb{x}_0}$ of the particle is an eigenstate of this mass operator, with eigenvalue $\newcommand{\mb}{\mathbf} M(\mb{x} | \mb{x}_0)$ given by

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle M(\mb{x}) \ket{\mb{x}_0} = M(\mb{x} | \mb{x}_0) \ket{\mb{x}_0} = \frac{m}{4\pi R_0^3/3} \theta(R_0 - |\mb{x}-\mb{x}_0|) \ket{\mb{x}_0}. \nonumber \end{align} \tag{ 44 }$$

With these assumptions in hand, one finds that the off-diagonal position-space density matrix elements decay exponentially in time

$$\begin{align} \renewcommand{\braket}[1]{\langle#1\rangle} \renewcommand{\bra}[1]{\langle#1|} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle \rho_{\mb{x}\mb{y}}(t) = \braket{\mb{x} | \rho(t) | \mb{y}} \sim \rho_{\mb{x}\mb{y}}(0) {\rm e}^{-\Gamma_{\rm PD}(\mb{x},\mb{y}) t} \nonumber \end{align} \tag{ 45 }$$

with rate

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle \label{PDrate} \Gamma_{\rm PD}(\mb{x},\mb{y}) = \frac{G_{\rm N}}{2 \hbar} \int {\rm d}^3\mb{x'} {\rm d}^3\mb{y'} \frac{\left[M(\mb{x}' | \mb{x}) - M(\mb{x}' | \mb{y}) \right] \left[M(\mb{y}' | \mb{x}) - M(\mb{y}' | \mb{y}) \right]}{|\mb{x}'-\mb{y}'|}. \nonumber \end{align} \tag{ 46 }$$

We see that superpositions between different positions $\newcommand{\mb}{\mathbf} \mb{x} \neq \mb{y}$ will be damped in time at a rate proportional to the Newton constant.

The decoherence rate (46) has some simple features that can be read off directly. For one thing, if there is no superposition-i.e. $\newcommand{\mb}{\mathbf} \mb{x} = \mb{y}$-we clearly have no decoherence, that is $\Gamma = 0$. Another feature is the behavior in the limit of a 'wide' superposition $\newcommand{\mb}{\mathbf} | \mb{x} - \mb{y} | \to \infty$. In (46), there are two types of terms, self-interactions involving only one branch of the wavefunction $\newcommand{\mb}{\mathbf} M(\mb{x}' | \mb{x}) M(\mb{y}' | \mb{x})$, similarly for $\newcommand{\mb}{\mathbf} \mb{y}$, and cross-interactions between the $\newcommand{\mb}{\mathbf} \mb{x}$ and $\newcommand{\mb}{\mathbf} \mb{y}$ branches $\newcommand{\mb}{\mathbf} M(\mb{x}' | \mb{x}) M(\mb{x}' | \mb{y})$, similarly for $\newcommand{\mb}{\mathbf} \mb{y}'$. The self-interaction terms comes with a positive sign while the cross-terms come with a minus and thus actually supress the decoherence. As we take the two branches to be well separated $\newcommand{\mb}{\mathbf} |\mb{x} - \mb{y}| \to \infty$, these cross-terms vanish, and so the decoherence rate is maximized. This is quite reasonable; two widely-separated states are much easier for the background 'noise' to distinguish, and thus this state should decohere faster, much as it would due to decoherence from random interactions with a thermal background (see e.g. [141]). Indeed, precisely as in that case, the maximal decoherence rate limits to a constant-it does not grow indefinitely with the spatial separation of the wave function.

For example, consider our matter-wave experiment in figure 2. Assuming that the particles are well-separated with respect to our cutoff scale $\Delta x/R_0 \gg 1$, the decoherence rate (46) can be easily estimated. The cross-terms simply vanish, and we are left with only the self-interaction terms, leaving

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle \Gamma = \frac{G_{\rm N}}{2 \hbar} \int \int {\rm d}^3\mb{x'} {\rm d}^3\mb{y'} \frac{M(\mb{x}' | \mb{x}) M(\mb{y}' | \mb{x}) + M(\mb{x}' | \mb{y}) M(\mb{y}' | \mb{y})}{|\mb{y}'-\mb{y}'|} \approx \frac{G_{\rm N} m^2}{\hbar R_0} \nonumber \end{align} \tag{ 47 }$$

where the approximation holds up to some dimensionless geometric factor of order one. Note that, by the argument in the previous paragraph, the dependence on the spatial separation $\Delta x$ between the beam arms has dropped out-the dominant part of the integral is the self-interaction terms. Observation of coherent oscillations in a matter-wave interferometry experiment then give lower bounds on this decay rate, in turn giving a bound on the free parameter R₀; for example, experiments using large organic molecules [32] with and give . One can do better with different types of interferometry; we refer the interested reader to [43] for the state of the art.

