After a simple outline of the local zeta method, we exemplify it in. Space sciences zeta functions, the casimir effect, and. For many years the casimir effect was little more than a theoretical curiosity. Local zeta regularization and the casimir effect pdf paperity. Naively, computation of the casimir pressure leads to infinite sums and therefore requires regularization.
In this paper i study the casimir effect caused by a charged and massive scalar field that breaks lorentz invariance in a cpteven, aetherlike manner. The casimir effect for the scalar and elko fields in a. The results agree with the ones obtained by more popular methods, such as point splitting regularization. Zeta function regularization in casimir effect calculations and j. Zeta function regularization is an effective method to extract physical significant quantities from infinite ones. A tool to explore the quantum world genova, january 2017 based on joint work with c. The casimir effect holds for any sufficiently gentle regularization function.
In regularizing the casimir energy for a massless scalar. Pdf local zeta regularization and the casimir effect. The dimensionally regularized sum is then expressed as a power series in am by means of zeta. Casimir 1 was the beginning of a whole branch of research which aims at answering, both from a fundamental and from a practical perspective, very profound questions on the vacuum structure of quantum field theory. Zeta function regularization is unambiguously used from the start and the properties of bessel and related zeta functions are central.
A new regularization procedure for calculating the casimir. Sometimes in mathematics and physics, we must evaluate divergent series of the form 1 k n n. Leonhard euler already considered the above series in 1740, but for positive integer values of s, and later chebyshev extended the definition to. Quantum mechanicscasimir effect in one dimension wikiversity. In part i of this series of papers we have described a general formalism to compute the vacuum effects of a scalar field via local or global zeta regularization. May 31, 2012 zeta function regularization in casimir effect calculations and j. Derivation of casimir effect assuming zetaregularization. We use the definition of the gamma function and have. For instance, the casimir effect in the parallel plate and spherical shell. The description ends with the seminal contributions of stephen hawking and stuart dowker and collaborators, considered by many authors as the actual starting point of the introduction of zeta function regularization methods in theoretical physics, in particular, for quantum vacuum fluctuation and casimir effect calculations. This approach can be applied to the stressenergy tensor of the casimir effect, and works as well on curved spacetimes.
Zeta regularization is a method to treat the divergent quantities appearing in several areas of mathematical physics and, in particular, in quantum field theory. We find that the casimir energy is intimately related to, but not identical to, the oneloop effective energy. Nontrivial results concerning these functions are given. It is due to quantum vacuum fluctuation of the electromagnetic field. The analytic regularization zeta function method and the cutoff method in casimir effect svaiter, n. A main class of sequences that admit a zeta regularization is given by the spectrum of a class of linear operators.
Pdf local zeta regularization and the scalar casimir. Abstract in this work, we obtain the casimir energy for the real scalar. Casimir effect for a perfectly conducting wedge in terms. In 19, this appeared in the work of a very famous mathematician from india, srinivasa. Apart from the standard procedure, we emp verschiedene regularisierungsschemata vexwendet. In the case of large scalar mass, simple analytic expressions are obtained for the zeta function and casimir energy of the scalar field when it is confined inside the spherical surface, and when it is confined. Analytic regularization for the casimir effect for rectangular systems inone, two and threedimensions, as well as for parallel conducting plates, is discussed. Spherical casimir effect for a massive scalar field on the. Zeta regularization and products of primes mathematics. Local zeta regularization and the casimir effect progress of. Several regulators can be used that yield the same answer. See wikiversity for an elementary calculation in one dimension. Zeta function regularization in casimir effect calculations and js. Equivalence of zeta function technique and abelplana formula.
In the original calculation done by casimir, he considered the space between a pair of conducting metal plates at distance apart. The usual spectral eigenmode zetafunctions will be denoted by ms x n. The scalar field satisfies dirichlet boundary conditions on a pair of very large plane parallel plates. However, physically, there must be a limit on how high these momenta can get. Pdf zeta functions and the casimir energy semantic scholar. Zeta function regularization in casimir effect calculations and stuart. A25 1992 979990 cbpfnf04290 33 connection between zeta and cutoff regularizations of casimir energies beneventano, c. We show that in general the casimir energy depends on a normalization scale. Does the fact that zeta function regularization works in. These finite results can then be used to do calculations and make predictions, as long as the final predictions are independent of the way in which do regularization. In this case, the standing waves are particularly easy to calculate, because the transverse. To develop a zetafunction definition of the casimir. Zeta function regularization might be the most mathematically convenient way of computing the casimir force, but it is probably also the most confusing and least physical. Pdf zeta function regularization in casimir effect.
Download pdf 559 kb abstract in part i of this series of papers we have described a general formalism to compute the vacuum effects of a scalar field via local or global zeta regularization. The gamma function and the zeta function can be defined for values d 0 by. The local zeta regularization allows to treat local divergences appearing in quantum field theory. We obtain the casimir energy in terms of the dimensional parameter and the lp. Local zeta regularization and the scalar casimir effect. The other technical type is the dimensional regularization method that involves the mathematical complications of the riemann zeta function and the analytical. Iris institutional research information system air. While part of their application agrees with previous knowledge, new results like the zetaregularized electromagnetic casimir energy for a circular wire. The casimir e ect making sense of in nity in nity conclusion history quantum electrodynamics zeropoint energy shift student of ehrenfest, worked with pauli and bohr retarded vanderwaalsforces e 23 4. Dos, zeta functions, determinants, and traces wodzicki residue, multiplicative or noncommutative anomaly, or defect singularities of. The usual spectral eigenmode zetafunctions will be denoted by ms x ns n.
