1 Introduction

While this paper is mostly on QFT and its behavior on singular spacetimes describing some models of Big Crunch/Big Bang its reason has roots in string theory. String theory, as a promising candidate for a theory of quantum gravity, is supposed to provide a satisfactory description of Big Bang/Big Crunch type singularities, or at least a S matrix in asymptotically flat spaces.

We want therefore to construct and study stringy toy models capable of reproducing a space-like (or null) singularity which appears in space at a specific value of the time coordinate and then disappears.

The easiest way to do so is by generating singularities by quotienting Minkowski with a discrete group with fixed points, i.e. orbifolding Minkowski. In this way it is possible to produce both space-like singularities and supersymmetric null singularities [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] (see also [16,17,18] for some reviews). Another possible way which is a generalization of the previous orbifolds with null singularity is consider gravitational shock wave backgrounds [19,20,21,22,23,24,25,26,27,28,29,30].

It happens that in these orbifolds the four tachyon closed string amplitude diverges in some kinematical ranges, more explicitly for the Null Shift Orbifold (which may be made supersymmetric and has a null singularity) we have

$$\begin{aligned} {{\mathcal {A}}} ^{(closed)}_{4 T} \sim \int _{q\sim \infty } \frac{d q}{|q|} q^{ 4- \alpha ' \textbf{p}^2_{\perp \, t}} , \end{aligned}$$
(1.1)

so the amplitude diverges for \(\alpha ' \textbf{p}^2_{\perp \, t}<4\) where \(\textbf{p}_{\perp \, t}\) is the orbifold transverse momentum in t channel. Until recently this pathological behavior has been interpreted in the literature as “the result of a large gravitational backreaction of the incoming matter into the singularity due to the exchange of a single graviton”. This is not very promising for a theory which should tame quantum gravity.

What has gone unnoticed is that if we perform an analogous computation for the four point open string function we find

$$\begin{aligned} {{\mathcal {A}}} ^{(open)}_{4 T} \sim \int _{q\sim \infty } \frac{d q}{|q|} q^{ 1- \alpha ' \textbf{p}^2_{\perp \, t}} tr\left( \{T_1, T_2\} \{T_3, T_4\}\right) , \end{aligned}$$
(1.2)

which is also divergent when for \(\alpha ' \mathbf \mathrm{}{p}^2_{\perp \, t}<1\) [31, 32]. This casts doubts on the backreaction as main explanation since we are dealing with open string at tree level. This is further strengthened by the fact that three point amplitudes with massive states may diverge [31] when appropriate polarizations are chosen. For example for the three point function of two tachyons and the first level massive state we find for an appropriate massive string polarization

$$\begin{aligned} {{\mathcal {A}}} ^{(open)}_{T T M} \sim \int _{u\sim 0} \frac{d u}{|u|^{5/2}} tr\left( \{T_1, T_2\} T_3\right) . \end{aligned}$$
(1.3)

In [31] this was interpreted as a non existence of the underlying effective theory. We now revisit this assertion and argue that the effective theory does exist but the usual approach based on the perturbative expansion in the interaction picture completely breaks down.

In this paper we consider what happens when we use perturbation theory in a time dependent background with a space singularity. It is somewhat obvious that we do not expect to find a well behaved perturbation theory because of the singularity. One could expect some kind of pathology like the series being asymptotics. We find a much worse behavior: a complete breakdown of perturbation theory, i.e. perturbation theory does not exist. Let us be more precise. We consider as unperturbed theory the free, non interacting QFT in the given singular time dependent background and then add interactions. We then use the usual interaction picture approach. This approach when used perturbatively naturally leads to Feynman diagrams and a nice particle interpretation of interactions. In the backgrounds we consider all of this suffers from a complete breakdown. There is no perturbative expansion in the usual sense. This prompts the question whether it is perturbation theory which fails or it is the very interacting theory which does not exists. To answer this question we consider the minisuperspace approach, i.e. the consider the QFT reduced to the spacially homogeneous configurations (see [33] for review). In this limit the theory reduces to Quantum Mechanics. We then show that these models do exist. One could wonder whether this reduction is a big limitations and the answer is no since it has been shown [14, 31] that the troubles in perturbation theory stem from these configurations. The main difference with the work from the 80s and 90s is that we are interested in going through the singularity and not giving the boundary conditions at the Big Bang.

This result stresses the importance of treating some sectors as exactly as possible in order to get a perturbation theory for the remaining sectors. Nevertheless it is noteworthy that from our analysis the original Krasner background, which is also a string background admits a good perturbation theory.

Even so we are left with the unanswered question whether it is really consistent to treat QFT on a given singular background without considering the backreaction. It is somewhat likely that the gravitational background and the matter should evolve together, especially in a background which has space singularities. Given the results of this paper it could be sufficient to consider the minisuperspace approximation to get a reasonable approximation. In any case this route is fraught with subtleties like the “problem of time” (see [34] for a review).

The paper is organized as follows.

In Sect. 2 we discuss the background of interest, the generalized Kasner metrics (of which the Boost Orbifold is a very special case) and the simplest interacting field theory, i.e. the scalar field and its minisuperspace approximation.

In Sect. 3 we discuss the simplest example where the perturbative interaction picture breaks completely down: the time dependent harmonic oscillator with \(\Omega ^2(t) = \omega ^2 + \frac{k}{t^2}\) (for reasons we are going to explain \(k\le \frac{1}{4}\) so that \(\Omega ^2\) may become negative). While this model is natural since it corresponds to, for example, de Sitter modes in conformal time the splitting we perform between the unperturbed Hamiltonian and the perturbative part is somewhat artificial but it is chosen in order to get the simplest example as possible.

In Sect. 4 we consider the interacting theory and we show that generically the perturbation theory of the interacting minisuperspace model does not exist. We then study the minisuperspace model non perturbatively and show that it does exist. The model exhibits two different behaviors: either it is dominated by the combination of kinetic and interaction terms or it is dominated by the time dependent harmonic oscillator (in the appopriate variable) term alone.

Finally in Sect. 5 we discuss what this means for the divergences in string theory. In nuce string theory is well, at least at tree level but the non Hamiltonian perturbation theory has troubles. Moreover we point out that the usual approach to orbifolds used in string theory is not on very sound basis when temporal orbifolds are considered since the orbifold generators are dynamical generators, except for Null Shift Orbifold in light-cone gauge.

2 The background

Our starting point is to consider a class of backgrounds which have a space-like singularity and on these backgrounds write down the simplest interacting scalar theory.

Previous results from the analysis of issues in open string amplitudes in these backgrounds [14, 31] hint toward the fact the all troubles derive from special field configurations to which we restrict. In particular this means that we restrict these theories to space independent but time dependent fields in the space-like singularity case.

More precisely this paper we are going to consider the following family of backgrounds.

2.1 Kasner-like metrics

The metric we consider is a generalization of the original Kasner metric and reads

$$\begin{aligned} d s^2 = -d t^2 + \sum _{i=1}^{D-1} |t|^{2 {p_{(i)}}} R_{(i)}^2 (d x^i)^2 ,~~~~ 0\le x^i < 2\pi , \end{aligned}$$
(2.1)

where we consider \(t\in {\mathbb {R}}\) and not only \(t>0\) and therefore we have written |t| since \({p_{(i)}}\in {\mathbb {R}}\). We have also considered the \(x^i\) to be compact in order to get a well defined minisuperspace approximation of the scalar field as in Eq. (2.3).

The original Kasner metric corresponds to the case where \(\sum _i {p_{(i)}}= \sum {p_{(i)}}^2 =1\) and space is not compact. It requires that at least one \({p_{(i)}}\) is negative when at least two \({p_{(i)}}\) are different from zero and corresponds to an empty space-time. Another special case is when only \(p_{(1)}=1\) and corresponds to Milne space.

All these metrics have a singularity at \(t=0\) which is the target of our investigation. They have generically also a singularity for \(|t| \rightarrow \infty \) when some p is negative. When all p are positive the metric requires repulsive matter.

For generic \({p_{(i)}}\), i.e. not the original Kasner metric this metric is not a consistent string backgroundFootnote 1 since \(Ric \ne 0\).

