Particle creation and energy conditions for a quantized scalar field in thepresence of an external, time-dependent, Mamev-Trunov potential.

Particle creation and energy conditions for a quantized scalar field in the
presence of an external, time-dependent, Mamev-Trunov potential.

Michael J. Pfenning Michael.Pfenning@usma.edu Department of Physics, United States Military Academy,West Point, New York, 10996-1790, USA
July 14, 2019
Abstract

We study the behavior of a massless, quantized, scalar field on a two-dimensional cylinder spacetime as it responds to the time-dependent evolution of a Mamev-Trunov potential of the form . We begin by constructing mode solutions to the classical Klein-Gordon-Fock equation with potential on the whole spacetime. For a given eigen-mode solution of the IN region of the spacetime (), we determine its evolution into the OUT region () through the use of a Fourier decomposition in terms of the OUT region eigen-modes. The classical system is then second quantized in the canonical quantization scheme. On the OUT region, there is a unitarily equivalent representation of the quantized field in terms of the OUT region eigen-modes, including zero-frequency modes which we also quantize in a manner which allows for their interpretation as particles in the typical sense. After determining the Bogolubov coefficients between the two representations, we study the production of quanta out of the vacuum when the potential turns off. We find that the number of “particles” created on the OUT region is finite for the standard modes, and with the usual ambiguity in the number of particles created in the zero frequency modes. We then look at the renormalized expectation value of the stress-energy-tensor on the IN and OUT regions for the IN vacuum state. We find that the resulting stress-tensor can violate the null, weak, strong, and dominant energy conditions because the standard Casimir energy-density of the cylinder spacetime is negative. Finally, we show that the same stress-tensor satisfies a quantum inequality on the OUT region.

pacs:
04.62.+v, 03.50.-z, 03.70.+k, 11.25.Hf, 14.80.-j

I Introduction

i.1 Quantum Inequalities

In quantum field theory (QFT), it is well known that the renormalized expectation value of the energy-density operator for a free, quantized field can be negative. Epstein, Glaser, and Jaffe Epstein et al. (1965) demonstrated this to be a generic property of QFTs under relatively weak assumptions. Also, negative energies seem to be a generic property for the vacuum state of a QFT in many curved spacetimes, and additionally, for the vacuum state in both flat and curved spacetimes with boundaries. The effect of a nonzero value for the renormalized expectation value of the vacuum state is often referred to as vacuum polarization, vacuum energy, zero-point energy, or equivalently, as the Casimir energy Casimir (1948). The study of this phenomenon has driven extensive research throughout the later half of the twentieth century which has continued into the twenty-first.

Negative energy densities also occur for multi-particle states where interference terms arise in the expectation value of the stress-energy tensor which have sufficient magnitude to overpower any positive constant positive terms. (See Pfenning (1998) for an easy example and Ford (1978) for a thorough discussion.) It was noted by Ford Ford (1978), that unrestrained negative energies can be used to violate the second law of thermodynamics. In the same paper, he argues that no such breakdown would occur in two dimensions if a negative-energy flux obeys an inequality of the form , where is the duration over which the flux occurs. In a subsequent paper Ford (1991), Ford was able to derive such an inequality constraining negative-energy fluxes directly from QFT which applies to all possible quantum states for the massless Klein-Gordon scalar field in flat spacetimes. In particular, if the flux is smeared in time against a normalized Lorentzian sampling function of characteristic width , then, in two dimensions

(1)

and in four dimensions

(2)

A few years later, Ford and Roman Ford and Roman (1995) extended this analysis to the energy-density observed along the worldline of a geodesic. Their analysis begins with with the derivation of a difference quantum inequality on the two-dimensional, spatially-compactified, cylinder spacetime (). Consider a timelike geodesic parameterized by proper time , whose tangent vector is denoted by . Letting be an arbitrary quantum state and be the Casimir vacuum state on the cylinder spacetime, they define the difference in the expectation value of the energy-density between these states as

(3)

On their own, each of the two terms in the difference are formally divergent, but both have the same singular structure, thus the difference is finite. (This is a typical “regularization” process employed in QFT.)

