Explosive particle creation by instantaneous change of boundary condition

We investigate the dynamic Casimir effect (DCE) of a $1+1$ dimensional free massless scalar field in a finite or semi-infinite cavity for which the boundary condition (BC) instantaneously changes from the Neumann to Dirichlet BC or reversely. While this setup is motivated by the gravitational phenomena such as the formation of strong naked singularities or wormholes, and the topology change of spacetimes or strings in quantum gravity, the analysis is quite general. For the Neumann-to-Dirichlet cases, we find two components of diverging flux emanate from the point where the BC changes. We carefully compare this result with that of Ishibashi and Hosoya (2002) obtained in the context of a quantum version of cosmic censorship hypothesis, and show that one of the diverging components was overlooked by them and is actually non-renormalizable, suggesting to bring non-negligible backreaction or semiclassical instability. On the other hand, for the Dirichlet-to-Neumann cases, we reveal for the first time that only one component of diverging flux emanates, which is the same kind as that overlooked in the Neumann-to-Dirichlet cases. This result suggests not only the robustness of the appearance of diverging flux in instantaneous limits of DCE but also that the type of divergence sensitively depends on the combination of initial and final BCs.


Introduction
Particle creation plays important roles in many contexts of gravitational and high-energy physics. For example, the particle creation during the gravitational collapse leading to black holes and the inflation of early universe play crucial roles in the evolution of spacetime, and completely changed our classical picture of spacetime [1]. Besides the particle creation by dynamical spacetimes, that by time-dependent boundary conditions and topology changes of spacetime or medium is also expected to play important roles. For example, the topology change of spacetime and of strings, on which quantum fields live, is expected to happen in quantum gravity and string theory, resulting in intensive particle creation [2,3,4,5].
Although the above examples are ones in high-energy physics, it should be stressed that the particle creation by time-dependent boundary condition is expected to happen in low-energy physics [6], and indeed had been realized for the quantum electromagnetic field in a laboratory experiment [7].
No one will disagree that more radical change of boundary condition, more particles are excited from the vacuum generically. And, it is worth asking what happens in the extreme of rapid change of boundary condition, namely, what happens when the change of boundary condition occurs instantaneously. To ask so is not only interesting but also important in that most models of particle creation by the change of boundary conditions have to reproduce the result of instantaneous-change model in the limit of infinitely rapid change, whenever such a limit exists in the models.
Thus, in this paper, we suppose the extreme situation in which the boundary condition for a massless scalar field in a one-dimensional (1D) finite or semi-infinite cavity changes instantaneously, i.e., without an intermediate interval of time, from Neumann to Dirichlet or reversely. Under this assumption, we quantize the scalar field and estimate the vacuum expectation value of its energymomentum tensor. Thanks to the idealization of physical situation, the analysis enjoys the use of various mathematical formulas mainly regarding summation and integral, which enables us to obtain the results in completely analytic form for every case. It is added that there is no adjustable parameter in our model.
As the results, we find that a thunderbolt-like null diverging flux emanates from the world point at which the boundary condition changes. Furthermore, we reveal that there are two components in the diverging flux. Interestingly, there is a kind of asymmetry between the Neumann-to-Dirichlet (N-D) case and Dirichlet-to-Neumann (D-N) case. Namely, the diverging energy flux in the N-D case consists of two components, while the flux in the D-N case consists of only one. In any case, the emergence of such a diverging flux suggests that the back-reaction of the quantum field to the background spacetime and cavity cannot be ignored. The back-reaction can destroy the system or prohibit the change of boundary condition.
Here, we introduce briefly past related works. Anderson and DeWitt examined the particle creation by the fission of a 1 + 1 universe, in which the spatial topology changes from S 1 to S 1 + S 1 [3]. They argued that a diverging flux appears at the world point of fission, though detailed calculation was not presented (see also [4]). A work most relevant to the present work is that by Ishibashi and Hosoya [5], who estimated the particle creation due to the instantaneous change of boundary condition from Neumann to Dirichlet at the both ends of a finite cavity. Although their position that they regard the change of boundary condition as the appearance of a strong naked singularity [8] is different from ours, the essential part of their computation seems the same as ours. Nevertheless, there seems a discrepancy in result between [5] and the present work. Therefore, we will revisit the analysis of [5] in Sec. 3 to look for the origin of discrepancy. The particle creation by the rapid appearance and/or disappearance of a wall in a 1D finite cavity were studied in [9] and [10]. In particular, the system with the instantaneous appearance and disappearance of a Dirichlet wall studied in [10] is more complex than but similar to the system in Sec. 2 of the present paper.
The organization of this paper is as follow. In Sec. 2, we investigate the particle creation due to the instantaneous change of boundary condition in a finite 1D cavity, for the N-D case (Sec. 2.2.1) and the D-N case (Sec. 2.2.2). The origin of discrepancy between the result in Sec. 2 and Ref. [5] is clarified in Sec. 3. In Sec. 4, the case of semi-infinite cavity is analyzed. We conclude in Sec. 5. The proof of consistency between different quantizations, called the unitarity relations, and some integration formulas are presented in Appendices A and C, respectively. The result for the semi-infinite cavity in Sec. 4 is reproduced in Appendix B with the Green-function method, which naturally involves the regularization of the vacuum expectation value of energy-momentum tensor. We work in the natural units in which c = = 1.
2 Finite cavity I

