Smooth and sharp creation of a Dirichlet wall in 1+1 quantum field theory: how singular is the sharp creation limit?

We present and utilize a simple formalism for the smooth creation of boundary conditions within relativistic quantum field theory. We consider a massless scalar field in $(1+1)$-dimensional flat spacetime and imagine smoothly transitioning from there being no boundary condition to there being a two-sided Dirichlet mirror. The act of doing this, expectantly, generates a flux of real quanta that emanates from the mirror as it is being created. We show that the local stress-energy tensor of the flux is finite only if an infrared cutoff is introduced, no matter how slowly the mirror is created, in agreement with the perturbative results of Obadia and Parentani. In the limit of instantaneous mirror creation the total energy injected into the field becomes ultraviolet divergent, but the response of an Unruh-DeWitt particle detector passing through the infinite burst of energy nevertheless remains finite. Implications for vacuum entanglement extraction and for black hole firewalls are discussed.


Introduction
Within the realm of relativistic quantum field theory, both in flat and curved spacetimes, the study of time-dependent boundary conditions has been a staple exercise in understanding particle-creation phenomena [1]. A non-inertially moving mirror, for example, induces the production of real particles out of the vacuum. Within a cavity setting this is commonly referred to as the dynamical Casimir effect [2], in which rapidly varying the length of an optical cavity can dynamically generate photons. This effect has been experimentally verified with a cQED analogue system [3]. Recently, there has been an increasing interest in utilizing the effect for quantum information processing and quantum metrology [4,5,6].
The majority of the existing literature is focused on the effects of moving boundaries. Here, we wish to properly examine a somewhat different case, and one that has been gaining interest in a number of areas. Rather than moving a boundary, we will instead create one. In particular, we take a 1 + 1 dimensional massless scalar field and consider at the origin a self-adjointness boundary condition that transitions smoothly in time between there being no boundary to there being a two-sided Dirichlet wall. Physically, one can imagine such a procedure being implemented via a reflectivity-tunable barrier [7]. Unsurprisingly, such a procedure also generates quanta out of the vacuum that radiate away from the creation event. Our goal in this paper is to examine the stress-energy contribution to the field and the response of a particle detector. As part of this exposition we will take the limit of instantaneous wall creation.
There are several motivations behind studying such a scenario. For example, as has been pointed out by Unruh [8], the act of instantaneously creating a mirror produces regions of spacetime between which a field is entirely uncorrelated. On the horizon separating these regions (the future lightcone of the creation event) there is expected to be a flux of quanta of diverging energy density and diverging total energy (as we will confirm). Interestingly, this phenomenon is very analogous to the much-debated black hole firewall [9,10,11,12,13,14,15] and related constructs [16,17,18] in which lack of correlation between the inside and outside of a black hole is proposed to induce a violent horizon. Indeed, artificially constructing uncorrelated spatial regions has been used as a simplified firewall model [19,20]. By considering the instantaneous limit of mirror creation within our formalism we are able to gain further insight into the nature of the divergence associated with firewalls.
The rapid creation of a mirror has recently gained further interest in studying the nature of vacuum entanglement [21,22,23]. It was shown in [21] that the two bursts of quanta produced by introducing a mirror are entangled with each other, and that this entanglement derives exactly from the previously present vacuum entanglement. The UV-divergent energy of these bursts is seen to be equivalent to the UV-divergence of the entanglement entropy between connected regions. This protocol has been proposed as a means of experimentally verifying vacuum entanglement. In any real experiment, however, the introduction of the mirror will take place over a finite time interval. In addition to theoretical insights into the sharp limit, considering a smooth transition (as we do here) may therefore prove vitally important for the development of such a program.
We have several goals in the current work, and give several different results of interest. First, we wish to present a simple formalism for considering the smooth cre-ation of boundary conditions in quantum field theory by time-dependent self-adjointness conditions [24,25,26], building on previous treatments in a variety of contexts [27,28,29,30,31,32,33,34,35,36]. We use this formalism to consider the smooth creation of a Dirichlet boundary both in free Minkowski space as well as in a Dirichlet cavity. We then compute the renormalized stress-energy expectation value associated with the induced quanta. Given this, we take the instantaneous limit and demonstrate that the energy density diverges stronger than in any distributional sense, and thus that this procedure inputs an infinite amount of energy into the field; these outcomes are consistent with the instantaneous wall creation discussion in [8], with the instantaneous topology change discussion in [27,28] and with the conformal field theory discussion in [22]. We also find a surprising result that is independent of how quickly the wallcreation takes place: any change in reflectivity of the barrier causes the stress-energy to become infrared-divergent! Due to this we will take an infrared cutoff (for example by considering a cavity). This result is intuitively challenging, however, and a deeper investigation into the nature and physicality of this divergence is warranted.
We go on to compute the response of an Unruh-DeWitt particle detector [37,38] that couples linearly to the proper time derivative of the field [19,39,40,41,42,43,44,45], choosing the derivative coupling because it is less sensitive to the infrared ambiguity of the Wightman function of a (1 + 1)-dimensional massless field [44]. Working within first order perturbation theory, we find that in the instantaneous wall creation limit the detector's response has two surprising properties. First, the response remains finite, despite the divergent total energy through which the detector passes. Second, the response depends on the infrared cutoff, even though the response in a number of other states, including the Minkowski vacuum, is independent of the infrared ambiguity [44]. These results corroborate well with those of [19,20], which consider the response of a detector due to a Minkowski space model of a firewall. This paper is organized as follows. We begin in Section 2 with an introduction to the formalism and fully work out the evolution of the quantum field for wall creation that takes place smoothly over a finite interval of time in Minkowski space. We compute the stress-energy associated with this process, and we show that the total energy diverges in the sharp creation limit. In Section 3 we perform the same analysis in the case of a Dirichlet cavity, demonstrating that the IR divergence associated with wall-creation is mitigated by the cavity. In Section 4 we show that similar properties hold for creating a wall in Minkowski space over an infinite interval of time with a specific profile that allows computations to be done in terms of elementary functions. In Section 5 we go on to use this specific profile to analyse an inertial particle detector and to demonstrate, among other results, the response to be finite even in the sharp-creation limit.
We denote complex conjugation by an overline. O(x) denotes a quantity such that O(x)/x remains bounded as x → 0, O ∞ (x) denotes a quantity that goes to zero faster than any positive power of x as x → 0, and O(1) denotes a quantity that remains bounded in the limit under consideration.

