Diffeomorphism invariance and quantum mechanical paradoxes

1. Introduction

Thought experiments play a large role in quantum mechanics, both in popular treatments, such as the prototypical Schrödinger’s cat, and in professional discourse. Quantum mechanics is often counterintuitive to our classically trained intuition and thought experiments have a long history of illuminating nonclassical logical results and fallacies. One such line of thought experiments is that of “Wigner’s Friend”¹ and various extensions, such as the Frauchiger–Renner (FR)² experiment, which has recently generated interest.^2,3,4 In the FR thought experiment, which we will use as a prototypical example throughout this paper, the authors argue that under some fairly reasonable assumptions in the framework of nonrelativistic quantum mechanics, one can construct an experiment where two observers measure conflicting results. They then use this result to argue that nonrelativistic quantum mechanics is not self-consistent.

We note as an aside for the reader that this experiment is a matter of some debate purely in the quantum information community, cf. Refs. 4,5,6,7 and others. In this work, we do not argue for or against the paradox or similar ones that come up in the literature. Rather we merely use the construction of such thought experiments to examine the subtle role of gravity, in particular in the construction of local observers, and how gravity interacts with putative paradoxical outcomes. A related exploration that differentiates between observations and inferences that must be compatible with physical theories can be found in Ref. 4.

As is typical in quantum information approaches, the FR thought experiment ignores gravitational effects when making claims of the self-consistency of quantum mechanics. Of course, we live in a world with gravity, so from a fundamental perspective the quantum theory that needs to be self-consistent is the full theory of quantum gravity—whatever it turns out to be. Structurally the physical Hilbert space and dynamics of quantum gravity are heavily constrained by the requirement of diffeomorphism invariance, cf. Ref. 8, which is typically not imposed in nonrelativistic quantum mechanics. However, when using paradoxes to make claims about underlying fundamental theories, such as quantum mechanics, it is necessary to at least ensure the paradoxical observables are gauge invariant. In the gravitational case, this would mean introducing the gravitational field as a dynamical object and imposing diffeomorphism invariance. While this may be difficult to do nonperturbatively, one tractable approach would be to match the field content of general relativity and impose diffeomorphism invariance perturbatively to first-order in Newton’s constant G. If such an approach can be made self-consistent and generate only small corrections to the nonrelativistic results then ignoring gravity is justified. Our goal in this paper is to examine whether one can build a “paradoxical” thought experiment that involves local observables while maintaining gauge invariance to first order in G in a self-consistent manner (up to the paradoxical result occurring at some time t).

We will find that perturbative gauge invariance can’t be self-consistently imposed if the experiment is to eventually generate a paradox—higher order corrections in G and fundamental aspects of full quantum gravity become important. This is perhaps surprising as it seems then that a low-energy thought experiment is somehow sensitive to quantum gravity. However, this is actually already a known occurrence in the gravitational literature for precisely those thought experiments that incorporate paradoxes and quantum mechanics. Classically, spacetimes that allow Closed Timelike Curves (CTCs) may have low curvature yet contain grandfather paradoxes. Quantum mechanically, these spacetimes lead to the breakdown of semi-classical gravity due to unstable energy–momentum tensor fluctuations⁹ and the necessity of the chronology protection conjecture,¹⁰ a statement about quantum gravity. Similarly, the Almheiri–Marolf–Polchinski–Sully (AMPS) or infaller paradox in black hole thermodynamics, which is built in spacetime regions with low curvature, generates extreme gravitational responses (e.g. firewalls)¹¹ once quantum mechanics is taken into account. Firewalls have also been argued to be one manifestation of a more generic paradox, due to the violation of boundary unitary,¹² that can occur in scattering experiments¹³ with local bulk operators. In all these scenarios, the existence of the paradox leads to a strong gravitational and even quantum gravitational response—one cannot stay in the weak field regime self-consistently and still assume there exists a quantum mechanical paradox.

The structure of the paper is as follows. In Sec 2, we discuss the FR thought experiment in detail. We lay out the original assumptions, the refinements necessary for a gravitationally coupled theory, detail the experimental setup and logic, briefly mention the purely quantum information-based objections in the literature, and explain why gravity may play a role. In Sec. 3, we discuss the gauge invariance problem of the observables in the context of Quantum ElectroDynamics (QED) and the idea of electromagnetic dressing, as that is more familiar outside the gravitational community. In Sec. 4, we describe the necessary gravitational dressings in perturbative gravity, show how quantum gravity comes into play due to the necessary existence of nonanalytic solutions and how this relates to boundary unitarity. Finally, we conclude in Sec. 5.