Quantization of massless scalar field
We consider a free massless scalar field in a 1D cavity of which length is L, At the right boundary x = L, we assume the homogeneous Dirichlet boundary condition all the time, At the left boundary x = 0, we consider two kinds of boundary conditions. One is the homogeneous Neumann boundary condition, Another is the Dirichlet boundary condition, During boundary conditions (2) and (3) are imposed, a natural set of positive-energy mode functions {f n } is given by In the rest of this paper, we suppose that n and n ′ entirely denote odd natural numbers, otherwise denoted. The above mode functions satisfy the following orthonormal conditions, where the asterisk denotes the complex conjugate and , denotes the Klein-Gordon inner product [1], During boundary conditions (2) and (4) are imposed, a natural set of positive-energy mode functions {g m } is given by In the rest of this paper, we suppose that m and m ′ entirely denote natural numbers, otherwise denoted. The above mode functions satisfy the following orthonormal conditions, Associated with the above two sets of mode function, {f n } and {g m }, there are two ways to quantize the scalar field. One is to expand the scalar field by f n , and impose the commutation relations, [a n , a † n ′ ] = δ nn ′ , [a n , a n ′ ] = 0.
By imposing the above commutation relations, the following equal-time canonical commutation relation is realized, Then, a n and a † n are interpreted as the annihilation and creation operators, respectively. The vacuum state in which no particle corresponding to mode function f n exists is defined by a n |0 f = 0, n = 1, 3, 5, · · · , 0 f |0 f = 1. (13) Another is to expand the field by g m , and impose the commutation relations, The vacuum state in which no particle corresponding to g m exists is defined by Later, we will estimate the vacuum expectation value of energy-momentum tensor for the scalar field. The energy-momentum tensor operator is written as T µν = ∂ µ φ∂ ν φ − 1 2 η µν (∂φ) 2 , where η µν = Diag.(−1, 1) is the 1 + 1 dimensional flat metric. Introducing double null coordinates, nonzero components of this tensor are Note that the energy density and momentum density in the original Cartesian coordinates are T tt = T −− + T ++ and T tx = T −− − T ++ , respectively.

Particle creation by instantaneous change of boundary condition
Given the above quantization schemes, we investigate how the vacuum is excited when the boundary condition at left boundary x = 0 is instantaneously, say at t = 0, changed from Neumann to Dirichlet (Sec. 2.2.1) and reversely (Sec. 2.2.2).

