Resonant particle creation by a time-dependent potential in a nonlocal theory

Considering an exactly solvable local quantum theory of a scalar field interacting with a $\delta$-shaped time-dependent potential we calculate the Bogoliubov coefficients analytically and determine the spectrum of created particles. We then show how these considerations, when suitably generalized to a specific nonlocal"infinite-derivative"quantum theory, are impacted by the presence of nonlocality. In this model, nonlocality leads to a significant resonant amplification of certain modes, leaving its imprint not only in the particle spectrum but also in the total number density of created particles.

Considering an exactly solvable local quantum theory of a scalar field interacting with a δ-shaped time-dependent potential we calculate the Bogoliubov coefficients analytically and determine the spectrum of created particles. We then show how these considerations, when suitably generalized to a specific nonlocal "infinite-derivative" quantum theory, are impacted by the presence of nonlocality. In this model, nonlocality leads to a significant resonant amplification of certain modes, leaving its imprint not only in the particle spectrum but also in the total number density of created particles. The effect of quantum particle creation in timedependent backgrounds has many interesting applications in all branches of physics. In particular, it provides a mechanism for the production of density fluctuations during the inflationary stage of the Early Universe [1][2][3], giving rise not only to the present large-scale structures in the Universe but also to the measured cosmic microwave background anisotropies. It is widely expected that gravity theory is to be modified by suitable UV completions at least at the Planck scale, though experimentally this modification is not ruled out for scales less than 50 µm [4]. The existence of a new fundamental scale can have important implications for our understanding of the Early Universe. In the literature there are many different modifications of gravity involving a minimal length scale, see Ref. [5] for a review.
Here we consider a particular class of nonlocal generalizations of relativistic field theories wherein the scale of nonlocality is encoded in a Lorentz invariant form factor depending on the scale of nonlocality ; these theories have recently been studied quite extensively [6][7][8][9][10][11]. Such form factors improve the UV behavior of theories [12] without introducing spurious degrees of freedom, which is why this type of theories are called ghost-free. Nonlocal form factors naturally appear in the context of quantum gravity and can help to resolve cosmological singularities [13,14] as well as black hole singularities [7,15,16].
Here we study imprints of nonlocality on the quantum effect of particle creation. While nonlocality complicates computations of quantum effects considerably, it is nevertheless possible to address some questions by considering an exactly solvable model. Exact solutions provide us with analytical results for scattering amplitudes, and allow the analysis of non-perturbative and non-analytical aspects of these problems. The results obtained for such an idealized model reveal nevertheless robust qualitative effects one can expect in more realistic systems.
A well known problem, both in the non-relativistic and * jboos@wm.edu † vfrolov@ualberta.ca ‡ zelnikov@ualberta.ca the relativistic case, is the quantum-mechanical scattering of a particle on a δ(x)-like potential. The scattering problem on such a potential in the framework of a nonlocal scalar field theory was recently studied in Ref. [17]. Beyond this simple model, in General Relativity the gravitational perturbations on the background of an expanding FLRW universe obey an effective scalar field equation with a time-dependent potential. For this reason a scalar field theory is a good starting point to study the evolution of quantum perturbations of gravitational fields. In the nonlocal case, however, the situation is more complicated and requires further analysis.
In this Letter we hence consider the effect of particle creation by a time-dependent δ(t)-like potential in a nonlocal scalar field theory. Our results provide some intuition of what one can expect in a more realistic setup: We show that nonlocality can lead to a significant resonant amplification of particle creation for some wavelengths defined by the scale of nonlocality and the potential strength λ. This effect is absent for the local theory with the same δ-potential, hence providing an interesting new effect solely due to the presence of nonlocality.