Clearly there are several problems with such stochastic theories. A simple objection is that they are entirely non-relativistic. Worse, the prescription in (43) for the mass operator is highly arbitrary; by varying the mass distribution one can vary the quantitative predictions of the theory over a very wide range. Finally, these models are ad hoc. They address a single problem, the measurement problem, in isolation; no general reason is given for introducing the stochastic fields, whose physical nature is not clarified, and whose effect on other physical phenomena is hardly discussed. Extra fields introduced in this way will likely conflict with other parts of physics; even if this is not the case, they will have testable effects on many other physical phenomena.

5.2.2. Penrose collapse model.

A quite different line of thought which leads in the simplest approximation to a very similar result to that above, was given by Penrose. The reasoning behind Penrose's approach can be seen by going back to the state $\renewcommand{\ket}[1]{|#1\rangle} \ket{\Psi}$ in (35), in which the gravitational field is entangled with the matter field. Now let us consider what happens when we take the inner product $\langle \Psi | \Psi \rangle$, in which there will be interference terms between the two branches of the wavefunction. Since the state includes the quantum state of the gravitational field, we are faced with interference terms of the form

$$\begin{align} \renewcommand{\braket}[1]{\langle#1\rangle} \renewcommand{\bra}[1]{\langle#1|} \newcommand{\e}{{\rm e}} \displaystyle \braket{\Phi_L; g^{(L)}_{\mu \nu} | \Phi_R; g^{(R)}_{\mu \nu}} \label{GRint} \nonumber \end{align} \tag{ 48 }$$

between the two states of the matter and metric. How are we to evaluate this expression? In particular, the two metrics g^(L) and g^(R) have in general different causal structures, and matter fields propagating on these background metrics would have different vacua, so it is not clear how to assign a meaning to this interference term.

Penrose has proposed that, in fact, there is an inherent difficulty in this problem precisely because of the differing causal structures related to the two metrics. He considered a special case where the two interfering states are stationary states, and found the difference in Newtonian gravitational potentials between the two metrics to be

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta E^g_{LR} = 4 \pi G \int {\rm d}^3r \int {\rm d}^3r' {\Delta M_{LR}({\bf r})\Delta M_{LR}({\bf r'}) \over |{\bf r} - {\bf r'}|} \label{penrose} \nonumber \end{align} \tag{ 49 }$$

where $\Delta M_{LR}({\bf r}) = M_L({\bf r}) - M_R({\bf r})$, and the M_j are the mass density distributions associated with each component of the wave function.

We notice the similarity in form of this result to that found for the stochastic decoherence rate, and we may analyze it in the same way as we did equation (46). However, as it stands $\Delta E^g_{LR}$ is an energy uncertainty, related to a time uncertainty $\Delta t_{LR}^g = \hbar/\Delta E^g_{LR}$. Penrose then makes the key step of identifying $\Delta t_{LR}^g$ as a decoherence time, to be viewed as an intrinsic decoherence existing in nature, implying a breakdown of quantum mechanics.

Given the radical difference between the underlying physical arguments leading to (49) and those leading to (46), we feel that it is a mistake to treat the Penrose result as equivalent to the stochastic result, as is sometimes done. In particular, while Penrose's model is limited to the collapse of a massive superposition, the noise-based models are more general; it seems implausible that the noise would only lead to collapse of massive superpositions. Nevertheless, at the level of the simple decoherence experiments considered in this paper, both models seem to give the same predictions.

Both collapse models suffer from the same imprecision in the definition of the mass density $M({\bf r})$. This point was made very clearly by Kleckner et al [142], who compared the results for $\Delta t_{LR}^g$ obtained by either (a) using the a Gaussian form for $M({\bf r})$ representing the ground state wave-packet, or (b) a density profile concentrated in a lattice of atomic nuclei, each of radius 10¹⁵m; in the first case one finds $\Delta t_{LR}^g \sim 1~$ s, and in the second, $\Delta t_{LR}^g \sim 10^{-3}~$ s.