Classical field
We work in (1 + 1)-dimensional Minkowski spacetime, with standard global Minkowski coordinates (t, x), in which the metric reads ds 2 = −dt 2 + dx 2 . In the global null coordinates u = t − x and v = t + x, the metric reads ds 2 = −du dv.
We consider a real massless scalar field φ. Without a wall, the field equation is the Klein-Gordon equation, where −∂ 2 x has its usual meaning as an essentially self-adjoint positive definite operator on L 2 (R).
To introduce a wall at x = 0, we replace (2.1) with The special case −∆ π/2 is that of the unique self-adjoint extension of −∂ 2 x on L 2 (R), corresponding to no wall at x = 0. The special case −∆ 0 is that of an impermeable wall at x = 0 with the Dirichlet boundary condition on each side. For the intermediate values of θ, −∆ θ interpolates between these two extremes, involving no boundary conditions for spatially odd wave functions but a two-sided Robin boundary condition [equation (A.3) in Appendix A] for spatially even wave functions.
The spectrum of each −∆ θ is the positive continuum. The wave equation (2.2) is hence free of tachyonic instabilities and provides a viable starting point for quantisation.
In physics terms, the wave equation (2.2) can be written for 0 < θ ≤ π/2 as where δ(x) is Dirac's delta-function and the positive constant L of dimension length is as introduced in Appendix A. The wall at x = 0 corresponds hence to a potential term proportional to δ(x) with a θ-dependent coefficient. The coefficient is positive for 0 < θ < π/2, and it tends to 0 in the no-wall limit θ → (π/2) − and to +∞ in the Dirichlet wall limit θ → 0 + . In the rest of this section we assume that θ(t) interpolates between no wall and a fully-developed Dirichlet wall over a finite interval of time. We may assume without loss of generality that the wall creation begins at t = 0, and we write the moment at which the Dirichlet wall is fully formed as t = λ −1 where λ > 0. We parametrise θ(t) as where h : R → R is a smooth function such that h(y) = π/2 for y ≤ 0 , (2.5a) 0 < h(y) < π/2 for 0 < y < 1 , Over the interval 0 < t < λ −1 , the boundary condition (A.3) then reads The parametrisation (2.4) hence means that λ −1 is the length of the time interval over which the boundary condition (2.6) evolves into Dirichlet, while the dimensionless function h specifies the shape of the evolution in (2.6) over this time interval. The limit in which a wall is created rapidly but the shape of the evolution is held fixed is the limit of large λ with fixed h.