From Neumann to Dirichlet
First, we assume that the boundary condition at x = 0 is Neumann (3) for t < 0 and Dirichlet (4) for t > 0, and that the quantum field is in vacuum |0 f in the Heisenberg picture. See Fig. 1 for a schematic picture of this situation. Then, we investigate how the vacuum is excited due to the change of boundary condition by computing the spectrum and energy flux of created particles.
Let us expand f n by g m , where the expansion coefficients, called the Bogoliubov coefficients, are computed by Using the explicit form of mode functions (5) and (8), we obtain Diri. Diri. z =0 f n Substituting Eq. (21) into Eq. (15) and using Eq. (11), we obtain ∞ n=1 n:odd which should be satisfied for the two quantizations, Eqs. (10) and (14), to be consistent. In Appendix A.1, these consistency conditions, which we call unitarity relations, are shown to be satisfied by Bogoliubov coefficients (20). The spectrum of created particles is given by the vacuum expectation value of number operator b † m b m , Note that this is finite but its summation over m, the total number of created particles, is divergent. This implies that the Fock-space representation associated with a n is unitarily inequivalent to that associated with b m [11]. The vacuum expectation value of energy-momentum tensor before the change of boundary condition at t = 0 is computed by substituting Eq. (10) into Eq. (17), and using Eqs. (11), (13), and (5) as This represents the Casimir energy density [12], which can be made finite with standard regularization schemes [1]. The most interesting quantity is the vacuum expectation value of energy-momentum tensor after t = 0. Substituting Eq. (14) into Eq. (17) and using Eq. (21), we obtain To derive Eq. (25), we symmetrize it with respect to dummy indices m and m ′ , and use the fact that α nm and β nm are real. Using the explicit expressions of Bogoliubov coefficients (20) and mode function (8), we obtain This is an even function of z ± with period 2L since it is invariant under reflection z ± → −z ± and translation z ± → z ± + 2L. Therefore, it is sufficient to calculate it in 0 ≤ z ± < 2L, and then generalize the obtained expression appropriately to one valid in the entire domain.
The first and second summations over m in Eq. (26) can be computed to give which is valid in 0 ≤ z ± < 2L, using the following formulas, For 0 < z ± < 2L, the rest summation over m in Eq. (27) can be computed to give using the following formula [13, p. 730 Combining Eqs. (30) and (31), we obtain This is the expression for 0 ≤ z ± < 2L, what we wanted to know. Extending the domain of Eq. (33), we obtain Let us consider the meaning of two terms in Eq. (34). The first term, the delta function squared multiplied by the logarithmically divergent series, represents the diverging flux emanating from the origin (t, x) = (0, 0) and localizing on the null lines (Fig. 2). The dependence of energy density on the delta function squared implies also the divergence of total energy emitted. This component of flux is similar to that predicted in the topology change of 1D universe [3] and the same as that predicted in the formation of a strong naked singularity [5].
The second term, at first glance, seems to represent the ambient Casimir energy just like Eq. (24), which is negative and finite after a regularization, and its vanishing on the null lines. As will be explicitly shown in the semi-infinite cavity case (Sec. 4 and Appendix B), however, this is not the case. A regularization corresponds to subtracting the spatially uniform diverging energy density due to the zero-point oscillation. If one subtracts such a uniform diverging quantity from Eq. (34), leading to the regularization of ambient Casimir term, a divergence appears on the null lines z ± = 2ℓL (ℓ ∈ Z). As far as the present author knows, this kind of flux component was first found in the particle creation due to the instantaneous appearance of Dirichlet wall in a cavity [10]. It was confirmed in the same paper that such a divergence appears in the instantaneous limit of smooth formation of a Dirichlet wall in cavity analyzed in [9].
It is suspicious that the second kind of flux component does not appear in the analysis of Ishibashi and Hosoya [5], since their system is quite similar to the present one. Thus, we will revisit their analysis in Sec. 3 and see that the component was just overlooked.