Testing either type of collapse theory proceeds most easily by preparing a single object in some kind of superposition state, and looking for it to decohere with the rates determined here. Experimental tests of the Penrose theory have been discussed, notably by Marshall et al [143] (see also [142]). They propose an interesting design in which a photon exists in a superposition of states located in two different optical cavities; in one of these cavities, the mirror is on an elastic spring, and is displaced when the photon is in the cavity, thereby entangling photon and mirror, and putting the mirror in a superposition of states at different locations. Original estimates for a mirror mass of $5 \times 10^{-12}~$ kg and oscillation frequency $500~$ Hz (giving a zero point spreading  ∼10⁻¹³m), show that experiments to look for gravitational decoherence here would be feasible—they have not yet been done.

In sections 5.1 and 5.2, we studied models in which gravity has been invoked as a source of non-linear quantum evolution. These models have been formulated in the language of canonical quantization and in a non-relativistic limit. In the Schrödinger–Newton approach, the gravitational field is treated as a fundamentally classical degree of freedom; in the collapse models of Penrose et al, the actual dynamics of the gravitational field are left unspecified. In neither case was any attempt made to set up a real theory—the goal was to investigate the putative collapse process in isolation.

The difficulties of setting up a new theory have already been highlighted in our discussion of the efforts of Kibble and Weinberg, from which we learned that any semiclassical treatment (in which the gravitational field is not quantized like the other fields) leads to apparently insuperable problems, and that these are compounded if we try to keep the usual theoretical superstructure of measurements, observers, operators, etc, that characterize conventional quantum mechanics and quantum field theory.

Thus it seems reasonable to attempt a new theory involving wholesale reconstruction, rather than just tinkering with isolated bits of the theory, while at the same time keeping essential elements of quantum mechanics and general relativity. And—a key point—the new theory has to be internally consistent as well as being consistent with general physical principles and existing experiments.

An attempt at such a theory has recently been outlined in several papers [144–146], and given the name 'Correlated Worldline' (CWL) theory. Although the formalism is rather complex in parts, the basic ideas are simple enough that we can give a brief outline here. A more detailed introduction to the rationale and assumptions behind the theory is also available [145]. From a purely theoretical point of view, one of the main attractions of the CWL theory is that it shows that a consistent theory of quantum gravity in which quantum mechanics is non-linearly modified is actually possible. However it also has experimentally testable implications, as we shall see.

The CWL theory is formulated in the language of path integrals, as this affords a very direct route to one thing we want to keep in the theory, which is the connection between action and quantum phase. To explain CWL theory in outline we first recall the standard path integral representation for the dynamics of a single particle, which propagates from position $\newcommand{\mb}{\mathbf} \mb{x}$ at time t to position $\newcommand{\mb}{\mathbf} \mb{x}'$ at time $t'$ with amplitude

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle U(\mb{x}',t';\mb{x},t) = \int_{\mb{q}(t)=\mb{x}}^{\mb{q}(t')=\mb{x}'} Dq \ {\rm e}^{-{\rm i} S[q]}. \nonumber \end{align} \tag{ 50 }$$

Here S[q] is the action functional and the integral is a sum over all paths $\newcommand{\mb}{\mathbf} \mb{q}(t)$ with the specified boundary conditions. We note that this is just a linear sum, with one term for each path; this linear summation expresses the superposition principle in the path integral, and it makes $\newcommand{\mb}{\mathbf} U(\mb{x}',t';\mb{x},t)$ unitary.

If we include the gravitational field dynamics into the theory, and look at a matter field with action S_M instead of just a particle, this generalizes to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle \label{1paction} U(\mb{x}',t',g^{(3)'}_{ab};\mb{x},t,g^{(3)}_{ab}) = \int_{x}^{x'} Dq \int_{g^{(3)}_{ab}}^{g^{(3)'}_{ab}} Dg_{\mu\nu} \ {\rm e}^{-{\rm i} S_{M}[q,g_{\mu\nu}] - {\rm i} S_{G}[g_{\mu\nu}]}. \nonumber \end{align} \tag{ 51 }$$

Here, in analogy with the matter path integral, the gravitational path integral is over all configurations of the metric $g_{\mu\nu}(x)$ interpolating between the three-dimensional spatial geometries $\newcommand{\mb}{\mathbf} g^{(3)}_{ab}(\mb{x})$ and $\newcommand{\mb}{\mathbf} g^{(3)'}_{ab}(\mb{x})$ on the spacelike hypersurfaces S, $S'$ representing the initial and final times¹¹. This expression is still just a linear sum over configurations of the particle and metric. This action is the one from which we would ordinarily derive the effective field theory studied in section 3; it is the standard action for conventional quantum gravity. We note that there is, implicit in this path integral, some sort of UV cutoff to deal with UV divergences.