2.2 Interacting scalar models

It is the immediate to write down the action for an interacting real scalar field as

$$\begin{aligned} S&= \int d t \prod _i d x^i\, \prod _i R_{(i)}|t|^{\sum _i {p_{(i)}}}\nonumber \\&\quad \times \Biggl [ \frac{1}{2}{\dot{\phi }} ^2 - \frac{1}{2}\sum _i \frac{1}{R_{(i)}^2 |t|^{2 {p_{(i)}}}} (\partial _i \phi )^2\nonumber \\&\quad - \frac{1}{2}m^2 \phi ^2 - \frac{1}{n} g_n \phi ^n \Biggl ] ,~~~~ n\in {4,6,\dots } . \end{aligned}$$
(2.2)

According to the analysis of string theory on Boost Orbifold  [14, 31] the problems for this theory derive from the field configurations where the field depends on time only. Restricting to this configuration we get the quantum mechanical model

$$\begin{aligned} S =&\prod _i(2\pi R_{(i)}) \int d t |t|^{2 A} \Biggl [ \frac{1}{2}{\dot{\phi }}^2 - \frac{1}{2}m^2 \phi ^2 - \frac{1}{n} g_n \phi ^n \Biggl ] , \end{aligned}$$
(2.3)

where we have defined \(2 A = \sum _i {p_{(i)}}\) for compactness. We consider only the case where \(A>0\).

3 The simplest example of failure of the perturbative expansion in interaction picture: the time dependent harmonic oscillator

In this section we would like to discuss how the usual perturbative expansion in interaction picture may completely break down when the interaction Hamiltonian has time singularities. This may happen despite the complete model is well defined.

In particular the model we want to consider is

$$\begin{aligned} L_R= |t|^{2 A} \left( \frac{1}{2}\dot{y} ^2 - \frac{1}{2}\omega ^2 y^2 \right) , \end{aligned}$$
(3.1)

which corresponds to the non interacting scalar on Kasner metrics. Two special cases are \(A=0\) and \(A=\frac{1}{2}\) and both correspond to the flat space but in Minkowski and Milne (Boost orbifold) coordinates. Upon a change of coordinates as

$$\begin{aligned} x = |t|^A y , \end{aligned}$$
(3.2)

we get

$$\begin{aligned}{} & {} L_B= \frac{1}{2}\dot{x} ^2 - \frac{1}{2}\left( \omega ^2 + \frac{k}{t^2} \right) x^2 + \frac{d}{d t}\left( \frac{1}{2}\frac{A}{t} x^2 \right) , \nonumber \\{} & {} \quad k=A (1-A)\in \left( -\infty ,\frac{1}{4}\right) \end{aligned}$$
(3.3)

The total derivative is uninfluential at the classical level while at the quantum it implies a relative time dependent phase for the wave function in the two coordinate systems see Eq. (3.30).

Notice that when k is negative (\(A>1\) or \(A<0\)) the potential is unbounded from below but despite this the full model is well defined. That this may happen is not a surprise since the hydrogen atom exists and has an unbounded potential. On the other side in the flat space \(A=0,\frac{1}{2}\) the potential is always bounded from below. In particular the \(A=\frac{1}{2}\) case is the Milne space which is a subset of Minkowski space and even so the model has a singular potential when there is only one space dimension otherwise it is the original Krasner solution.

This model emerges besides the obvious case of the non interacting scalar in Kasner-like metrics mentioned above also in the following cases :

  1. 1.

    The particle or the string in a certain pp-wave background in Brinkmann coordinates that is described by the metric

    $$\begin{aligned} d s^2_B= & {} - 2 d u\, d v + \sum _{I=1}^{D-2} A_I (A_I - 1) ( x^I)^2 \frac{1}{u^2} d u^2 \nonumber \\{} & {} + \sum _{I=1}^{D-2} ( d x^I)^2 . \end{aligned}$$
    (3.4)

    Notice however that a purely gravitational string background, i.e. with trivial dilaton and Kalb–Ramond, must be a Ricci flat background so we need to impose \(\sum _I A_I (A_I - 1) = 0\) if we want a consistent model propagating in this background. The particle action in light-cone gauge \(u=\tau \) reads

    $$\begin{aligned} S_{LC} = \int d \tau \, \left[ \frac{-1}{e} \dot{v} + \sum _{I=1}^{D-2} \left( \frac{-1}{e} (\dot{x}^I)^2 + \frac{-1}{e} \frac{A_I (A_I - 1)}{\tau ^2} (x^I)^2 \right) \right] .\nonumber \\ \end{aligned}$$
    (3.5)

    Since e is constant on shell, any \(x^I\) has the action (3.3) with \(\omega ^2=0\). The case with \(\omega ^2\ne 0\) is recovered when string is considered. In facts the previous \(x^I\) are the string zero modes and the string non zero modes \(x^I_n\) have \(\omega ^2 \propto n^2\).

  2. 2.

    The modes of the scalar field in de Sitter universe in conformal time. If we consider the FLRW metric

    $$\begin{aligned} d s^2= & {} d t^2 - a^2(t) \sum _{i=1}^{D-1} ( d x^i)^2 \nonumber \\= & {} a^2(\eta ) \left( d \eta ^2 - \sum _{i=1}^{D-1} ( d x^i)^2 \right) , \end{aligned}$$
    (3.6)

    with \(d \eta = \frac{1}{a(t)}d t\). For de Sitter we have \(a_{d S}(t)=e^{H t}\) so that \(a_{d S}(\eta )= - \frac{1}{H \eta }\) with \(-\infty< \eta < 0^-\). The real scalar action is then

    $$\begin{aligned} S_{FLRW}&= \int d t\, d^{D-1} x\, a(t)^D\, \left[ \frac{1}{2}({\dot{\phi }})^2 \right. \nonumber \\&\quad \left. - \frac{1}{2}a(t)^{-2} (\partial _i \phi )^2 - \frac{1}{2}m^2 \phi ^2 \right] \nonumber \\&= \int d \eta \, d^{D-1} x \Biggl [ \frac{1}{2}({\dot{\chi }})^2 - \frac{1}{2}(\partial _i \chi )^2\nonumber \\&\quad - \frac{1}{2}\left( m^2 a^2 - \frac{D-2}{2} \frac{a''(\eta )}{a(\eta )}\right. \nonumber \\&\quad \left. - \frac{(D-2)(D-4)}{4} \left( \frac{a'(\eta )}{a(\eta )} \right) ^2 \right) \chi ^2 \Biggr ] , \end{aligned}$$
    (3.7)

    where we defined \(\phi (t,x) = a^{1- \frac{D}{2}}(\eta ) \chi (\eta ,x^i)\) and \(a'(\eta ) = \frac{d a(\eta )}{d \eta }\). Performing the Fourier transform w.r.t. to the space coordinates we get

    $$\begin{aligned} S_{FLRW}&= \int d \eta \, d^{D-1} k \Biggl [ \frac{1}{2}|{\tilde{\chi }}'(\eta ,k)|^2\nonumber \\&\quad - \frac{1}{2}\left( k_i^2 + m^2 a^2 - \frac{D-2}{2} \frac{a''(\eta )}{a(\eta )}\right. \nonumber \\&\quad \left. - \frac{(D-2)(D-4)}{4} \left( \frac{a'(\eta )}{a(\eta )} \right) ^2 \right) |{\tilde{\chi }}(\eta ,k)| \Biggr ] , \end{aligned}$$
    (3.8)

    which in de Sitter space becomes

    $$\begin{aligned} S_{d S}&= \int d \eta \, d^{D-1} k \left[ \frac{1}{2}|{\tilde{\chi }}'(\eta ,k)|^2 \right. \nonumber \\&\quad \left. - \frac{1}{2}\left( k_i^2 + \frac{m^2}{H^2} \frac{1}{\eta ^2} - \frac{D(D-2)}{4} \frac{1}{\eta ^2} \right) |{\tilde{\chi }}(\eta ,k)| \right] , \end{aligned}$$
    (3.9)

    which shows that the modes again have action (3.3) but with \(\eta <0\) so the model we consider is a kind of cyclic de Sitter.

  3. 3.

    The particle in Vaidya metric with linear mass.

3.1 Failure of the perturbative expansion of the evolution operator in the interaction picture

Let us now consider the Hamiltonian corresponding to (3.3) as the sum of the usual harmonic oscillator and a quadratic time dependent interaction term. The splitting we perform between the unperturbed Hamiltonian and the perturbative part is somewhat artificial but it is chosen in order to get the simplest example as possible and then discuss the issues in the simplest context.