From Dirichlet to Neumann
We assume that the boundary condition at x = 0 is Dirichlet (4) for t < 0 and Neumann (3) for t > 0, and that the quantum field is in vacuum |0 g . See Fig. 3 for a schematic picture of the situation. Since this situation is a kind of time reversal of that in Sec. 2.2.1, most parts of calculation can be reused but the results are different.
Let us expand g m by f n , where the expansion coefficients are given by Here, α nm and β nm are given by Eq. (20).
which should be satisfied again for the two quantization, Eqs. (10) and (14), to be consistent. It is shown in Appendix A.2 that the Bogoliubov coefficients given by Eq. (36) indeed satisfy unitarity relations (38).
The vacuum expectation value of number operator a † n a n , representing the energy spectrum of created particles, is computed as This and its summation over odd n, i.e., the total number of created particles, are divergent. This implies that the Fock-space representation associated with b m is unitarily inequivalent to that associated with a n [11]. The vacuum expectation value of energy-momentum tensor before the change of boundary condition at t = 0 is computed by substituting Eq. (14) into Eq. (17), and using the explicit expression of mode function (8), This represents the Casimir energy density, which can be made finite by standard renormalization procedures [1]. The vacuum expectation value of energy-momentum tensor after t = 0 is computed by substituting Eq. (10) into Eq. (17), and using Eq. (37), as which we symmetrize with respect to dummy indices n and n ′ , and use the fact that ρ mn and σ mn are real.
Using the explicit form of Bogoliubov coefficients and mode function, Eqs. (36), (20), and (5), we obtain This is an even function of z ± with period 2L, since it is invariant under reflection z ± → −z ± and translation z ± → z ± + 2L. Therefore, it is sufficient to calculate it in 0 ≤ z ± < 2L, and generalize it appropriately to one valid in the entire domain. The first summation over odd n in Eq. (42) can be computed to give which is valid in 0 ≤ z ± < 2L. Here, we have used the following formula [13, p. 733], It is noted here that there are typos in Ref. [13, p. 733] about formulas (44) and (47) (see below). For z ± = 0, from Eq. (43), we have For 0 < z ± < 2L, the rest summation over odd n in Eq. (43) can be computed to give using the following formula [13, p. 733], Combining Eqs. (45) and (46), and extending the domain periodically into the entire domain, we have Comparing the above result with that in the N-D case (34), one sees that there is no flux component of delta function squared in this case. As will be explicitly shown in the semi-infinite cavity case (Sec. 4 and Appendix B), Eq. (48) represents the non-renormalizable diverging flux localized on the null lines z ± = 2ℓL (ℓ ∈ Z) and the ambient Casimir energy. Thus, the diverging flux emanates from origin (t, x) = (0, 0) and propagates along the null lines in a similar way to Fig. 2. 3 Finite cavity II: Revisit Ishibashi-Hosoya [5] As seen in Sec. 2, the vacuum expectation value of energy-momentum tensor has two components in the N-D case as Eq. difference between the N-D and D-N cases will be discussed in Conclusion. Here, let us consider the consistency between these results and a relevant past work. In Ref. [5], the authors considered the instantaneous change of boundary condition at the both sides of finite cavity. The boundary conditions for t < 0 are Neumann at the both sides and those for t > 0 are Dirichlet at the both sides, which we call the NN-DD case. Since this NN-DD case resembles the N-D case, one can expect the similar results. Namely, we expect that two diverging flux components appear in the NN-DD case. Reference [5], however, concludes the flux involves only the component of delta function squared. Therefore, we will reconsider here the system adopted in [5].

Quantization of massless scalar field
We consider the situation that the Neumann boundary condition is imposed at x = 0 and x = L for t < 0, while the Dirichlet boundary condition is imposed at x = 0 and x = L for t > 0 (see Fig. 4).
In this case, a normalized positive-energy mode function for t < 0 is given by A normalized mode function for t > 0 is given by Eq. (8).
The scalar field is quantized by expanding it by set of mode functions {h k } and an additional zero-mode function h 0 , being spatially uniform, as Here, Q and P are Hermitian (Q † = Q, P † = P ), and the following commutation relations are imposed Note that zero-mode h 0 , which exists because the boundary conditions are Neumann at the both ends, is indispensable to realize the equal-time commutation relation (12) using commutation relations (51).

Particle creation by instantaneous change of boundary condition: From Neumann-Neumann to Dirichlet-Dirichlet
Let us expand h 0 and h k by g m , where the Bogoliubov coefficients are given by Using the explicit form of mode functions (8) and (49), and Eq. (50), Bogoliubov coefficients (53) are computed as Here, we have introduced the following symbols, Substituting Eq. (52) into Eq. (50), and comparing it with Eq. (14), we have Substituting Eq. (57) into Eq. (15) and using Eq. (51), we obtain the unitarity relations, In Appendix A.3, we will show that the operators given in Eqs. (54) and (55) satisfy unitarity relations (58). We define the vacuum in which particle corresponding to h 0 and h k does not exist, Then, the spectrum of created particles are given by the expectation value of number operator b † m b m , The vacuum expectation value of energy-momentum tensor before the change of boundary conditions at t = 0 is computed by substituting Eq. (50) into Eq. (17), and using explicit form of mode function (49) as This represents the Casimir energy density, which can be made finite by standard regularization schemes such as the ζ-function regularization, the point-splitting regularization, and so on [1]. The vacuum expectation value of energy-momentum tensor after t = 0 is computed by substituting Eq. (14) into Eq. (17), and using Eq. (57), which we symmetrize with respect to dummy indices m and m ′ , and we have used the fact that ξ km and ζ km are real. Using explicit form of mode functions (8) and Bogoliubov coefficients (55), we obtain The summations over odd m in the first two terms of Eq. (63), both of which are the contributions of the zero-mode, are computed using the following formulas, where Π b a (x) is the rectangular function defined as The rest summations over odd and even m in Eq. (63) are computed using formulas (28), (29), (32), (44), and (47) in addition to the above formulas, to obtain After setting L = π and regularizing the diverging summation as ∞ k=1 k = − 1 12 by the ζ-function regularization, Eq. (67) should be equal to Eq. (31) of Ref. [5]. The vanishing of Casimir energy on the null lines in Eq. (67), however, has no counterpart in Eq. (31) of Ref. [5].
While we have derived Eq. (67) with keeping the parallelism with the other analyses in the present paper, it is unclear from where the discrepancy comes. In the next subsection, therefore, we will re-derive Eq. (67) with a method similar to one in Ref. [5].

