One-loop analysis with nonlocal boundary conditions

TL;DR

This paper performs a detailed one-loop analysis of quantum mechanical models with nonlocal boundary conditions, revealing that the zeta(0) value matches that of Robin boundary conditions, and discusses potential applications in quantum field theory and gravity.

Contribution

It introduces a detailed technique for one-loop calculations with nonlocal boundary conditions, applicable to quantum field theory and gravity.

Findings

zeta(0) matches Robin boundary conditions

technique applicable to quantum field theory

potential use in quantum gravity

Abstract

In the eighties, Schroder studied a quantum mechanical model where the stationary states of Schrodinger's equation obey nonlocal boundary conditions on a circle in the plane. For such a problem, we perform a detailed one-loop calculation for three choices of the kernel characterizing the nonlocal boundary conditions. In such cases, the zeta(0) value is found to coincide with the one resulting from Robin boundary conditions. The detailed technique here developed may be useful for studying one-loop properties of quantum field theory and quantum gravity if nonlocal boundary conditions are imposed.

Equations

$$B_R\equiv \left\{x,y:x^2+y^2\le R^2\right\},$$

$$q_R(x)\equiv \frac{1}{2\pi R}\sum _{l=-\infty }^{\infty }{e}^{ilx/R}{\int }_{-\infty }^{\infty }{e}^{-ily/R}q(y)dy,$$

$$-\left(\frac{{\partial }^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{{\partial }^2}{\partial {\phi }^2}\right)u=Eu,\forall r<R,$$

$$\frac{\partial u}{\partial r}+R{\int }_{-\pi }^{\pi }{q}_R(R(\phi -\theta ))u(R,\theta )d\theta =0.$$

$$u_{l,E}(r,\phi )=J_l(r\sqrt{E})e^{il\phi },l\in Z,$$

$$G_l(h)\equiv h{J}_l^{\prime }(h)+R\tilde{q}\left(\frac{l}{R}\right)J_l(h)=0,$$

$$\tilde{q}\left(\frac{l}{R}\right)={\int }_{-\infty }^{\infty }q(x)e^{-i\frac{l}{R}x}dx.$$

$${\zeta }_A(s)\equiv {\mathrm{Tr}}_{L^2}(A^{-s})=\sum _n{\lambda }_n^{-s},$$

$${\beta }_l\equiv R\tilde{q}\left(\frac{l}{R}\right).$$

$$K(t)={\int }_{B_R}K(\xi ,\xi ;t)={\mathrm{Tr}}_{L^2}{e}^{-tA},$$

$$K(t)\sim \sum _{n=0}^{\infty }{B}_n{t}^{\frac{n}{2}-1}.$$

$$\zeta (s,h^2)\equiv \sum _n({\lambda }_n+h^2{)}^{-s},$$

$${\int }_0^{\infty }t{e}^{-h^{2}t}K(t)dt=\Gamma (2)\zeta (2,h^2).$$

$$\Gamma (2)\zeta (2,h^2)\sim \sum _{n=0}^{\infty }{B}_n\Gamma \left(1+\frac{n}{2}\right)h^{-n-2}.$$

$$\Gamma(2)\zeta(2,h^{2})=\sum_{l}^{\infty}(-N_{l})\left({1\over 2h}{d\over dh}\right)^{2}\log\Bigr{[}(ih)^{-l}G_{l}(ih)\Bigr{]},$$

$$\zeta (s)$$

$$J_{-l}(z)=(-1{)}^l{J}_l(z),$$

$$G_{-l}(z)=(-1{)}^l{G}_l(z).$$

$$\sum _l^{\infty }=2\sum _{l=1}^{\infty }+\mathrm{contribution}\mathrm{of}(l=0)=2\sum _{l=0}^{\infty }-\mathrm{contribution}\mathrm{of}(l=0).$$

$$\displaystyle\log\Bigr{[}(ih)^{-l}G_{l}(ih)\Bigr{]}\sim C-l\log(l+\alpha_{l})+{1\over 2}\log(\alpha_{l})+\alpha_{l}$$

