Watching the Clocks: Interpreting the Page–Wootters Formalism and the Internal Quantum Reference Frame Programme

Abstract

We discuss some difficulties that arise in attempting to interpret the Page–Wootters and Internal Quantum Reference Frames formalisms, then use a ‘final measurement’ approach to demonstrate that there is a workable single-world realist interpretation for these formalisms. We note that it is necessary to adopt some interpretation before we can determine if the ‘reference frames’ invoked in these approaches are operationally meaningful, and we argue that without a clear operational interpretation, such reference frames might not be suitable to define an equivalence principle. We argue that the notion of superposition should take into account the way in which an instantaneous state is embedded in ongoing dynamical evolution, and this leads to a more nuanced way of thinking about the relativity of superposition in these approaches. We conclude that typically the operational content of these approaches appears only in the limit as the size of at least one reference system becomes large, and therefore these formalisms have an important role to play in showing how our macroscopic reference frames can emerge out of wholly relational facts.

Appendices

Appendix A: Other Relational Formalisms

In addition to the PW formalism, there are several other formal frameworks which enable us to obtain relative states from a global state in a similar manner. One approach uses relational Dirac observables [71], which encode how one observable evolves relative to another along the flow generated by the Hamiltonian constraint. Relational Dirac observables are often invoked in the context of classical Hamiltonian approaches, but we can obtain corresponding variables in the quantum case by first constructing variables in the kinematical Hilbert space which describe some system property conditional on some reference frame being in some orientation, and then mapping them to Dirac observables on the physical Hilbert space using a G-twirl [18]. We also have the relational Heisenberg picture [27], where we begin by trivializing the constraint so it acts only on the degrees of freedom of the reference frame (e.g. the clock) - that is, if T is the trivialization map and C the Hamiltonian constraint operator, we have $T H T^{-1} = (H_C - \epsilon ) \otimes I_S$ for any real $\epsilon$. This transforms physical states into product states with a fixed redundant clock factor, and we then project onto the classical gauge-fixing condition to remove the redundant reference frame degrees of freedom. These various approaches differ in their technical details but all present a roughly similar picture in which temporal notions are understood in terms of states of the universe relative to some physical reference frame. Indeed ref [27] demonstrates that the PW formalism, the relational Dirac observable formalism, and the relational Heisenberg formalism are all in fact equivalent. Ref [27] also demonstrates that in all three cases, the procedure for changing temporal reference frames amounts to transforming from a state relative to some system A to a global state which encodes all the possible relative states and then finally applying a reduction procedure to arrive at the state relative to some other system B. This demonstrates the vital importance of the global perspective-neutral picture over and above all the individual relative-state pictures, and indeed ref [19] argues that it offers a new way of interpreting the notion of the ‘quantum state of the universe’ as an encoding of all these relative states.

We should also distinguish between the IQRF approach and the related reference frame research programme in quantum information [72,73,74]. In the QI context, reference frames typically come into play in the operational setting where we have a pair of observers who wish to exchange information but who do not share any common reference frame, which is to say we have no guarantee that they will perform measurements in the same basis. The solution is to encode information into relational degrees of freedom- for example, we could encode a bit into two qubits with 0 associated with the state $\frac{1}{\sqrt{2}} (| 00 \rangle + |11 \rangle )$ and 1 associated with the state $\frac{1}{\sqrt{2}} (| 01 \rangle + |10 \rangle )$, because the answer to the question ‘are the states of the two qubits the same or different?’ will be the same regardless of what basis we measure in. The key difference between these two approaches is that the QI formalism employs observers who are external to the quantum system in question, rather than seeking to define reference frames from within the system as the IQRF programme does: hence in the QI approach the set of states which are regarded as being legitimate in the absence of an external reference frames are the incoherently group-averaged states, rather than the coherently group-averaged states as in the IQRF approach, reflecting the fact that the QI approach uses external observers who cannot themselves be entangled with parts of the system in question [17]. The QI approach therefore poses no particular interpretational challenges over and above the usual measurement problem, as all interpretations of quantum mechanics will make the same predictions for the results of the measurements performed by these external observers. Whereas the correct way to model observers as internal features of a quantum system remains highly controversial [75, 76], and therefore the internal IQRF approach lies right at the meeting point of a number of thorny conceptual issues in quantum foundations. We therefore focus in this article on the internal reference frame programme rather than the related QI approach.

