Conditions for Unitarity in Timeless Quantum TheoryPreprint: APS/123-QED

The concept of time as an external parameter in conventional quantum theory clashes with the background-independent character of general relativity and suffers from several explanatory and technical issues, generally termed the “problem of time” 26; 25; 1. A promising solution to this problem is to remove any appeal to an external time parameter and recover a notion of dynamics relative to some systems acting as clocks in an overall stationary quantum Universe 31. Crucially, quantum theory interpreted as a universal physical theory must already contain an explanation for the relative states underlying such an approach to time 10. Therefore, we can gather the explanations of time that start from a stationary quantum Universe (some of which have been proven to be equivalent 23) under the term “timeless quantum theory” 25. This approach usually leads to the ordinary dynamics of quantum theory only when the clocks are isolated from the rest of the Universe. The dynamics is substantially altered when the clocks interact with the rest of the Universe 17; 29; 37; 38; 8; 30; 36; 35; 15; 14; 27; 6, and is even non-unitary for some types of interaction 32; 33; 4. Non-unitarity was also observed in semiclassical approaches to time 3; 2; 20. The timekeeping of physical clocks is clearly affected if they interact with other systems in specific ways—for instance, when a watch smashes on the ground or melts under intense heat. Such interactions should therefore be prevented. However, some interactions cannot be shielded 7; 37; 8 or are required to measure the time of the clock 27, while others seem to be inevitable in relativistic settings 38. Therefore, the effect of interactions with the clock in timeless approaches cannot be neglected. Moreover, since unitarity is central to quantum theory, it is crucial to explore the origin of non-unitarity and its role in timeless quantum theory.

In this work, I apply Page and Wootters’ approach 31 to derive necessary and sufficient conditions on the Universe’s Hamiltonian for its relative dynamics to be unitary. These conditions are linked to some physical properties of the clock. Specifically, unitary dynamics is associated with clock rates that are constant in time and independent of the clock’s internal structure.

The starting point is the stationarity condition given by the Wheeler-DeWitt equation 11

where $\hat{H}_{U}$ and $\left|\left.\kern-1.2pt\Psi_{U}\right>\right>$ are the Hamiltonian and the state vector of the Universe, respectively, and the identification of a subsystem $\mathcal{C}$ of the Universe that acts as a clock 31. The double-ket notation $\left|\left.\kern-1.2pt\cdot\right>\right>$ is a visual aid to denote that the state vector belongs to the Hilbert space of the Universe 17. In a non-relativistic setting, we can decompose the Hilbert space of the Universe as $\mathscr{H}_{U}=\mathscr{H}_{\mathcal{C}}\otimes\mathscr{H}_{\mathcal{R}}$, where $\mathscr{H}_{\mathcal{C}}$ and $\mathscr{H}_{\mathcal{R}}$ are the Hilbert spaces of $\mathcal{C}$ and of the rest of the Universe $\mathcal{R}$, respectively, and write $\hat{H}_{U}$ as

with $\hat{V}$ an interaction term between $\mathcal{C}$ and $\mathcal{R}$.¹¹ 1 In the rest of the paper, the tensor products with the identity operators will often be dropped to improve the readability of the equations. The Hilbert space acted upon by the operators should be clear from the indices and the context. I will later relax this assumption on the form of $\hat{H}_{U}$. In the rest of the discussion, $\mathcal{C}$ is assumed to be an ideal clock 23, meaning that it has a self-adjoint operator $\hat{T}_{\mathcal{C}}$ such that $\left[\hat{T}_{\mathcal{C}},\hat{H}_{\mathcal{C}}\right]=i\hbar$, with its eigenstates ${\left|\phi_{\mathcal{C}}(t)\right>}=e^{-i\hat{H}_{\mathcal{C}}(t-t^{\prime})/\hbar}{\left|\phi_{\mathcal{C}}(t^{\prime})\right>}$ $\forall t,t^{\prime}\in\mathds{R}$ corresponding to the different time readings.²² 2 In the rest of the paper $\hbar=1$. $\mathcal{C}$ is ideal in the sense that the set of states $\left\{{\left|\phi_{\mathcal{C}}(t)\right>}\right\}_{t\in\mathds{R}}$ are orthogonal and so $\mathcal{C}$ can perfectly distinguish any time instant. Such a clock is usually deemed unphysical because $\hat{H}_{\mathcal{C}}$ is unbounded from below. Nevertheless, ideal clocks can be thought of as the limiting case of ever-better realistic clocks. Finally, the states $\left\{{\left|\phi_{\mathcal{C}}(t)\right>}\right\}_{t}$ are not normalizable and thus do not represent physical states. This is analogous to the eigenstates of the position operator in ordinary quantum mechanics and can be mathematically dealt with using the Rigged Hilbert Space formalism 24 or introducing a “physical inner product” 37.