In the specific case of an inertial observer, and again using a Lorentzian weighting function with characteristic width , they derive the lower bound

(4)

The important thing to note, which is true for most all forms of inequalities, is that the lower bound is on the difference between the expectation values between two different states. The difference quantum inequality can be converted to bounds on the renormalized value of the energy-density by noting that

(5)

Thus, an absolute quantum inequality takes the form

(6)

where we have made use of the time independence and symmetry properties of the renormalized stress-tensor in the Casimir vacuum state on the cylinder spacetime.

In the same paper, Ford and Roman go on to derive a quantum inequality in four-dimensional Minkowski spacetime;

(7)

Here, the colons denote normal ordering with respect to the standard Minkowski space vacuum; in other words, it is again a lower bound on the difference between expectation values between two states.

Since their initial discovery, quantum inequalities have been developed for an assortment of QFTs, both massless and massive, in a variety of spacetimes, both flat and curved. Additionally, they have been proven for a large class of weighting function beyond just the Lorenzian; first by Flanagan Flanagan (1997) for the scalar field in two-dimensional Minkowski spacetime, and followed by Fewster and Eveson Fewster and Eveson (1998) for the massive scalar field in -dimensional Minkowski spacetime. For example, let be a smooth, strictly positive function on the real line with unit area under the curve (one can relax the unit area condition), then Flanagan’s lower bound states

(8)

In the same paper, Flanagan also derives a spatial quantum inequality with nearly identical form,

(9)

Both bounds are derived in the rest frame of an inertial observer and are the optimal lower bound over all possible states. The colons again means normal ordering with respect to the Minkowski vacuum state.

Significant improvements in the mathematical rigor for the derivation of quantum inequalities were made by Fewster Fewster (2000) by employing microlocal analysis in the context of algebraic QFT in curved spacetime. The pairing quantum inequality now serves as an umbrella term, of which the the most frequently studied type is the quantum weak energy inequality (QWEI), which typically takes the form Fewster (2000); Fewster and Pfenning (2003)

(10)

Here, is a smooth compactly-supported test function, and are Hadamard states, and the colon with the subscript denotes normal ordering with respect to , thus these are again a form of difference inequality, with serving as the reference state. Finally, the functional is independent of the state , and microlocal analysis is used to prove that it is finite. These can again be recast in terms of the renormalized expectations values, resulting in

(11)

In applications, it is commonplace to take the reference state to be the Casimir vacuum state, although this is by no means a requirement.

i.2 Claims of Violations of Quantum Inequalities

In two recent papers, Solomon Solomon (2011, 2012) puts forth models of a massless, quantized, scalar field in two-dimensional Minkowski spacetime with the presence of an external, time-dependent potential of the form . Here is the standard Heaviside unit-step-function and is the coupling constant between the potential to the field. The scalar field obeys the Klein-Gordon-Fock wave equation

(12)

Such models can be interpreted as a quantum field which transitions from a field interacting with the potential to being a free field at the Cauchy surface. We call the causal past and future of this Cauchy surface the IN and OUT regions, respectively.

The classical wave equation associated with the equation above can be solved independently in both regions using standard PDE techniques for the potentials chosen by Solomon. For the IN region, one assumes harmonic time dependence, such that positive-frequency modes are given by

(13)

where the ’s are a complete set of orthonormalized eigenfunctions to the equation

(14)

and is a label for uniquely identifying an eigenfunction.

The transition across the Cauchy surface is then handled by assuming continuity conditions in time, i.e.,

(15)

From a physical standpoint, this is reasonable; one evolves a solution to the wave equation with potential up to the Cauchy surface, at which point and serve as the Cauchy data for the continued evolution of the wave into the causal future of the Cauchy surface. In two-dimensional Minkowski spacetime, where Solomon is working, the future evolution is easily determined using d’Alembert’s solution to the wave equation,

(16)

Thus, we have mode solutions to the classical wave equation on the whole spacetime of the form

(17)

Using canonical quantization, one can then lift the general solution to the classical wave equation to a self-adjoint operator,

(18)

where is an appropriate measure for the labeling set of the ’s, and are the standard creation and annihilation operators, respectively, with the usual commutation relations, and we use the standard QFT Fock space on which these operators act. In particular, the IN vacuum state is defined such that for all .