Appendix B: Final-Measurement Interpretation

The final-measurement interpretation allows us to say more about the notion that time is just gauge. It is still true in this picture that specifying a state at a time or a single relative state is enough to determine the whole universal quantum state, so in that sense it remains true that time is a form of gauge, at least with respect to the universal quantum state. But in the final-measurement picture the relative states don’t really refer to anything physically real, so the fact that they are all related by gauge transformations may seem less surprising: we simply have a choice of gauge in which to express the probability rule which is used to determine the appropriate probability distribution for the final measurement. On the other hand, if we now look at the actual course of history selected by this measurement, we will see apparent indeterminism corresponding to the probabilistic results of quantum measurements, so it is not true for this actual course of history that the state at one time is sufficient to determine the state at all other times. Therefore time is not merely gauge for the real timeline as defined by the register readings - genuinely new information comes into being at different times along this timeline. This suggests that the problem of time in quantum gravity is to some extent an artefact of an approach which reifies the universal quantum state whilst sweeping the measurement problem under the rug: if one includes some actual measurement results the picture looks very different and there is less reason to be concerned about ‘timelessness.’ Note that this consequence is not unique to the final-measurement interpretation: as observed by Callender and Weingard [37], ‘contrary to the static situation described at the outset, a Bohmian approach to cosmology admits nontrivial evolution of the dynamical variables.’ So it seems that quite generally, as soon as we we add to the universal quantum state some variables describing what actually occurs, we no longer have the problem of timelessness. Indeed, from the point of view of approaches like the final-measurement interpretation and the Bohmian interpretation, the universal wavefunction is in a sense simply describing a set of counterfactual possibilities, and thus its timelessness is not at all surprising - for presumably one would not expect a set of counterfactual possibilities to undergo change or evolution in any case.

Given that the final measurement on the register will necessarily take the state outside the physical Hilbert space, one might perhaps have concerns that this interpretation would break the symmetries which have been used to set up the physical Hilbert space. And indeed, it is true that if we define a post-measurement state in the usual way this state will no longer belong to the physical Hilbert space; but by definition the final-measurement occurs after all meaningful dynamical evolution is complete, so we are not required to actually perform a projection into a post-measurement state since nothing happens after the final measurement. The constraint quantisation ensures that the probability distribution defined over possible register readings by the quantum state is invariant under diffeomorphisms and global time translations, and therefore the process of selecting one set of register readings according to that probability distribution is also necessarily invariant under diffeomorphisms and global time translations. Moreover, the constraints also ensure that the ‘course of history’ thus selected is defined only up to diffeomorphisms and global time translations, which ensures that we are not ‘double counting’ - i.e. two courses of history related by a global time translation are not treated as distinct possibilities by the probability distribution, they are assigned the some probability and the sum over histories required to normalise the probability distribution counts this probability only once. So the final-measurement is compatible with the symmetries used to define the quantised theory, and therefore the final-measurement approach does not violate the founding idea of the IQRF approach that all physical quantities are ultimately relational: the course of history that we select in the final measurement is definite and observer-independent, but it does not presuppose an absolute background spacetime.

Appendix C: Penrose

Ref [9] suggests an application for the quantum equivalence principle in the context of Penrose’se argument for gravitationally induced collapse [66]. That is, ref [9] contends that Penrose’s argument can be blocked by observing that he is using the wrong Equivalence Principle - he is working with the classical Equivalence Principle whereas he should in fact be using the Quantum Equivalence Principle. Now if this were correct, it would clearly be a counterexample to our worry that the quantum equivalence principle fails to have operationally meaningful content that goes beyond the classical SEP. But let us consider in more detail the way in which Penrose’s argument uses the Equivalence principle. Penrose’s fundamental concern is that in cases where we have a superposition of two different spacetimes there is no unique mapping from the points of one spacetime to the points of another, since the principle of general covariance prevents us from assigning primitive identities to spacetime points which would identify them across branches of the wavefunction. Penrose then worries that the global time-translation operator does not seem to be well-defined unless we can find a natural way of mapping the points of one superposed spacetime to points of the other, and therefore asks how we might go about constructing such a map. This is where Penrose invokes the Principle of Equivalence - he notes that ‘In accordance with the principle of equivalence, it is the notion of free fall which is locally defined, so the most natural local identification between a local region of one space-time and a corresponding local region of the other would bef that in which the free falls (i.e. spacetime geodesics) agree.’