Despite the Universe being stationary, we can recover a notion of dynamics by considering the states of the Universe relative to the clock $\mathcal{C}$ showing different times 31. When the clock reads the time $t$, the corresponding state of the Universe is given by the following Everettian 13 relative state

where $\hat{\Pi}_{t}\coloneqq{\left|\phi_{\mathcal{C}}(t)\right>}{\left<\phi_{\mathcal{C}}(t)\right|}\otimes\hat{\mathds{1}}_{\mathcal{R}}$ is the projector on the state of the clock showing the time $t$. $\hat{\Pi}_{t}$ is an “improper” projector because $\hat{\Pi}_{t}^{2}$ diverges due to the states $\left\{{\left|\phi_{\mathcal{C}}(t)\right>}\right\}_{t}$ being not normalizable. Again, this problem can be dealt with using the Rigged Hilbert Space formalism and would not arise for realistic clocks.

If there are no $\mathcal{C}$-$\mathcal{R}$ interactions, that is, $\hat{V}=0$, then Eqs. (1)–(2) imply that

recovering the unitary evolution of ordinary quantum theory for a system with Hamiltonian $\hat{H}_{U}$ and initial state $\hat{\Pi}_{0}\left|\left.\kern-1.2pt\Psi_{U}\right>\right>$. In this case, there is the freedom to choose an initial state of the form ${\left|\phi_{\mathcal{C}}(0)\right>}{\left|\psi^{\prime}_{\mathcal{R}}(0)\right>}$ for any ${\left|\psi^{\prime}_{\mathcal{R}}(0)\right>}\in\mathscr{H}_{\mathcal{R}}$ 35. Note that Eqs. (3) and (4) are slightly different from those discussed in the literature, which usually focus on the evolution of $\mathcal{R}$ alone by considering the “partial” inner product ${\left|\psi_{\mathcal{R}}(t)\right>}\coloneqq\left.\kern-1.2pt\left<\phi_{\mathcal{C}}(t)\left|{\Psi_{U}}\right>\right>\right.\kern-1.2pt$ 17. Instead, Eqs. (3) considers the state of the whole Universe at time $t$, which can also be written as $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>={\left|\phi_{\mathcal{C}}(t)\right>}{\left|\psi_{\mathcal{R}}(t)\right>}$. Eq. (4) implies that ${\left|\psi_{\mathcal{R}}(t)\right>}=e^{-i\hat{H}_{R}t}{\left|\psi_{\mathcal{R}}(0)\right>}$ $\forall t\in\mathds{R}$, so ${\left|\psi_{\mathcal{R}}(t)\right>}$ also evolves unitarily. Unlike $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$, ${\left|\psi_{\mathcal{R}}(t)\right>}$ can be chosen to be normalisable and with norm $\norm{\ket{\psi_{\mathcal{R}}(t)}}=1$, $\forall t\in\mathds{R}$ 37.

