Lecture Notes in Physics. New Series m: Monographs. m35. Berlin: Springer-Verlag. xii, 224 p. DM 62.00; öS 452.60; sFr 60.00 (1995).

The name “zeta-function regularization” has become known within broader circles of theoretical physicists than those who actually use this elegant technique. Most users are quantum field theorists who need to excise the ultraviolet divergences from their formulae. Zeta-function regularization is known as one of the better tools for doing this. There exist other ultraviolet regularization methods: the cutoff method and dimensional regularization to name just two. It is fair to say these other methods are not “deep” in a mathematical sense. However the zeta function method does directly involve very deep and beautiful mathematics. This is the first thing to understand when learning about zeta-function methods in physics.
Roughly half of the present book is what its title indicates: a collection of physical applications of the zeta-function method. These examples are chosen to highlight the method. The other half of the book is devoted to technique. The mechanics of zeta-function regularization forms the central theme of this book. The author makes clear at the outset his goal of bringing the reader to a certain level of proficiency with the method. He chooses to do this within important self-imposed restrictions. One limitation (quite appropriate for an introductory book) is the choice of spectra: only known spectra are considered -- i.e. one-dimensional and multidimensional spectra constructed explicitly from the integers. This excludes general zeta-function theory but includes many interesting and important spectra, ones likely to interest the reader. He presents a substantial body of technique for these spectra culled from various original sources. In some of the ten applications an effort is made to explain the underlying physics. In others there is little or no explanation. This is neither a book on physics nor a book on field theory. All applications reviewed in it are {\it global} in nature. There is no analysis of local quantities such as the vacuum stress tensor of the quantum field; no discussion of field nonuniformity caused by any of the backgrounds considered.
Before going on to a more detailed discussion of the book’s subject and contents it seems a good idea to display -- for the benefit of possible nonspecialist readers -- some of the basic mathematical structure of quantum field theory (hereafter: QFT).
Perturbation continuum QFT on a spacetime manifold ${\cal M}$ begins with a fundamental wave equation including boundary conditions for each field $\widehat {\phi}$ of the theory (or possibly a Schrödinger-type equation for $\widehat {\phi}$). The field theorist uses this equation to select a complete orthonormal set of modes $\{\phi_n \}$ for $\widehat {\phi}$ from all possible functions on ${\cal M}$. Associated with this set of modes is an infinite spectrum $\{ \lambda_n \}$. For space-times on which global momentum is a useful concept $\lambda_n$ is constructed from the momentum vector of the mode $\phi_n$. To have tractable mathematics it is essential that $\lambda_n\to \infty$ as label $n\to \infty$ with no infinite degeneracies along the way, and that $\lambda_n$ grows with $n$ sufficiently rapidly that spectral series like $$Z(s) \equiv \sum_n \lambda_n^{-s}= \int_{\cal M} \,dx Z(s|x,x), \qquad \text{Re}\, s> B> 0, \tag 1$$ and the bilocal version of this series $$Z(s|x,y) \equiv \sum_n \lambda_n^{-s} \phi_n (x) \overline {\phi} (y) \qquad \aligned &x,y\in {\cal M},\\ &x=y, \text{Re}\,s> B> 0,\\ &x\ne y \text{ all }s\endaligned \tag 2$$ with $x=y$ converge uniformly for $\text{Re}\,s>B> 0$. (Here we view $s$ as complex. The well-defined real number $B$ depends on the dimension of spacetime.) However these series always diverge for $\text{Re}\,s<B$ and this is precisely where the ultraviolet (UV) divergences of QFT originate. Physical quantities are expressed by the theory in terms of divergent sums. An excellent example is the stress-energy-momentum tensor of $\widehat {\phi}$ which is expressed in terms of the series $Z(s|x,x)$ and other series obtained from (2) by differentiation -- with $s$ values $s\leq B$.
Nearly 50 years ago it was discovered that, for a class of QFT’s called “renormalizable”, it is possible to perform a systematic and physically-motivated subtraction of all UV divergences throughout the entire theory. These divergences can be pushed into vacuum renormalization and into certain physical quantities: the masses and couplings. The latter thereby assume their “renormalized” final physical values -- the values accessible to physical measurement. In this way physicists grew accustomed to viewing the infinities in renormalizable theories as being troublesome but nonpathological. When one has reached this point it becomes natural to detach the idea of renormalization from physics and regard it as an abstract mathematical device -- a prescription for assigning finite values to divergent spectral series and integrals. Mathematicians have more-or-less separately evolved their own rules for doing similar things.