$$\tau \equiv \frac{l}{{\alpha }_l}=\frac{l}{\sqrt{l^2+h^2}},$$

$$u_0(\tau )=1,u_1(\tau )=\frac{\tau }{8}-\frac{5}{24}{\tau }^3,$$

$$u_2(\tau )=\frac{9}{128}{\tau }^2-\frac{77}{192}{\tau }^4+\frac{385}{1152}{\tau }^6,$$

$$v_0(\tau )=1,v_1(\tau )=-\frac{3}{8}\tau +\frac{7}{24}{\tau }^3,$$

$$v_2(\tau )=-\frac{15}{128}{\tau }^2+\frac{99}{192}{\tau }^4-\frac{455}{1152}{\tau }^6,$$

$$v_3(\tau )=-\frac{105}{1024}{\tau }^3+\frac{5577}{5120}{\tau }^5-\frac{6545}{3072}{\tau }^7+\frac{95095}{82944}{\tau }^9,$$

$$a_k(\tau )=\frac{u_k(\tau )}{{\tau }^k},b_k(\tau )=\frac{v_k(\tau )}{{\tau }^k},$$

$$\frac{u_k}{l^k}=\frac{a_k}{({\alpha }_l{)}^k},\frac{v_k}{l^k}=\frac{b_k}{({\alpha }_l{)}^k}.$$

$$\frac{1}{2h}\frac{d{\alpha }_l}{dh}=\frac{1}{2{\alpha }_l},$$

$$\left({1\over 2h}{d\over dh}\right)^{2}\log(l+\alpha_{l})=-{1\over 4}(l+\alpha_{l})^{-2}\Bigr{[}2\alpha_{l}^{-2}+l\alpha_{l}^{-3}\Bigr{]},$$

Full text

I Introduction

In the late eighties, motivated by physical models of Bose condensation and mathematical study of Schrödinger operators, the work in Ref. Schroder studied spectral properties of the Laplace operator with nonlocal boundary conditions. Within this framework, on considering the region

[TABLE]

one builds, out of a function $q$ which is both Lebesgue summable and square-integrable on the real line, the periodic function

[TABLE]

which has period $2\pi R$ and approaches $q$ if $R\rightarrow\infty$. On using polar coordinates $(r,\varphi)$, the nonlocal boundary-value problem studied in Ref. Schroder reads as

[TABLE]

[TABLE]

The resulting spectrum has both a positive ($E>0$) and a negative ($E<0$) part. For $E>0$, the solutions regular at the origin $r=0$ are factorized in the form

[TABLE]

the $J_{l}$ being Bessel functions of first kind. On defining the dimensionless variable $h\equiv R\sqrt{E}$, Eq. (1.4) takes eventually the form Schroder

[TABLE]

where ${\tilde{q}}$ is the Fourier transform of $q$, i.e. [1]

[TABLE]

We note from (1.6) that ${\tilde{q}}$ must have dimension ${\rm length}^{-1}$, and hence $q$ must have dimension ${\rm length}^{-2}$.

In the present paper, we have tried to work out the one-loop properties pertaining to the problem defined by Eqs. (1.3) and (1.4). In the physics-oriented literature, one-loop calculations are more frequently performed in the case of quantum field theories, but the quantum mechanical framework is already of interest Barvinsky , and may provide valuable information on the behaviour of solutions of elliptic equations under a scale dilation. Such a property is neatly described by the $\zeta(0)$ value, where $\zeta$ is the spectral (or generalized) $\zeta$-function of the elliptic operator $A$ under consideration, defined by

[TABLE]

where each eigenvalue $\lambda_{n}$ is counted with its degeneracy. For this purpose, section $2$ outlines the analytic technique of Ref. Moss and its application to our boundary-value problem (1.3) and (1.4). Section $3$ evaluates the first $3$ sums that contribute to the $\zeta(0)$ value, whereas section $4$ studies the fourth and last sum contributing to $\zeta(0)$, for three choices of the function $q$ and hence of the dimensionless coefficients $\beta_{l}$ from Eqs. (1.6) and (1.7), i.e.

[TABLE]

Concluding remarks are presented in section $5$, whereas relevant details are given in the appendix.