Explicitly in Schroedinger picture we have

$$\begin{aligned} H_S(t)&= H_{S 0}(t) + H_{S 1}(t)\nonumber \\ H_{S 0}(t)&= \frac{p_S^2}{2} + \frac{1}{2}\omega ^2 x_S^2, ~~~ H_{S 1}(t) = \frac{k}{t^2} x_S^2 . \end{aligned}$$
(3.10)

Obviously the perturbation Hamiltonian is dominant for small t and therefore one can expect that perturbation theory be asymptotic as it happens in Stark effect. However we find a complete breakdown of perturbation theory and not an asymptotic series.

The interaction picture is obtained from Schroedinger equation

$$\begin{aligned} i \frac{\partial }{\partial t} |\psi _S(t,t_0)\rangle = H_S(t) |\psi _S(t,t_0)\rangle , \end{aligned}$$
(3.11)

by defining

$$\begin{aligned} |\psi _I(t,t_0)\rangle= & {} U_{0 S}(t_0, t) |\psi _S(t,t_0)\rangle ,~~~~ U_{0 S}(t_0, t)\nonumber \\= & {} T e^{-i \int ^{t_0}_t d t' H_{0 S}(t')} , \end{aligned}$$
(3.12)

where \(U_{0 S}\) is the evolution operator for the “free” Hamiltonian \(H_{0 S}\). The new state \(|\psi _I(t,t_0)\rangle \) then evolves as

$$\begin{aligned} i \frac{\partial }{\partial t} |\psi _I(t,t_0)\rangle&= H_I(t, t_0) |\psi _I(t,t_0)\rangle ,\nonumber \\ H_I(t, t_0)&= U_{0 S}(t_0, t) H_{1 S} (t) U_{0 S}(t, t_0) . \end{aligned}$$
(3.13)

The Schroedinger equation in interaction picture has then formal and perturbative solution

$$\begin{aligned}{} & {} |\psi _I(t,t_0)\rangle = T e^{-i \int ^{t}_{t_0} d t' H_{I}(t', t_0)}|\psi _I(t_0,t_0)\rangle \nonumber \\{} & {} \quad = \left( 1 -i \int ^{t}_{t_0} d t' H_{I}(t', t_0) +\dots \right) |\psi _I(t_0,t_0)\rangle . \end{aligned}$$
(3.14)

If we apply this formalism to our specific case we obtain the interaction Hamiltonian

$$\begin{aligned} H_{I}(t,t_0){} & {} = - \frac{k}{4 \omega t^2} \left( e^{2 i \omega (t-t_0)} a_S^{\dagger 2}\right. \nonumber \\{} & {} \quad \left. + e^{-2 i \omega (t-t_0)} a_S^{ 2} - a_S^{\dagger } a_S - a_S a_S^{\dagger } \right) , \end{aligned}$$
(3.15)

where we have as usual

$$\begin{aligned} a_S{} & {} = \frac{p_S -i \omega x_S}{\sqrt{2 \omega }} ,~~~~ [a^\dagger _S, a_S]=1 , \nonumber \\ U_{0 S}(t, t_0){} & {} = e^{ -i \omega (a_S^{\dagger } a_S + \frac{1}{2})(t-t_0)} . \end{aligned}$$
(3.16)

We can then build a basis for the Hilbert space \(\{|n\rangle \}_{n\in {\mathbb {N}}}\) as

$$\begin{aligned} a_S |0\rangle =0 ,~~~~ |n\rangle = \frac{a^{\dagger n}_S}{\sqrt{n!}} |0\rangle . \end{aligned}$$
(3.17)

It is then immediate to see that the first order in perturbative expansion for the evolution operator from a negative \(t_0<0\) time to a positive time \(t_1>0\) is infinite. Explicitly, if we evolve perturbatively from \(|\psi _I(t_0,t_0)\rangle = |n\rangle \) to \(|\psi _I(t_1,t_0)\rangle \) and we try to expand \(|\psi _I(t_1,t_0)\rangle \) on the basis \(\{|m\rangle \}\) we have

$$\begin{aligned}&\langle m | \int ^{t_1}_{t_0} d t' H_{I}(t', t_0) | n \rangle \nonumber \\&\quad = - \frac{k}{4 \omega } \delta _{m, n} (2 m +1) \int ^{t_1}_{t_0} d t \frac{1}{t^2}\nonumber \\&\quad \quad - \frac{k}{4 \omega } \delta _{m, n+2} \sqrt{m (m-1)} \int ^{t_1}_{t_0} d t \frac{e^{2 i \omega (t-t_0)}}{t^2}\nonumber \\&\quad \quad - \frac{k}{4 \omega } \delta _{m, n-2} \sqrt{(m+2) (m+1)} \int ^{t_1}_{t_0} d t \frac{e^{-2 i \omega (t-t_0)}}{t^2} . \end{aligned}$$
(3.18)

This shows that not only the amplitude is divergent but that we cannot expand \(|\psi _I(t_1,t_0)\rangle \) on the Hilbert basis moreover the divergence cannot be reabsorbed into a c-number shift of the Hamiltonian since all coefficients depend on the states. For later use we notice that to this order of perturbation we have

$$\begin{aligned} \langle m | \int ^{t_1}_{t_0} d t' H_{I}(t', t_0) | n \rangle = \int ^{t_1}_{t_0} \langle m_S(t', t_0) | H_{1 S}(t') | n_S(t', t_0) \rangle ,\nonumber \\ \end{aligned}$$
(3.19)

i.e. we can actually use the Schroedinger states and Hamiltonian without actually computing the corresponding objects in the interaction picture.

3.2 The complete theory is well defined: the \(H_B\) case

Given the previous failure of the perturbative expansion one can wonder whether the theory exists across the singularity. The answer as we show is affirmative. The same problem has been considered before in [9,10,11,12, 22,23,24,25, 28, 35] but our point of view is slightly different since this is not the final research target of this paper but we want anyhow to show that we can traverse the singularity and then use this solution for the interacting models.

Even if we are actually interested in adding quartic and higher interactions to \(L_R\) we will perform the analysis for \(L_B\) since it looks more familiar and then map it to \(L_R\) using a time dependent unitary transformation.

The time dependent harmonic oscillator

$$\begin{aligned} i \partial _t \psi (x,t) = -\frac{1}{2}\partial _x^2 \psi (x,t) + \frac{1}{2}\left( \omega ^2 + \frac{k}{t^2} \right) \psi (x,t) , \end{aligned}$$
(3.20)

can be solved exactly using complex classical solutions with a well defined normalization. We review the derivation for completeness in Appendix 1 where we give also more details which are not relevant for the present discussion. The main result is then that the generating function of a possible complete setFootnote 2 of wave functions is

$$\begin{aligned}&\sum _{n=0}^\infty \frac{z^n}{\sqrt{n!}} \psi _{ n\{t_0\}}(x, t, t_0)\nonumber \\&\qquad = \root 4 \of {\frac{1}{2 \pi }} \frac{1}{\sqrt{{{\mathcal {X}}} (t)}} e^{i \frac{1}{2} \frac{ {\dot{{{\mathcal {X}}} }}(t) }{ {{\mathcal {X}}} (t)} x^2 + \frac{1}{{{\mathcal {X}}} (t)} x z - \frac{1}{2}\frac{{{\mathcal {X}}} ^*(t)}{{{\mathcal {X}}} (t)} z^2 } , \end{aligned}$$
(3.21)

where we have introduced the complex classical solution \({{\mathcal {X}}} (t)\) and its normalization condition

$$\begin{aligned}&\ddot{{\mathcal {X}}} (t) + \Omega ^2(t) {{\mathcal {X}}} (t) = 0 ,\nonumber \\&{{\mathcal {X}}} ^* {\dot{{{\mathcal {X}}} }} - {{\mathcal {X}}} {\dot{{{\mathcal {X}}} }}^* = i . \end{aligned}$$
(3.22)

We can now solve perturbatively the classical equations of motion around \(t=0\).

An issue which arises is the continuation across the singularity but the normalization condition required for the quantum model and “continuity” fix it (see also [36] for the case \(A=\frac{1}{2}\)).