Penrose does seem to be invoking a version of the SEP, but note that he does not appear to be insisting that the principle of equivalence will be violated if there does not exist a way of identifying the two spacetimes such that the geodesics agree. Rather, he is making use of the operational construal of the principle of equivalence, which he understands as the statement that the metric can be accessed by means of local observables in the form of geodesic motions: if this is accepted, presumably any sensible mapping between the two spacetimes should map the geodesic motions of one onto the geodesic motions of the other. Note that Penrose in fact already seems to be employing ‘quantum reference frames’ in some sense - that is, he seems to take it that the correct way to use the SEP in this situation is to apply it separately within each branch of the wavefunction to draw conclusions about our local access to the metric inside each branch. This application of the equivalence principle is entirely separate to his questions about whether there exists a way of mapping the locally accessible features of the metric of one onto the locally accessible features of the metric of the other. He does not say anywhere that the existence of such a mapping is required by the equivalence principle, or that he thinks there must be one unified Lorentzian frame of reference encapsulating both branches of the wavefunction - indeed, such a thing would be meaningless from the operational point of view where frames of reference are associated with actual or possible measurements, since measurements surveying two macroscopically distinct metrics must necessarily take place in one branch or another, so it would be nonsensical to postulate a frame of reference associated with measurements surveying both metrics at once. This indicates that Penrose’s argument can’t be blocked by simply altering the equivalence principle to apply separately within different branches of the wavefunction, because that is exactly how Penrose is already using it.

Note that it is not our intention to claim that Penrose’s argument is right. There are other possible objections to it - for example, ref [9] also notes (correctly, in our view) that there is no obvious reason why it should be possible to define a universal time-translation operator which applies across all the different branches of a spacetime superposition, since we can simply allow that time flows at different rates within the different branches. But whether or not there exists such an operator is nothing to do with the principle of equivalence: the idea that the stability of a state depends on the existence of a well-defined time evolution operator is a further assumption of Penrose’s argument which is a consequence of the standard quantum-mechanical definition of a stationary state, so it is actually an assumption that comes from quantum mechanics rather than general relativity, and therefore it is not related to his use of the principle of equivalence to identify the local observables providing access to the metric.

Appeindix D: Relational Superposition

On top of the claims about relational superposition made in the IQRF research programme, there are several other ways in which superposition might be said to be relational. First, of course superposition is relational in a trivial sense of being relative to a basis: for example, a qubit in the state $|0\rangle$ is not in a superposition relative to the computational basis $\{ | 0 \rangle , |1 \rangle \}$, but it is in a superposition relative to the Hadamard basis $\{ | + \rangle , | - \rangle \}$, since $|0 \rangle = \frac{1}{\sqrt{2}} ( | + \rangle + |- \rangle )$. In that sense, the assertion that a quantum system is ‘in a superposition’ is meaningless without the specification of a basis. But the claims about the relativity of superposition in the IQRF research programme are stronger than that: for there we find that given a certain fixed basis, whether or not a system is in a superposition relative to that basis is itself relative to a choice of reference frame. For example, in ref [5] we are given an example where relative to particle C, the particles A and B are in the entangled state $\int dx |x \rangle _A |x + X \rangle _B$ so neither of them has a sharp state in the position basis relative to C, but when we switch to A’s reference frame we find the state of B is now sharp in the position basis.

Second, relativistic effects can cause superpositions to look different from different reference frames in settings where frames are moving quickly relative to one another. For example, in ref [77] it is noted that a Hamiltonian is defined with respect to a given slicing of space-time in equal-time surfaces, so observers in different references frames will assign different interaction Hamiltonians to a given experiment and as a result will come to different conclusions about the amount of superposition and/or entanglement present in a given physical situation. While these effects are very interesting and suggestive, they don’t seem to require any major conceptual revolutions, as the ‘reference frames’ in question are the ordinary kinds of reference frames commonly invoked within special relativity, and we are already very familiar with the idea that unusual frame-relative effects may occur when such frames are moving quickly relative to one another. On the other hand, the relativity of superposition invoked by the IQRF research programme occurs even in the absence of relative motion, so it can’t be understood in the same way as relativistic frame-dependence.

Third, we can also see frame-dependence of superposition and entanglement in the QI approach to reference frames based on the existence of an external reference frames; in this setting frame-dependence arises because the state of a system undergoes decoherence when we discard one reference frame during the switch to another reference frame [72, 78]. ‘Reference frame’ is here being understood in a similar way as in the IQRF reference frame, i.e. reference frames are associated with individual physical objects, and therefore much of what we are about to say regarding the relativity of superposition in the IQRF framework also apply here. However, it’s important to note that the frame-dependent effects in the QI case can be understood as a consequence of a system (the initial reference frame) being discarded, whereas in the IQRF framework this explanation is not available since we are dealing with the whole universe at once and therefore nothing can be discarded.