The proposal in [144, 145] is to now enlarge the theory by allowing for correlations between various paths $q,q'$ in the path integral. This automatically violates the superposition principle. It is then further argued that these correlations originate in gravity. This gives a special place to gravity in the theory, but we note that $g^{\mu\nu}(x)$ is still treated as a quantum field. By fairly lengthy arguments [145] one can start with the most general possible way of correlating the different paths, and then arrive at a specific form for gravitational correlations, using physical arguments based on the equivalence principle. Crucially, it turns out there are two different possible CWL theories [146], but one of them fails certain consistency tests. In what follows we explain the basic idea, and discuss how the CWL predictions for experiment can be made, and how they differ from conventional quantum mechanics. These formulations are:

• (i)  
  Summation form: In this form, we sum over all possible correlations between different paths in the propagator—see [145] for a derivation. The CWL propagator is then an infinite sum $K = \sum\nolimits_n K_n$, of form

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle K_{\rm CWL}(\mb{x}',t',g^{(3)'};\mb{x},t,g^{(3)}) = \sum_{n=1}^{\infty} \int_{x}^{x'} \prod_{i=1}^{n} Dq_{i} \int_{g^{(3)}}^{g^{(3)'}} Dg \ {\rm e}^{-{\rm i} \sum\nolimits_i S_{M}[q_i,g] - {\rm i} S_{G}[g]}. \nonumber \end{align} \tag{ 52 }$$

  Note that K is not in general unitary. The key point here is that the path integral over the metric configurations correlates all the paths. The n  =  1 term is equivalent to (51); it is just standard quantum gravity, and so reproduces the predictions of the effective field theory picture in section 3. The next n  =  2 term is

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\mb}{\mathbf} \displaystyle \label{CWL2} K_{2}(\mb{x}',t',g^{(3)'};\mb{x},t,g^{(3)}) = \int_{x}^{x'} Dq_1 Dq_2 \int_{g^{(3)}}^{g^{(3)'}} Dg \ {\rm e}^{-{\rm i} S_{\rm mat}[q_1,g] - S_{\rm mat}[q_2,g] - {\rm i} S_{EH}[g]} \nonumber \end{align} \tag{ 53 }$$

  in which each path q₁ in the path integral couples to every other path q₂ via the gravitational interaction. The higher order terms $n=3, 4, \ldots$ allow more complex correlations to develop. Figure 7 shows a diagrammatic representation of calculations of the propagator in an expansion in gravitons.As long as we are working with weak gravitational sources, for example particles in a laboratory, we can ignore CWL correlations beyond n  =  2, since these will be suppressed by powers of the Planck mass. The n  =  2 term comes in at the same order in perturbation theory in the Planck mass as the n  =  1 term: both terms come from single-graviton exchange diagrams (see figure 7), in the n  =  1 term from graviton exchange between two different particles, whereas in the n  =  2 term from graviton exchange between different copies of the same particle. This then leads to an interesting pattern of correlations, in which we begin with the standard correlations from the n  =  1 term and progressively destroy them through the $n \geqslant 2$ terms.There is a very concise way of formulating the summed CWL theory, which is to write a generating functional (the analogue of the partition function in statistical mechanics) from which all propagators, correlation functions, etc, can be derived. For the summed CWL theory, this takes the form [145]

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathbb{Q}[J] = \oint {{\mathcal D}}g \; {\rm e}^{{{\rm i} \over \hbar} S_G[g]} \sum_{n=1}^{\infty} {1 \over n!}\;Q_n[g,J] \;\;\;\;\; {\rm (summed)} \;\;\; \label{Q-sch1} \nonumber \end{align} \tag{ 54 }$$

  in which $Q_n[g,J]$ is that contribution to the sum coming from an n-tuple of paths, and $J(x)$ is an external 'source' current coupled to the matter field. One then obtains propagators and correlators automatically, just by functionally differentiating $\mathbb{Q}[J]$ with respect to $J(x)$; for details see [145, 146].However, it turns out there is a fly in the ointment—a recent more detailed investigation [146] has found that the summed version of CWL theory has an inconsistency in the semiclassical limit—thus it has to be discounted, although many useful lessons can be learnt from it.
• (ii)  
  Product form: It turns out that one can derive a CWL theory which passes all consistency tests: it has sensible semiclassical and perturbative expansions, and satisfies all Ward and Noether identities. We can compare this with summed CWL very simply by writing a new generating functional