Let us start considering the asymptotic behavior for \(t\rightarrow 0^+\) as \({{\mathcal {X}}} \sim t^a\) with \(t>0\). It is immediate to find the equation

$$\begin{aligned} a^2-a +k = 0 \Longleftrightarrow a\in \{A, 1-A\} , \end{aligned}$$
(3.23)

so that the leading behavior is

$$\begin{aligned} {{\mathcal {X}}} (t) = c_0 (\omega t)^A (1+ O(t^2) ) + c_1 (\omega t)^{1-A} (1+ O(t^2) ) , t>0 .\nonumber \\ \end{aligned}$$
(3.24)

The normalization condition then implies

$$\begin{aligned} -(2 A - 1) \omega |c_1|^2 \Im \left( \frac{c_0}{c_1} \right) = - \frac{1}{2}. \end{aligned}$$
(3.25)

Let us consider the case \(A>\frac{1}{2}> 1-A\) since \(A<\frac{1}{2}<1-A\) is obtained by swapping \(A \leftrightarrow 1-A\). Then the previous condition implies that the wave functions are normalizable since (\(t>0\))

$$\begin{aligned} \psi _0(x,t) \sim&\frac{1}{\sqrt{|\omega t|^{1-A}}} e^{ i \frac{1}{2}\left( \frac{1-A}{t} + (2 A-1) \omega \frac{c_0}{c_1} |\omega t|^{2(A-1)} \right) x^2 }\nonumber \\&\Longleftrightarrow |\psi _0(x,t)|^2 \sim \frac{1}{|\omega t|^{1-A}} e^{ - (2 A-1) \omega \Im \left( \frac{c_0}{c_1}\right) |\omega t|^{2(A-1)} x^2 } . \end{aligned}$$
(3.26)

As discussed in appendix around Eq. (A.27) this is not by chance: the normalization condition on \({{\mathcal {X}}} \) always implies the normalizability of the wave functions.

Let us exam the solution for \(t<0\). One would be tempted to write exactly the same Eq. 3.24 with the substitution \(t \rightarrow -t\). However this would lead to a different normalization condition (3.25). The difference being an overall sign in the left hand side of the normalization equation, i.e. \(+\frac{1}{2}\) in stead of \(-\frac{1}{2}\). Therefore the proper asymptotic behavior valid for all t is either

$$\begin{aligned} {{\mathcal {X}}} (t) = c_0 |\omega t|^A (1+ O(t^2) ) + c_1 \omega t |\omega t|^{-A} (1+ O(t^2) ) , \end{aligned}$$

or

$$\begin{aligned}{} & {} {{\mathcal {X}}} (t)\nonumber \\{} & {} \quad = c_0 \omega t |\omega t|^{A-1} (1+ O(t^2) ) + c_1 |\omega t|^{1-A} (1+ O(t^2) ) .\nonumber \\ \end{aligned}$$
(3.27)

Since this is a classical solution we may expect that the trajectory is continuous then for \(A>1\) comparing \(t |t|^{-A}\) and \(|t|^{1-A}\) we realize that only the latter is continuous. Hence the true solution is (3.27). Because of this the previous expression for the wave function (3.26) where we took care of distinguish between t and |t| is valid for all t values.

As discussed in Appendix 1 the previous choice can also be obtained regularizing the time dependent pulsation \(\Omega ^2(t)=\omega ^2+ \frac{k}{t^2}\).

It is also possible and instructive to use the WKB expansion. We write \(\psi (x,t) = e^{i S(x,t) }\) so that we have to solve the equation

$$\begin{aligned} \partial _t S(x,t) +\frac{1}{2}(\partial _x S(x,t))^2 + \frac{1}{2}\left( \omega ^2 + \frac{k}{t^2} \right) - i \frac{1}{2}\partial _x^2 S(x,t) = 0 .\nonumber \\ \end{aligned}$$
(3.28)

This is done in Appendix 1.

Notice that (3.26) has two completely different behaviors as \(t \rightarrow 0\).

$$\begin{aligned} |\psi _0(x,t)|^2 \sim _{t \rightarrow 0} \left\{ \begin{array}{l c} 0 &{} A>1 \\ \infty &{} A<1 \end{array} \right. . \end{aligned}$$
(3.29)

This can be understood considering the classical trajectory which behaves as \(x\sim |t|^{min(A, 1-A)}\). For \(A>1\) it diverges but the direction depends on the initial \(\dot{x}\) which quantum mechanically cannot be fixed therefore the quantum state is spread over all the possible values of x. This is shown in Fig. 1a, b. Notice that the classical trajectory (not the complex one used in computing the quantum wave function) is not well defined through \(t=0\) since we can require the continuity of the trajectory but it is difficult if not impossible to relate the velocity before and after the singularity. On the contrary the quantum theory is well defined since we can find a well defined basis of wave functions.

Fig. 1
figure 1

Classical motion for \(L_B\) with \(A>1\) has two possible asymptotic behaviors

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Differently for \(A<1\) the classical solution has a fixed point \(x(0)=0\) and therefore the wave function is a \(\delta (x)\). This is shown in Fig. 2a, b.

Fig. 2
figure 2

Classical motion for \(L_B\) with \(A<1\) has only one possible asymptotic behavior

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Finally notice that the wildly oscillating phase in (3.26) is not an issue as hypothesized in [9,10,11,12], on the contrary as shown in [31] it is a virtue since it helps the convergence of the integrals in the distributional sense (see also [37]).

3.3 Relation between \(L_R\) and \(L_B\) non interacting models

While at the classical level the two models are related as described before by a simple change of coordinates and a boundary term, at the quantum level we have

$$\begin{aligned} \psi _B(x, t) = |t|^{-\frac{1}{2}A} e^{ i \frac{1}{2}A \frac{x^2}{t}} \psi _R(y = |t|^{-A} x, t) . \end{aligned}$$
(3.30)

This can be obtained in two different ways. Both start from the Hamiltonians

$$\begin{aligned} H_R&= \frac{p_y^2}{2 |t|^{2 A} } + \frac{1}{2}\omega ^2 |t|^{2 A} y^2\nonumber \\ H_B&= \frac{p^2}{2 } + \frac{1}{2}\left( \omega ^2 + \frac{k}{t^2} \right) x^2 . \end{aligned}$$
(3.31)

The first method is a sequence of transformations on the Schroedinger equation. We first change variables from

$$\begin{aligned} \left\{ \begin{array}{c} x= |t|^{A} y\\ {\tilde{t}} = t \end{array} \right. ~~\Rightarrow ~~ \left\{ \begin{array}{c} \frac{\partial }{\partial t} = \frac{\partial }{\partial {\tilde{t}}} + \frac{A x}{{\tilde{t}}} \frac{\partial }{\partial x}\\ \frac{\partial }{\partial y} = |{\tilde{t}}|^A \frac{\partial }{\partial x} \end{array} \right. . \end{aligned}$$
(3.32)

Then the \(H_R\) Schroedinger equation becomes

$$\begin{aligned} i \frac{\partial }{\partial {\tilde{t}}} {\hat{\psi }}(x, {\tilde{t}}) = \left( -\frac{1}{2}\partial _x^2 + \frac{1}{2}\omega ^2 x^2 - i \frac{A x}{{\tilde{t}}} \partial _x \right) {\hat{\psi }}(x, {\tilde{t}}) , \end{aligned}$$
(3.33)

with \(\psi _R(y, t) = {\hat{\psi }}(x, {\tilde{t}})\). However this equation is not a Schrodinger equation since the would be Hamiltonian is not Hermitian because of the term \(- i \frac{A x}{{\tilde{t}}} \partial _x\). To get an Hermitian Hamiltonian we redefine \({\hat{\psi }}(x, {\tilde{t}})= |{\tilde{t}}|^{\frac{1}{2}A} \psi _I(x, {\tilde{t}})\). Notice that the factor \(|{\tilde{t}}|^{\frac{1}{2}A}\) is the factor one could expect from the measure due to the change \(x= |t|^{A} y\). We get then the intermediate Schroedinger equation