Mode functions
As preparation for quantisation, we need to find the mode solutions that reduce to the usual Minkowski modes for t ≤ 0, where the wall has not yet started to form.
Since the spatially odd solutions to the field equation (2.2) do not feel the wall, it suffices to consider the spatially even solutions. It further suffices to write down the expressions for these solutions in the half-space x > 0; by spatial evenness, the expressions at x < 0 follow by (t, x) → (t, −x), or in terms of the null coordinates, by (u, v) → (v, u).
We work in the null coordinates (u, v) and look for the mode solutions with the ansatz where k > 0 and E k is to be found. Each term in (2.7) satisfies the wave equation at x > 0, and the left-moving part of U k has the standard form proportional to e −ikv . Requiring (2.7) to satisfy (A.3a) with θ = θ(t) gives With θ(t) parametrised by (2.4), the solution is where B(y) is the solution to for 0 ≤ y < 1 with the initial condition B(0) = 1. An alternative expression for R K (y) for 0 < y < 1 is Using (2.5) and the smoothness of h, it follows from (2.11) that 1/B(y) and all of its derivatives tend to zero as y → 1 − , and this can be used to show from (2.12) that R K (y) is a smooth function of y everywhere, including y = 1. It follows that E k (u) is a smooth function of u. At u ≤ 0 and u ≥ λ −1 , the mode functions U k reduce respectively to (2.13) At u ≤ 0, U k have not yet been affected by the wall, and they coincide with the usual spatially even mode functions in Minkowski, positive frequency with respect to ∂ t . At u ≥ λ −1 , U k feel the fully-developed Dirichlet wall, and they coincide with the half-space mode functions with the Dirichlet boundary condition. In the interpolating region, 0 < u < λ −1 , U k are given by (2.7) with (2.9)-(2.11). The different regions are illustrated in Figure 1.
Recalling that the above formulas hold for x > 0 and the corresponding formulas for x < 0 are obtained by spatial evenness, it can be verified that U k satisfy the usual Klein-Gordon orthonormality relations where ( · , · ) is the Klein-Gordon (indefinite) inner product [1]. Figure 1: (1 + 1)-dimensional Minkowski spacetime with a wall evolving at x = 0. The wall starts to evolve at (t, x) = (0, 0) and becomes a fully-developed two-sided Dirichlet wall at (t, x) = (λ −1 , 0). The wall sends a pulse of energy that travels to the right in the null strip 0 < u < λ −1 and to the left in the null strip 0 < v < λ −1 . The figure shows also the world line of an inertial detector at x = d > 0.
wall −a/2 a/2 t x Figure 2: (1 + 1)-dimensional Dirichlet cavity of length a with a wall evolving at the centre, x = 0. The wall evolution is as in Figure 1, but the reflections from the boundaries at x = ±a/2 affect the evolution of the mode functions for sufficiently late times. The figure shows the case a > 2/λ, in which the Dirichlet wall at x = 0 has fully formed before the changes in the field due to the wall evolution reach the boundaries at (t, x) = (a/2, ±a/2).