2. The Frauchiger–Renner Thought Experiment

2.1. The assumptions

The FR thought experiment² works in the framework of nonrelativistic quantum mechanics and makes the following assumptions, labeled (Q), (C) and (S) in its original construction:

(Q)	Every observer evolves under the rules of standard quantum mechanics on their local Hilbert spaces, in particular the Born rule. Within that framework, there is a process that allows observers to make conclusive, classically stable measurements. Specifically, let a system S be in state $|\psi {〉}_S$ at time $t_0$, and let there exist a family of Heisenberg projection operators, ${\{{\pi }_x\}}_{x\in ℝ}$, associated with the possible outcomes x of the measurement of some observable X which occurs at time $t>t_0$. If the state after measurement is $|\psi 〉$ and $〈\psi |{\pi }_{\xi }|\psi 〉=1$ for $\xi$ being some possible value of x, then one can make the definitive statement at time t that $x=\xi$. We will refer to the ability to make such definitive statements about the outcomes of measurements as with certainty. Furthermore, since the measurement is stable, $x=\xi$ for all time after t.
(C)	Inferred knowledge is transferable between two observers. If observer A knows that observer $A^{\prime }$ has concluded that $x=\xi$ at some time tand that $A^{\prime }$ is reasoning using the exact same theory as A, then A can conclusively state that $x=\xi$ at time t.
(S)	Contradictory logical statements cannot be true simultaneously. If A believes that $x=\xi$ at time t is true, then they cannot also believe that $x\ne \xi$ at time t is true.

With the above assumptions in hand, we now turn to the thought experiment itself.

2.2. The experimental setup

The setup for the FR thought experiment is similar to constructions employed in Hardy’s and Wigner’s friend thought experiments. Consider four quantum observers, Alice, Bob, Charlie and Diane. Charlie and Diane are each locked away in different local, isolated laboratories, while Alice and Bob are outside observing Charlie’s and Diane’s laboratories, respectively. The experiment proceeds as follows, in rounds $n=0,1,2,\dots .$ The number on the right indicates the timing of the steps and it is assumed that each step takes one unit of time:

$n:00$	Charlie measures the state of a quantum coin toss, $|coin〉\equiv (|heads〉+\sqrt{2}|tails〉)/\sqrt{3}$, in the $\{|heads〉,|tails〉\}$ basis, and records the result as $r\in \{heads,tails\}$. He then sets the spin state S of a particle to $|\downarrow {〉}_S$ if the result is $r=heads$ and to $|\to {〉}_S\equiv (|\uparrow {〉}_S+|\downarrow {〉}_S)/\sqrt{2}$ if the result is $r=tails$. Charlie then sends the particle to Diane.
$n:10$	Diane measures S with respect to the $\{|\uparrow {〉}_S,|\downarrow {〉}_S\}$ basis and records the result as $z\in \{-\frac{1}{2},+\frac{1}{2}\}$.
$n:20$	Alice measures Charlie’s laboratory with respect to a basis containing the state $|\overline{ok}〉\equiv (|\overline{h}〉-|\overline{t}〉)/\sqrt{2}$, where the states $\{|\overline{h}〉,|\overline{t}〉\}$ are the possible states of Charlie’s laboratory after he has obtained the result of the coin toss. If the result associated with the $|\overline{ok}〉$ state occurs Alice records $\overline{w}=\overline{ok}$ otherwise she records $\overline{w}=\overline{fail}$.
$n:30$	Bob measures Diane’s laboratory with respect to a basis containing the state $|ok〉\equiv (|-\frac{1}{2}〉-|\frac{1}{2}〉)/\sqrt{2}$, where the states $\{|-\frac{1}{2}〉,|\frac{1}{2}〉\}$ are the possible states of Diane’s laboratory after she has obtained the result of the spin measurement. If the outcome associated with the $|ok〉$ state occurs, Bob records $w=ok$, otherwise, he records $w=fail$.
$n:40$	If both Alice and Bob measure $\overline{w}=\overline{ok}$ and $w=ok$, respectively, the experiment is stopped, otherwise it continues to the next round.

2.3. The paradox

Alice and Bob can construct an entangled qubit from the correlations between the measurements of Charlie and Diane,

where the first qubit is associated with Alice and the second qubit is associated with Bob. Alice and Bob can calculate the probability they will measure $\overline{w}=\overline{ok}$ and $w=ok$, respectively, by acting the states $|\overline{ok}〉$ and $|ok〉$ on $|\psi 〉$, Another analysis can be performed by examining the inferences that each observer—Alice, Bob, Charlie and Diane—can construct via assumptions (Q), (C) and (S). Examination of (1) shows that if Diane measures $z=-1/2$, then Alice can only measure the state $(|\overline{h}〉+|\overline{t}〉)/\sqrt{2}$. Hence, from Alice’s perspective, if she measures $\overline{w}=\overline{ok}$, then she can infer Diane has measured $z=+1/2$. From Diane’s perspective, if she measures $z=+1/2$, then she can infer that Charlie has measured $r=tails$ since there is no $|\overline{h}{〉}_A\otimes |+\frac{1}{2}{〉}_B$ component in (1). From Charlie’s perspective, if he measures $r=tails$, then he can infer that Diane will be in the $(|-\frac{1}{2}〉+|+\frac{1}{2}〉)/\sqrt{2}$ state, implying that Bob will measure $w=fail$. Thus, if Alice measures $\overline{w}=\overline{ok}$, then via the inference chain above she knows with certainty that “Diane knows that Charlie knows that Bob measured $w=fail$”, even though Bob and Alice will have a nonzero probability of “agreement”. From this analysis Frauchiger and Renner have concluded that a contradiction arises, yielding a paradox.