If $\hat{V}\neq 0$, ${\left|\psi_{\mathcal{R}}(t)\right>}$ obeys a modified Schrödinger equation 37. This equation can also be written as (Appendix A)

where $\hat{V^{\prime}}$ is an operator-valued function of $t$ and derivatives with respect to $t$. Similarly, the dynamical equation for $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$ is given by

These equations are linear, and the maximum order of the time derivatives appearing in them is equal to the maximum power of $\hat{H}_{\mathcal{C}}$ in $\hat{H}_{U}$. The solutions of these equations are known in closed form only for some types of interaction 17; 37; 38; 8; 35; 27; 14; 6 and can lead to a non-unitary evolution 32; 33. Note that if ${\left|\psi_{\mathcal{R}}(t)\right>}$ evolves unitarily, then so does ${\left|\Psi_{U}(t)\right>}$, and vice versa (Appendix B). Incidentally, this shows that non-unitarity can also appear at the level of the Universe, where no external entangled system has been traced out. Therefore, this effect is different from the non-unitary dynamics of open quantum systems, and is related to the structure of the state of the Universe $\left|\left.\kern-1.2pt\Psi_{U}\right>\right>$ and the way the clock’s time states “slice” it.

The central result of this work is the following characterization of the types of interaction that lead to a unitary dynamics. First, consider the self-adjoint “rate” 33 or “redshift” 8 operator³³ 3 More precisely, one needs to assume that $\hat{H}_{U}$ and $\hat{T}_{\mathcal{C}}$ admit a common invariant dense domain on which the commutator in Eq. (7) is defined, and that $\hat{\alpha}$ admits a self-adjoint extension. Also, note that due to the algebra of $\mathcal{C}$, the commutator $i\left[\cdot,\hat{T}_{\mathcal{C}}\right]$ can be interpreted as a derivative along the diagonal of $\hat{H}_{\mathcal{C}}$ 28; 35.

The following two requirements are sufficient for the dynamics to be unitary:

Note that these conditions do not rely on the specific form of the Hamiltonian of the Universe of Eq. (2) and so apply to more general types of constraint (excluding some pathological constraints discussed in Appendix D). Since we are dealing with unbounded operators, in the rest of the work Eqs. (8)-(9) will be assumed to hold in the strong sense, i.e., that $\hat{\alpha}$ commutes strongly with $\hat{T}_{\mathcal{C}}$ and $\hat{H}_{U}$ 34. Strong commutativity ensures the existence of a joint spectral measure and hence a joint functional calculus, allowing us to define products of strongly commuting operators unambiguously and control their domains. All the operator equalities are understood on the natural domains defined by the joint functional calculus of the strongly commuting operators involved. Eq. (8) in the strong sense implies that $\hat{\alpha}$ must be of the form $\hat{\alpha}=\hat{\alpha}(\hat{T}_{\mathcal{C}})=\int\text{d}t\,{\left|\phi_{\mathcal{C}}(t)\right>}{\left<\phi_{\mathcal{C}}(t)\right|}\otimes\hat{\alpha}_{\mathcal{R}}(t)$, where $t\rightarrow\hat{\alpha}_{\mathcal{R}}(t)$ is a measurable operator-valued function.

Using Eq. (8), $\hat{\Pi}_{t}\hat{H}_{U}\left|\left.\kern-1.2pt\Psi_{U}\right>\right>=0$, which follows directly from Eq. (1), leads to (Appendix C)

which is a type of generalized Schrödinger equation. There are now two cases. When $\hat{\alpha}$ is invertible, Eq. (10) becomes

meaning that $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>=\hat{U}(t)\left|\left.\kern-1.2pt\psi_{U}(0)\right>\right>$, with $\hat{U}(t)=e^{-i\hat{\alpha}^{-1}\hat{H}_{U}t}$. The operator $\hat{\alpha}^{-1}\hat{H}_{U}$ is defined via the joint functional calculus of the strongly commuting operators $\hat{\alpha}$ and $\hat{H}_{U}$, which ensures its self-adjointness and thus the unitarity of $\hat{U}(t)$.