The stress-tensor operator associated with the quantized scalar field for the Klein-Gordon-Fock equation can be separated into two parts,

(19)

where, using the terminology of Solomon, the kinetic-tensor is defined as the portion of the stress-tensor that is explicitly free of the potential, i.e.,

(20)

while the potential-tensor is everything in the stress-tensor explicitly involving the potential, i.e.,

(21)

It is important to note that the support of the kinetic-tensor is the whole spacetime, while the support of the potential-tensor is restricted to the support of the potential. Thus, if the support of the potential is closed or compact, then the support of the potential-tensor will be closed or compact. Solomon’s kinetic energy-density is just the component of the kinetic-tensor111Solomon uses the letter to represent both the stress-tensor and the kinetic-tensor. We choose the alternate notation of and to avoid any unintended confusion between them.. In regions of space where the potential vanishes, the stress-tensor is equal to the kinetic-tensor. Because of the potential, all three of the tensors defined above have nontrivial traces and nontrivial divergences222The traces are given by , , and , where is the dimension of the spacetime. The divergences are , and . .

In his papers, Solomon calculates the expectation value of the kinetic energy-density for the IN vacuum state on the IN region, finding

(22)

where the prime denotes differentiation of the function with respect to the argument. After a lengthy calculation, the expectation value of the energy-density for the IN vacuum state on the OUT region is

(23)

where the basis eigenfunctions are chosen to be real valued333If the basis of eigenfunctions is not real valued, then the expression for the energy-density in the OUT region would be . In the regions of the spacetime where the potential is zero, Solomon conjectures that we may use any of the standard renormalizations schemes to determine the renormalized values of both of these expressions. Thus, if there is a stationary Casimir effect due to the potential in any portion of the IN region of the spacetime, this will become a left and right moving pulse of energy on the OUT region of the spacetime. For example, one model that Solomon presents is that of a double-delta-function potential of the form

(24)

for which Mamev and Trunov Mamaev and Trunov (1981) have shown that there is a constant, negative-valued, Casimir effect for the vacuum expectation value of the energy-density in the region of space between the two delta-functions and vanishing outside. Explicitly,

(25)

where is a positive function of the coupling constant and separation given by

(26)

Mamev and Trunov are silent on what the renormalized expectation value of the energy-density is at the locations of the delta-function potentials () . They do state in Mamev and Trunov (1982) that additional renormalization terms are required that depend on the potential and its derivatives to determine .

Solomon uses the Mamev and Trunov double-delta-function potential on the IN region of his spacetime, which he rewrites as

(27)

For the OUT region of the spacetime, Solomon then posits

(28)

As was the case with Mamev and Trunov, Solomon is silent about the value of the renormalized kinetic-tensor on the IN region at , and consequently for the renormalized stress-tensor at points along the future-pointing null rays emanating from the points on the OUT region. Solomon then goes on to show that this particular expression for the vacuum expectation value of the energy-density would indeed violate the quantum inequalities of Flannagan Flanagan (1997) on the OUT region of the spacetime.

Unfortunately, Solomon’s conclusions are incorrect, as Eqs. (27) and (28) are incomplete expressions for both the IN-region kinetic energy-density and the OUT-region energy-density, respectively. In the case of static Minkowski spacetime with a time-independent double-delta-function potential, it has been shown by Graham and colleagues Graham et al. (2002), in the context of a massive scalar field, that the renormalized energy-density has nonzero contributions at the points . Unfortunately, there is no straightforward way to take the limit of the Graham et. al. results and then separate the renormalized kinetic energy-density out of the expression for the renormalized energy-density.

However, for the massless field we can conjecture that the renormalized IN-region kinetic-tensor will have a -component of the form

(29)

where is another function of the coupling constant and separation . This yields an OUT-region energy-density of the form

(30)

Physically, this describes two square-wave pulses of negative energy with amplitude traveling outward at the speed of light from the initial location of the potential; one moving to the left and one moving to the right. Additionally, on the leading and trailing edges of the square-wave pulses are delta-function spikes of energy, with magnitude which, as we will see below for a related model, are positive. The positive energy comes from the creation of particles out of the vacuum by the quantum field in response to the shutting off of the potential.

