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Vacuum Energy of Quantum Fields in Classical Background Configurations

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Vacuum Energy of Quantum Fields in Classical Background Configurations I. Drozdov∗ University of Leipzig, Institute for Theoretical Physics Augustusplatz 10/11, 04109 Leipzig, Germany February 1, 2008 Der Fakult¨at f¨ur Physik und Geowissenschaften der Universit¨at Leipzig eingereichte DISSERTATION zur Erlangung des akademischen Grades arXiv:hep-th/0311199v1 21 Nov 2003 Doctor rerum naturalium (Dr.rer.nat) vorgelegt von Diplom Physiker (UA) Igor Drozdov geboren am 06.02.1970 in Dnepropetrowsk, Ukraine Leipzig, den 3.11/2003 ∗e-mail: [email protected] 1 Abstract The ground state energy of a quantum field in the background of clas- sical field configurations is considered. The subject of the ground state energy in framework of the quantum field theory is explained. The short review of calculation methods (generalized zeta function and heat kernel expansion ) and its mathematical foundations is given. We use the zeta-functional regularization and express the ground state energy as an integral involving the Jost function of a two dimensional scat- tering problem. We perform the renormalization by subtracting the contri- butions from first several heat kernel coefficients. The ground state energy is presented as a convergent expression suited for numerical evaluation. The investigation for three models has been carried out: scalar quantum field on the background of scalar string with rectangular shape, spinor vac- uum polarized by magnetic string of the similar shape and spinor vacuum interacting with the Nielsen Olesen vortex. Using the uniform asymptotic expansion of the special functions entering the Jost function we are also able to calculate higher order heat kernel coefficients. Several features of vacuum energy have been investigated numerically. We discuss corresponding numerical results. 2 Contents 1 Introduction 6 1.1 Generalremarks............................ 6 1.2 Motivation............................... 7 1.3 Historicalreview ........................... 8 1.4 Examlpes ............................... 10 1.5 Theaimofthework ......................... 10 1.6 Themethod.............................. 11 2 The ground state energy of quantized field 13 2.1 Theeffectiveaction.......................... 13 2.2 The ground state energy in the frameworkof canonical quantization 15 2.2.1 Quantizedscalarfield. 15 2.2.2 QuantizedDiracfield. 18 3 The equations of motion and solutions 21 3.1 The scalar field with the scalar background . 22 3.2 The Dirac spinor field with the magnetic background . .. 23 3.3 The Dirac spinor field with the abelian Higgs background . .... 27 3.3.1 Nielsen-Olesenvortex. 29 3.3.2 Interaction with the spinor field . 30 4 The zeta functional regularization of the ground state energy 32 4.1 TheRiemannzetafunction . 32 4.2 The generalized local zeta function . 34 4.3 Theheatkernel ............................ 36 4.4 Heatkernelcoefficients . 38 4.4.1 Scalar quantum field in scalar background . 39 4.4.2 Scalar field in constant vector background . 40 4.4.3 Spinor field interacting with gauge vector field . 41 4.4.4 Spinor field interacting with an electroweak background.. 42 4.4.5 Other approaches to the calculation of the heat kernelcoefficients: the expansion of the 4.4.6 Some general properties of heat kernel coefficients . 45 4.5 Zetafunctionalregularization . 47 4.5.1 The space dimension and structure of poles . 47 4.5.2 Theeffectiveaction. 48 4.5.3 Thegroundstateenergy . 50 4.5.4 The large mass asymptotics for the zeta-regularizedground state energy of massive fields 5 Analytical treatment of theground state energy 54 5.1 The regularized sum over the spectrum of hamiltonian . ... 54 5.1.1 Scalarfield .......................... 55 3 5.1.2 Diracspinorfield ....................... 60 5.2 TheexactJostfunctions . 64 5.2.1 Scalar field with rectangular scalar background . 64 5.2.2 Spinor field with rectangular magnetic background . 65 5.2.3 Spinor field with background of Nielsen-Olesen Vortex .. 