If $\hat{\alpha}$ is not invertible, then its spectrum must contain 0, that is, $0\in\sigma(\hat{\alpha})$. Calling $\hat{P}^{(0)}=\int\text{d}t\,{\left|\phi_{\mathcal{C}}(t)\right>}{\left<\phi_{\mathcal{C}}(t)\right|}\otimes\hat{P}^{(0)}_{\mathcal{R}}(t)$ the (possibly improper) projector onto the generalized 0-eigenspace of $\hat{\alpha}$ and $\hat{P}^{(+)}=\hat{\mathds{1}}_{U}-\hat{P}^{(0)}=\int\text{d}t\,{\left|\phi_{\mathcal{C}}(t)\right>}{\left<\phi_{\mathcal{C}}(t)\right|}\otimes\hat{P}^{(+)}_{\mathcal{R}}(t)$ the projector onto its orthogonal complement, so that $\hat{P}^{(0)}+\hat{P}^{(+)}=\hat{\mathds{1}}_{U}$ and $\hat{P}^{(0)}\hat{P}^{(+)}=\hat{P}^{(+)}\hat{P}^{(0)}=0$, we can multiply Eq. (10) by $\hat{P}^{(0)}$ and $\hat{P}^{(+)}$ to get, respectively,

The first equation gives the dynamics for the state vector $\hat{P}^{(+)}\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>={\left|\phi_{\mathcal{C}}(t)\right>}\hat{P}^{(+)}_{\mathcal{R}}(t){\left|\psi_{\mathcal{R}}(t)\right>}$; the second imposes a constraint. Note that due to Eq. (8) (in the strong sense) the projection preserves the form of the states of the Universe at time $t$ as product states of ${\left|\phi_{\mathcal{C}}(t)\right>}$ and ${\left|\psi_{\mathcal{R}}(t)\right>}$.

In general, the dynamics of $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$ is undetermined. However, Eq. (9) implies that $\hat{P}^{(+)}$ and $\hat{P}^{(0)}$ strongly commute with $\hat{H}_{U}$ so they share a complete set of eigenstates.⁴⁴ 4 More precisely, their spectral measures commute. Excluding the pathological cases when $\hat{H}_{U}$ has 0-eigenstates of the form ${\left|\phi_{\mathcal{C}}(t)\right>}{\left|\chi_{\mathcal{R}}\right>}$ (see Appendix D), Eq. (13) implies that $\hat{P}^{(0)}{\left|\Psi_{U}(t)\right>}={\left|\phi_{\mathcal{C}}(t)\right>}\hat{P}^{(0)}_{\mathcal{R}}(t){\left|\psi_{\mathcal{R}}(t)\right>}=0$, effectively restricting the allowed states of $\mathcal{R}$ in time. This restriction is well-defined in time because $\hat{P}^{(+)}$ and $\hat{P}^{(0)}$ commute with $\hat{H}_{U}$. Having excluded such states from $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$, Eq. (12) can be written as

where $\hat{\alpha}^{+}$ is the Moore-Penrose inverse of $\hat{\alpha}$ (defined via spectral calculus). The solution of this equation evolves unitarily because $\hat{\alpha}^{+}$ is self-adjoint and if $\hat{\alpha}$ strongly commutes with $\hat{H}_{U}$ then so does $\hat{\alpha}^{+}$ and thus $\hat{\alpha}^{+}\hat{H}_{U}$ is self-adjoint.

We have thus seen that Eqs. (8)–(9) imply that $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$ evolves with the unitary operator

after possibly restricting $\left|\left.\kern-1.2pt\psi_{\mathcal{R}}(t)\right>\right>$ as prescribed by Eq. (13) if $\hat{\alpha}$ is not invertible ($\hat{\alpha}^{+}=\hat{\alpha}^{-1}$ if $\hat{\alpha}$ is invertible). Consequently, ${\left|\psi_{\mathcal{R}}(t)\right>}$ also evolves unitarily, although its evolution has a more complicated expression; namely,

with $\hat{H}_{\mathcal{R}}^{eff}(s)\propto{\left<\phi_{\mathcal{C}}(s)\right|}\hat{\alpha}^{+}\hat{H}_{U}-\hat{H}_{\mathcal{C}}{\left|\phi_{\mathcal{C}}(s)\right>}$ (see Appendix E).