There are counterarguments to the logic of the FR thought experiment.^14,15,16 One counterargument, proposed by Ref. 5, highlights the issues of using inferred information before and after a measurement is performed. The inference—if Charlie is in the $|\overline{t}〉$ state then Bob measures $w=fail$—only holds for the original qubit state in (1). However, Alice first measures (1) in the $|\overline{ok}〉$ bases, thus transforming (1), implying Charlie no longer has access to the information required to make his inference. Therefore Charlie could never conclude that Bob has measured $w=fail$, since the nonmeasured version of (1) does not exist after Alice’s measurement. The rejoinder of Renner¹⁷ is that no information dependent on Alice’s measurement is used in Charlie’s logic and hence the paradox is logically sound. We are not concerned with this type of argument, rather we are interested in the consequences of gauge invariance, in particular diffeomorphism invariance, in the construction of the thought experiment.

The FR thought experiment, as constructed, requires classically certain measurements, local observers, and logical inferences to generate a paradox, as codified in assumptions (Q), (C) and (S). Under assumption (Q) Alice and Bob are defined to be local observers with separable Hilbert spaces. This is, however, in tension with general covariance (which should be preserved in low energy quantum gravity as it is a symmetry of general relativity), as constructing separable Hilbert spaces in generally covariant theories is nontrivial.^18,19 In other words, the formulation is not a priori gauge invariant.

Any claim that asserts quantum mechanics is not self-consistent, based on nongauge-invariant observables, must be eyed with caution. Such a claim is equivalent to observers who use different $U(1)$ gauges to claim that electromagnetism is not self-consistent because they each record different values for the vector potential. Certainly, it is possible that there is a physical difference but the difference could be pure gauge. Remedying this in the gravitational context means dressing any local observables so that they are gauge invariant under diffeomorphisms. Only then can one explore the ramifications for quantum mechanics. The key question is whether or not making the construction gauge invariant can be done while remaining in a weak gravity regime. Before we get to the gravitational case we analyze the case in electromagnetism, which is familiar to more readers.

3. QED Gauge Invariant Observables and the FR Thought Experiment

3.1. Background

Following Ref. 20, in N-dimensional systems with gauge symmetries there exist a set of primary constraints, functions ${\varphi }_m(p,q)$ such that,

where $m=1,\dots ,M$, which do not arise from the equations of motion. The set of constraints in (3) define a constraint hypersurface in phase space, on which the solutions to the equations of motion are unique and the canonical Hamiltonian is well defined. Since the constraints vanish on the hypersurface the canonical Hamiltonian remains unchanged under the replacement, where the $u^m$ are arbitrary functions of the configuration space coordinates and velocities and ensure that the Legendre transformation is invertible.

A basic consistency requirement is that the primary constraints are constants in time and thus have vanishing Poisson brackets with (4). The resulting equations of motion are either independent of the $u^m$, and thus only involve the canonical coordinates, or they may impose restrictions on the $u^m$. If the equation of motion for some ${\varphi }_m$ is independent of the $u^m$ and its vanishing imposes a relationship, ${\varphi }_k$ with $k>M$, between the canonical coordinates, independent of the other primary constraints, then ${\varphi }_k$ is considered a secondary constraint. Note that secondary constraints also vanish on the constraint hypersurface, however, their defining feature is that they only hold when the equations of motion are satisfied.

Again consider the consistency requirement, that both the primary and secondary constraints are constants in time and thus have weakly vanishing Poisson brackets with (4). The resulting equations of motion impose restrictions on the $u^m$. Using the equations of motion for the constraints to find solutions for the $u^m$, it follows that,

where $U^m$ denotes particular solutions, $V_a^m$ denotes general solutions, and the coefficient functions, $v^a$, are totally arbitrary. One can now construct a total Hamiltonian defined everywhere in phase space.

A first-class constraint is any constraint that has a weakly vanishing Poisson bracket with all other constraints, i.e.

where $j=1,\dots ,M+K$. The importance of the first-class constraints is fully realized in the following statement, all first-class, primary constraints generate gauge transformations, i.e. where $\delta F$ is the gauge transformation of a canonical function F, and $v^a$ is the difference between two choices of arbitrary functions at different times, related to the first-class, primary constraints ${\varphi }_a$. It should be noted, as counterexamples to the Dirac conjecture have shown, that only some first-class, secondary constraints generate gauge transformations. Any gauge transformation allows one to determine the value of canonical functions F on different hypersurfaces in phase space, allowing for the construction of dressings that, when applied to observables, make them gauge invariant.

As a simple example of the above, consider the Lagrangian density for vacuum electrodyamics,

where $F_{ab}={\partial }_a{A}_b-{\partial }_b{A}_a$. One can show that the temporal component of the canonical momentum ${\pi }^a(x)$ vanishes, thus defining the primary constraint hypersurface, i.e. ${\varphi }_1(p,q)={\pi }^0=0$, for the theory. The Hamiltonian density for this theory becomes, where Roman indices sum over spatial quantities and Greek indices sum over spacetime quantities. The last Lagrange multiplier term enforces the constraint and vanishes on the constraint hypersurfaces. The form of the second to last term is achieved via integration by parts.

The equation of motion for $A_0$ yields the secondary constraint, ${\varphi }_2(p,q)={\partial }_i{\pi }^i=0$, which is Gauss’ law. Since the Poisson bracket between constraints ${\varphi }_1$ and ${\varphi }_2$ vanishes, both constraints are first-class.