Using this new expression for the renormalized energy-density, we can again consider Flanagan’s quantum inequality on the OUT region. To do this, we use unit-area test functions with the constraint that they only have support on the OUT region of the spacetime. Then, substituting the above energy-density into the quantum inequality, and using a geodesic parameterized by , where and , results in

(31)

To determine if the quantum inequality is violated will depend on the relative strength of the delta-function contributions to the negative-energy contribution of the square wave part of the energy-density.

We will put off definitively settling whether or not Flanagan’s quantum inequality is violated for a follow-up paper. Instead, for the remainder of this paper, we determine the renormalized kinetic-tensor on the IN region and the renormalized stress-tensor on the OUT region of a the two-dimensional cylinder spacetime with a single delta-function potential that is abruptly shut off at . We find that particle creation in our model causes a left- and right-moving delta-function of positive energy in the OUT region stress-tensor. We also show that all of the classical point-wise energy conditions fail on this spacetime because of a negative-energy Casimir effect, but that the positive-energy pulses are sufficiently large to ensure that the quantum inequality for this spacetime is satisfied for all inertial observers on the OUT region of the spacetime, and for all values of the coupling constant .

i.3 Outline

We begin by considering a massless, quantized, scalar field coupled to a scalar potential on the spatially-compact, two-dimensional, cylinder spacetime . We use the standard coordinates with the identification of points such that . Here, is the circumference of the spatial sections of the universe and we use the standard metric . The choice of spacetime is made such that the mathematics which follows is tractable. Similar calculations could be performed in other spacetimes.

The quantized scalar field obeys the Klein-Gordon-Fock wave equation, Eq. (12), with a Mamev-Trunov-type potential of the form

(32)

where is a positive coupling constant and is the Dirac-delta-function. The factor of 2 is included solely for convenience. The potential is a delta-function of strength that is abruptly turned off at time .

The Mamev-Trunov-type potential breaks the spacetime into two regions: a static IN region for where the scalar field is coupled to a non-zero delta-function potential, and a static OUT region for where the scalar field is free from the potential. A graphical representation of this spacetime with the potential is presented in Fig. 1.

In Sect. II, we determine the mode solutions to the Klein-Gordon-Fock wave equation for both regions. It is advantageous to separate the modes based on their spatial symmetry/antisymmetry properties about . On the whole spacetime, both the IN and OUT regions, there exists a complete set of antisymmetric, orthonormal, positive-frequency, modes solutions of the form

(33)

with and . There are also negative-frequency antisymmetric mode solutions given by the complex conjugate. Because these modes vanish at the origin, they do not experience or interact with the potential.

There also exist symmetric, orthonormal, positive-frequency, mode solutions to the wave equation, which are sensitive to the potential, of the form

(34)

The IN portion of this mode solution is given by the expression

(35)

where , is the -th positive root of the transcendental equation

(36)

and is a normalization constant defined in Eq. (62) below. The IN portion of the modes solutions have a corner at the location of the delta-function potential, while the OUT portion have corners that propagate outward from the origin of the spacetime at the speed of light.

The OUT portion of the symmetric mode solution is given by a Fourier series, Eq. (92), in terms of the “standard” basis of symmetric modes for the potential-free Klein-Gordon equation, of which there are two kinds: a) an infinite family of time-oscillatory mode solutions, with the positive-frequency solutions given by

(37)

where , and the negative-frequency solutions given by the complex conjugate, and b) two topological, zero-frequency, mode solutions given by

(38)

and its complex conjugate. The topological modes exist because the spatial sections of the cylinder spacetime are compact. Furthermore, they are necessary to have a complete basis set to represent a solution to the Cauchy problem for all initial data. The antisymmetric, mode solutions, given by Eq. (33), do not appear in the Fourier representation for the OUT portion of the symmetric mode solutions on the whole spacetime.