Quantisation and the rapid wall creation limit
We quantise the field in the usual fashion, adopting U k as the positive norm mode functions in the spatially even sector and the usual spatially odd Minkowski mode functions in the spatially odd sector. The spatially even part of φ is expanded as where the nonvanishing commutators of the annihilation and creation operators are a k , a † k = δ(k − k ). We denote by |0 M the normalised state that is annihilated by all a k and by all the usual Minkowski annihilation operators of the spatially odd sector. |0 M is indistinguishable from the usual Minkowski vacuum in the region t < |x| which is outside the causal future of the wall.
We are interested in the energy that is transmitted into the quantum field by the evolving wall. Recall first that the classical stress-energy tensor of a massless minimally coupled scalar field is given by We point-split the quantised versions of these espressions and express their expectation values in |0 M in terms of the Wightman function of the field, using (2.7) and (2.15). Subtracting the Minkowski contribution and taking the concidence limit, we find that the renormalised stress-energy tensor T ab is given by where the constant µ is an infrared cutoff which we have inserted by hand. When µ > 0, T uu is well defined for all u, and vanishing for u ≤ 0 and u ≥ λ −1 , as is seen from (2.9) and (2.10). The convergence of the integral in (2.16b) at k → ∞ for 0 < u < λ −1 follows because E k (u) 2 = k 2 + O k −2 at large k, as can be verified by repeated integration by parts in (2.10), integrating the exponential factor [46]. When µ = 0, T uu is still well defined and vanishing for u ≤ 0 and u ≥ λ −1 , but it is infrared divergent for 0 < u < λ −1 : this follows because for 0 < u < λ −1 (2.9) and In words, this means that a positive infrared cutoff is required to make T ab finite on the light cone of each wall point where the wall has started to form but has not yet reached the Dirichlet form. Where T ab is nonzero, it corresponds to null radiation travelling away from the wall.
The total energy transmitted into the quantum field during the creation of the wall where for Σ we may take any a constant t hypersurface in the region t > λ −1 , and the last expression in (2.17) follows using (2.16) and by including the contribution from x < 0. Inserting the solution (2.9)-(2.11) in (2.16), we find For rapid wall creation, we consider the limit of large λ with fixed h. Recall from (2.12) that for 0 < y < 1 we have , where the first term is bounded because 1/B(y) and its derivatives tend to zero as y → 1 − . From (2.18) we hence obtain We conclude that in the rapid wall creation limit the energy transmitted into the quantum field diverges proportionally to λ ln(λ/µ). The energy comes out as an increasingly narrow pulse near the light cone of the point (t, x) = (0, 0) but the magnitude of the pulse grows so rapidly that the stress-energy tensor does not have a distributional limit and the total energy diverges.

Wall creation within a Dirichlet cavity
In this section we adapt the analysis of Section 2 to a wall that is created at the centre of a static cavity whose left and right walls have time-independent Dirichlet boundary conditions. The main point of this adaptation is to verify that there is no need to introduce an infrared cutoff by hand since such a cutoff is already provided by the cavity.

Classical field and mode functions
Following the notation of Section 2, we confine the field φ to a static cavity whose walls are at x = ±a/2, where the positive constant a is the length of the cavity. We take φ to satisfy the Dirichlet boundary condition at x = ±a/2.
At the centre of the cavity, x = 0, we introduce the time-dependent boundary condition as in Section 2, with the same assumptions about θ(t). Again, the boundary condition does not affect the spatially odd part of the field, and it suffices to consider the spatially even part. We write down the formulas assuming 0 < x < a/2, with the spatial evenness providing the formulas for −a/2 < x < 0.
We look for the mode solutions with the ansatz where the index n is an odd positive integer and the function F n is to be found. This ansatz satisfies the wave equation at 0 < x < a/2, and it satisfies V n (u, a + u) = 0, which is the Dirichlet boundary condition at x = a/2.
We again parametrise θ = θ(t) by (2.4). We choose the solution that for u < min(a, λ −1 ) is given by where R K is given by (2.10) and (2.11). For u ≤ 0 this implies so that at early times V n are the standard spatially even mode functions in the Dirichlet cavity. The domain u < min(a, λ −1 ), where the solution (3.3) holds, is where the timedependence due to the evolving wall has not yet come back to x = 0 after being reflected from x = a/2.
To evolve F n further to the future, one needs to account for the reflections of the time-dependence that start to arrive to x = 0. The case of main interest for us is when λ > a −1 , which occurs when a is considered fixed and we consider a rapid wall formation. In this case the Dirichlet wall at x = 0 is fully formed when the first reflection due to the wall evolution arrives back to x = 0. Equation (3.3) then holds for u < λ −1 , so that F n (u) = −e −iπnu/a for λ −1 ≤ u ≤ a, and the evolution of F n (u) to u > a is given just by successive Dirichlet reflections from x = 0 and x = a/2. The case in which λ > 2/a is illustrated in Figure 2.