We now consider an arbitrary linear combination of both first-class constraints, i.e.

where $f,g$ are arbitrary functions. Calculating the Poisson bracket of C and $A_0,A_i$ yields These then have exactly the form of the gauge transformation for $A_a$ at some instant of time t. The Poisson bracket between $A_a(x)$ and linear combinations of the first-class constraints therefore generates the typical gauge transformation between instantaneous states of electrodynamics, i.e. $A_a(x)\to A_a(x)-{\partial }_a\zeta (x)$, where $\zeta (x)$ is arbitrary. One can promote this to a gauge transformation over histories as well via the extended Hamiltonian, see Ref. 21 for further discussion on this point.

Similarly if we had treated full QED and included a Dirac field in the Lagrangian the Poisson bracket between the local Dirac field $\mathrm{\Psi }(x)$ and linear combinations of the first-class constraints (with the Gauss law appropriately modified as there are now sources) would generate the gauge transformation of $\mathrm{\Psi }(x)$,

In other words, $\mathrm{\Psi }(x)$ is not by itself a gauge invariant object.

3.2. Gauge invariant fields and asymptotic dependence

A gauge-invariant field can be constructed by promoting $\mathrm{\Psi }(x)\to \tilde{\mathrm{\Psi }}(x)=\mathrm{\Psi }(x)e^{iq\zeta (x)}$. A key aspect is that $\zeta (x)$ does not have to have compact support. Following Dirac’s original analysis in Ref. 22, one could choose $\zeta (x)=\int d^3{x}^{\prime }{c}^a(x^{\prime },x)A_a(x^{\prime })$. The new field, $\tilde{\mathrm{\Psi }}(x)$, is then gauge invariant as long as ${\partial }_i{c}^i(x^{\prime },x)={\delta }^{(3)}(x^{\prime }-x)$ is satisfied. The choice of $c^i$, or dressing of the local, electron field, is arbitrary up to the condition on $c^i$. Physically, this corresponds to the fact that there are infinitely many gauge-invariant dressings of the local field, all of which are equivalent up to solutions of the source-free Maxwell’s equations.²³

Not only can $\zeta (x)$ be noncompact, but in certain cases it must be. Specifically, if O is a local field with compact support that transforms nontrivially under $U(1)$ gauge transformations, then the corresponding gauge invariant field $Õ$ depends on the asymptotic field configuration. To see this, expand $Õ$ in a power series in q,

Since $Õ$ is gauge invariant, its Poisson bracket with the constraints vanishes. In particular, let us consider the integrated Gauss’ law constraint, ${\varphi }_2$, on a hypersurface in Minkowski space where $J_b$ is the current for some matter distribution and ${\xi }^b$ is the timelike normal to the three-dimensional hypersurface. Without loss of generality, we can take ${\xi }^b=(1,0,0,0)$. Integrating by parts yields The first term is just a boundary term while the second is the total charge $Q=q$. We can therefore write the constraint as $C_G=C_S-Q$ where $C_S$ is a pure boundary term. We now consider the Poisson bracket of $C_G$ with $Õ$. Since $Õ$ is gauge invariant, its Poisson bracket with $C_G$ must vanish, i.e. is equal to zero.

Since O is compact, $\{C_S,O\}=0$. Moreover, Q is the generator of gauge transformations and hence $\{Q,O\}\ne 0$. Therefore the remaining term must be canceled by the $q\{C_S,O_1\}$ term, which implies that $O_1$ must depend on the asymptotic electromagnetic field, i.e. the dressed gauge-invariant field $Õ$ always has noncompact support.

Gravitationally, a physical observer will transform nontrivially under at least one of the transformations of the Poincare group—there is no ability, for example, to set the total energy of a system to zero. The electromagnetic parallel is the total electric charge of the observing system that is nonvanishing. We shall now examine the consequence of this assumption for the construction of the FR thought experiment as a pedagogical introduction to the gravitational case.

3.3. Consequences for the FR thought experiment

Let us consider a possible implementation of a measurement system for Alice and Bob that parallels the gravitational case and illuminates the difficulty. Assume messages and information between observers, such as the particle of defined spin in $n:00$ of the FR thought experiment, are sent using polarized photons. Alice and Bob measure the polarization via a QED system that correlates the polarization with an electron, i.e. a net charge being present in their local detectors. This can be done as the scattering amplitude in QED is dependent on both photon and electron polarizations.²⁴ The system is designed so that the measurement results, $ok,\overline{ok}$, correlate with the electron state $|e〉$ while $fail,\overline{fail}$, correlate with the vacuum $|0〉$.

The local field associated with the electron, ${\mathrm{\Psi }}_e(x)$, transforms nontrivially under $U(1)$ gauge transformations, and hence a local excitation in Alice or Bob’s detector is not gauge invariant. To maintain gauge invariance there cannot only be a local excitation but also an electromagnetic field that extends to infinity. Prosaically, this is just Gauss’ Law applied to the measurement system above. For a related derivation see Ref. 25.