$\hat{\alpha}^{+}$ can be seen as modifying the normal rate of the clock, hence the name (inverse) “rate operator”. It is also easy to see that $\hat{U}(t)$ leaves $\hat{H}_{U}$ invariant, $\hat{U}^{\dagger}(t)\hat{H}_{U}\hat{U}(t)=\hat{H}_{U}$, meaning that the energy of the Universe is conserved in time, and acts on $\hat{T}_{\mathcal{C}}$ in the following way $\hat{U}^{\dagger}(t)\hat{T}_{\mathcal{C}}\hat{U}(t)=\hat{T}_{\mathcal{C}}+t\cdot\hat{P}^{(+)}$. Lastly, there is the freedom to choose any initial state of the form ${\left|\phi_{\mathcal{C}}(0)\right>}{\left|\chi_{\mathcal{R}}(0)\right>}$ that does not lie in the kernel of $\hat{\alpha}$ (Appendix F).

This section shows that while the conditions of Eqs. (8) and (9) are not necessary, they are necessary up to constraint equivalence. First, a simple example shows that the conditions are not strictly necessary. Consider the interacting Hamiltonian $\hat{H}^{\prime}_{U}=\left(\hat{H}_{\mathcal{C}}^{2}+\hat{H}_{\mathcal{R}}^{2}+\delta\right)\cdot\left(\hat{H}_{\mathcal{C}}+\hat{H}_{\mathcal{R}}\right)$, expressed as a product of two commuting factors, with $\delta>0$. It is easy to check that $\left[\hat{T}_{\mathcal{C}},\hat{\alpha}\right]\neq 0$ and that ${\left|\psi_{\mathcal{R}}(t)\right>}$ evolves with the unitary operator $e^{-i\hat{H}_{\mathcal{R}}t}$. This means that Eq. (8) is not necessary to recover a unitary dynamics. Similar considerations hold for Eq. (9). Here the reason is easy to understand: the factor $\hat{H}_{\mathcal{C}}^{2}+\hat{H}_{\mathcal{R}}^{2}+\delta$ in $\hat{H}^{\prime}_{U}$ has no physical relevance for the relative dynamics of the Universe because it is positive definite, so the 0-eigenstates of $\hat{H}^{\prime}_{U}$ are all and only the 0-eigenstates of $\hat{H}_{\mathcal{C}}+\hat{H}_{\mathcal{R}}$. Therefore, $\hat{H}^{\prime}_{U}$, as a constraint, acts equivalently to $\hat{H}_{\mathcal{C}}+\hat{H}_{\mathcal{R}}$, for which Eqs. (8)-(9) do hold.

This is not just a coincidence. It can be proved, more generally, that if $\hat{H}_{U}$ leads to unitary relative dynamics, then $\hat{H}_{U}$ is physically equivalent to a constraint $\hat{\mathscr{C}}$ for which Eqs. (8)–(9) hold in the strong sense. This is proven in Appendix G assuming that ${\left|\Psi_{\mathcal{R}}(t)\right>}$ evolves with a strongly continuous unitary operator $\hat{U}(t,t_{0})$ admitting a self-adjoint generator $\hat{X}_{\mathcal{R}}(t)$ which is bounded, $\sup\limits_{t}\norm{\hat{X}_{\mathcal{R}}(t)}<\infty$. Here, physically equivalent means that the physical Hilbert space $\mathscr{H}_{phy}(\hat{H}_{U})$, i.e. the 0-eigenspace of $\hat{H}_{U}$, is the same as $\mathscr{H}_{phy}(\hat{\mathscr{C}})$, and consequently $\hat{H}_{U}$ and $\hat{\mathscr{C}}$ must lead to the same dynamics for $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$ and ${\left|\Psi_{\mathcal{R}}(t)\right>}$.

As a result, if $\hat{H}_{U}$ is not physically equivalent to a constraint with such properties, then the relative evolution must be non-unitary. For example, the non-interacting Hamiltonian of the Universe $\hat{H}^{\prime\prime}_{U}=\left(\hat{H}_{\mathcal{C}}+\hat{H}_{\mathcal{R}}\right)\cdot\left(\hat{H}_{\mathcal{C}}-\hat{H}_{\mathcal{R}}\right)$ is not physically equivalent to a constraint satisfying Eqs. (8)-(9) and it indeed leads to a non-unitary relative dynamics. Specifically, ${\left|\psi_{\mathcal{R}}(t)\right>}$ obeys a dynamical equation analogous to the Klein-Gordon equation 12 and so it evolves, in general, non-unitarily. Some particular states ${\left|\psi^{\prime}_{\mathcal{R}}(t)\right>}$ evolve unitarily, but further conditions are required to restrict to only such states 22.