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathbb{Q}[J]=\prod\limits_{n=1}^\infty \tilde{Q}_n[\,J\,] \;\;\;\;\;\; {\rm (\,product)} \nonumber \end{align} \tag{ 55 }$$

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tilde{Q}_n[\,J\,]= \oint {{\mathcal D}}g \; {\rm e}^{{{\rm i} \over \hbar} S_G[g]} \,\left({{\mathcal Z}}[g, J/c_n]\right)^n \label{bbQ-J01} \nonumber \end{align} \tag{ 56 }$$

  where ${{\mathcal Z}}$ is just the particle generating functional in conventional field theory (with a regulator c_n that we do not discuss here); we are now summing over logarithms, and $\ln \mathbb{Q}[J]$ is now used to generate correlation functions.

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At first glance all one seems to have done here is substitute a logarithm of a product in place of a sum—why is there any difference? But it turns out that the correlations between paths that are generated come with a different weighting, and this cures the problems in the original summation form. We still have correlations between paths and so the superposition principle is still violated. To see how this works in the same kind of perturbative expansion that we just described for summed CWL theory is a little more difficult in the product version. If one evaluates the lowest order graphs, one still sees that correlations develop between different paths. However they now appear as a kind of gravitational attraction of the different graphs towards the vacuum energy generated by each graph. See [146] for more on the technical details.

The principal change in the physics caused by the correlations in CWL theory is what is called "path bunching"; the attractive gravitational interaction between the paths causes them to start clustering together once the mass of the propagating object is large enough [145, 146]. This is quite different from standard quantum mechanics, where one sums over independent paths, with the differences between the phases of the different paths giving the usual interference phenomena. Thus a new mass scale $m_{\rm CWL}$ emerges in CWL theory. Below $m_{\rm CWL}$, gravitational correlations have little effect on the propagator, and standard interference phenomena prevail. However once the mass exceeds $m_{\rm CWL}$, path bunching prevents paths from separating widely, at which point double-slit interference no longer becomes possible—the paths cannot separate to encompass both slits, and the mass starts to behave classically. We can therefore think of $m_{\rm CWL}$ as a 'crossover mass' in the theory, between quantum and classical behavior (see figure 8).

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The physics of path bunching is illustrated in figure 8, which contrasts the kinds of path contributing in a double-slit experiment to the amplitude to propagate to a given point on a screen in conventional quantum mechanics, with that prevailing in CWL theory for a sufficiently large mass. In the latter case, a set of paths passing through one slit cannot be accompanied by paths traveling through the other, since the latter paths are too strongly attracted by the former.

Note there is no decoherence involved here in the 'quantum-classical transition' between purely quantum behavour and the classical behavior for large masses; no external environments or noise are involved. Another key feature of the theory is that if a microscopic system interacts with a massive measuring system, so that they become entangled, then path bunching occurs for the 'combined system  +  apparatus', and the pair of them now show classical behavior. As a general result the crossover between quantum and classical dynamics happens very quickly as one increases the mass of the objects involved [146]; unless the mass happens to be very close to $m_{\rm CWL}$, one sees either quantum dynamics or classical dynamics.

Experiments: the CWL theory has no adjustable parameters, and so testable predictions can in principle be made. However we emphasize that to date there is still no reliable result for the crossover mass $m_{\rm CWL}$ in the product version of CWL theory. From calculations in summed CWL theory, it is known that the crossover mass depends on the detailed composition of the object—its shape and density, and even its phonon spectrum—but these are all readily determined in advance. It is expected to be rather large for typical solid bodies. In the summed CWL theory one finds that the crossover mass is typically about 10⁻⁹ kg, not much less than the Planck scale of $2.2 \times 10^{-8}$ kg. The crossover mass in the product CWL theory is likely to be somewhat larger—a detailed evaluation has yet to be given.

One class of experiments involves Talbot–Lau interferometry, where we only have one particle to deal with. In this case, the prediction would be a disappearance of the interference pattern above a certain mass scale. We refer the reader to [144, 145] for some more detailed discussion of this type of experiment. Another experiment may be more likely to succeed in the near term. The set-up of Marshall et al [143] mentioned earlier (or some variant of it), involving superpositions of mirrors on springs, should allow spatial superpositions of much more massive objects than a Talbot–Lau set-up. If superpositions involving masses  ∼10⁻⁹ kg can be formed, this would offer an ideal route towards testing CWL theory.