Now, consider Alice’s role in the original FR thought experiment. With certainty implies that all measurements accessible to Alice are consistent with her inference that Bob measured $w=fail$ (while Bob actually measured $w=ok$) and the existence of a vacuum between them. Her measurements of the electromagnetic field, at her position, must therefore be consistent with Bob measuring $w=fail$ and the rest of the assumptions of the experiment. By superposition, we can split the electric field measured by Alice as ${\overrightarrow{E}}_{AA}+{\overrightarrow{E}}_{AB}$, where ${\overrightarrow{E}}_{AA}$ is the self-consistent electric field generated by Alice’s apparatus measuring $\overline{w}=\overline{ok}$ and ${\overrightarrow{E}}_{AB}$ is the electric field at Alice generated by Bob’s measurement. Since Alice’s measurements must be consistent with her inference that Bob measured $|fail〉$ and vacuum in between them, ${\overrightarrow{E}}_{AB}=0$.

We now see the immediate problem when implementing the Gauss’ Law constraint on the electric fields. Since the measurements are stable, the measured electric fields must be time-independent. Therefore, via Maxwell’s equations, the electric field is curl-free and the derivative of a potential $\mathrm{\Phi }$. Therefore the Gauss’ law constraint reduces to Poisson’s equation, ${\nabla }^2\mathrm{\Phi }=\rho$, where $\rho$ is the charge density. By superposition, Alice can definitively state that the potential due to Bob and its derivative vanish in her region, i.e. ${\mathrm{\Phi }}_{BA}=0,\overrightarrow{\nabla }{\mathrm{\Phi }}_{BA}=0$. However, the unique solution to this is that the potential due to Bob everywhere is, ${\mathrm{\Phi }}_B(r)=0$. Therefore if Bob measures ${\mathrm{\Phi }}_{BB}\ne 0$, the Gauss’ law constraint is violated. There is no way to even define the paradox in a gauge invariant sense.

The escape from this conclusion, in the electromagnetic case, is simple: use detectors/measurement devices that don’t carry a net $U(1)$ charge. This is easy to do since electromagnetism permits screening²² due to the existence of both positive and negative charges. In this case, either outcome of Bob’s measurement would leave no electric field imprint for Alice to measure, allowing the FR thought experiment to go through as it was originally proposed. This leaves higher moments (e.g. electric or magnetic dipoles or quadrupoles) that could still be detected by Alice, but one could in principle cancel those as well out to some finite order. At the monopole level, however, since there are no negative masses available in general relativity to screen the gravitational field, we don’t have measurement systems that are uncharged under the Poincaré group. Therefore the key question is whether such a screening procedure can work at all for gravitationally charged measurement devices? We now turn to this question.

4. Gravitationally Gauge Invariant Observables and the FR Thought Experiment

4.1. Background

Similar to the electromagnetic case, gravity also possesses a gauge symmetry: diffeomorphism invariance. Local fields are not diffeomorphism invariant. For example, let $\mathrm{\Phi }(x)$ be a local, massive, scalar field. Under a diffeomorphism $f:{ℳ}\to {ℳ}$, $\mathrm{\Phi }(x)$ transforms as,

Under an infinitesimal transformation generated by a vector field ${\xi }^a(x)$, the diffeomorphism takes the form, $f^a(x)=x^a+\kappa {\xi }^a$ and the variation of the transformed field is,

$$\begin{array}{ccc}\delta \mathrm{\Phi }(x)\hfill & =\hfill & \mathrm{\Phi }(f^{-1}(x))-\mathrm{\Phi }(x)\hfill \\ \hfill & =\hfill & \kappa {\xi }^a{\partial }_a\mathrm{\Phi }(x)+\frac{1}{2!}{\kappa }^2{\xi }^a{\xi }^b{\partial }_a{\partial }_b\mathrm{\Phi }(x)\hfill \\ \hfill & \hfill & +\frac{1}{3!}{\kappa }^3{\xi }^a{\xi }^b{\xi }^c{\partial }_a{\partial }_b{\partial }_c\mathrm{\Phi }(x)+{𝒪}({\kappa }^4),\hfill \end{array}$$(20)

up to second-order in $\kappa =\sqrt{32\pi G}$. Thus, local fields are not inherently diffeomorphism invariant since the variation of the transformed field is nonvanishing. And, similar to the electromagnetic case, one must construct diffeomorphism-invariant measurements to claim that the theory can make physical predictions.

One can examine the full diffeomorphism-invariance of general relativity but, since the fields are weak in the FR thought experiment, it will suffice to consider diffeomorphism invariance in linearized gravity, up to first-order in Newton’s constant G. We, therefore, work with perturbations around Minkowski space such that the metric is given by

where ${\eta }_{ab}$ is the background Minkowksi metric and $h_{ab}$ is the metric perturbation.

Any field excitation that is diffeomorphism invariant must have a vanishing variation under diffeomorphisms, i.e. for a scalar field

under a diffeomorphism. One may construct such an excitation from a dressing $V^a(x)$ and the local field excitation $\mathrm{\Phi }(x)$. Generally, the dressing V expanded to nth order in $\kappa$ will contain higher order derivatives, however, let us follow²⁶ and consider the following construction, up to first-order in $\kappa$Combining the transformation laws from (20) and (22) implies that, at first-order in $\kappa$, the dressing must transform as under diffeomorphisms.