To determine the Fourier coefficients for the OUT portion of the symmetric mode solution, we require, like Solommon, continuity (in time) of the complete mode solution across the Cauchy surface. Essentially, we are using the known behavior of the IN symmetric mode functions at time as the Cauchy data to determine a unique solution of the wave equation in the OUT region. The resulting Fourier series has non-zero Fourier coefficients, Eqs. (86) and (88), for both the topological modes and the positive and negative-frequency even mode solutions. Thus, the initially positive-frequency even mode solution on the IN region of the spacetime develops both positive and negative-frequency components at the moment that the potential turns off which persist through the OUT region.

In Sec. III, we second-quantize our system, following the standard canonical quantization scheme in literature (see, for example, Birrell and Davies Birrell and Davies (1982)). In this scheme, one promotes the real-valued classical field to a self-adjoint operator on a Hilbert space of states. The typical Hilbert space is usually given by a standard Fock space. For a Bosonic field theory, the field operator and its conjugate momenta also satisfy a standard set of equal time commutation relations. This process works well for our spacetime because it has a convenient timelike Killing vector.

On the IN region of the spacetime, the Fock space associated with the field algebra has the usual form, and we define the IN vacuum state to be the state destroyed by all of the annihilation operators of the field algebra, Eq. (101). The subscripted is included in the notation to remind us that this is the ground state on a spatially-closed spacetime of circumference , and not the standard Minkowski-space vacuum state, which we will denote by . States with higher particle content can be constructed in the usual way by acting with the creation operators.

On the OUT region of the spacetime, there exists an unitarily equivalent field algebra based upon the “standard” mode solutions to the potential-free wave equation. So we also present the second-quantization of this equivalent system. However, we do make one modification to the standard quantization procedure; along with the time-oscillatory modes, we also second-quantize the topological modes using the method developed by Ford and Pathinayake Ford and Pathinayake (1989). At the classical level, the topological modes given by Eq. (38) have nonzero conjugate momenta, therefore they can be included in the classical symplectic form that gets lifted to the commutator relation of the field algebra. It is found that such a process produces an algebra with a non-trivial center Dappiaggi and Lang (2012).

Because the OUT region had two equivalent field algebras and Fock spaces, we determine the Bogolubov transformation between the elements of the algebras. Since the OUT portion of the symmetric mode solutions is already given by a Fourier series in terms of the “standard” modes, determining the explicit form of the Bogolubov coefficients is simply a task of identifying the correct Fourier coefficient.

Working in the Heisenberg picture, we then calculate the number of “standard” quanta created on the OUT region of the spacetime for the IN vacuum state . We find that (a) no quanta are created in the odd modes, (b) a finite, non-zero number of quanta are created in the topological modes, Eq. (119), (c) a finite, non-zero number of quanta are created in the time-oscillatory even modes, Eq. (120), and (d) the total number of quanta created is finite. All the quanta created in this model come into existence at the moment the potential is shut off, i.e., at .

In Sec. IV, we determine the renormalized expectation value of the stress-tensor for the IN ground state on both the IN and OUT regions of the spacetime. For the IN region of the spacetime we find

(39)

which holds everywhere except at the location of the delta-function potential. The part of this expression is the standard Casimir energy-density for the cylinder spacetime. The is the correction to the ground state energy-density due to the presence of the potential. Here, both coefficients and are positive functions of , given by infinite summations over the transcendental eigenvalues, Eqs. (140) and (154) respectively, and are plotted in Fig. 2. We explicitly prove that both are convergent, and we determine that the difference between them always satisfies

(40)

We also determine the renormalized expectation of the stress-tensor on the OUT region for the same state;

(41)

which holds for all spacetime locations to the future of the Cauchy surface. It is covariantly conserved, i.e., , and we find the standard Casimir energy-density for the cylinder spacetime followed by a correction to the ground state energy-density given by the term. The remaining terms in the above expression are the contributions to the stress-tensor due to the quanta excited (i.e. particle creation) from the shutting off of the potential. The remarkably simple expression of two classical, point-like particles moving outward from the origin to the left and right with equal amplitude is the result of a very detailed analysis of the properties of the Bogolubov coefficients and identities, and their application to the very complicated expression for the “moving” parts of the stress-tensor given by the Fourier series in Eq. (142).