Quantisation and the rapid wall creation limit
We again quantise the field in the usual fashion and denote by |0 c the vacuum with the above choice for the above positive norm mode functions. |0 c is indistinguishable from the usual Dirichlet cavity vacuum in the region t < |x|, where its renormalised stress-energy tensor has the expectation value [1] T uu (early) = T vv (early) = − π 96 a 2 , (3.5a) T uv (early) = 0 . (3.5b) To examine the stress-energy tensor due to the wall creation, we assume λ > 2/a, and we consider the region 0 < x < a/2 and t < a/2, as illustrated in Figure 2. In this region the solution (3.3) holds, and the v-dependent part of V n has still the standard form proportional to e −inv/a . Writing we find where the convergence of the sum in (3.7b) at large n can be verified as in Section 2, and there is no infrared divergence because the sum starts at n = 1. ∆ T uu is vanishing for u ≤ 0 and for u ≥ λ −1 . The total energy transmitted into the quantum field is given as in (2.17) but with T ab replaced by ∆ T ab , and Σ being now any constant t hypersurface at λ −1 < t < a/2. Using (3.7b) with (3.3), we obtain (3.8) In the limit of large λ, we may approximate the sum by an integral, and using the properties of R K as in Section 2 gives The energy diverges proportionally to λ ln(λa/π), and comparison with (2.19) shows that π/a plays the role of an infrared cutoff. The divergence implies that the stress-energy tensor does not have a distributional limit at λ → ∞.

Wall creation in Minkowski space over infinite time
In this section we adapt the Minkowski space analysis of Section 2 to a specific oneparameter family of wall evolution profiles for which the evolution is nontrivial at all finite times but reduces to no wall in the asymptotic past and to a wall with nonvanishing reflection and transmission coefficients in the asymptotic future. The main point is to verify that passing to an appropriate limit within this one-parameter family allows us again to model a rapid creation of a Dirichlet wall, and the results for the stress-energy tensor agree with those in Section 2. These properties will justify our use of this oneparameter family of evolution profiles with a particle detector in Section 5. We take the boundary condition to be as in (2.6) with λ a positive parameter and so that Since 0 < θ(t) < π/2, the wall exists for all t, and it is never Dirichlet. Since θ(t) → π/2 as t → −∞, the wall disappears in the asymptotic past, and the wall formation starts exponentially slowly. Since θ(t) → arccot(λL) as t → ∞, the end state of the wall in the asymptotic future is not Dirichlet, but it can be made arbitrarily close to Dirichlet by taking λL large. The parameter λ has hence a dual role: it determines both how rapid the wall formation is and how close the wall is to Dirichlet in the asymptotic future. In the limit λ → ∞, we approach the instantaneous creation of a Dirichlet wall at t = 0.
We proceed as in Section 2. Equation (2.10) is now replaced by where (2.11) and the initial condition B(y) → 1 as y → −∞ give B(y) = 1 + e y .

(4.4)
We find that U k is given by (2.7) with For the stress-energy tensor, (2.16) gives where the positive infrared cutoff µ is again needed to make T uu finite.
When λ → ∞, T uu vanishes for u = 0 and diverges for u = 0. To examine the strength of this divergence, we write and observe that f λ (u) → δ(u) as λ → ∞. The divergence is hence too strong for T uu to have a distributional limit. The total energy transmitted into the quantum field is where Σ t is a hypersurface at constant t, and the final expression comes using (4.7) and observing that ∞ −∞ f λ (u) du = 1. In the limit λ → ∞, the energy diverges proportionally to (6π) −1 λ ln(λ/µ) and comes out as a narrow burst near the light cone of (t, x) = (0, 0).