Origin of discrepancy
Substituting Eq. (14) into Eq. (17), and using Eq. (57), the vacuum expectation value of the energymomentum tensor after t = 0 is written as Using explicit form of Bogoliubov coefficients (54) and (55), and mode function (8), this quantity is rewritten in a compact form, The summations over odd m and m ′ in Eq. (69) can be evaluated with the following formulas, ∞ k=−∞ k:odd which are equivalent to Eqs. (64) and (65), respectively. Finally, in order to obtain the final result, it is necessary to use the following relation, Then, we obtain Eq. (67). It seems that Ref. [5] overlooked the left-hand side of Eq. (72) to vanish on null lines z − = 0 and z + = L. This would be the origin of the discrepancy between our result and their result.

Semi-infinite cavity
In the rest of this paper, we investigate the particle creation by the instantaneous change of boundary condition in a semi-infinite cavity, which correspond to the limit L → +∞ of the finite-cavity model in Sec. 2. We will see that some simplifications happen in such a limit. Namely, one needs just some simple integral formulas rather than the non-trivial summation formulas in Sec. 2. The analysis in semi-infinite space x ∈ [0, +∞) can be a footing to generalize the present analysis, for example, to higher-dimensional models by regarding the spatial coordinate x as a radial coordinate of higherdimensional spaces (see [14] for a relevant higher-dimensional consideration). While the Bogoliubov transformation will be used in this section again in order to keep the parallelism with the previous sections, the results will be re-derived in Appendix B with an independent method using the Green functions, which naturally involves the point-splitting regularization of the vacuum expectation value of energy-momentum tensor.

Quantization of massless scalar field
We consider a free massless scalar field in the semi-infinite cavity, of which equation of motion is given by Eq. (1) with L → +∞. At left boundary x = 0, we consider two kinds of boundary conditions. One is the Neumann boundary condition (3). Another is the Dirichlet boundary condition (4).
During Neumann boundary condition (3) is satisfied, a natural set of positive-energy mode functions {f p }, which is labeled by continuous parameter p, is given by This mode function satisfies the following orthonormal conditions, where the integration range of Klein-Gordon inner product, Eq. (7), is from 0 to +∞. During Dirichlet boundary condition (4) is satisfied, a natural set of positive-energy mode functions {g q } is given by This mode function satisfies the following orthonormal conditions, Associated with the above two sets of mode functions, {f p } and {g q }, there are two ways to quantize the scalar field. Namely, we can expand the scalar field by two sets of mode functions, where the expansion coefficients are imposed the commutation relations, Operators a p and b q (resp. a † p and b † q ) are interpreted as annihilation (resp. creation) operators. Accordingly, we can define two normalized vacuum states, Then, |0 f (resp. |0 g ) is the state where no particle corresponding to f n (resp. g m ) exists.

Particle creation by instantaneous change of boundary condition
Given the above quantization of scalar field in the semi-infinite cavity, we investigate how the vacuum is excited when the boundary condition at x = 0 instantaneously changes from Neumann to Dirichlet