$$\begin{aligned}{} & {} i \frac{\partial }{\partial {\tilde{t}}} \psi _I(x, {\tilde{t}}) = \left[ \frac{1}{2}\left( -i \partial _x + \frac{A x}{{\tilde{t}}}\right) ^2\right. \nonumber \\{} & {} \quad \left. +\frac{1}{2}\left( \omega ^2 - \frac{A^2}{2 {\tilde{t}}^2} \right) x^2 \right] \psi _I(x, {\tilde{t}}) , \end{aligned}$$
(3.34)

with \(\psi _R(y, t) = |{\tilde{t}}|^{\frac{1}{2}A} \psi _I(x, {\tilde{t}})\). Finally we make a further redefinition as \(\psi _I(x, {\tilde{t}}) = e^{-i \frac{1}{2}\frac{A^2}{{\tilde{t}}^2} x^2} \psi _B(x, {\tilde{t}})\) in order to have a canonical kinetic term. We finally get the desired result

$$\begin{aligned} i \frac{\partial }{\partial {\tilde{t}}} \psi _B(x, {\tilde{t}}) = \left[ - \frac{1}{2}\partial _x^2 + \frac{1}{2}\left( \omega ^2 + \frac{A-A^2}{2 {\tilde{t}}^2} \right) x^2 \right] \psi _B(x, {\tilde{t}}) , \end{aligned}$$
(3.35)

where the relation between \(\psi _R\) and \(\psi _B\) is the one given above in (3.30).

The second method is operatorial. The first step is to use a unitary transformation which implements

$$\begin{aligned} \left\{ \begin{array}{c l} x= |t|^{A} y &{}= U_{R\rightarrow I}^\dagger \, y\, U_{R\rightarrow I}\\ p = \frac{p_y}{|t|^{A}} &{}= U_{R\rightarrow I}^\dagger \, p_y\, U_{R\rightarrow I} \end{array} \right. ~~ \Rightarrow ~~ U_{R\rightarrow I} = e^{ i \ln (|t|^{A})\, \frac{1}{2}\{y, p_y\} } = |t|^{\frac{1}{2}A} |t|^{i A y p_y} . \end{aligned}$$
(3.36)

We then get the intermediate HamiltonianFootnote 3

$$\begin{aligned} H_I&= U_{R\rightarrow I}^\dagger \, H_R\, U_{R\rightarrow I} + i \dot{U}_{R\rightarrow I}^\dagger \, U_{R\rightarrow I}\nonumber \\&= \frac{1}{2 } \left( p_y - \frac{A}{t} y \right) ^2 + \frac{1}{2}\left( \omega ^2 - \frac{A^2}{t^2} \right) y^2 . \end{aligned}$$
(3.37)

With a further unitary transformation

$$\begin{aligned} \left\{ \begin{array}{l} y = U_{I\rightarrow B}^\dagger \, y\, U_{I\rightarrow B}\\ p_y - \frac{A}{t} y = U_{I\rightarrow B}^\dagger \, p_y\, U_{I\rightarrow B} \end{array} \right. ~~ \Rightarrow ~~ U_{I\rightarrow B} = e^{ -i \frac{1}{2}\frac{A}{t} y^2} , \end{aligned}$$
(3.38)

used to make the kinetic term canonical we finally get the desired result. Explicitly

$$\begin{aligned} H_B&= U_{I\rightarrow B}^\dagger \, H_I\, U_{I\rightarrow B} + i \dot{U}_{I\rightarrow B}^\dagger \, U_{I\rightarrow B}\nonumber \\&= \frac{1}{2 } p_y^2 + \frac{1}{2}\left( \omega ^2 + \frac{A-A^2}{t^2} \right) y^2 , \end{aligned}$$
(3.39)

so that

$$\begin{aligned} |\psi _B(t)\rangle = U_{I\rightarrow B}^\dagger U_{R\rightarrow I}^\dagger |\psi _R(t)\rangle , \end{aligned}$$
(3.40)

which again reproduces (3.30).

3.4 Explicit mapping of the quantum \(H_B\) solutions to \(H_R\) solutions

Using the explicit mapping in (3.30) we can write the generating function for a complete set of solutions for \(H_R\) as

$$\begin{aligned}&\sum _{n=0}^\infty \frac{z^n}{\sqrt{n!}} \psi _{R\, n\{t_0\}}(y, t, t_0) \nonumber \\&\quad =\root 4 \of {\frac{1}{2 \pi }} \frac{1}{\sqrt{{{\mathcal {X}}} _R(t)}} e^{i \frac{1}{2} \frac{ {\dot{{{\mathcal {X}}} }}_R(t) }{ {{\mathcal {X}}} _R(t)} x^2 + \frac{1}{{{\mathcal {X}}} _R(t)} x z - \frac{1}{2}\frac{{{\mathcal {X}}} _R^*(t)}{{{\mathcal {X}}} _R(t)} z^2 } , \end{aligned}$$
(3.41)

where we have introduced the complex classical solution \({{\mathcal {X}}} _R(t)= |t|^{-A} {{\mathcal {X}}} (t)\) in analogy to \(y= |t|^{-A} x\). Its e.o.m and normalization condition follow from the \({{\mathcal {X}}} \) ones and read

$$\begin{aligned}&|t|^{-2A} \frac{d}{d t} \left( |t|^{2A} {\dot{{{\mathcal {X}}} }}_R(t) \right) + \omega ^2 {{\mathcal {X}}} _R(t) = 0 ,\nonumber \\&{{\mathcal {X}}} _R^* {\dot{{{\mathcal {X}}} }}_R - {{\mathcal {X}}} _R {\dot{{{\mathcal {X}}} }}_R^* = i |t|^{-2A} . \end{aligned}$$
(3.42)

In particular the “ground state” behaves as

$$\begin{aligned} \psi _{R\,0}(y,t) \sim&|t|^{\frac{1}{2}(A-1)} e^{ i \frac{1}{2}\left( (1-A) sgn(t) |t|^{2A-1} + (2 A-1) \frac{c_0}{c_1} \omega ^{2A-1} |t|^{2(2 A-1)} \right) y^2 }\nonumber \\&\Longleftrightarrow |\psi _{R\,0}(y,t)|^2 \sim |t|^{2A-1} e^{ - (2 A-1) \omega \Im \left( \frac{c_0}{c_1}\right) \omega ^{2A-1} |t|^{2(2 A-1)} y^2 } . \end{aligned}$$
(3.43)

The wave functions always vanish for \(t\rightarrow 0\) while still being normalizable because the classical particle is diffused on the entire y axis since \(y\sim |t|^{-2 A}\). This diverges but the direction depends on the initial \(\dot{y}\) which quantum mechanically cannot be fixed.

4 Interacting quantum and classical mechanical models

We can now pass to exam what happens when we add interactions to the Kasner metrics. The corresponding quantum mechanical models are

$$\begin{aligned} L_R = |t|^{2 A} \left( \frac{1}{2}\dot{y}^2 -\frac{1}{2}\omega ^2 y^2 - \frac{g}{n} y^n \right) ,g>0 , n\in \{4,6,\dots \} ,\nonumber \\ \end{aligned}$$
(4.1)

which become in x coordinate

$$\begin{aligned} L_B = \frac{1}{2}\dot{x}^2 - \frac{1}{2}\left( \omega ^2 + \frac{k}{t^2} \right) x^2 - \frac{g}{n} \frac{1}{ |t|^{A (n-2)} } x^n . \end{aligned}$$
(4.2)

These models show a strange time dependence in the interaction term which can be explained by noticing that the change from y to x in quantum mechanical models cannot be implemented on the metric.

The B models suggest that the interaction is dominant for small \(\omega t\). This is not evident in R models and it is not always true.