One can now determine the form of $V^a(x)$ that allows local field excitations to be diffeomorphism invariant. Applying the logic from Ref. 22, let $V^a(x)$ be a linear functional of the metric perturbation over the spacetime volume. The dressing takes the form,

where the tensor f is symmetric in the second and third indices. Enforcement of the transformation law in (24) and the transformation laws for the metric perturbation imply that f must satisfy, The function f determines the structure of the gravitational field that is created by excitations of the $\varphi$ field. There is freedom in the choice of f, however, this freedom does not allow for physically inequivalent dressings. For examples of the different choices for f see Ref. 26.

4.2. Consequences for the FR thought experiment

As in the electromagnetic case, the question for the construction of the FR thought experiment is whether there exists an f that has local support given some measurement apparatus. In general relativity, there is also a dressing theorem,²⁶ which states that if a matter distribution in a compact region has nonvanishing Poincaré charge, then the gravitational field depends on the asymptotic metric, i.e. there are no local functions f such that $V^a(x)$ and hence $\tilde{\mathrm{\Phi }}(x)$, the gauge-invariant operator, is of compact support. Therefore any measurement device that stores information in, e.g. the energy eigenstate of a detector or the spin of a particle, would generate a gravitational field that extends to infinity (with some falloff). In this situation, the problem is the same as the charged electromagnetic case—there is no gauge-invariant construction of the paradox since the field measured by Alice must be compatible with her belief that Bob measures $w=fail$ and this would violate at least one of the constraint equations of general relativity.

The key question is then: can Bob make a measurement and store information in such a way that, outside his local region, the gravitational field is indifferent to the results of the measurement? In other words, is it possible to screen the gravitational field? Gravitational screening has been investigated by a number of authors. Carlotto and Schoen proved that one can construct classical solutions to the vacuum Einstein equations that satisfy the constraint equations and have support only in conical regions.²⁷ This allowed them to build global solutions by gluing; where two regions of nontrivial gravitational field are separated by a region of flat space for an arbitrarily long time T. Similar vacuum solutions have been found that are identical to the Schwarzschild-AdS spacetime outside a bounded region on a spacelike slice but differ inside.²⁸ However, such solutions share a generic feature: they are nonanalytic somewhere in the spacetime and, as pointed out in Ref. 13, quantum gravity theories that have unitarity can forbid nonanalytic solutions. For our purposes, the relevant arena to examine this line of reasoning is in the perturbative regime, which has been explored in Ref. 29 and which we paraphrase below.

Consider a neighborhood U containing a local state $|\psi 〉$ and an $ϵ$-extended neighborhood $U_{ϵ}$, such that $U\subset U_{ϵ}$. One can construct, up to first-order in $\kappa$, a split gravitational dressing that shields information from observers outside of $U_{ϵ}$. Specifically, outside $U_{ϵ}$, the gravitational field is only a function of the total Poincaré charges of the fields inside U. Therefore if two states $|{\psi }_1〉$ and $|{\psi }_2〉$ share the same Poincaré charge, they are indistinguishable to an outside observer, i.e.

for all observers outside $U_{ϵ}$.

We now ask if such a pair of states can contain information. We have already seen the issue with charged fields, so let us consider uncharged fields that contain information strictly in the gravitational sector. We will let the region U be Bob’s local measuring device. Assign the states $|{\psi }_1〉,|{\psi }_2〉$ to $|ok〉,|fail〉$, and let the information stored in the classically stable measuring device be stored in two differing energy–momentum tensors inside Bob’s region, call them $T_{ab}^n=〈{\psi }_n|{\widehat{T}}_{ab}|{\psi }_n〉$ for $n=1,2$. This then also implies that inside Bob’s region, the gravitational fields must also differ, i.e. inside $U_{ϵ}$,

Furthermore, since the information must be stable according to the assumptions in the construction of the FR thought experiment, the time derivative ${ḣ}_{ab}$ must vanish. As long as such a situation is possible then there are no consequences for the FR thought experiment and gravity, quantum or otherwise, can indeed be neglected.

4.3. Inconsistency of the weak field general relativistic approximation

The argument above is not self-consistent, however. In order for the linearized, weak-field approximation to properly reflect gauge invariance, via imposition of the general relativistic constraint equation, we need two things to hold. First, the perturbations $h_{ab}$ that are solutions to the quadratic action need to be small so that the full action is well approximated by the quadratic action. Second, the derivatives of the perturbations also must remain small. If they do not, then one cannot use the quadratic approximation to the general relativistic action, and corresponding constraint equations, but instead must use a quantum-corrected effective action that includes higher curvature terms. Note there are higher curvature corrections that are still quadratic in $h_{ab}$. As an example, consider the addition of a putative, quantum gravity-induced, subdominant Weyl squared term to the Einstein–Hilbert action

where $W^2=W_{abcd}{W}^{abcd}$ and $W_{abcd}$ is the Weyl tensor.

The quadratic action in the Newtonian gauge³⁰ is then

where $h_{ij}$ are the usual spatial excitations of the metric. Similar expressions hold for the scalar and vector sectors of $h_{ab}$ as well. Note the presence of terms that are quadratic in the field, yet have higher derivatives in both space and time. If derivatives of a putative quadratic solution in general relativity ($\alpha \to 0$) become large compared to the Planck scale, then these additional terms will contribute to the action and the general relativity approximation is then not self-consistent for these solutions. Quadratic in field terms containing arbitrary higher derivatives exist as well, e.g. via the insertion of a term in the Lagrangian with a d’Alembertian between the Weyl tensors.