Particle creation: free fields.-Consider D = (d + 1)dimensional Minkowski spacetime and let X µ = (t, x) be its Cartesian coordinates in which the metric reads Consider first a local free massless scalar quantum field ϕ(X) obeying the equation We write this field in the form Here we denote ω k = |k|, the spatial basis as andâ k andâ † k are annihilation and creation operators, respectively, obeying the canonical commutation relations

arXiv:2011.12929v1 [hep-th] 25 Nov 2020
Let us discuss now a nonlocal theory of the scalar massless field which is obtained by substituting the -operator in (2) by the operator The function f which enters this expression is called a form factor and > 0 is the scale of nonlocality. We assume that f (z) does not have zeroes in the complex plane of the complex variable z. In this case, the inverse of the operator (6) does not introduce new poles besides those of the -operator, indicating that such a theory does not have new spurious degrees of freedom (ghosts).
To construct such a theory it is sufficient to choose is an entire function. For example, one can take g(z) to be a polynomial, and in order to facilitate analytical calculations we shall use the form factor f (z) = exp(z 2 ).
In the absence of sources and external potentials, the nonlocal field equation has the same solutions as the local equation (2), which can be written in the form (3), implying that on-shell there is no difference between the models (2) and (7). Time-dependent potential: local case.-Let us consider now a local theory of a scalar field with a time-dependent potential V (t), Its solutions can be expanded in modeŝ In the local case each modeφ k (t) obeys the equation and Ω 2 k (t) = ω 2 k + V (t). Let us assume that the potential V (t) vanishes or becomes very small outside some time interval. In these past and future time domains the frequency Ω k (t) is constant and coincides with ω k .
Denote by ϕ k (t) a complex solution of the equation which has the distant past and future asymptotics The complex coefficients α ω k and β ω k are called Bogoliubov coefficients and can be obtained by solving Eq. (11). Let us emphasize that these coefficients depend on the frequency ω k rather than wave vector k, making them invariant under the reflection k → −k. The Bogoliubov coefficients satisfy the relation Denoting the number of particles in the mode k created by the time-dependent potential as n ω k and the total density of particles as N one has Equation (11) coincides with that of a harmonic oscillator with a time-dependent frequency. If the corresponding quantum oscillator is initially in its ground state, as a result of such parametric excitations it can jump into excited levels, where the expression (14) describes the probability of such a process. Time-dependent potential: nonlocal case.-The nonlocal field equation is of the form where is the scale of nonlocality. We assume again that the potential V (t) vanishes or becomes very small outside some time interval. The effects of nonlocality are controlled by the length scale parameter , such that in the remote past and remote future the asymptotic solutions of Eq. (16) coincide with the solutions of the local equation. Hence one can define creation and annihilation operators in these past and future asymptotic domains.
In order to find a relation between them one can proceed as in the local case. Namely, one can write a solution in the form (9) where now the equation for ϕ k (t)-modes is By solving this nonlocal equation one can extract the coefficients α ω k and β ω k via Eq. (12) by relating the inand out-asymptotics of the solutions, and thereby calculate the number of particles in the mode k created by the time-dependent potential in the nonlocal model. Exactly solvable model: local case.-It is possible to determine the Bogoliubov coefficients analytically in some cases. Let us first demonstrate this in the local theory. In the absence of the time-dependent potential, the retarded Green function One can use it to obtain the Lippmann-Schwinger integral representation [18] for the solution of Eq. (11), Here, ϕ 0 k (t) is a free solution which satisfies [∂ 2 t + ω 2 k ]ϕ 0 k (t) = 0, and we may choose it to correspond to the asymptotic past of Eq. (12). In the special case of a δ-shaped potential, V (t) = λδ(t), this integral collapses: where ϕ 0 Using Eq. (12) we can read off the Bogoliubov coefficients The spectral density and the total number density of particles are given by the expressions For d ≥ 2 the number density of created particles N formally diverges at large momenta k, which is due to the infinitely small width of the δ-potential. For potentials of finite time duration τ , there appears an effective cut-off k max ∼ 1/τ at high momenta [19] and the number density of created particles is indeed finite. Exactly solvable model: nonlocal case.-Let us now consider a nonlocal model and study how the presence of nonlocality affects the Bogoliubov coefficients and, ultimately, the particle spectrum as well as total number density of produced particles. In the limiting case of → 0 one recovers the local theory, but for > 0 the above differential operator can lead to nonlocal behavior [20][21][22][23]. The corresponding nonlocal retarded Green function, in the absence of the potential, is a solution of It can be decomposed as It has been shown that at late and early times ∆G k vanishes and one has [21] G k t→±∞ = G R k (t) .
