Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems

Abstract

The standard formulation of quantum theory relies on a fixed space-time metric determining the localisation and causal order of events. In general relativity, the metric is influenced by matter, and is expected to become indefinite when matter behaves quantum mechanically. Here, we develop a framework to operationally define events and their localisation with respect to a quantum clock reference frame, also in the presence of gravitating quantum systems. We find that, when clocks interact gravitationally, the time localisability of events becomes relative, depending on the reference frame. This relativity is a signature of an indefinite metric, where events can occur in an indefinite causal order. Even if the metric is indefinite, for any event we can find a reference frame where local quantum operations take their standard unitary dilation form. This form is preserved when changing clock reference frames, yielding physics covariant with respect to quantum reference frame transformations.

Figures (6)

• Figure 1: Description of the operational meaning of the framework. $\mathrm{A}$ and $\mathrm{B}$ perform experiments on $\mathrm{S}$ in two stages. In the preparation stage ($\bf{\mathsf{a}}$), $\mathrm{A}$ and $\mathrm{B}$ specify the states of their clocks, subsystems (depicted by a blue ball) and ancillas (depicted by a red ball, $\mathrm{a}$, for $\mathrm{A}$ and by a green ball, $\mathrm{b}$, for $\mathrm{B}$). They do so by freely choosing the knob settings that control these systems. $\mathrm{S}$ can be a composite system entangled between $\mathrm{A}$ and $\mathrm{B}$. This is indicated by the line joining $\mathrm{A}$ and $\mathrm{B}$'s subsystems. $\mathrm{A}$ and $\mathrm{B}$ can program their clocks so that an interaction between $\mathrm{S}$ and the ancillas, $\mathrm{a}$ and $\mathrm{b}$, is turned on at a specific local time, say $t^*$. In the detection stage ($\bf{\mathsf{b}}$), $\mathrm{A}$ and $\mathrm{B}$ read the measurement results by looking at the outcome of their clocks and ancillas. By assumption, the choices made by $\mathrm{A}$ and $\mathrm{B}$ in the preparation stage do not depend on the results obtained in the detection stage, even if the experiments take place in an indefinite causal structure. As the figure shows, we assume that both parties have access to all the data of the experiment. With these data, $\mathrm{A}$ and $\mathrm{B}$ “map” the set of events into “space-time,” which we depict as a manifold foliated by surfaces of constant time according to $\mathrm{A}$ (dotted lines) and according to $\mathrm{B}$ (solid lines). The event produced by $\mathrm{A}$ ($\mathrm{B}$) is depicted by a red (green) star. In the case depicted here, the two events are sharply localised in both $\mathrm{A}$ and $\mathrm{B}$'s time reference frames. This feature is consistent with a fixed space-time background, where the time localisation of events is absolute. However, in this work we show that there are situations involving gravitating quantum systems which lead to an indefinite metric background. In such backgrounds, whether an event is sharply localisable in time or not depends on the time reference frame.
• Figure 2: Relative localisability of events for non interacting clocks with unsharp initial states. This figure describes the history state $\mathop{\left|\Psi\right>}\nolimits$, solution to Eq. (\ref{['constraintwmeasurements']}), in the time reference frame of $\mathrm{A}$ ($\bf{\mathsf{a}}$) and of $\mathrm{B}$ ($\bf{\mathsf{b}}$). There are two events, (depicted by a red and a green star), taking place in the experiment. In the event depicted by a red star, the system $\mathrm{S}$, (depicted by the blue ball), interacts with $\mathrm{A}$'s ancilla, $\mathrm{a}$ (depicted by a red ball). In the event depicted by a green star, the system $\mathrm{S}$ interacts with the ancilla $\mathrm{b}$ (depicted by a green ball), initially under the control of $\mathrm{B}$. In $\mathrm{A}$'s time reference