PHYSICS REPORTS (Review Section of Physics Letters) 134, Nos. 2 & 3 (1986) 87—193. North-Holland, Amsterdam THE CASIMIR EFFECT Günter PLUNIEN, Berndt MULLER and Walter GREINER Institutfür Theoretische Physik der I. W. Goethe Universitüt Frankfurt; 60(X) Frankfurt/Main, Fed. Rep. Germany Received 8 August 1985 Contents: 1. Introduction 89 4.3. Quarks and gluons in a bag 139 2. Energy of the vacuum state 91 5. The Dirac vacuum in external electromagnetic fields 147 2.1. Zero-point energies in field quantization and the 5.1. Vacuum energy and vacuum polarization 147 definition of the physical vacuum energy 91 5.2. The supercritical vacuum 159 2.2. Implications of the vacuum energy 102 5.3. The vacuum energy in nuclear scattering 167 3. Mathematical formulation and evaluation methods for 6. Casimir energy at finite temperature 172 vacuum energies 106 6.1. Partition functions and free energy 172 3.1. Method of mode Summation 106 6.2. Finite temperature propagators 174 3.2. The local formulation and Green function methods 110 6.3. Casimir energy 178 33. The multiple-scattering expansion for Green functions 7. Applications 180 and the Casimir energy 121 7.1. Casimir energy in dielectric media and the relation to 3.4. Phase-shift representation 130 the bag model of hadronic particles 180 4. Casimir effect 132 7.2. Boundary problems and Casimir energy in gauge 4.1. The Casimir effect between perfectly conducting plates 132 theory 186 4.2. The Casimir energy of a massive scalar field in a finite 8. Concluding remarks 189 cavity 137 References 190 Abstract: This report gives an introduction to the Casimir effect in quantum field theory and its applications. The interaction between the vacuum of a quantized field and an external boundary or a classical external field is investigated laying particular emphasis on Casimir’s concept of measurable change in the vacuum energy. The various methods for evaluation and regularization of the Casiinu- energy are discussed in detailfor specificfield configurations. Recent applications of the Casimir effect in supercritical fields, QCD bag models and electromagnetic media are reviewed. Singleorders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 134, Nos. 2 & 3 (1986) 87—193 Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfi. 72.00, postage included. 0 370-1573/86/$37.45 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) THE CASIMIR EFFECT Günter PLUNIEN, Berndt MULLER and Walter GRE1NER Institut für Theoretische Physik der J. W. Goethe Universitât Frankfurt, 6000 Frankfurt/Main, Fed. Rep. Germany NORTH-HOLLAND - AMSTERDAM G. Plunien et a!., The Casimir effect 89 1. Introduction The occurrence of divergent zero-point energies is an inherent feature of quantum field theory. It appears as a direct consequence of canonical field quantization, which permits to establish the correspondence between classical (observable) quantities and quantum operators. Zero-point energies arise because the canonical quantization scheme does not fix the ordering of non-commuting operators in the field Hamiltonian. Thus, quantum field theory based on the operator concept, in principle, needs additional prescriptions in order to become a well-defined theory. Normally, this ambiguity is removed by requiring a certain ordering of operator products, Wick’s normal ordering. For the canonical field Hamiltonian this prescription implies a formal subtraction of the infinite zero-point energy and, per definition, the vacuum expectation value of the normal-ordered field Hamiltonian is then equal to zero. Accordingly, this formally introduced (mathematical) vacuum state has peculiar properties, namely, it carries neither energy nor momentum nor angular momentum. The standard arguments for such a subtraction lay particular emphasis on the fact that, in practice, it is not possible to measure the absolute value of the energy, but only differences in energy, allowing for an arbitrary choice of the origin on the energy scale. This statement is certainly valid for the microscopic Systems. However, the question remains whether the zero-point fluctuations of quantum fields must be taken seriously or not. For systems with a finite number of degrees of freedom zero-point energies have never been envisaged as problematic, because they are finite and measurable (e.g., the zero-point oscillations of crystals at zero temperature are observable) and, thus, they represent a real property of the ground state. That the situation should be different in the case of quantized fields is not a priori evident. The absolute value of the vacuum energy is, in principle, a measurable quantity, because it gravitates. The energy density of the vacuum must therefore be expected to contribute to some extent to the total energy density of the universe. However that may be, one would like to know how to deal with infinite zero-point energies in a meaningful way within a field theory based on the canonical field Hamiltonian and, in close connection with that, whether one