Using the nonlocal retarded Green function we can express a solution of Eq. (16) as Here, ϕ 0 k (t) is again a free solution, and due to (28) we may choose it identical to the local case. For the δ-shaped potential we then find Now substitute here t = 0 and solve the resulting algebraic equation for ϕ k (0) in terms of ϕ 0 k (0), yielding Using the future asymptotic Eq. (12) as well as (28) we can read off the Bogoliubov coefficients at t → ∞, The number of particles in the mode k created by the time-dependent potential is given by Recall that, as in the local case, the total number of created particles diverges at large momenta. However, the integral over the momenta is truncated at high frequencies k max ∼ 1/τ if the potential has a finite width (duration of time) τ . However, if one solely considers the nonlocal contribution to the particle number density the UV cutoff problem does not appear at all. Resonant particle creation.-As a new nonlocal effect, the denominator in (34) may vanish. At this wavelength some kind of resonant particle creation takes place, leading to a huge amplification of modes with the wave vector k that satisfies the condition This condition defines a resonant frequency ω = |k |. Inserting t = 0 into Eq. (27) one can show [20] ∆G where we defined the dimensionless wave number κ = k . Introducing the dimensionless coupling Λ = λ , the resonance condition (35) takes the simple form F = 1/Λ with F = ∆G k (0)/ . It has a solution κ provided At very high frequencies, the nonlocal modification decreases and hence the density of created particles asymptotically approaches the local theory for those high frequencies. For Λ → Λ crit the resonant nonlocal amplification happens at When Λ → ∞ it corresponds to While the local particle spectrum is invariant under the transformation λ → −λ, this is no longer the case in the nonlocal theory. In fact, the resonant amplification occurs solely for positive λ ≥ Λ crit / . At the level of the local Bogoliubov coefficients, a transformation λ → −λ merely induces a phase shift of π in the far future modes. This is no longer true in the nonlocal theory, and could be related to the fact that nonlocality smears the phases of incoming plane waves in vicinity of the source.
For wave numbers close to the resonance k the density of created particles (34) diverges as ∼ (k − k ) −2 and their integral number diverges. This can be the result of idealization of the potential as a δ-function, which necessarily includes infinitely high wave numbers. In a realistic system the potential has a finite width, and for such a potential of the characteristic width τ only the modes with k < k max ∼ 1/τ are created, which means that one can expect resonant particle number amplification for potentials satisfying the approximate condition (λ ) −1/3 < τ / < 1. It is natural to expect for realistic localized potentials that the width of the resonant band will be finite and the total number density of created particles also will be regularized.
Discussion.-We computed the rate of scalar particle creation in the framework of ghost-free nonlocal field theory in the presence of a time-dependent potential. Our particular model of nonlocality was dictated by absence of instabilities in this model [24] as well as because of its analytically known Green function, allowing for an exact study of the problem including non-perturbative effects.
We found a new, nonlocal effect: there appears a new feature of resonant particle creation at some frequency ω = κ / . This effect depends on the sign and the amplitude λ of the δ(t)-potential. In the limit when the scale of nonlocality tends to zero, we naturally recover the predictions for particle creation at all frequencies except a very narrow peak at high frequencies ω , which grows with decreasing . In other words, in the local limit the resonant frequency is shifted to infinitely high frequencies.
We believe that qualitative effects found in our exactly solvable model are robust and are applicable to a wide variety of more realistic nonlocal theories. An interesting question is: Can the nonlocal parametric resonant amplification effect, discussed in this Letter, manifest itself as a potentially observable imprint on the spectrum of the primordial perturbations in early inflationary cosmology and be tested in observations? We shall leave this question for future work.