Response of an Unruh-DeWitt detector to rapid wall creation
In this section we consider the response of an inertial Unruh-DeWitt particle detector to the creation of a wall. We work in Minkowski spacetime with the wall creation profile (4.2). We are interested in the limit of large λ, in which the burst of energy from the wall diverges on the light cone of (t, x) = (0, 0). We ask what happens in the limit of large λ to the response of a detector that crosses this light cone.

Detector and its trajectory
We consider a version of the Unruh-DeWitt detector [37,38] that couples linearly to the proper time derivative of the field [19,39,40,41,42,43,44,45]. Following the notation of [44], we denote by x(τ ) the detector's worldline, parametrised by the proper time τ . We assume that the coupling to the field is proportional to a real-valued function χ(τ ) that specifies how the interaction is turned on and off. We call χ the switching function and assume it to be smooth with compact support. In first-order perturbation theory, the detector's probability to make a transition from a state with energy 0 to a state with energy ω is proportional to the response function, given by where the correlation function W is the pull-back of the Wightman function to the detector's worldline, and |ψ is the state to which the field was initially prepared. The superscript (1) in (5.1) is a reminder that the detector couples to the (first) derivative of the field. The derivatives in (5.1) are understood in the distributional sense, and integration by parts gives the alternative expression where Q ω (τ ) := e −iωτ χ(τ ). F (1) is hence well defined whenever W is a well-defined distribution.
We take the detector's trajectory to be where d is a positive constant. The detector is inertial and it crosses the light cone of the origin at (t, x) = (d, d). The zero of the proper time has been chosen to occur at this crossing. The geometry is shown in Figure 1.

Preliminaries: Minkowski vacuum and Dirichlet half-space
For comparisons to be made below, we record here the response in Minkowski vacuum and in Minkowski half-space with the Dirichlet boundary condition. When there is no wall and the field is in the usual Minkowski vacuum, the response function is given by [19] F (1) Mink is independent of the infrared cutoff, and its asymptotic form at large |ω| is [19] F (1) When there is a static wall at x = 0 and the field is in the usual vacuum state with Dirichlet conditions at this wall, we show in Appendix B that the response function is which is again independent of the infrared cutoff. We also show that the asymptotic large |ω| form of ∆ Dir F (1) is

Evolving wall
When the wall is present with the profile (4.2), we write the response function as We show in Appendix C that ∆F (1) λ has a finite limit as λ → ∞, given by where ∆W(τ , τ ) is given by the following expressions: Here µ is the infrared cutoff and is assumed positive. E 1 is the exponential integral in the notation of [47], taking values on its principal branch in the sense of → 0 + . We further show in Appendix C that when ω + µ = 0, ∆F (1) can be put in the form Four observations are in order. First, given that µ is assumed positive, equations (5.11) and (5.12) show that ∆F (1) is manifestly finite. The detector's response remains finite when the wall creation becomes instantaneous, even though the detector passes through an infinite pulse of energy.
Second, ∆F (1) has a finite µ → 0 limit if and only if χ(0) = 0. This is seen from (5.13a) where the only potential divergence at µ → 0 comes from the first term. The infrared cutoff can hence be removed if and only if the detector does not operate at the moment of crossing the light cone of the wall creation event.
Third, as a consistency check, we note that if χ(τ ) vanishes for τ ≤ 0, the first three terms in (5.13a) vanish, and comparison of (5.13) and (5.8) shows that ∆F (1) reduces to ∆ Dir F (1) if µ is taken to zero. If the detector operates only after crossing the light cone of the wall creation event, the response is identical to that in a half-space with a static Dirichlet wall.
Fourth, we verify in Appendix C that the asymptotic form of ∆F (1) at large energy gap is 14) The terms proportional to ω cos(2dω) and sin(2dω) are as expected from the corresponding terms in ∆ Dir F (1) Z (5.9). The additional term, proportional to [χ(0)] 2 , comes strictly from the moment of crossing the light cone of the wall creation event. This term is dominant for ω → ∞ and subdominant for ω → −∞.