Using the results from the previous section on the behavior of the wave function at \(t=0\) we can now see that the perturbative expansion of the evolution matrix in interaction picture does not exist. Since B models are unitarily equivalent to R models as explicitly shown in (3.40) the results we get for B models are valid for R models. Explicitly for B models we get

$$\begin{aligned}&\int d t' \langle \psi _B(t') | H_{B\, S 1}(t') |\psi _B(t')\rangle \nonumber \\&\quad \sim \int d t' \frac{1}{ |t|^{A (n-2)} } \int d x\, x^n\, |t'|^{-\alpha } e^{ - |t'|^{-2\alpha } x^2 }\nonumber \\&\quad \sim \int d t' \frac{1}{ |t'|^{A (n-2)} } \left( \frac{1}{|t'|^{-2\alpha }}\right) ^\frac{n}{2} , \end{aligned}$$
(4.3)

which has an unavoidable divergence for \(A>1\) and \(-\alpha =A-1>0\). More precisely the integral is divergent for \(2A> \frac{n+1}{n-1}\). Anticipating the results (discussed below Eq. (4.10) for the classical case and around Eq. (4.21) for the quantum case) this means that when the behavior is dominated by the interaction, i.e. \(2 A > \frac{n+2}{n-2}\) the integral is divergent. This integral may also be divergent when the theory is dominated by the unbounded time dependent harmonic oscillator (in the appropriate variable), i.e. \(\frac{n+1}{n-1}< 2A < \frac{n+2}{n-2}\) (see Eq. (4.16) and Eq. (4.24)). The results are summarized in Table 1. It is noteworthy that the original Krasner background, which is also a string background admits a good perturbation theory since \(2 A =1\).

Table 1 Summary of results for the interacting models (n even, \(n\ge 4\))
Full size table

4.1 The classical motion

The classical e.o.m for the R models reads

$$\begin{aligned} |t|^{-2 A} \frac{d}{d t} \left( |t|^{2 A} \frac{d y}{d t} \right) + \omega ^2 y + g y^{n-1} =0 . \end{aligned}$$
(4.4)

This equation is very close to the Emden-Fowler equation

$$\begin{aligned} \frac{d}{d t} \left( t^{\mu } \frac{d y}{d t} \right) + t^\nu y^{m} =0 . \end{aligned}$$
(4.5)

This equation is treated in [38] with the result that (with the appropriate range of the parameters \(\mu , \nu \) which can be easily obtained from our treatment) the solution exhibits an oscillating behavior with maxima and minima diverging with a power law. Instead of the analysis presented there we introduce a different approach which is simpler and clearer based on the action. We apply immediately this approach to the R models whose action is

$$\begin{aligned} S_R = \int _I d t\, |t|^{2 A} \left( \frac{1}{2}\dot{y}^2 -\frac{1}{2}\omega ^2 y^2 - \frac{g}{n} y^n \right) , \end{aligned}$$
(4.6)

where I is the integration interval. We look for a change of variables as

$$\begin{aligned} t = sgn({\tilde{t}} ) |{\tilde{t}}|^\beta ,~~~~ y = |{\tilde{t}}|^\alpha z , \end{aligned}$$
(4.7)

so that the kinetic term and the interaction term \(z^n\) have coefficients independent of the new time \({\tilde{t}}\). Explicitly we get

$$\begin{aligned} S_R&= \int _{{\tilde{I}}} d {\tilde{t}}\, \Bigl \{ \frac{1}{2}\frac{1}{\beta } |{\tilde{t}}|^{(2A-1)\beta + 2\alpha +1} \left( \frac{d z}{d {\tilde{t}}} - \frac{\alpha }{{\tilde{t}}} z \right) ^2\nonumber \\&\quad - \frac{1}{2}\beta \omega ^2 |{\tilde{t}}|^{(2A+1)\beta + 2\alpha -1} z^2 - \beta \frac{g}{n} |{\tilde{t}}|^{(2A+1)\beta + n\alpha - 1} z^n \Bigr \} , \end{aligned}$$
(4.8)

where \({\tilde{I}}\) is the image of the interval I. We can now require a time independent kinetic and \(z^n\) term imposing

$$\begin{aligned} (2A-1)\beta + 2\alpha +1=0 ,~~~~ (2A+1)\beta + n\alpha - 1=0 , \end{aligned}$$
(4.9)

which can be solved as

$$\begin{aligned} \alpha = \frac{4 A}{ 2(n-2) A - (n+2)} , \beta = -\frac{n+2}{ 2(n-2) A - (n+2)} ,\nonumber \\ \end{aligned}$$
(4.10)

and get

$$\begin{aligned} S_R&= \int _{{\tilde{I}}} d {\tilde{t}}\, \Bigl \{ \frac{1}{2}\frac{1}{\beta } \left( \frac{d z}{d {\tilde{t}}} - \frac{\alpha }{{\tilde{t}}} z \right) ^2 - \beta \frac{1}{2}\omega ^2 |{\tilde{t}}|^{(2-n)\alpha } z^2 - \beta \frac{1}{n} g z^n \Bigr \} . \end{aligned}$$
(4.11)

The previous action can be recast in a more standard form by integrating by part the term proportional to \(\frac{d z}{d {\tilde{t}}} z = \frac{1}{2}\frac{d z^2}{d {\tilde{t}}}\) to get

$$\begin{aligned} S_R&= \left. +\frac{1}{2}\frac{\alpha }{\beta } \frac{1 }{{\tilde{t}}} z^2 \right| _{{\tilde{I}}}\nonumber \\&\quad + \int _{{\tilde{I}}} d {\tilde{t}}\, \left\{ \frac{1}{2}\frac{1}{\beta } \left( \frac{d z}{d {\tilde{t}}} \right) ^2 + \Bigl [ \frac{1}{2}\frac{ \alpha \left( \alpha + 1 \right) }{\beta } - \beta \frac{1}{2}\omega ^2 \frac{1}{|{\tilde{t}}|^{(n-2)\alpha }} \Bigr ] z^2 - \beta \frac{g}{n} z^n \right\} . \end{aligned}$$
(4.12)

If \(\alpha>0>\beta \), i.e. \(2A>\frac{n+2}{n-2}\) the interval around the singularity \(t=0\) \(I=[-\epsilon _1, +\epsilon _2]\) is mapped into an interval around \(|{\tilde{t}}|=\infty \) as \({\tilde{I}} = [-\infty , -\frac{1}{\epsilon _1}] \cup [\frac{1}{\epsilon _2}, +\infty ]\) then the \(z^2\) terms are subdominant since \(|{\tilde{t}}|^{(2A-1)\beta + 2\alpha +1} = \frac{1}{{\tilde{t}}^2} \) and \(|{\tilde{t}}|^{(2A+1)\beta + 2\alpha -1} = \frac{1}{|{\tilde{t}}|^{(n-2) \alpha }}\). Moreover the boundary term is finite.

Under the previous choice of \(\alpha , \beta \) we can approximate the action \(S_R\) for the I around the singularity simply asFootnote 4

$$\begin{aligned} S_R&\sim \int _{{\tilde{I}}} d {\tilde{t}}\, \left\{ \frac{1}{2}\frac{1}{\beta } \left( \frac{d z}{d {\tilde{t}}} \right) ^2 - \beta \frac{g}{n} z^n \right\} . \end{aligned}$$
(4.13)

Hence the trajectory \(z({\tilde{t}})\) is simply oscillating with period

$$\begin{aligned} \frac{1}{2}P = \frac{1}{\sqrt{2 |\beta | E_z}} \left( \frac{n E_z}{|\beta | g} \right) ^{\frac{1}{n} } \int _{-1}^{+1} d \zeta \frac{1}{\sqrt{1 - \zeta ^n}} , \end{aligned}$$
(4.14)

where \(E_z\) is the system energy.

Despite this nice feature the crossing of the singularity is not very well defined at the classical level since \(t=0^\pm \) is mapped to \({\tilde{t}}= \pm \infty \) and there the particle is spread over the interval \([-\left( \frac{n E_z}{|\beta | g} \right) ^{\frac{1}{n} } , \left( \frac{n E_z}{|\beta | g} \right) ^{\frac{1}{n} } ]\) in z coordinate and it is not obvious how to match the position at \({\tilde{t}}= +\infty \) with the position at \({\tilde{t}}= - \infty \). This is shown in Fig. 3a, b for \(t\rightarrow 0^-\), i.e. for \(\tilde{t}\rightarrow -\infty \). And in a smoother case in Fig. 4a, b.