II $\zeta(0)$ value from the $\zeta$-function at large $h$

For the Laplacian $A$ on the left-hand side of Eq. (1.3), the associated heat equation has a heat kernel $K(\xi,\eta;t)$, whose diagonal $K(\xi,\xi;t)$ yields, upon integration over the whole region $B_{R}$ in (1.1), the integrated heat kernel (for gauge theories, the trace to be integrated is instead the fibre trace of the heat-kernel diagonal)

[TABLE]

which has, as $t\rightarrow 0^{+}$, the asymptotic expansion Moss ; Esposito98 ; Gilkey

[TABLE]

In our $2$-dimensional region $B_{R}$, the method used in Ref. Moss considers the so-called spectral $\zeta$-function at large $h$, i.e. ($\lambda_{n}$ and $h$ being dimensionless in (2.3))

[TABLE]

which is related to the integrated heat kernel (2.1) by the identity

[TABLE]

If one now inserts into the left-hand side of (2.4) the asymptotic expansion (2.2), one finds

[TABLE]

On the other hand, on considering the equation (1.6), which is the equation obeyed by the eigenvalues $E={h^{2}\over R^{2}}$ by virtue of the boundary conditions, one has also the identity (see Moss and our appendix)

[TABLE]

where the degeneracy $N_{l}$ equals $2$, as is clear from (1.5) (for each value of $l$, there exist $2$ linearly independent eigenfunctions proportional to $\cos(l\varphi)$ and $\sin(l\varphi)$, respectively). In light of (2.5) and (2.6), the desired $\zeta(0)$ is the coefficient of $h^{-4}$ in the asymptotic expansion of the right-hand side of Eq. (2.6), because $\zeta(0)=B_{2}$ from the formulae [4]

[TABLE]

In the course of performing sums over all positive and negative values of $l$, it is helpful to exploit the identity

[TABLE]

as well as the even nature of $\beta_{l}$ as a function of $l$ (see section $4$). This implies that a real root of $G_{l}$ with positive $l$ is also a real root of $G_{l}$ with negative $l$, because

[TABLE]

We can therefore limit ourselves to summing over positive values of $l$, writing that, in (2.6),

[TABLE]

Following Ref. Moss , we use in (2.6) and (2.10) the uniform asymptotic expansion of $J_{l}$ and its first derivative $J_{l}^{\prime}$, which involve the polynomials $u_{k}$ and $v_{k}$ occurring below and in the appendix. On denoting by $C$ a constant, and defining $\alpha_{l}(ih)\equiv\sqrt{l^{2}+h^{2}}$, we obtain

[TABLE]

where, having defined the variable

[TABLE]

and exploiting the following among the many Debye-Olver polynomials Olver :

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

the polynomials $a_{1},a_{2},b_{1},b_{2},b_{3}$ are evaluated according to the definition

[TABLE]

which implies the simple but very helpful relations

[TABLE]

III Contributions independent of $\beta_{l}$

By virtue of (2.6) and (2.11) the three contributions independent of $\beta_{l}$ are obtained by applying twice the operator ${1\over 2h}{d\over dh}$ to the first line on the right-hand side of (2.11). For this purpose, we need the following identities:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus, upon applying the split (2.10), the terms independent of $\beta_{l}$ are obtained by taking twice (from the factor $2$ multiplying $\sum_{l=0}^{\infty}$ in (2.10)) the following sums:

[TABLE]

[TABLE]

[TABLE]

III.1 Contribution of $\sigma_{1}$

The sums (3.5)-(3.7) can be studied in a thorough way with the help of the Euler-Maclaurin summation formula Wong . This states that, if $f$ is a real- or complex-valued function defined on $[0,\infty)$, and if its derivatives of even order are absolutely integrable on $(0,\infty)$, one has, for $n=1,2,...$,

[TABLE]

where the Bernoulli numbers ${\tilde{B}}_{s}$ are defined by the expansion

[TABLE]

while the remainder $R_{m}(n)$ is majorized according to Wong

[TABLE]

As $n$ approaches $\infty$, Eq. (3.8) provides a very useful asymptotic expansion for the desired sum of the series, i.e.