Let us examine the localization procedure on gravitational states relevant for the FR thought experiment, i.e. those defined in (28), in this context. First, since we assume $h_{ab}$ is time-independent, we can ignore any time derivatives. Now consider the difference $\mathrm{\Delta }{h}_{ab}=〈ok|h_{ab}|ok〉-〈fail|h_{ab}|fail〉$ over the whole spacetime. The function $\mathrm{\Delta }{h}_{ab}$ is of compact support, as it is nonzero inside $U_{ϵ}$ and identically zero outside, by construction. Therefore, it is nonanalytic by construction. In the linearized approximation, superposition holds and thus $\mathrm{\Delta }{h}_{ab}$ is also a solution to the linearized constraint equations. If one is to be able to neglect higher curvature terms then the nth derivative of $\mathrm{\Delta }{h}_{ab}$ must grow no faster than a polynomial in n. In other words, there exists some mass scale $M<M_{Pl}$, such that ${\partial }^n{h}_{ab}/\partial x^n<M^n$. Only if this is true will higher curvature terms (with generic $O(1)$ coefficients) evaluated on the lowest order solutions remain subdominant to the Einstein–Hilbert term. Only then can the general relativistic approximation and use of the corresponding constraint equations remain valid.

However, since $\mathrm{\Delta }{h}_{ab}$ is a function of compact support and nonanalytic, its derivatives must violate precisely this condition! Real analytic functions, $f(x)$, satisfy the following property³¹: for each point $x_0$ within the domain of $f(x)$, there exists a neighborhood ${𝒱}$ of $x_0$ such that

for all $x\in {𝒱}$, where $C\in {ℝ}_{+}$ and $m\in \mathbb{N}$. Nonanalytic functions must have at least one derivative, of order m, that violates (31). In fact, an infinite number of derivatives violate this bound for $\mathrm{\Delta }{h}_{ab}$, our bump function solution to the constraint equations. To see this, work by contradiction and assume that only a finite number of derivatives violate this bound. Thus, there exists some number q such that all qth and higher derivatives of $\mathrm{\Delta }{h}_{ab}$ are analytic. However, the derivative of a bump function is another bump function. Hence, the qth and higher derivatives cannot be analytic. Therefore, there are an infinite number of derivatives of $\mathrm{\Delta }{h}_{ab}$ that violate the bound (31).

The consequence of this argument is that one cannot state that a nonanalytic solution to linearized general relativity of compact support is an actual solution of quantum gravity. In other words, there is no way to consistently implement a weak gravity, gauge invariant construction of a paradox within the framework of general relativity, neglecting higher curvature quantum gravity corrections.

4.4. Boundary unitarity versus nonanalytic, localized solutions

The above argument is analogous to the path leading to the chronology protection conjecture where a paradox even in low curvature regions leads unavoidably to quantum gravity. Of course, this does not, by itself, lead to any conclusion on whether or not the construction is possible in quantum gravity. One has to invoke additional assumptions about quantum gravity to make progress. We examine below how the principle of boundary unitarity, a proposed property of holographic quantum gravity, sheds light on this latter question.

The FR thought experiment generates a paradox by isolating two observers, allowing each to make measurements with certainty, stably record the results, and then meet. We have shown that whether one can formulate this procedure in a gauge-invariant manner is sensitive to the behavior of higher curvature terms and is hence a question for quantum gravity. It is illuminating to ask whether there are any other resolutions to paradoxes in gravitational physics that involve diffeomorphism invariance, nonanalytic behavior, and unitary evolution. Marolf¹² has examined the consequences of diffeomorphism invariance, which yields a property of quantum gravity known as boundary unitarity. This property has been proposed as a solution to the infaller paradox of black hole physics.¹³

Consider the classical, diffeomorphism-invariant, general relativistic Hamiltonian

where $N_a$ are the lapse and shift, $C^a$ are the constraints, and $H_{\partial }$ is the boundary Hamiltonian. At asymptotic times (32) generates time translations via Hamilton’s equations. Let O be an element of the boundary algebra ${{𝒜}}_{\partial }$—the algebra of diffeomorphism-invariant observables that exist on the past/future null infinities. Since ${{𝒜}}_{\partial }$ is closed under the Poisson bracket, the Poisson bracket of O with the constraints vanishes. Hence, the evolution of diffeomorphism-invariant observables on the past/future null boundary is governed by the boundary Hamiltonian As pointed out in Ref. 13, (33) admits solutions that are nonanalytic in time, which through simple propagation would also lead to nonanalytic in space solutions. Therefore, one cannot classically forbid nonanalytic solutions.

Can the same be said for quantum gravity? Assuming the quantum gravitational Hamiltonian—whatever it may be—is diffeomorphism invariant, mirroring the structure of (32), and the constraint algebra is closed under the commutator, then the evolution of boundary observables is given by

Thus, given the above requirements, the evolution of boundary observables is governed by the boundary Hamiltonian of the theory.