From Neumann to Dirichlet
We assume that the boundary condition at x = 0 is Neumann (3) for t < 0 and Dirichlet (4) for t > 0, and that the quantum field is in vacuum |0 f , defined by Eq. (81). See Fig. 5 for a schematic picture of the situation.
Let us expand f p by g q as, where the expansion coefficients are given by Using Eqs. (73) and (75), we obtain Substituting Eq. (86) into Eq. (80), and using Eq. (79), we obtain the unitarity relations, In Appendix A.4, we prove that Bogoliubov coefficients (85) satisfy Eq. (87). The spectrum of created particles are computed as This and its integration over q are divergent due to the contribution from the infrared regime. The vacuum expectation value of energy-momentum tensor before the change of boundary condition at t = 0 is computed by substituting Eq. (77) into Eq. (17), and using Eqs. (79) and (73), as Unlike the finite-cavity case, there is no Casimir energy in this semi-infinite case. The above result just represents the divergent energy density due to the zero-point oscillation. Thus, the renormalized vacuum expectation value obtained by subtracting such a zero-point contribution identically vanishes everywhere as Eq. (146). The vacuum expectation value of energy-momentum tensor after t = 0 is computed by substituting Eq. (78) into Eq. (17), and using Eq. (86), as To derive Eq. (90), we symmetrize it with respect to integration variables q and q ′ , and use the fact that α pq and β pq are real. Using explicit expressions of Bogoliubov coefficients (85) and mode function (75), we obtain The integration over q in Eq. (91) can be computed to give where sgn denotes the sign function, Note that we have used the following integration formulas, ∞ 0 x sin(ax) See Appendix C for the derivation of the second and third formulas. Let us consider the meaning of two terms in Eq. (92). The first term, the delta function squared multiplied by a divergent integral, represents the diverging flux emanating from the origin (t, x) = (0, 0) and localizing on the null line z − = 0. The divergent factor involves the infrared divergence too since there in no infrared cutoff introduced by finite L. The dependence of energy density on the delta function squared implies also the divergence of total energy emitted.
The second term, at first glance, seems to represent an ambient divergent energy density and its vanishing on the null line emanating from the origin (note that sgn(0) = 0). As will be seen below, however, this is not the case. Namely, the divergence at z ± = 0 just represents the energy due to the zero-point oscillation just like Eq. (89). Therefore, the regularized vacuum expectation value of energymomentum tensor should be defined by subtracting such a diverging quantity distributing uniformly in space and time. As the result of such a subtraction, the divergence appears on the null line z − = 0. Such a renormalized vacuum expectation value of energy-momentum tensor is computed in Appendix B with the Green-function method, which naturally involves the point-splitting regularization. The result is Here, z ± and z ′ ± are the coordinates of two points on which the Green functions are evaluated. As explained above, the second term diverges on the null line and vanishes elsewhere. Thus, there remain the two components of diverging flux even after the renormalization to propagate along the null line z − = 0.

From Dirichlet to Neumann
We assume that the boundary condition at x = 0 is Dirichlet (4) for t < 0 and Neumann (3) for t > 0, and that the quantum field is in vacuum |0 g , given by Eq. (82). See Fig. 6 for a schematic picture of the physical situation. Then, we investigate how the vacuum is excited by computing the spectrum and energy flux of created particles.
Let us expand g q by f p as, Here, the expansion coefficients are given by where α pq and β pq are given by Eq. (85). Substituting Eq. (98) into Eq. (78), and comparing it with Eq. (77), we obtain Substituting Eq. (100) into Eq. (79), and using Eq. (80), we obtain the unitarity relations, In Appendix A.5, it is shown that Bogoliubov coefficients (99) indeed satisfy Eq. (101). The spectrum is computed as which is divergent. The expectation value of energy-momentum tensor before the change of boundary condition at t = 0 is computed by substituting Eq. (78) into Eq. (17), and using Eqs. (80) and (75), as This represents the divergence due to the zero-point oscillation, and the regularized value vanishes as given by Eq. (160). The expectation value of energy-momentum tensor for t > 0 is computed by substituting Eq. (77) into Eq. (17), and using Eq. (100), as where we symmetrize it with respect to integration variables p and p ′ , and use the fact that ρ qp and σ qp are real. Substituting explicit form of the Bogoliubov coefficients, given by Eqs. (99) and (85), and mode function (75) into Eq. (104), we have The integrations over p in Eq. (105) are evaluated using formulas (95) and (96) to obtain Again, result (106) seems to represent a diverging flux and its vanishing on the null line emanating from the origin. After subtracting the uniform contribution from the zero-point oscillation, however, the divergence appears on the null line. This is explicitly shown by adopting the Green-function method in Appendix B. The result is given by Here, z ± and z ′ ± are the coordinates of two points on which the Green functions are evaluated. The flux diverges on the null line and vanishes elsewhere. Thus, there remains only one component of diverging flux after the renormalization to propagate along the null line z − = 0.