Fig. 3
figure 3

Classical motion with \(\alpha >0\)

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Fig. 4
figure 4

Another classical motion with \(\alpha >0\) with a smoother behavior

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For the case \(\alpha<0<\beta \), i.e. \(0<2A<\frac{n+2}{n-2}\) the behavior of the classical motion is dictated by

$$\begin{aligned} S_R&\sim \int _{-|{\tilde{\epsilon }}_1|}^{-|{\tilde{\epsilon }}_2|} d {\tilde{t}}\, \frac{1}{\beta } \left\{ \frac{1}{2}\left( \frac{d z}{d {\tilde{t}}} \right) ^2 + \frac{1}{2}\alpha (\alpha +1) \frac{1}{|{\tilde{t}}|^2} z^2 \right\} \nonumber \\&\sim \int _{-|{\tilde{\epsilon }}_1|}^{-|{\tilde{\epsilon }}_2|} d {\tilde{t}}\, \frac{1}{\beta } \left\{ \frac{1}{2}\left( \frac{d z}{d {\tilde{t}}} \right) ^2 + \frac{1}{2}\frac{4A ( 2 n A - (n+2) )}{ ( 2(n-2) A - (n+2) )^2} \frac{1}{|{\tilde{t}}|^2} z^2 \right\} , \end{aligned}$$
(4.15)

because the boundary term does not contribute to the e.o.m we find again a time dependent harmonic oscillator as in Eq. (3.3) but with \(A_{eff}\) (where \(k_{eff}= - \alpha (1+\alpha ) = A_{eff} (1-A_{eff})\), i.e. \(A_{eff}=-\alpha \)) which is always real, explicitly

$$\begin{aligned} \left\{ \begin{array}{c c c} k_{eff}> \frac{1}{4} &{} A_{eff}\in {\mathbb {C}}&{} \text{ not } \text{ possible }\\ 0< k_{eff}< \frac{1}{4} &{} 0\le A_{eff}\le 1 &{} 2 A< \frac{n+2}{n}\\ k_{eff}< 0 &{} A_{eff}>1 &{} \frac{n+2}{n}< 2 A < \frac{n+2}{n-2} \end{array} \right. , \end{aligned}$$
(4.16)

As usual numerics can be tricky and give the wrong impression: compare the Fig. 5a, b with the same solution extended closer to the origin given in Fig. 6a, b. Both for \(t\rightarrow 0^-\), i.e. for \({\tilde{t}}\rightarrow 0^-\)

Fig. 5
figure 5

Classical motion with \(\alpha <0\) with a too short integration range to show the expected behavior

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Fig. 6
figure 6

Classical motion with \(\alpha <0\) with a proper integration range to show the expected behavior

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4.2 The quantum interacting models exist

We can now exam the question of what happens to the quantum model. We treat only the wave function approach because it is more intuitive.

Despite the fact the classical motion is not very well defined the quantum system seems to be perfectly fine and generically better behaved than the non interacting one. The last sentence means that we can write a normalizable wave function which generically vanishes at \(t=0\) but at slower rate that the non interacting, i.e. time dependent quadratic R theory. The adverb generically refers to the fact that there is a “small” range of parameters where system behavior can be mapped to a time dependent harmonic oscillator with unbounded potential.

Another point to stress is that we have found a possible continuation through the singularity it may be that there are other possibilities as in the free case [36].

In order to show that we start with Schroedinger equation for R model

$$\begin{aligned} i \partial _t \psi (y,t) = \left[ -\frac{1}{2}\frac{1}{|t|^{2A}} \partial _y^2 + |t|^{2A} \left( \frac{1}{2}\omega ^2 y^2 + \frac{g}{n} y^n \right) \right] \psi (y,t) , \end{aligned}$$
(4.17)

and following the previous section on the classical motion we perform the same change of variables as in the classic case (4.7)

$$\begin{aligned} \left\{ \begin{array}{c} {\tilde{t}} = sgn(t) |t|^{\frac{1}{\beta }}\\ z= |t|^{-\frac{\alpha }{\beta }} y \end{array} \right. ~~\Rightarrow ~~ \left\{ \begin{array}{c} \frac{\partial }{\partial t} = \frac{|{\tilde{t}}|^{-\beta +1}}{\beta } \left( \frac{\partial }{\partial {\tilde{t}}} - \frac{\alpha z}{{\tilde{t}}} \frac{\partial }{\partial x} \right) \\ \frac{\partial }{\partial y} = |{\tilde{t}}|^{-\alpha } \frac{\partial }{\partial z} \end{array} \right. , \end{aligned}$$
(4.18)

along with setting \(\psi (y,t)= |{\tilde{t}}|^{-\frac{1}{2}\alpha } {\tilde{\psi }}(z,{\tilde{t}})\). The choice of the \({\tilde{t}}\) power is made considering the invariance of the probability density \(|\psi (y,t)|^2 d y = |{\tilde{\psi }}(z,{\tilde{t}})|^2 d z\). The Schroedinger equation then becomes

$$\begin{aligned} i \frac{1}{\beta } \frac{\partial }{\partial {\tilde{t}}} {\tilde{\psi }}(z,{\tilde{t}}) =&-\frac{1}{2}|{\tilde{t}}|^{-(2A-1)\beta -2\alpha -1} \frac{\partial ^2}{ \partial z^2 } {\tilde{\psi }}(z,{\tilde{t}})\nonumber \\&+ \frac{g}{n} |{\tilde{t}}|^{(2A+1)\beta +n\alpha -1} z^n {\tilde{\psi }}(z,{\tilde{t}})\nonumber \\&+ \frac{1}{2}\omega ^2 |{\tilde{t}}|^{(2A+1)\beta +2\alpha -1} z^2 {\tilde{\psi }}(z,{\tilde{t}})\nonumber \\&+i \frac{\alpha }{2 \beta } \frac{1}{{\tilde{t}}} \left( z \frac{\partial }{\partial z} + \frac{\partial }{\partial z} z\right) {\tilde{\psi }}(z,{\tilde{t}}) . \end{aligned}$$
(4.19)

If we require the kinetic and \(z^n\) terms be time independent we get exactly the same solution for \(\alpha , \beta \) as in the classical case (4.10) and the Schroedinger equation becomes

$$\begin{aligned} i \frac{1}{\beta } \frac{\partial }{\partial {\tilde{t}}} {\tilde{\psi }}(z,{\tilde{t}}) =&-\frac{1}{2}\frac{\partial ^2}{ \partial z^2 } {\tilde{\psi }}(z,{\tilde{t}})\nonumber \\&+ \frac{g}{n} z^n {\tilde{\psi }}(z,{\tilde{t}}) + \frac{1}{2}\omega ^2 \frac{1}{|{\tilde{t}}|^{(n-2) \alpha }} z^2 {\tilde{\psi }}(z,{\tilde{t}})\nonumber \\&+ i \frac{\alpha }{2 \beta } \frac{1}{{\tilde{t}}} \left( z \frac{\partial }{\partial z} + \frac{\partial }{\partial z} z\right) {\tilde{\psi }}(z,{\tilde{t}}) , \end{aligned}$$
(4.20)

which is exactly the Schroedinger equation associated with Eq. (4.11).

If \(\alpha>0>\beta \) the \(z^n\) term is dominating for \({\tilde{t}}\rightarrow \pm \infty \) (\(t\rightarrow 0^\pm \)) as in the classical motion then we get a complete set of wave functions as

$$\begin{aligned} \psi _k(y, t) \sim _{t\rightarrow 0} |t|^{ \left| \frac{\alpha }{2 \beta } \right| } \exp \left( - i E_k \beta \frac{sgn(t)}{ |t|^{ \left| \frac{1}{\beta } \right| }} \right) {\tilde{\psi }}_k(z= |t|^{ \left| \frac{\alpha }{\beta } \right| } y) , \end{aligned}$$
(4.21)

where \(E_k\) it the k-th energy eigenvalue of the effective Hamiltonian \(H_{eff}= \frac{1}{2}p_z^2+ \frac{1}{n} g z^n\) and effective time \(t_{eff}=\beta {\tilde{t}}\).

The wave functions are normalizable and vanish for \(t\rightarrow 0\) allowing for a nice and “smooth” crossing of the singularity. The vanishing of the wave function can be again interpreted as the fact that the classical particle is spread over all the possible values of y. Since \(\left| \frac{\alpha }{2 \beta } \right| = \frac{2 A}{n+2}\) the wave functions vanish (generically) slower than the non interacting case and this can be interpreted as the fact that interactions has a better behavior than the non interacting case. Better means that classical interacting particle goes to infinity slower than the free one.