Discussion
The purpose of this work has been to demonstrate a straightforward formalism for discussing the smooth creation of boundary conditions, and to highlight some preliminary findings of interest. Specifically, we have examined several properties of the particle flux resulting from the smooth creation of a Dirichlet boundary condition for a massless (1 + 1)-dimensional scalar field and the resulting response of a particle detector. We have paid particular attention to the sharp creation limit of such a procedure. This type of scenario has gained interest recently from a number of different perspectives, and is markedly different from the more standard setting of a moving boundary condition. Our primary findings from this work are the following. First, we have discovered that the creation of a wall induces an energy flux that is infrared divergent, independent of how slowly and smoothly the creation unfolds. While the Wightman function of the (1 + 1)-dimensional massless field is well known to be infrared divergent, it may be surprising that in our situation the infrared divergence shows up also in the stress-energy expectation value, which involves the Wightman function only through its derivatives. The upshot seems to be that in our time-dependent situation the infrared divergence of the Wightman function can no longer be thought of as an infinite additive constant but must be regarded as an infinite function, which does not drop out on taking a derivative. It should be interesting to give this phenomenon a more precise mathematical description, especially given its surprising and unintuitive nature.
Second, we have demonstrated that in the sharp creation limit (i.e. instantaneously producing a mirror) the resulting energy density flux is UV divergent, and diverges stronger than in any distributional sense. Thus, such a process would input an infinite amount of energy into the field. Indeed such a result is to be expected [8,22,27,28], and as demonstrated in [21] is related to the fact that the entanglement entropy between the two regions on either side of the created wall is UV divergent.
Third, we have considered the response of an inertial derivative-coupling Unruh-DeWitt detector that crosses the energy flux emitted from the wall creation. We showed that the detector's response remains finite in the limit of instantaneous wall creation, despite the infinite amount of energy that the sharp creation injects into into the field. We also showed that in this sharp wall creation limit the detector's response depends on the infrared cutoff, even though the derivative-coupling detector is known to be insensitive to the infrared ambiguity of the Wightman function in a number of other quantum states. Both of these properties are similar to the response of an inertial detector in a Rindler firewall [19], and they lend support to the notion that perhaps a firewall is not, after all, an impenetrable barrier.
Our detector results were obtained in Section 5 under a specific one-parameter family of wall creation profiles. We conjecture that the same results for the sharp creation limit ensue within the full family of profiles introduced in Section 2. It is straightforward to verify that within this full family the pointwise sharp creation limit of the Wightman function is still given by (5.12); to justify the conjecture, it would remain to show that the sharp creation limit in the response function (5.3) can be taken pointwise under the integral. This question warrants further consideration.
An interesting next step would be to examine the entanglement structure between the bursts of particles generated by smooth wall creation, with the aim of showing how the formalism and results of [21] emerge in the sharp creation limit and comparing with the conformal field theory treatments of [22,23]. Another next step would be to examine how this entanglement may be harvested by particle detectors. Conversely, it would be interesting to examine how pre-existing entanglement between particle detectors is affected by the wall creation, in the formalism that was applied to a Rindler firewall in [20].
Finally, we have throughout maintained that the quantum field lives on a nondynamical Minkowski metric even when the energy in the quantum field became infinite. Allowing the metric to become dynamical and to respond to the growing stress-energy could provide a model for a firewall in an evaporating black hole spacetime, in which the gravitational aspects near the horizon have had time to become significant.
A −∂ 2 x on a line with a distinguished point In this appendix we collect relevant properties about the self-adjoint extensions of the operator −∂ 2 x on L 2 (R \ {0}). The general theory can be found for example in [24,25] and a pedagogical summary in [26].
We take the coordinate x to have the physical dimension of length. The self-adjoint extensions of −∂ 2 x form a U (2) family, specified by the boundary condition [26] Lψ where ψ is the (generalised) eigenfunction, ψ ± := lim x→0 ± ψ(x), ψ ± := lim x→0 ± ψ (x), L is a positive constant of dimension length and U ∈ U (2). The constant L has been introduced for dimensional convenience and its value is considered fixed. The extensions are then uniquely parametrised by the matrix U ∈ U (2). Physically, U encodes the reflection and transmission coefficients across x = 0. We specialise to the one-parameter subgroup of U (2) given by On the odd subspace of L 2 (R\{0}), −∆ θ reduces to the unique self-adjoint extension of −∂ 2 x on the odd subspace of L 2 (R). The generalised eigenfunctions are proportional to sin(kx) where k > 0, and the spectrum is the positive continuum.
On the even subspace of L 2 (R \ {0}), −∆ θ is determined by the Robin boundary condition (A.3) on each side of x = 0. When 0 ≤ θ ≤ 1 2 π, the spectrum is the positive continuum, and the generalised eigenfunctions are proportional to sin(k|x| + δ k ) where k > 0 and δ k may be found in terms of θ from (A.3). When 1 2 π < θ < π, the spectrum consists of the positive continuum, with the generalised eigenfunctions as above, together with the single negative proper eigenvalue − cot 2 (θ)/L 2 [26].
We may summarise: • On the odd subspace of L 2 (R \ {0}), −∆ θ involves no boundary condition and coincides with the unique self-adjoint extension of −∂ 2 x on the odd subspace of L 2 (R).
The following two cases have special interest. When θ = π/2, (A.3) reduces to Neumann on each side of x = 0. −∆ π/2 hence coincides with the essentially self-adjoint operator −∂ 2 x on L 2 (R). There is no boundary condition and the point x = 0 has no special role.
When θ = 0, (A.3) reduces to Dirichlet on each side of x = 0. Since the Dirichlet boundary condition is identically satisfied by odd wave functions, this means that R + and R − are completely decoupled by an impermeable two-sided Dirichlet wall at x = 0.
Finally, we note that when θ = 0, we may informally write where δ(x) is Dirac's delta-function. In physics language, the boundary condition (A.1) with (A.2) can hence be described as a delta-function potential at x = 0, with the θdependent coefficient shown in (A.4). Our reason to describe −∆ θ in terms of θ, rather than in terms of the coefficient of the Dirac delta in (A.4), is that this will allow us to control in the main text the regularity of the Dirichlet limit θ → 0 + , in which the coefficient of the Dirac delta in (A.4) tends to +∞.