In Sec. V, we evaluate the energy conditions from general relativity on the OUT region of the spacetime, using the expression above for the renormalized stress-tensor. For a timelike geodesic worldline, the renormalized expectation value of the energy-density is given by Eq. (195), and for a null geodesic worldline by Eq. (200). We find that the null energy condition (NEC), weak energy condition (WEC), the strong energy condition (SEC), and the dominant energy condition (DEC) all fail on some region of the space-time for the OUT-region stress-tensor because the difference , and is therefore insufficiently large to overcome the usual term of the Casimir energy. We then calculate the total energy in a constant-time Cauchy surface on the OUT region, finding

(42)

We note that the total energy is a constant, independent of time, further indicating that the renormalized stress-tensor on the OUT region is conserved for all time . Additionally, because of the dependance of on the value of , the total energy in the Cauchy surface is negative for values of , positive for values of , and it passes through zero somewhere in the range .

In the final part of Sec. V, we use our normal-ordered expectation value of the energy-density for the IN vacuum state on the OUT region in a QWEI for the two-dimensional cylinder spacetime without potential, given by

(43)

where is any Hadamard state on the cylinder spacetime, and is a smooth, real-valued, compactly-supported test function on real line. The derivation of this QWEI, with the inclusion of the topological modes, is contained in Appendix E. We can use this inequality on the OUT region of our spacetime if we restrict the set of test functions to only those which have compact support to the future of the Cauchy surface.

Evaluating the left-hand side of the inequality for the state yields

(44)

Notice that only the first of the four terms in the result for the left-hand side is negative, and that it is identical to the first term on the right-hand side of the QWEI. The remaining terms on the right-hand side are negative. Thus, the QWEI is satisfied by the stress-energy tensor of the IN vacuum state on the OUT region of the spacetime for all allowed test functions with support to the future of the Cauchy surface, and for all values of .

The main body of the paper concludes with some comments and conjectures in Sec. VI. In addition to the main body, there are five appendices containing technical information necessary for the paper to be complete, and to which we refer throughout the document. The appendices include: a proof of the equivalence of the IN and OUT region mode functions on a bow-tie shaped domain surrounding the Cauchy surface; the construction of the advanced-minus-retarded Green’s function on the cylinder spacetime when topological modes are included; the convergence and properties of certain summations over the eigenvalues of the transcendental equation; notes on an alternative way to determine the IN vacuum stress-tensor on the IN region and why it fails; and finally the derivation of the QWEI on the cylinder spacetime.

i.4 Mathematical Notation

We use units in which , and are set to unity throughout the paper. The complex conjugate of a complex number is denoted by , and similarly for functions. For complex-valued functions and , we use the standard L inner-product,

(45)

The normalization for mode solutions of the wave equation is chosen such that the modes are pseudo-orthonormal with respect to the standard bilinear product used in QFT Birrell and Davies (1982),

(46)

Operators will be typeset in bold face to distinguish then from variables and functions. The Hermitian conjugate of an operator will be denoted by .

We define the Fourier transform on a Schwartz class function , the space of smooth functions that decay at infinity, as

(47)

Since the Fourier transform is an automorphism on Schwartz class functions, we have that the inverse Fourier transform is

(48)

For this choice of definition of the Fourier transform, the convolution theorem states

(49)

which has as a corollary Parseval’s theorem,

(50)

Ii The Classical Formalism

Let be an -dimensional, globally hyperbolic, Lorentzian spacetime with smooth metric of signature . On this spacetime we have a real-valued scalar field , which interacts with a scalar potential . This situation is described by the action

(51)

where is the spacetime metric, , is the inverse of the metric, and is the partial derivative. Variation of the action with respect to the scalar field yields the standard Klein-Gordon-Fock wave equation

(52)

or, more succinctly, . Similarly, the stress-tensor is found by varying the action with respect to the inverse-metric. When considered with the gravitational action Wald (1984), the stress-tensor for minimal coupling has the form

(53)

We now make a two choices so that the mathematics which follows is more tractable. First, we choose to work on the the standard two-dimensional cylinder spacetime . This is done for two reasons: a) the spactime is boundaryless so there are no boundary conditions to consider, and b) the spectrum of the Laplace operator on , with and without the potential, is discreet. We use the standard Minkowski space coordinates with the identification of points such that . Here, is the circumference of the spatial sections of the universe. Secondly, on this spacetime we have a scalar, Mamev-Trunov-type potential Mamev and Trunov (1982) given by Eq. (32).