[TABLE]

The integral on the right-hand side of (3.11) can be evaluated or studied in a qualitative way, while the derivatives of odd order at [math] and at $\infty$ can be obtained in a systematic way. We refer the reader to the last chapter of the book by Hardy Hardy for a thorough analysis of the Euler-Maclaurin formula.

For our purposes, after having re-expressed $\sigma_{1}$ in the form

[TABLE]

where we have set

[TABLE]

[TABLE]

we now take the limit as $n\rightarrow\infty$ in Eq. (3.8), with $f(l)$ replaced by $F(l;h)+H(l;h)$. We then find that, in (3.11), only the first derivative of $F$ at $l=0$ gives a contribution proportional to $h^{-4}$, and indeed equal to

[TABLE]

III.2 Contributions of $\sigma_{2}$ and $\sigma_{3}$

We now rely again upon the limit as $n\rightarrow\infty$ of Eq. (3.8). By virtue of (3.11), only half the value at $l=0$ of ${1\over 2}\alpha_{l}^{-4}$ contributes to the $h^{-4}$ term in $\sigma_{2}$, i.e.

[TABLE]

whereas $\sigma_{3}$ gives a vanishing contribution to the term proprtional to $h^{-4}$

[TABLE]

IV Contribution from the $\beta_{l}$ coefficients

We now aim at studying the contribution of the second line of the asymptotic expansion (2.11) to Eq. (2.6). For this purpose, on the one hand we denote by $\Omega$ all terms added to $1$ within the square brackets in (2.11), and consider the asymptotic expansion

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

On the other hand, it is clear that no further progress can be made without explicit forms of the $\beta_{l}$ coefficients. For example, we find from (1.7) and (1.9)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here we consider first the choice of $\beta_{l}$ in Eq. (4.6), and exploit the formulae (2.6), (2.11)-(3.1) and (4.1)-(4.4), arriving therefore at the sums (see details below)

[TABLE]

[TABLE]

supplemented, in principle, by infinitely many other terms, i.e.

[TABLE]

In the formula (4.10), it is helpful to use (4.3) where we re-express $a_{1},b_{1}$ and $b_{2}$ in the form

[TABLE]

where the numerical coefficients $a_{1r},b_{1r}$ and $b_{2r}$ can be read off from (2.13), (2.15) and (2.16). Thus, a patient calculation shows that (the superscript $(l)$ denotes here dependence on $l$)

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and hence, by repeated application of (3.1), we obtain from (4.10) and (4.13)

[TABLE]

IV.1 Effect of Eq. (4.9)

In Eq. (4.9), by virtue of the remarkable formula Moss

[TABLE]

$\Gamma$ being the standard notation for the $\Gamma$-function, we find

[TABLE]

a result which agrees with the application of Eq. (3.11). Moreover, the asymptotic expansion (3.11) implies that the first sum on the right-hand side of (4.9) is equal to

[TABLE]

where, on defining $Y\equiv{y\over h}$, we find

[TABLE]

It is clear once more that $h$ plays the role of regularizing parameter, since without it the integral (4.21), and many of the integrals below, would not exist. Last, the third sum on the right-hand side of (4.9) is again studied with the help of (3.11), and we find

[TABLE]

where

[TABLE]

and hence no contribution to $h^{-4}$ arises from (4.22) at large $h$.

IV.2 Contribution of (4.17) and $\zeta(0)$ value

In Eq. (4.17), by virtue of (4.18), the sums

[TABLE]

do not contribute to $h^{-4}$, while Eq. (3.11) tells us that

[TABLE]

and hence (4.24) does not contribute to $h^{-4}$ either. Furthermore, the last two sums on the right-hand side of (4.17), which contain the effect of $\beta_{l}$, with the particular choice (4.6) for this coefficient are found to involve

[TABLE]

[TABLE]

none of which contains terms proportional to $h^{-4}$ at large h.

The contributions to $h^{-4}$ arising from all terms in (4.11) are found to vanish with the same procedure just adopted in studying all terms in Eq. (4.17), and hence we find from the second line of (2.11) a contribution to $h^{-4}$ equal to

[TABLE]

Eventually, we obtain from Eqs. (2.10), (3.15)-(3.17) and (4.27)

[TABLE]

where $-{1\over 2}$ is the term denoted in (2.10) by minus the contribution of ($l=0$), and arises from $\sigma_{2}$ in (3.6).