What is the physical interpretation of Eqs. (8) and (9)? To understand this, consider the meaning of $\hat{\alpha}$. This operator was defined in 33 as the “time rate of the clock” $\mathcal{C}$ from the perspective of another non-interacting clock $\mathcal{C}_{2}$. As $\hat{\alpha}$ is not a c-number, it should be more aptly called a rate operator. To see how it is connected to the rate of $\mathcal{C}$ from the perspective of $\mathcal{C}_{2}$, assume that $\mathcal{C}_{2}$ is an ideal clock and consider the Hamiltonian of the Universe $\hat{H}_{U}=\hat{H}_{\mathcal{C}}+\hat{H}_{\mathcal{R}}+\hat{V}_{\mathcal{C}\mathcal{R}}+\hat{H}_{\mathcal{C}_{2}}$, with $\hat{V}_{\mathcal{C}\mathcal{R}}$ an interaction term between $\mathcal{C}$ and $\mathcal{R}$ only. Similarly to Eqs. (3)–(4), the state of $\mathcal{CR}$ when $\mathcal{C}_{2}$ shows the time $t$ is given by ${\left|\psi_{\mathcal{CR}}(t)\right>}=e^{-i\hat{H}_{\mathcal{CR}}t}{\left|\psi_{\mathcal{CR}}(0)\right>}$, with $\hat{H}_{\mathcal{CR}}\coloneqq\hat{H}_{\mathcal{C}}+\hat{H}_{\mathcal{R}}+\hat{V}_{\mathcal{C}\mathcal{R}}$. Due to $\mathcal{C}$’s interaction with $\mathcal{R}$, ${\left|\psi_{\mathcal{CR}}(t)\right>}$ will be, in general, an entangled state consisting of a superposition of states with different time readings of $\mathcal{C}$, ${\left|\psi_{\mathcal{CR}}(t)\right>}=\int_{-\infty}^{+\infty}\text{d}\tau\,{\left|\phi_{\mathcal{C}}(\tau)\right>}\innerproduct{\phi_{\mathcal{C}}(\tau)}{\psi_{\mathcal{CR}}(t)}$. The average time reading of $\mathcal{C}$ when $\mathcal{C}_{2}$ reads $t$ is given by

so the rate of $\mathcal{C}$ from the perspective of $\mathcal{C}_{2}$ is (Appendix H)

This equation connects the rate operator $\hat{\alpha}$ of Eq. (7) to the real-valued, time-dependent rate $\alpha(t)$ (in special relativity we would call this quantity $1/\gamma(t)$).⁵⁵ 5 Both the numerator and the denominator in these equations are divergent due to the states of the clock $\mathcal{C}$ being non-normalizable. Here, I assume that these divergencies cancel out, as they would if $\mathcal{C}$ were a realistic clock.

Now, Eq. (9) implies that (Appendix H)

meaning that the rate of the clock is constant in time, $\alpha(t)=\alpha(0)$ for all $t\in\mathds{R}$. The converse is also true: if $\alpha(t)$ is constant in time for any choice of initial state ${\left|\psi_{\mathcal{CR}}(0)\right>}$ (this is allowed because $\mathcal{CR}$ does not interact with $\mathcal{C}_{2}$), then $\left[\hat{H}_{U},\hat{\alpha}\right]=0$. Eq. (19) is also equivalent to

showing that the times of $\mathcal{C}$ and $\mathcal{C}_{2}$ are related by a linear transformation. The same would hold for any other clock not interacting with $\mathcal{C}$.