Conclusion
We have investigated the particle creation due to the instantaneous change of boundary condition in the one-dimensional (1D) finite cavity (Secs. 2 and 3) and semi-infinite cavity (Sec. 4) by computing the vacuum expectation value of energy-momentum tensor for the free massless Klein-Gordon scalar field. The boundary condition changes from Neumann to Dirichlet (N-D) in Secs. 2 and 4, from Neumann-Neumann to Dirichlet-Dirichlet (NN-DD) in Sec. 3, and from Dirichlet to Neumann (D-N) in Secs. 2 and 4.
Although any actual change of boundary condition takes a finite interval of time, we believe that these models are capable of extracting the essence of phenomenon when the boundary condition changes rapidly enough compared to typical time scales in the system. In addition, the choice of Dirichlet and Neumann boundary conditions introduced no adjustable parameters into the system, which made the whole analysis simple to be a good starting point for succeeding considerations. Most models of the particle creation due to time-dependent boundary conditions would have to reproduce the results in this paper in the limit of infinitely rapid change.
Thanks to the above simplifications made in our model, we could obtain almost all the results in completely analytic form. For the finite cavity N-D (resp. D-N) case, the vacuum expectation value of energy-momentum tensor was obtained as Eq. (34) (resp. (48)). Our result that the flux in the N-D and D-N cases consist of two terms and only one term, respectively, seemed to contradict the result in Ref. [5], which analyses the NN-DD case. Therefore, we revisited the NN-DD case in Sec. 3 to obtain Eq. (67), which is consistent with the result in Sec. 2. The flux in the N-D and NN-DD cases consist of terms of δ 2 (z ± ) and 1/(z ± − z ′ ± ) 2 , while the flux in the D-N case consists of only term of 1/(z ± − z ′ ± ) 2 . Although we cannot argue which term is stronger to dominate at this point, it will be the case that not only the flux but also the total energy radiated becomes large since the integration of flux cross z ± = 0 diverges.
While the results in the semi-infinite cavity for the N-D case (92) and D-N case (106) are quite similar to their respective counterparts in the finite cavity, the analysis for the infinite cavity is much simpler than the finite-cavity case in that non-trivial mathematical formulas such as summation formulas of Eqs. (29), (44), and so on, are not necessary. This is a technical but an important point for succeeding studies such as the generalizations of this work (future works will be mentioned later). In addition, the vacuum expectation value of energy-momentum tensor in the semi-infinite cavity was re-derived by the Green-function method in Appendix B. This method not only naturally involves the point-splitting regularization but also involves only simpler calculations than the Bogoliubov method in the text. Again, this is a technical but an important point. Finally, the analysis for the semi-infinite cavity confirmed that the divergence of flux due to the change of boundary condition is nothing but an ultraviolet effect rather than an infrared one, and that the divergence of the flux has nothing to do with the Casimir effect, which exists only when L is finite.
Let us discuss the origin of asymmetry between the N-D and D-N cases, of which related conjecture was already proposed in the previous paper of the present author and his collaborators [10]. The δ 2term seems to stem from a temporal discontinuity of mode function f n and f p . For instance, in the finite-cavity N-D case, mode function f n is given by Eq. (5) for t < 0, having a non-zero value at x = 0, but given by Eq. (18) for t > 0, vanishing at x = 0. Therefore, f n (t, 0) is discontinuous as a function of time at t = 0. On the other hand, in the finite-cavity D-N case, mode function g m is given by Eq. (8) for t < 0 and Eq. (35) for t > 0, both of which vanish at x = 0. Therefore, g m (t, 0) is continuous as a function of time at t = 0. In a similar way, h k (t, 0) and h k (t, L) are discontinuous as functions of time at t = 0 in the NN-DD case, and f p (t, 0) (resp. g q (t, 0)) is discontinuous (resp. continuous) at t = 0 in the semi-infinite N-D (resp. D-N) case. We conjecture that such a discontinuity, which would create a shock in the classical mechanics point of view, is the origin of the delta function squared.
Naively speaking, the results in this paper suggest that the back-reaction of created particles to the spacetime and/or the cavity cannot be ignored. However, the analysis is based on the test-field approximation, therefore, it is too early to assert such an implication of the results. As a next step, it is natural to investigate the back-reaction through, say, the semi-classical Einstein equation, where the right-hand side of Einstein equation is replaced by the regularized vacuum expectation value of energy-momentum tensor of quantized fields [1].
Given the results in this paper, there would be several directions to proceed besides investigating the back-reaction mentioned above. Firstly, it is natural to generalize the present analysis to higherdimensional spacetime (see Ref. [14] for a highly relevant study). Secondly, it would be important to generalize the boundary condition in the present paper (i.e., Dirichlet and Neumann) to the Robintype boundary condition (see, e.g., [15]), which takes the form of φ(t, x) − a∂ x φ(t, x)| x=0 = 0. Taking different values of constant a before and after t = 0, one can generalize the present analysis. By such a generalization, we would be able to verify the above conjecture about the origin of asymmetry between the N-D and D-N cases, and understand more deeply how the time-dependent boundaries make the quantum vacuum excite in general.