frame ($\bf{\mathsf{a}}$), the event depicted by a red star is sharply localised in time. In contrast, the event depicted by a green star has an uncertainty, characterised by $\sigma$, in its time localisation, due to the “fuzzy” initial state of clock $\mathrm{B}$ in $\mathrm{A}$'s time reference frame (see main text). The roles are reversed in the time reference frame of $\mathrm{B}$ ($\bf{\mathsf{b}}$). In this frame, it is the event depicted by a green star which is sharply localised in time (as defined by clock $\mathrm{B}$), whereas the event depicted by a red star exhibits some time uncertainty in $\mathrm{B}$'s reference frame. This uncertainty is due to the fact that, in $\mathrm{B}$'s frame, the initial state of clock $\mathrm{A}$ is “fuzzy.”
• Figure 3: Gravitating quantum clocks from the point of view of $\mathrm{C}$. In a thought experiment, $\mathrm{A}$ ($\bf{\mathsf{b}}$) sets up an event, consisting in an interaction between $\mathrm{S}$ (blue ball) and $\mathrm{a}$ (red ball), when her clock shows a certain time $t_\mathrm{A}^*$. $\mathrm{A}$'s clock is influenced by a gravitational field sourced by the energy of $\mathrm{B}$'s clock ($\bf{\mathsf{a}}$). The initial quantum state of $\mathrm{B}$'s clock (depicted by the green Gaussian) has a characteristic width $\sigma$, which specifies its accuracy (the smaller $\sigma$, the higher the accuracy). As a consequence, the energy of $\mathrm{B}$'s clock is not well defined ---it has an uncertainty of $1/\sigma$. Therefore, the gravitational field sourced by $\mathrm{B}$ is not well defined either. As a consequence, the time dilation of clock $\mathrm{A}$ becomes uncertain from the point of view of $\mathrm{C}$. This is shown by the “fuzzy” red wave packets representing $\mathrm{A}$'s clock state. By Eq. (\ref{['pninthiststate']}), this uncertainty leads to an indefiniteness in the localisation of $\mathrm{A}$'s event, as depicted by the “fuzzy” red stars on the bottom of the wave packets.
• Figure 4: Experimental set up of the gravitational switch. The mass $\mathrm{M}$ can be either close to $\mathrm{A}$ ($\bf{\mathsf{a}}$) or to $\mathrm{B}$ ($\bf{\mathsf{b}}$). $\mathrm{A}$ ($\mathrm{B}$) applies an operation on the system $\mathrm{S}$ by means of an ancilla $\mathrm{a}$ ($\mathrm{b}$). $\mathrm{A}$'s operation is in the causal future (past) of $\mathrm{B}$'s operation when the mass is on the left (right). This fact is depicted by the system traveling from $\mathrm{B}$ to $\mathrm{A}$ ($\bf{\mathsf{a}}$) and from $\mathrm{A}$ to $\mathrm{B}$ ($\bf{\mathsf{b}}$). In the quantum switch experiment, the mass is put in a superposition between being close to $\mathrm{A}$ and close to $\mathrm{B}$, leading to a superposition of causal orders.
• Figure 5: Space-time diagram for gravitational switch thought experiment as described in the time reference frame of $\mathrm{C}$. The event in which $\mathrm{A}$ ($\mathrm{B}$) acts on $\mathrm{S}$ is depicted by a red (green) star. These events are delocalised in time from $\mathrm{C}$'s point of view. In the spacetime $\mathcal{M}_\mathrm{R}^{(\mathrm{C})}$ ($\mathcal{M}_\mathrm{L}^{(\mathrm{C})}$), the mass $\mathrm{M}$ is on the right (left), implying that $\mathrm{A}$ acts before (after) $\mathrm{B}$. (For clarity, the worldline of $\mathrm{M}$ is not shown.) In $\mathcal{M}_\mathrm{R}^{(\mathrm{C})}$ ($\mathcal{M}_\mathrm{L}^{(\mathrm{C})}$), $\mathrm{A}$'s action happens at time $t^*/\Delta^\mathrm{R}(\mathrm{A},\mathrm{C}) \approx t^*-\delta$ (resp. $t^*/\Delta^\mathrm{L}(\mathrm{A},\mathrm{C})\approx t^*+\delta$) and $\mathrm{B}$'s action happens at $t^*/\Delta^\mathrm{R}(\mathrm{B},\mathrm{C})\approx t^*+\delta$ (resp. $t^*/\Delta^\mathrm{L}(\mathrm{B},\mathrm{C})\approx t^*-\delta$), for $\delta = (\Phi^\mathrm{L}_\mathrm{C}-\Phi^\mathrm{L}_\mathrm{A})/t^*$. The dashed red (green) line joining $\mathrm{A}$'s ($\mathrm{B}$'s) event in both spacetimes is drawn to emphasise that this is the same event, even though we have used two red (green) stars, one per spacetime, to depict it.