can conclusively show possible con- sequences which indicate the presence of vacuum fluctuations. Such questions immediately lead to the Casimir effect. In 1948 Casimir showed [11that neutral perfectly conducting parallel plates placed in the vacuum attract each other. The attractive force can be considered as arising due to the change in the zero-point energy of the electromagnetic field when the plates are brought into position. The experimental verification [2—8]of this attraction has put the discussion about zero-point energies in field theory on firm ground. The general importance of the treatment of zero-point energy in the derivation of the original Casimir effect lies in the fact that it implies that the energy of the vacuum state of quantized fields cannot be correctly defined by normal ordering. The basic idea of what will later on be called “Casimir’s concept of vacuum energy” can be described like the following: A meaningful definition of the physical vacuum energy must take into account that in a real situation quantum fields always exist in the presence of external constraints, i.e. in interaction with matter or other external fields. An idealized description of such circumstances is obtained by forcing the field to satisfy certain boundary conditions. The zero-point modes are affected by the presence of these external constraints, and therefore, the zero-point energy is modified. Consequently, one is led to calculate the physical vacuum energy or Casimir energy of a quantized field with respect to its interaction with external constraints. It is generally defined as the difference between the zero-point energy referring to the distorted vacuum configuration and the free vacuum configuration, respectively. This formal definition only makes sense, if it is combined with appropriate regularization methods which guarantee a finite expression for this energy difference. Any variation of 90 G. Plunien et aL, The Casimir effect the external boundary configuration induces a change in the vacuum energy, which is now considered as a measurable quantity. The main consequence of this concept is that the physical vacuum of a quantum field no longer represents a state with trivial global properties. Instead, the Casimir vacuum must be considered as a physical object reacting against distortions, i.e., possible vacuum effects can be interpreted as the “response” of the vacuum due to the presence of external constraints. This line of reasoning is spelled out in detail in the first part of section 2 of the present review article, where it is shown how zero-point energies arise when quantizing the scalar field, the electromagnetic field and the Dirac field. This introductory section continues with a more specific definition of the vacuum energy. Depending on the particular kind of external constraints distorting the free vacuum configuration, the vacuum energy may be understood as a contribution to the self-energy or to the interaction among the given boundary itself. Possibly it may cause boundary effects due to surface energies contributing, e.g., to surface tension or stresses inside the boundary. Basic investigations have been made of the Casimir energy of the electromagnetic field under constraints, which demonstrate the physical im- plication of the zero-point energy. As already shown by Casimir [301the zero-point energy of the electromagnetic field can be usefully applied to explain van der Waals attraction. However, the existence of repulsive Casimir forces [37] of electromagnetic origin seems to contradict this inter- pretation. This example already reveals that the physical content of the concept of Casimir energy embraces more than its use as an alternative way to understand certain known effects that can also be derived by conventional approaches. Presently, Casimir energies of quantized fields are studied in connection with a variety of problems, ranging from applications in particle physics, e.g. in QCD bag models, to gravitational physics, where its possible influence on the structure of space-time is studied. Examples for physical implications of Casimir energies are reviewed in the second part of section 2. In practice, the evaluation of vacuum energies remains a problematic exercise, because the available methods, in most cases, only allow an approximate calculation. Mainly two methods can be dis- tinguished: The mode-summation method is based on the direct evaluation of infinite sums over energy eigenvalues of the zero-point field modes. Within the local formulation on the other hand, one examines the constrained propagation of virtual field quanta and considers the vacuum stress tensor, which can be expressed in terms of propagators. Both treatments can be shown to be formally equivalent. Still, ambiguities between the results obtained by these methods cannot be excluded a priori since they imply inherently different regularization schemes, but correct results for the Casimir energy should be independent of the applied methods.