The underlying reason why zeta-function regularization is relevant for physics has already been given. QFT lies within a broad area of mathematics one might call “spectral mathematics”. Any well-posed spectral problem yields spectral sums like (1), (2) which (a) converge for $\text{Re } s>B$ with $B$ finite and (b) can be analytically continued into $\text{Re } s<B$. Neither of these properties is to be taken for granted, and both are utterly crucial for the manageability of QFT and for renormalization. If the series (1), (2) converge nowhere in the $s$ plane there would be no possibility of analytic continuation. One would confront quite unstructured and unmanageable mathematics. However QFT and its spectral mathematics is very highly structured, sufficiently so to make regularization meaningful.
Consider $Z(s)$ and $Z(s|x,x)$ in (1), (2) as typical quantum functions in a field theory problem. Both functions converge uniformly in $\text{Re } s>B$, and both are therefore analytic in this half plane. Continuation into $\text{Re } s<B$ reveals these functions are meromorphic in $s$, with their rightmost poles on the real axis at $s=B$ and additional poles on the real axis to the left of $s=B$. There is, however, never a pole at $s=0$. The latter property enables one to write down the most-quoted and used formula in all of analytic regularization: $$\text{ln det } \widehat {A}= \ln \prod_n \lambda_n= \sum_n \ln \lambda_n=- {\textstyle {d\over ds}} Z(s) |_{s=0}+ \ln \mu Z(0). \tag 3$$ Here (i) $\{\lambda_n \}$ is the eigenvalue spectrum of the operator $\widehat {A}$; (ii) $Z(s) \equiv \sum_n (\mu/ \lambda_n)^s$ and (iii) $\mu$ is an arbitrary dimensional parameter having the same physical dimension as $\lambda_n$. (This parameter must be introduced to make the argument $\mu/ \lambda_n$ of $(\ )^s$ dimensionless. This should also have been done in (1), (2). In this way the arbitrary dimensional constants inseparable from UV renormalization enter the zeta-function mathematics.) Because both terms in the final line of (3) are finite, this formula assigns a finite, unique value to the divergent quantity $\text{ln det } \widehat {A}$.
Many people identify zeta-function regularization with (3). This is somewhat unfortunate because the zeta-function method is far more broad and interesting. Elizalde with his various examples and developed theory brings this out clearly. One should add that zeta-function regularization even in its broadest sense lies within a larger, more general approach to spectral divergent series and integrals based on the idea of analytic continuation. There exist infinitely many (weighted) continuous spectra which have associated meromorphic functions which are not zeta functions. In QFT one finds use of the analytic continuation idea for the regularization of divergent Feynman diagrams already in the early 1960’s -- a method called “analytic regularization” in those days. This is perhaps the best name for the entire approach to divergent series and integrals based on analytic continuation.
Returning to $Z(s)$, $Z(s|x,x)$ let us ask what it means for the defining series (1), (2) to be continued into the half-plane $\text{Re } s<B$. It means these series have been summed to explicit functions of $s$, or reorganized into sums or integrals over explicit functions of $s$, or in some other way have had their $s$ dependence explicitly revealed. This is the essential calculational step in zeta-function regularization. Most of the technique reviewed by Elizalde is precisely for accomplishing this step. Once $Z(s)$, $Z(s|x,x)$ are known meromorphic functions of $s$ a rather powerful (and fun) mathematical tool has been crafted. Look at what interesting things can be done with it.
Consider the “simple” example of the Riemann zeta-function $$\zeta (s)\equiv \sum^\infty_{m=1} m^{-s}, \qquad \text{Re}\,s>1. \tag 4$$ Suppose one wishes to know the value of the divergent series $$\align \zeta (-1) &= 1+ 2+ 3+ \cdots\\ &=\text{undefined using numerical summation} \tag 5\\ &=-{1\over 12} \text{ using analytic continuation}. \endalign$$ There seem to be only two things one can do with a series like $1+ 2+ 3+ \cdots$. One may decline to see any meaning at all in it. Alternatively one may interpret this series as the meromorphic function $\zeta (s)$ evaluated at $s=-1$ which has the known value $\zeta (-1)= -1/12$. Similar things can be said about $\zeta (-2)= 1^2+ 2^2+ 3^2+ \cdots =0$ and about any other special value of $\zeta (s)$ with $\text{Re}\, s<1$. These simple examples contain the entire idea of zeta-function regularization. Shall we perceive the series $\zeta (-1)= 1+ 2+ 3+ \cdots$ to be a meaningless expression, or shall we perceive this divergent series to be the name (this is meant literally) of the meromorphic function $\zeta (s)$ evaluated at $s= -1$? It is the mathematical properties of the function $\zeta (s)$ that matter, not the mathematical properties of the name $1^{-s}+ 2^{-s}+ \dots$ of the function. The latter perspective never gets one into trouble, not even when one is trying to evaluate $\zeta (1)= 1+ {1\over 2}+ {1\over 3}+ \dots =\infty$. The Riemann zeta function has its only pole at $s=1$; $$\zeta (1+ \varepsilon)= {\textstyle {1\over \varepsilon}}+ \gamma+ O(\varepsilon), \qquad \varepsilon\to 0. \tag 6$$ Thus for $s=1$ analytic continuation and numerical summation agree. But even here the analytic continuation method is superior: the infinity in (6) is cleanly split off from the well-defined finite part $\gamma=$ Euler’s constant. If one is prepared to throw away the pole and keep the finite part -- a kind of principal-part regularization -- then one can even use zeta functions at their poles. Essentially everything said in this paragraph can be immediately extended to arbitrary spectra and their zeta functions.