Basic zeta functions and some applications in physics. Equivalence of zeta function technique and abelplana. In classical physics, the definition of the vacuum is based on the pressure of the system. Ten physical applications of spectral zeta functions. The effect was predicted by the dutch physicist hendrick casimir in 1948. The zeta function regularization technique is used to study the casimir effect for a scalar field of mass m satisfying dirichlet boundary conditions on a spherical surface of radius a. We analyze the massless and the massive case for both.
Casimir effect, zeta functions, and cosmology casimir e. There are alternative ways of computing the casimir force that are le. With the hadamard complement, it has been shown to provide. This report gives an introduction to the casimir effect in quantum field theory and its. Regularization of the casimir effect physics stack exchange. The expansion of the casimir energy for a scalar field with mass m, in a space where one dimension has been compactified into a circle of length a, leads to a double. Use of this method yields, for instance, the vacuum energy corresponding to a quantum physical system with. Among the different methods, zetafunction regularization which is obtained by analytic continuation in the complex plane of the zetafunction of the relevant physical operator in each case might well be the most beautiful of all. Uses of zeta regularization in qft with boundary conditions. Regularization is the process by which an infinite result is replaced with a finite result in a way so that it keeps the same properties. We consider the analytic regularization by employing the riemann zeta function as well as the zeta functions introduced by epstein. Casimir effect for a perfectly conducting wedge in terms of local zeta function.
Zeta function regularization imperial college london. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta function regularization analytic continuation, asymptotic expansions, many of which appear here, in book format, for the first time are. Zeta function regularization of path integrals in curved spacetime, commun math phys 55, 3 1977. This book is the result of several years of work by the authors on different aspects of zeta functions and related topics. Exploration of the quantum casimir effect this result does not explicitly make sense because the sum in equation 2 is a divergent sum.
Our attention is devoted to the case of a neutral, massless scalar. Zeta regularization techniques with applications e. Local zeta regularization and the casimir effect article pdf available in progress of theoretical physics 1263 april 2011 with 92 reads how we measure reads. Abstract we use zeta function techniques to give a finite definition for the casimir energy of an arbitrary ultrastatic spacetime with or without boundaries. Zetafunction regularization 2 is a powerful and increasingly liked method, but to our knowledge it has never been completely applied to electromagnetic e. Space sciences zeta functions, the casimir effect, and cosmology. What is the relation between zeta and cutoff regularization. After a general outline of the local zeta method for the casimir effect, we exemplify it in the typical case of a dirichlet field between two parallel plates, or outside them.
The various methods for evaluation and regularization of the casiinu energy are. Zeta regularization has proven to be a powerful and reliable tool for the regularization of the vacuum energy density in ideal situations. The link between the zeta function, the heat kernel and the mellin transform is explained and the equivalence between zeta and dimensional regularizations is shown and explicitly derived for the case 4 theory. Frontiers zeta regularization of infinite products physics. The casimir effect is a consequence of the vacuum or zeropoint fluctuation, this results in the apparition of an observable force between material bodies that can be measured. The casimir effect is a small attractive force which acts between two closed parallel uncharged conducting plates. Applying the general framework for local zeta regularization proposed in part i of this series of papers, we compute the renormalized vacuum expectation value of several observables in particular, of the stressenergy tensor and of the total energy for a massless scalar field confined within a rectangular box of arbitrary dimension. How can the use of divergent series in calculating the. Complete zetafunction approach to the electromagnetic. This regularization completely eliminates divergent expressions in the course of calculations and gives rise to a finite. Dowkers contribution article pdf available in international journal of modern physics a 2715 may 2012 with 7 reads.
Derivation of casimir effect assuming zeta regularization see wikiversity for an elementary calculation in one dimension. Zeta regularization is performed in this framework for both local and global observables, like the stressenergy tensor and the total energy. Local zeta regularization and the casimir effect arxiv. Zetafunction method for regularization encyclopedia of. In the literature there are at least two methods to derive casimir effect. The calculation of the casimir effect involves a sum over all possible momenta of the waves between the plates. Local zeta regularization and the casimir effect progress. Applying the general framework for local zeta regularization proposed in part i of this series of papers, we renormalize the vacuum expectation value of the stressenergy tensor and of the total energy for a scalar field in presence of an external. The casimir force is the sum over all field modes of the difference between inner and outer radiation pressures using the causality properties of the scattering amplitudes, and the transparency of mirrors at highfrequencies, the casimir free energy can be written as a sum over matsubara frequencies cavity confinement effect. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. Zeta function regularization in casimir effect calculations. Applying the general framework for local zeta regularization proposed in part i of this series of papers, we compute the renormalized vacuum expectation value of several observables in particular, of the stressenergy tensor and of the. This technique agrees with dimensional regularization where one generalises to ndimensions by adding extra.
Iris institutional research information system air archivio istituzionale della ricerca. This is precisely what we report in the present note. Local casimir effect, renormalization, zeta regularization. The local zeta regularization allows to treat local divergences appearing in.
By contrast, abelplana formula method permits explicit separation of divergent terms. A simply regularized derivation of the casimir force. The casimir effect holds for any sufficiently gentle regularization function the choice of an exponential regularization function might seem arbitrary to some readers. In the original calculation done by casimir, he considered the space between a pair of conducting metal plates at distance a \displaystyle a apart. It is regarded as mathematically simple and elegant but the isolation of the physical divergency is hidden in its analytic continuation.
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