The classical mode functions to the wave equation can be solved for independently in both regions. To determine mode functions on the whole spacetime, we take each mode function from the IN region and require that the function and its first derivative match across the Cauchy surface to a general Fourier decomposition of the wave function in the OUT region, i.e. we require continuity in of the wave functions. This matching is used to determine the Fourier coefficients for the OUT solution of the wave solution. We now present the details of this process.

ii.1 Mode Solutions on the IN Region,

For our spacetime, and upon substitution of the potential, the wave equation for the IN region is

(54)

Using the standard techniques for separation of variables, we assume a solution of the form , such that the time dependence solves

(55)

while the space dependence leads to the Schrödinger-like equation

(56)

Here, is the separation constant, playing a role akin to the energy in ordinary quantum mechanics. The operator

(57)

is Hermitian, i.e., , with respect to the standard inner product on .

The spatial sections of the universe are compact, therefore the eigenvalues are discrete. Furthermore, the eigenvalues are real-valued and greater than or equal to zero. A convenient -orthonormalized basis of eigenfunction to Eq. (56) is given by (a) a family of antisymmetric eigenfunctions,

(58)

where , and , and (b) a family of symmetric eigenfunctions,

(59)

where , , and is the -th positive root of the transcendental equation

(60)

For any value of , the value of lays in the interval between and . For , the values of the ’s approach the poles of the cotangent function from above. A fairly good approximation for using the first two terms in the Taylor series of the cotangent function is

(61)

where . Strictly speaking, the exact value of is always less that the value of the approximation above.

The normalization constant for the symmetric eigenfunctions is

(62)

There do not exist any eigenfunctions with eigenvalue .

From the above -eigenfunctions, we can define positive-frequency mode solutions to the wave equation on the IN-region:

(63)

and

(64)

The normalization for these mode solutions has been chosen such that the modes are orthonormal with respect to the standard bilinear product used in QFT, Eq. (46). Negative-frequency mode solutions are given by the complex conjugate of the above expressions.

ii.2 Mode Solutions on the OUT Region,

The OUT region is simply the spacetime with no potential, i.e., it is the standard cylinder spacetime. Assuming a solution of the form , we find that the time dependence again solves Eq. (55), while the space dependence leads to

(65)

Here, is again the separation constant. The eigenvalues and eigenfunctions to the spatial equation are well known; There are (a) antisymmetric eigenfunctions

(66)

(b) symmetric eigenfunctions

(67)

and (c) a zero-eigenvalue topological solution

(68)

Both the symmetric and antisymmetric eigenfunctions have with . A generic function on the circle can be represented as a Fourier series in this basis as

(69)

where , and are all Fourier coefficients. In particular, the Dirac -function on has the representation

(70)

The positive-frequency mode solutions to the wave equation on the OUT region for the antisymmetric and symmetric eigenfunctions are simply

(71)

and

(72)

respectively. The negative-frequency solutions are given by the complex conjugate of the above expressions. The topological eigenfunction leads to an often neglected solution of the wave equation,

(73)

where is an arbitrary constant that sets a length scale Ford and Pathinayake (1989). Unlike the time oscillatory solutions, the topological solution is not an eigenfunction of the energy operator . The complex conjugate of the topological solution is also a linearly independent solution of the wave equation. All three types of solutions are orthonormal with respect to the bilinear product Eq. (46), i.e., they satisfy

(74)

where the labels and specify both the type of mode and the value of .

A generic, complex-valued, classical solution to the wave equation in the OUT region is given by the Fourier series

(75)

where , , , , , and are complex-valued constants.

ii.3 Mode Solutions on the Whole Spacetime

Next, we determine mode solutions on the whole of the spacetime for the time-dependent potential. Let be any solution to the wave equation on the IN region. We know that a general solution in the OUT region is given by Eq. (75) above. At the Cauchy surface where the potential abruptly turns off, we require continuity of the wave function and its first time derivative, i.e.,

(76)

Upon substitution, we find

(77)

and