The other case is \(\beta>0>\alpha \) as in the classical motion. In this case the y kinetic term is dominating for \({\tilde{t}}\rightarrow 0^\pm \) (\(t\rightarrow 0^\pm \)). In fact in this limit the Schroedinger equation is

$$\begin{aligned} i \frac{1}{\beta } \frac{\partial }{\partial {\tilde{t}}} {\tilde{\psi }}(z,{\tilde{t}}) \sim&-\frac{1}{2}\frac{\partial ^2}{ \partial z^2 } {\tilde{\psi }}(z,{\tilde{t}}) + i \frac{\alpha }{2 \beta } \frac{1}{{\tilde{t}}} \left( z \frac{\partial }{\partial z} + \frac{\partial }{\partial z} z\right) {\tilde{\psi }}(z,{\tilde{t}}) . \end{aligned}$$
(4.22)

Redefining \({\tilde{\psi }}(z,{\tilde{t}}) = e^{i \frac{1}{2}\frac{\alpha }{\beta } \frac{z^2}{{\tilde{t}}} }\Psi (z,{\tilde{t}})\) we get

$$\begin{aligned} i \frac{1}{\beta } \frac{\partial }{\partial {\tilde{t}}} \Psi (z,{\tilde{t}}) \sim&-\frac{1}{2}\frac{\partial ^2}{ \partial z^2 } \Psi (z,{\tilde{t}}) - \frac{1}{2}\frac{\alpha (\alpha +1) }{ ( \beta {\tilde{t}} )^2} z^2 \Psi (z,{\tilde{t}}) , \end{aligned}$$
(4.23)

which is the Schroedinger equation derived from (4.12) and can be seen as a time dependent harmonic oscillator with \(\Omega _{eff}^2 = - \frac{ \alpha (1+\alpha ) }{ t_{eff}^2 }\) (so that \(A_{eff}=-\alpha \) as in the classical case) and \(t_{eff}=\beta {\tilde{t}}\) and therefore it exists as a theory. In particular we get the leading behavior for the “ground state”

$$\begin{aligned} \psi (y,t)&= | {\tilde{t}}|^{-\frac{1}{2}\alpha } e^{i \frac{1}{2}\frac{\alpha }{\beta } \frac{z^2}{{\tilde{t}}} }\Psi (z,{\tilde{t}})\nonumber \\&\sim |t|^{ - \frac{2\alpha + 1}{\beta } } \exp \left\{ \frac{1}{2}i \left( \frac{2 \alpha + 1}{\beta } \frac{sgn(t)}{ |t|^{ \frac{2\alpha + 1}{\beta } } } \right. \right. \nonumber \\&\quad \left. \left. - \frac{2\alpha + 1}{ |\beta |^{2\alpha +1}} \frac{b_0}{b_1} |t|^{ - 2 \frac{2\alpha + 1}{\beta } } \right) y^2 \right\} , \end{aligned}$$
(4.24)

where \(b_0= c_0 \omega ^{A_{eff}}\) and \(b_1= c_1 \omega ^{1-A_{eff}}\), i.e. we have reabsorbed the \(\omega \) dependence in (3.27) into the coefficients which must therefore satisfy an equation corresponding to (3.25) without \(\omega \). Finally notice that \(\frac{2\alpha + 1}{\beta } = -\frac{ 2 n A - (n+2) }{ n+2 }\) so that when perturbation theory breaks down, i.e. when \(2 A > \frac{n+2}{n-2}\) (\(\alpha <-1\)) the wave function vanishes when \(t\rightarrow 0\) and the potential is unbounded. Notice that the wave function vanishes when \(t\rightarrow 0\) in a wider range of A values, i.e. \(2 A > \frac{n+2}{n}\) (\(\alpha <-\frac{1}{2}\)) but not all of them implies a perturbation theory breakdown because the potential is bounded (\(-1< \alpha <-\frac{1}{2}\)). See Table 1 for a summary of the behaviors.

5 Implications for string theory on temporal orbifolds

All the previous discussion is for the generic Kasner metrics of which the Boost Orbifold is a peculiar case. For the Boost Orbifold where \(A=\frac{1}{2}\) the QFTs considered do not suffer from any breakdown and this is apparently a puzzle because the string on Boost Orbifold has a divergence. The solution of this apparent puzzle is that divergences appear in QFT when higher derivatives interaction terms (induced by massive string states [31]) or non linear sigma model interactions are included.

The reason we did not discuss the quantum mechanical models associated with these QFTs is that either they suffer from Ostrogradskii instability or they are not renormalizable. In any case this is not a limitation since it is easy seen that we suffer of the same issues as the models discussed.

We have then a clear explanation of the origin of the divergences in four point amplitudes as a breakdown of the perturbative expansion. These divergences are also present in three point amplitudes with massive states, i.e. in the lowest order of perturbation theory. The fact that we need to consider the full theory was also partially guessed in [11].

In the full interacting open string at tree level this does not necessarily mean that gravitational backreaction is not going to play any role. In facts in the open string case when solved the issues at tree level it may be well reappear to one loop open string amplitudes. This is however not at all obvious since the previous argument on perturbation theory breakdown applies to closed string as well so the resolution of the issues at the sphere level with three or four punctures could suggest the resolution at the annulus level, i.e. the sphere with two punctures.

Another point worth mentioning is that we have discussed the Boost Orbifold only and not the Null Shift Orbifold. The reason in this case is technical. While for the Boost Orbifold and its generalization the Kasner metric we can reduce the QFT to a quantum mechanical model in the Null Shift Orbifold we can only reduce to a 2d QFT since we need keeping both \(x^\pm \). Nevertheless we expect the same mechanism to be in action for this case too.

An important point which is worth stressing is that divergences are present in Lagrangian approach, i.e. in the covariant one where the time is integrated over but there is no divergence in the light-cone formalism which is Hamiltonian and where the time is not integrated [39]. This is is the same as the previous quantum mechanical models: the Hamiltonian exists but the perturbation theory does not. Finally notice that this can be shown explicitly for the Null Shift Orbifold which is easily quantized on the light-cone [39]. This observation explains also why the matrix model with light like dilatonic profile [40] and the matrix model for the Null Shift Orbifold [41, 42] is well defined.

Since the problem is essentially Lagrangian this is also an issue for Witten string field theory and in general for all the covariant formulations.

So we are left with the issue on how treat this divergences. One possibility is to use the Hamiltonian formalism, for example the light-cone when available. Even if these backgrounds do not possess Poincaré symmetry and the light-cone formalism is well adapted (it is possible to use the light-cone formalism also in other less obvious cases [43]) one could desire to have a covariant formulation in this case too then a possible approach is [14]. Another possibility is to regularize the theory in some way, for example non commutativity can do the job [44].

Finally let us mention that the way of performing the orbifold projection in the temporal orbifold cases used in literature are not on very sound basis since the generators used to write the orbifold projector are dynamical and they change when interactions are switched on. The only clear cut case where this is not the case is the Null Shift Orbifold in light-cone quantization. If we were to use the proper interacting generators there could also be some cancellations which could give raise to finite amplitudes.

6 Conclusions

First of all let us discuss what the previous computations imply for QFT and then shortly for string theory since we have discussed string theory in the previous section.

The first and most important point is that interactions can drastically change the fate of the fields under a Big Crunch/Big Bang.

Secondly what happens seems to depend on the details of the interaction, in the models we studied the power of the interaction \(\phi ^n\) and the value of \(2A =\sum _i {p_{(i)}}\). For certain ranges there is no breakdown, in particular Iit is noteworthy that the original Krasner background, which is also a string background admits a good perturbation.

Thirdly the breakdown of the perturbation theory is a breakdown of Feynman diagram approach, i.e. of the concept of particle. Obviously this happens because of the spacetime region around the singularity and excluding this region, i.e. before and after it the perturbation theory is well defined. Nevertheless this result rises the question of how to treat the S matrix in these backgrounds, in facts the theory exists and spaces are asymptotically flat so we could expect to be able to define some kind of S matrix. Nevertheless it seems that the usual constraints from unitarity must be revisited since near the singularity the concept of particle breaks down.

Finally the previous results seem to point to the importance of minisuperspace approach and pose the question how to extend it to string theory.

For the string theory the main result is that, at least, at the tree level string theory is well for these backgrounds. Whether divergences from backreaction appear at loop level is by now unknown also because we have to find a good way of treating the tree level.