The two evolution equations, (33) and (34), are similar. However there is a key difference—while the classical evolution equation admits solutions that are nonanalytic in time, the quantum evolution equation does not. Indeed, the Hamiltonian would be an element of ${{𝒜}}_{\partial }$, as would be the commutator of the Hamiltonian with any other observable. Thus, (34) would be an element of ${{𝒜}}_{\partial }$ as well. This implies that the boundary algebra associated with quantum gravity is closed under unitary evolution. In particular, quantum mechanically, the solution to (34) for an observable O is

Assuming that $H_{\partial }$ is self-adjoint and O is bounded, (35) is an analytic function in time. The inner product of (35) with eigenstates of $H_{\partial }$ has the capacity to be analytic in time, however, there are infinite terms in the inner product, which could hide nonanalytic behavior. A remedy is to assume $H_{\partial }$ is bounded from below and then apply an energy cut-off as an upper bound.

Boundary unitarity forbids nonanalytic behavior in time, while the FR thought experiment yields nonanalytic behavior on a spatial slice. One can, of course, translate the nonanalytic, spacelike solutions of compact support to the boundary by considering one of the localized, weak field solutions necessary for the FR thought experiment and allowing it to propagate outwards. Since the excitation is localized, upon arrival at the boundary, or in some suitable asymptotic region, it would constitute new information and generate nonanalytic behavior in time. The gravitational degrees of freedom would be static and consistent with the background spacetime up until some time t, then, upon arrival of the excitation, suddenly change, i.e. behave nonanalytically, which would be forbidden by boundary unitarity. Hence, if boundary unitarity holds for quantum gravity, the local excitations necessary for the FR paradox to even be stated would be forbidden.

5. Conclusions

The FR thought experiment generates a paradox between two local observers in the absence of gravity. However, any thought experiment, especially those that draw deep conclusions about quantum mechanics, should employ gauge-invariant observables to ensure that the conclusions are gauge-invariant. We implemented the gauge symmetry of general relativity, diffeomorphism invariance, on the observables in the FR thought experiment under the assumptions of the experiment, at first order in Newton’s constant. Perhaps contrary to expectations, such a perturbative construction winds up not being self-consistent; the necessary solutions were sensitive to higher curvature and quantum gravity due to their nonanalytic nature. This result is similar to the results of other paradoxes in weak-field gravity. The paradox in the FR thought experiment can be argued to be ill-posed and irrelevant to our universe if quantum gravity forbids nonanalytic, bulk solutions, similar to how the principle of boundary unitarity forbids nonanalytic solutions in time.¹³ The paradox also technically requires a quantum gravitational framework, even starting from weak field gravity, analogous to how semi-classical gravity breaks down even in low curvature spacetimes with closed timelike curves and grandfather paradoxes.¹⁰

Note that no individual piece of what we used was inherently not self-consistent. Without gravity, the local excitations necessary for the FR experiment are perfectly valid. Similarly, the construction of diffeomorphism-invariant observables, at first order in Newton’s constant, is perfectly well defined. It is only the combination of the two that leads one to difficulty. Specifically, when one combines certainty, which implies exact knowledge of the gravitational field on a hypersurface, stability, which implies that the knowledge can be kept immutably in a time-independent manner, and a discrepancy in gravitational observations between two separated observers does one find that the resulting construction cannot be made diffeomorphism invariant in a way that is not sensitive to quantum gravity. Furthermore, if quantum gravity satisfies boundary unitarity, then there would be no possible construction of the FR thought experiment that is gauge invariant.

To conclude, we point out one more similarity with the infaller paradox. There is an additional way to “resolve” the paradox in the FR thought experiment with local excitations if one desired to enforce their existence. Using the same assumptions about certainty and stability as before, let Alice and Bob classically record two different measurements from the same experiment, resulting in their respective, local stress-energy tensors to differ. It follows that Alice’s and Bob’s spacetime regions are described by differing local metrics, $g_{ab}^A$ for Alice and $g_{ab}^B$ for Bob. Assuming a vacuum between Alice and Bob, one could solve the respective constraint equations to extend their metric solutions uniquely onto the three-dimensional spacelike hypersurface at some time t. One can’t do this globally, of course, and therefore along some (otherwise arbitrary) $2D$ surface $\gamma$ that intersects every curve between Alice and Bob one must have a discontinuity in the metric. The metric that describes the total spacetime is then $g_{ab}=\mathrm{\Theta }(x)g_{ab}^A+\mathrm{\Theta }(-\ell )g_{ab}^B$, where we have chosen for convenience the boundary surface $\gamma$ to be at $x=0$. Since the metric is discontinuous substituting it into the Einstein equations would generate a divergent stress energy tensor along $\gamma$, similar to a firewall. This would be the necessary energy–momentum configuration to keep local excitations while maintaining gauge invariance. We do not, however, espouse this “solution”, but rather argue instead that the construction itself cannot be made gauge invariant and that such a solution would amount to a misunderstanding of the quantum gravitational Hilbert space.

Acknolwedgments

This research was supported in part by DOE Grant DE-SC0020220. We thank Jon Cheyne for reading a draft of this paper.

ORCID

David Mattingly  https://orcid.org/0000-0002-6911-3383