Eq. (9) also implies that the variance of the time readings of $\mathcal{C}$ from the perspective of $\mathcal{C}_{2}$ at time $t$ is given by

with $\sigma^{2}_{\tau_{\mathcal{C}}}(0)$, $\sigma^{2}_{\alpha}(0)$ and $\text{Cov}(\alpha(0),\tau_{\mathcal{C}}(0))$ the variances and covariance of $\tau_{\mathcal{C}}$ and $\alpha$ at time $0$, respectively (Appendix H). When ${\left|\psi_{\mathcal{CR}}(0)\right>}$ is an eigenstate of $\hat{T}_{\mathcal{C}}$, this reduces to $\sigma^{2}_{\tau_{\mathcal{C}}}(t)=t^{2}\sigma^{2}_{\alpha}(0)$ and the relative uncertainty on the readings of $\mathcal{C}$ from the perspective of $\mathcal{C}_{2}$ is $\sigma_{\tau_{\mathcal{C}}}/t=\sigma_{\alpha}$, which is constant in time.

What about Eq. (8)? This condition can be interpreted as imposing that the rate operator $\hat{\alpha}$ must be independent of the internal structure of the clock. This is because the equation (in the strong sense) implies that $\hat{\alpha}$ is an operator-valued function of $\hat{T}_{\mathcal{C}}$ only and thus does not contain $\hat{H}_{\mathcal{C}}$, which describes the internal structure of the clock. This is similar to relativity, where time dilation effects are independent of the clocks chosen to measure the time.⁶⁶ 6 We can imagine a Hamiltonian of the Universe $\hat{H}_{U}=\gamma^{-1}\hat{H}_{\mathcal{C}}+\hat{H}_{\mathcal{R}}$ with $\gamma^{-1}\in\mathds{R}$, for which $\hat{\alpha}=\gamma^{-1}\hat{\mathds{1}}_{U}$. However, in this case we would have to rescale the time states ${\left|\phi_{\mathcal{C}}(t)\right>}$ with the same factor to preserve their interpretation. This would change $\hat{T}_{\mathcal{C}}$ to $\gamma\hat{T}_{\mathcal{C}}$ and restore the “natural” rate of the clock $\hat{\alpha}=\hat{\mathds{1}}_{U}$.

We have seen that the following statements are equivalent:

1. 1.

   $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$ and ${\left|\psi_{\mathcal{R}}(t)\right>}$ evolve unitarily.

2. 2.

   $\hat{H}_{U}$ is physically equivalent to a constraint for which Eqs. (8) and (9) hold (excluding the pathological constraints discussed in Appendix D).

3. 3.

   The rate of the clock $\mathcal{C}$ is constant with respect to any other non-interacting clock and does not depend on the internal structure of $\mathcal{C}$.

This characterizes the types of Hamiltonians of the Universe that lead to a unitary dynamics from both the mathematical and the interpretational point of view. This characterization relies on the clock $\mathcal{C}$ being an ideal clock, so it should also hold in the limiting case of ever more precise realistic clocks. Therefore, we can expect its insights on unitarity in timeless quantum theory to be general. Nevertheless, it would be useful to consider how these results extend to non-ideal clocks, which are often treated using POVMs rather than projective measurements 5; 38; 23; 9, and where further breakdowns of unitarity might appear 18. Closely related questions arise in the framework of quantum reference frames, where time is treated relationally 16; 19; 39. I leave these investigations for future work.

What to make of non-unitarity? The two conditions found in this work point to two different origins of this effect. Hamiltonians of the Universe that violate Eq. (8) make the rate of the clock depend on its internal structure, meaning that the flow of time changes with the choice of clock, and so the effect is not general but clock-specific. This might happen for some types of interactions with the clock, but we have also observed this for a non-interacting relativistic Hamiltonian of the Universe. In this latter case, non-unitarity is due to the relativistic quantum mechanical treatment, similarly to the non-unitarity of the Klein-Gordon equation, and the problem might be solved by adopting a quantum field theoretic timeless approach (see 21 for a first step in this direction).