B Detector response in static half-space
In this appendix we verify the properties quoted in subsection 5.2 about the response of the inertial detector (5.4) in Minkowski half-space with Dirichlet boundary conditions. In the Minkowski half-space x > 0 with the Dirichlet boundary conditions at x = 0, W(τ , τ ) consists of the Minkowski vacuum piece and the image contribution [44] ∆ Dir W(τ , τ ) = 1 2π where → 0 + . From (5.3) and (5.7) we then have After inserting (B.1) and writing out the τ -derivative, the inner integral may be evaluated using the identity lim →0 where P stands for the Cauchy principal value. Equations (5.8) in the main text then follow by writing out Q ω (τ ) = e iωτ χ (τ ) + iωχ(τ ) and performing straightforward integration variable changes.

C Detector response for a rapidly created Dirichlet wall
In this appendix we verify the properties quoted in subsection 5.2 about the response of the inertial detector (5.4) for a wall created in Minkowski space with the profile (4.2).

C.1 Rapid wall creation limit
At x > 0, the spatially even mode functions are given by (2.7) with (4.5), while without the wall the spatially even mode functions are given by (2.7) with E k (t) = e −ikt . From (5.10) we then have ∆τ := τ − τ , the positive constant µ is an infrared cutoff, and E 1 is the exponential integral in the notation of [47], taking values on its principal branch in the sense of → 0 + . We wish to take the limit λ → ∞ in (C.1). For the terms in (C.2) that contain λ in the argument of E 1 , we may use properties of E 1 from [47] [the integral representation (6.2.1) and the asymptotic expansion (6.12.1)] to show that the contribution from these terms vanishes in the limit λ → ∞. For the remaining terms in (C.2) the limit is elementary, leading to equations (5.11) and (5.12) in the main text.

C.2 Simplified expression (5.13) for the response function
We now express ∆F (1) , given by (5.11) with (5.12), in terms of integrals that do not involve special functions.