Acknowledgments
The author would like to thank T. Harada and S. Kinoshita for useful discussions. This work was partially supported by JSPS KAKENHI Grant Numbers 15K05086 and 18K03652.

A Proof of unitarity relations
where we define The summation over odd n in Eq. (110) can be computed to give using the following formulas [13, pp. 688-689], where we define The summations over m in Eq. (116) can be computed to give where Applying formulas (118) and (119) to W mm ′ , and formulas (112) and (113) to X mm ′ , we obtain Substituting Eq. (123) into Eqs. (120) and (121), we see that unitarity relation (58) holds.

A.4 Equation (87)
Using Eq. (85), the left-hand side of Eq. (87) is written as where we define By simple algebra, this is rewritten as Adapting the following formula [17, p. 488] to the first and second terms of Eq. (127), and noting the third term vanishes from Eq. (95), we have Substituting Eq. (129) into Eqs. (124) and (125), we see Eq. (87) to hold.
This is computed as

B Green-function method for semi-infinite cavity
We re-analyze the vacuum excitation by the change of boundary condition for the semi-infinite cavity using the Green-function method [1,10], which naturally incorporates the renormalization of zeropoint energy.

B.2 From Neumann to Dirichlet
The vacuum expectation value of energy-momentum tensor before the change of boundary condition is obtained by differentiating the Hadamard elementary function F (1) with respect to two points z and z ′ , and taking the same-point limit z ′ → z, From Eqs. (137) and (142), one obtains One can see that Eq. (143) reproduces Eq. (89) if one takes limit z ′ → z before the p-integration. Equation (144) shows that 0 f |T ±± |0 f Green t<0 contains the ultraviolet divergence ∼ 1/(∆z ± ) 2 , which is the vacuum energy due to the zero-point oscillation always existing even in a free Mankowski spacetime. Therefore, the renormalized energy-momentum is defined by subtracting this ultraviolet divergence as which reasonably vanishes before changing the boundary condition. The vacuum expectation value of energy-momentum tensor after the change the boundary condition has the same expression as Eq. (142). However, since the boundary condition is changed at t = 0, Hadamard elementary function F (1) before the change of boundary condition has to be propagated into t > 0 region using Pauli-Jordan function iG [10]. Thus, the energy-momentum is represented as where A := z and B := z ′ . Namely, in this abbreviated notation, let a capital Latin letters (except G and F ) denote a world point, e.g., φ(A, B) = φ(z, z ′ ). In addition, let a pair of repeated capital Latin letter denote the Klein-Gordon inner product at t = 0, e.g., φ(A)ψ(A) := φ, ψ | t=0 .
This represents the ultraviolet divergence due to the zero-point oscillation. The normalized energymomentum is defined by subtracting such a divergence, 0 g |T ±± |0 g ren t<0 := 0 g |T ±± |0 g Green t<0 − lim which reasonably vanishes before the change of boundary condition. The energy-momentum after the change of boundary condition is obtained by propagating G (1) by iF , 0 g |T ±± |0 g The inner product in Eq. (162) is written as Using Eq. (140), derivatives of F in Eq. (163) are computed as where we have used formulas (153) and (154). The combination of Eqs. (166) and (162) gives The renormalized energy-momentum is obtained by subtracting the zero-point energy (159) from Eq. (167), 0 g |T ±± |0 g ren t>0 := 0 g |T ±± |0 g x sin(ax) where −∞ < a < ∞, b > 0. Note that we always consider only principal values for improper integrals. These are written as We suppose two contours C + and C − drawn in Fig. 7 and use Cauchy's integral theorem and the residue theorem.
For a > 0, taking contour C ± for I ± and J ± , we have