Hamiltonians that satisfy Eq. (8) but violate Eq. (9) seem to lead to a different type of non-unitarity. Specifically, they are related to clock rates that change in time but do so independently of the clock choice. In this case, the origin of non-unitarity can be intuitively grasped in the following way. Assume for simplicity that $\hat{\alpha}$ is invertible. The evolution operator of $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$ is still given by $\hat{U}(t)=e^{-i\hat{\alpha}^{-1}\hat{H}_{U}t}$, but with $\hat{U}(t)$ non-unitary because $\hat{\alpha}^{-1}$ and $\hat{H}_{U}$ do not commute. $\hat{\alpha}^{-1}$ can be seen as changing the evolution rate of the different eigenstates of $\hat{\alpha}^{-1}$ appearing in $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$. If $\hat{\alpha}^{-1}$ and $\hat{H}_{U}$ commuted, then they would share a common set of eigenstates $\left\{\left|\left.\kern-1.2pt\chi_{j}\right>\right>\right\}_{j}$ (assuming the set is discrete for ease of notation) such that $\hat{\alpha}^{-1}\left|\left.\kern-1.2pt\chi_{j}\right>\right>=\alpha^{-1}_{j}\left|\left.\kern-1.2pt\chi_{j}\right>\right>$ and $\hat{H}_{U}\left|\left.\kern-1.2pt\chi_{j}\right>\right>=E_{j}\left|\left.\kern-1.2pt\chi_{j}\right>\right>$, and thus $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>$ could be expressed as $\left|\left.\kern-1.2pt\psi_{U}(t)\right>\right>=\sum_{j}\lambda_{j}e^{-i\alpha^{-1}_{j}E_{j}t}\left|\left.\kern-1.2pt\chi_{j}\right>\right>$, showing that $\hat{\alpha}^{-1}$ affects the evolution rate of the different eigenstates in a way that is non-trivial but preserves unitarity. However, if $\hat{\alpha}^{-1}$ and $\hat{H}_{U}$ do not commute, then $\hat{H}_{U}$ mixes the different eigenstates of $\hat{\alpha}^{-1}$, so a state that starts evolving with a certain rate ends up in a superposition of states evolving with different rates. The result is a non-unitary evolution.

Under the condition of Eq. (8), preventing the clock from coupling to other systems would ensure unitarity because we would have $\hat{\alpha}=\hat{\mathds{1}}_{U}$. If the clock were a composite system, it would be sufficient to prevent the coupling between the degrees of freedom responsible for the clock’s timekeeping and the rest of the Universe. However, some interactions with the clock’s internal degrees of freedom may be inevitable 7 while others lead to desirable relativistic time dilation effects 37. A straightforward generalization of these types of interactions leads to non-unitarity. Consider, for instance, $\hat{H}_{U}=\hat{H}^{int}_{\mathcal{C}}+\frac{\left(\hat{P}^{CM}_{\mathcal{C}}\right)^{2}}{2m_{\mathcal{C}}}+\hat{H}_{\mathcal{R}}+\Lambda(\hat{X}^{CM}_{\mathcal{C}})\hat{H}^{int}_{\mathcal{C}}\hat{H}_{\mathcal{R}}$, with $\hat{H}^{int}_{\mathcal{C}}$ the Hamiltonian of the internal degrees of freedom of the clock, $\hat{P}^{CM}_{\mathcal{C}}$ and $\hat{X}^{CM}_{\mathcal{C}}$ the momentum and position of its centre of mass, and the interaction term obtained using the mass-energy equivalence principle in a Newtonian-like potential between $\mathcal{C}$ and $\mathcal{R}$ 7 which is function of $\hat{X}^{CM}_{\mathcal{C}}$. The rate operator $\hat{\alpha}=\hat{\mathds{1}}_{U}+\Lambda(\hat{X}^{CM}_{\mathcal{C}})\hat{H}_{\mathcal{R}}$ does not commute with $\hat{H}_{U}$ and so the evolution is non-unitary despite $\hat{H}_{U}$ describing a simple and natural physical scenario of a clock moving under the influence of a Newtonian-like potential.

Overall, non-unitarity appears to be a genuine feature of timeless quantum theory related to well-defined physical properties of the clock. It is now crucial to understand how this effect can be reconciled with the physical interpretation of unitarity in standard quantum theory.