This reviewer -- and many others including Elizalde obviously -- find the situation described more than a little fascinating. The fundamental issue is really philosophical and touches upon the old and deep question of why and how abstract mathematics {\it can} match up so well with physical reality. This matchup seems amazingly tight in many areas of physics. However in QFT, where the physical mathematics equates highly structured but divergent expressions with physical quantities, there is some play. Is a divergent Feynman diagram a meaningless quantity? Is it the divergent “name” of an underlying analytic function which is revealed by analytic continuation? Is it something else? No one is going to give final and conclusive answers to these questions. It seems that a pragmatic attitude is called for. Does regularization work? Yes, so keep on using it -- but carefully and with many checks.
From time to time papers have appeared in which purported ambiguities in the zeta function method are announced. Invariably these are based on a misapplication of the method. A problem has its own spectrum. One had better not change spectra in mid-problem unless one knows exactly how to do this. Elizalde’s book performs a useful service by drawing attention to this kind of blunder.
We now review the Ten Physical Applications of this book.
1. The first utilizes equation (3) to compute the vacuum energy of a scalar field $\widehat {\phi}$ forced to exist on (and interact with) nontrivial manifolds such as $T^2 \times E^2$, or $E^d$ with parallel Dirichlet boundary planes arranged in $p$ orthogonal spatial directions (with $d$ and $p\leq d-1$ arbitrary). Actually one is interested in the Casimir energy of $\widehat {\phi}$ which is the energy shift experienced by $\widehat {\phi}$’s vacuum state when the nontrivial (topological or boundary) background replaces trivial (i.e. free, flat, infinite, empty) spacetime. Vacuum energies themselves are not defined in an absolute way. The Casimir calculations are standard ones, compactly reviewed.
2. Two more Casimir energy calculations follow, for a massless scalar field on $d$-dimensional spherical spacetime with $d= 3, 4$. These results are presented with useful discussion of the relevant zeta functions for spherical geometry. Such results are also known for massive fields.
3. Reference is made to a Kaluza-Klein model: a scalar field defined on spacetime $M^4 \times S^2$ ($M^4$ is 4D Minkowski space). The extra two compact dimensions $S^2$ are supposed to be tiny, but nonetheless may leave their imprint in the $M^4$ part of the theory. (This was the Kaluza-Klein discovery.) Unfortunately an intermediate formula from elsewhere is used as starting point. The reader will be unable to relate the zeta-function processing of this formula to physics.
4. Next follows the effective potential in a four-fermion theory at finite temperature, again quoted from elsewhere but at least a type of formula familiar to many readers. This effective potential is efficiently processed to a form suitable for critical-point analysis.
5. A functional determinant arising in a specific quantum gravity problem can be evaluated à la equation (3) using a zeta function discussed earlier in the book. Some properties of this zeta function and determinant are studied.
6. A study by Fujikawa of regularized current-operator matrix elements is the topic. However, the main analysis seems to be a derivation of a standard series representation for $\text{csch } x$ using zeta-function machinery.
7. Next comes an interesting physical system discussed at some length. The Casimir energy of a closed two-segment quantum string experiencing transverse vibrations is computed. In the very recent literature one can find more general versions of this system and calculation.
8. Spontaneous compactification in 2D quantum gravity is the topic here. For spacetime $E^1 \times S^1$ the gauge-invariant effective action is calculated. Spontaneous compactification to this manifold can occur. Few details are given.
9. In somewhat more detail a problem in the theory of quantum $p$-branes (in $d$-dimensional space) is considered. Casimir energies for rigid $p$-branes are basically quoted from elsewhere and then analysed.
10. The final Application is again more complete. The system is a mass-$M$ scalar field with $\phi^4$ interaction defined on partially compactified space $T^N \times E^n$ with $N+ n=4$. The conditions under which this system can experience spontaneous symmetry breaking are rederived. They are: $M=0$ and $n=0$ or $n=1$, with the lengths of the torus $T^N$ not too different.
An extensive but not exhaustive list of references is provided (172 references). This book, because of the body of technique presented, will be useful as a reference for those interested in learning zeta-function methods.