IV.3 Other choices of $\beta_{l}$

For a generic choice of $\beta_{l}$ coefficient, our Eq. (4.9) gets replaced by

[TABLE]

If $\beta_{l}$ is taken in the form (4.5), we find, by virtue of (3.11), the asymptotic expansion ($K_{n}$ being the standard notation for modified Bessel functions of second kind and order $n$)

[TABLE]

which has no term proportional to $h^{-4}$, while $\beta_{l}$ in the form (4.8) leads to

[TABLE]

and the integral on the right-hand side of (4.31) does not have a term proportional to $h^{-4}$ at large $h$, either. Moreover, the last two sums in (4.17) suggest introducing the notation

[TABLE]

[TABLE]

Now we find, for the two choices of $\beta_{l}$ according to (4.5) or (4.8),

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Among the integrals occurring in (4.34)-(4.37), only those on the right-hand side of (4.34) might give rise to a contribution proportional to $h^{-4}$, because

[TABLE]

However, the constant coefficients within square brackets in (4.38) add up to zero, so that no term proportional to $h^{-4}$ actually occurs at large $h$. Such a kind of integral may be studied with the help of complex integration, because

[TABLE]

This suggests considering the following integral:

[TABLE]

where $z=\rho e^{i\theta}$, and $\gamma$ is a rectangle with a side $\gamma_{1}$ given by the closed interval $[-\rho,\rho]$ on the real line, while the other three sides have equation ($\varepsilon$ being a positive parameter approaching [math], and we take $h$ positive as well)

[TABLE]

[TABLE]

[TABLE]

The resulting integrand $\varphi(z)$ has a third-order pole at $z=i{h\over 2}$, with residue

[TABLE]

which is one of the three terms occurring in (4.38).

V Concluding remarks

As far as we know, our paper has performed the first $\zeta(0)$ calculation with nonlocal boundary conditions in quantum mechanics. We have proved explicitly that at least three choices of kernel in the nonlocal boundary operator exist (see (4.5), (4.6) and (4.8)) for which the $\zeta(0)$ value coincides with the value resulting from local boundary conditions of the Robin type. Our $\zeta(0)$ value does not describe a one-loop conformal anomaly, as it would be the case in quantum field theory, but it remains relevant for the understanding of scaling properties of the quantum Hamiltonian operator.

It remains to be seen whether, for yet other choices of $\beta_{l}$ in (4.17) and (4.29), a contribution to $\zeta(0)$ can be found which is compatible with (1.7), (1.9), the assumption $q\in L_{1}({\bf R})\cap L_{2}({\bf R})$ and condition (A7) of the Appendix. This means having to study the sums

[TABLE]

The mere recourse to the formula Moss

[TABLE]

suggests a negative answer, because for example, upon requiring

[TABLE]

one finds the positive root

[TABLE]

which has a growth rate incompatible with (1.7) and (1.9), if one looks for functions $q\in L_{1}({\bf R})\cap L_{2}({\bf R})$. However, the general starting point should be the asymptotic expansion inspired by (3.11), i.e.

[TABLE]

and the application of (5.3) to (5.1) deserves further work.

Furthermore, with the help of the experience gained from our calculation, it should be possible in the near future to investigate the one-loop properties of Euclidean quantum gravity with nonlocal boundary conditions, along the lines of Refs. Esposito99a ; Esposito99b , where it was suggested that the Universe might become classical, after a quantum origin, by virtue of a wave function that decays as it occurs in the case of quantum mechanical surface states Schroder with nonlocal boundary conditions. In order to help the general reader and stress the relevance of our work, we find it also appropriate to write what follows.

The use of spectral $\zeta$-functions has led to several important developments over the last decades, with application to partition functions of strings and membranes, Casimir effect, relation between the generalized Pauli-Villars and covariant regularizations, spontaneous compactification in two-dimensional quantum gravity, stability of the rigid membrane, topological symmetry breaking in self-interacting theories, functional determinants for bosonic and fermionic fields, ground-state energy under the influence of external fields, Bose-Einstein condensation of ideal Bose gases under external conditions book1 ; book2 ; book3 . Furthermore, the work in Ref. JMAPA obtained heat-kernel coefficients of the Laplace operator on the $D$-dimensional ball, Ref. PHRVA evaluated Casimir energies for massive fields in the bag, while the Casimir energy for a massive fermionic quantum field with a spherical boundary was obtained in Ref. JPAGB . The regularization used in our paper, which relies as we said on the pioneering work in Ref. Moss , was applied successfully in Ref. localD to the first correct calculation of one-loop conformal anomaly for a massless fermionic field with local boundary conditions, at a time when the powerful geometric formulas in Ref. Gilkey were not in final form. It is therefore encouraging that, after almost thirty years, such a regularization technique is still useful in opening yet new perspectives. For example, it would be interesting to apply it to the Casimir effect book3 in cosmological backgrounds, and to establish a correspondence with yet other models in Ref. book3 . For the gravitational field, one cannot generalize the Schröder scheme by simply studying the eigenvalue problem for an operator of Laplace type on metric perturbations $h_{\mu\nu}$, subject to nonlocal boundary conditions. The reason is that boundary conditions invariant under infinitesimal diffeomorphisms on $h_{\mu\nu}$ take, in field-theoretic language, the form CMPHA

[TABLE]

[TABLE]

where $\Pi$ is a projection operator that picks out only spatial components of $h_{\mu\nu}$, while $\Phi_{\mu}(h)$ is the gauge-fixing functional. Thus, nonlocality in the boundary conditions can only result from the gauge-fixing term; but then the invertible operator on metric perturbations is no longer differential but it belongs to the broader class of pseudodifferential operators Esposito99a ; Esposito99b , for which a $\zeta(0)$ calculation is still a challenging task. Thus, new exciting goals are in sight in the area of physical applications of spectral $\zeta$-functions, bearing also in mind the enlightening assessment in Ref. Witten .

Acknowledgements.

The authors are grateful to Dipartimento di Fisica “Ettore Pancini” for hospitality and support.

Appendix A Concepts and formulas from complex analysis

We here summarize some concepts and results of complex analysis that are applied in our paper.

If $F$ is an entire function with a countable infinity of zeros $\left\{\mu_{i}\right\}$, the genus of the canonical product for $F$ Ahlfors is the minimum integer $n$ such that

[TABLE]

converges. In particular, if the genus is equal to $1$, this ensures that one can write, for some constant $\gamma$,

[TABLE]

where the function $g$ is entire. In particular, the function (1.6) occurring in the nonlocal boundary condition is an entire function of genus $1$. Such a property can be checked by pointing out that the zeros of $G_{l}(h)$ at large $h$ are given approximately by

[TABLE]

because for fixed $l$, as $h\rightarrow\infty$, one has

[TABLE]

[TABLE]

and hence

[TABLE]

which is independent of $\beta_{l}$.

We further recall that, if $M(r)$ is the maximum of $F(z)$ on $|z|=r$, the order of the entire function $F$ is defined to be Boas

[TABLE]

An example of an entire function of order $n$ is $e^{z^{n}}$. Our function $G_{l}$ in (1.6) can be also studied from the point of view of its order, as defined in (A6), and its relation with the genus Boas .

Moreover, we can apply to Eq. (1.6) a theorem studied, among the others, by Watson Watson , according to which, if $A$ and $B$ are real and $\nu>-1$, the function $AJ_{\nu}+BzJ_{\nu}^{\prime}$ has all its zeros real, except that it has $2$ purely imaginary zeros when ${A\over B}+\nu<0$. This implies that our $G_{l}(h)$ has only real roots if

[TABLE]

This condition is fulfilled by all forms (4.5)-(4.8) of $\beta_{l}$ considered in our paper.

The uniform asymptotic expansions of Bessel functions and their first derivative are due to Debye and Olver Olver , and are used extensively in Ref. Moss and in our paper. They read as

[TABLE]

[TABLE]

where (see also (2.13)-(2.19))

[TABLE]

[TABLE]

and, in particular,

[TABLE]

[TABLE]