67 5.3 The uniform asymptotics of the Jost function from theLippmann-Schwinger equation 73 5.3.1 The Lippmann-Schwinger equation . 73 5.3.2 Scalarfield .......................... 74 5.3.3 Spinor field in cylindric magnetic background . 77 5.3.4 Spinor field in arbitrary abelian Higgs background withcylindric symmetry 79 5.4 The representation of the renormalized ground state energy.... 84 5.4.1 The divergent part of ground state energy . 84 5.4.2 The finite and the asymptotic parts of energy . 87 5.4.3 The finite part Ef ofthegroundstateenergy . 89 5.4.4 The asymptotic part Eas oftheground stateenergy . 91 5.4.5 Cancellation of divergent contributions in the asymptoticenergy 94 5.5 Renormalizationandinterpretation . 97 5.5.1 Thecounterterms ...................... 97 5.6 Higher orders of the uniform asymptotic expansion of the Jost functionand the heat kernel coefficien 6 Graphics and numerical results 106 6.1 Scalarstring.............................. 106 6.1.1 Finite and asymptotic vacuum energy . 106 6.1.2 Parametrical dependence of vacuum energy . 107 6.2 Magneticstring ............................ 110 6.2.1 Finite and asymptotic vacuum energy . 110 6.2.2 Parametrical dependence of vacuum energy . 112 6.2.3 Some remarks on the numerical difficulties . 115 6.3 NOvortex............................... 117 7 Conclusions and Discussions 119 7.1 Scalar field in scalar background. 119 7.2 Spinor field in magnetic background. 119 7.3 Spinor field in NO vortex background. 120 7.4 Summary ............................... 121 8 Appendix 123 8.1 The uniform asymptotic expansions for logarithm of Jost function 123 8.1.1 The uniform asymptotics of modified Bessel functionsK and I123 8.1.2 The saddle point expansion . 124 8.1.3 Scalar field with scalar background . 125 8.1.4 Spinor field with magnetic background . 127 8.1.5 Spinor field with NO vortex background . 127 4 8.2 Performing of the spectral sum for Bag boundary conditions . 134 8.3 Calculation of the asymptotic part of ground state energy .... 136 8.3.1 Usefulrelations . 136 8.3.2 Scalar field with scalar background . 136 8.3.3 Spinor field with magnetic background . 138 8.3.4 Spinor field with NO vortex background . 139 8.4 The renormalization of the NO-vortex on-shell . 143 5 1 Introduction 1.1 General remarks The vacuum energy is a spectacular manifestation of the quantum nature of the real objects, in particular the fields of matter. In the framework of natural science it is also a powerful tool for investigation and of fundamental properties of our world. It is a really non-trivial fact, that quantum field can be observable even in the case that there are no real particles of it present. The reason for this phenomena is one of the main properties inherent basically to any quantum system, namely the existence of the ground state. Indeed, it is well known from such a simple example of a quantum mechanical system as the linear harmonic oscillator with the eigenfrequency ω, that the energy spectrum of this system possesses the lowest state and the energy of this state E0 is not zero but is proportional to the Planck constant 1 E0 = ~ω. (1.1) 2 This fact is generally also an intrinsic feature of any quantum object. Here we mean first of all quantum fields, namely matter fields (fermionic fields) and fields of interaction (bosonic fields). It means, one can establish the existence of a quantum field in the nature without to observe any real particles (quantums) of this field, but only through non zero vacuum values of observables, that are measurable, for example the vacuum energy. The main point of this phenomena is that the energy of the ground state of a quantum system is not defined absolutely, that applies by the way in general to the energy itself as a physical feature. It is defined generally with a relation to some other state. That is why the discussion about the vacuum energy of quantum field makes sense in correspondence to some external condition, under which it becomes visible (or measurable in physical sense). Such an external condition can be some boundary or some classical field interacting to the quantum one (background) The most famous manifestation of the vacuum energy was the Casimir ef- fect (H.B.G.Casimir 1948), which consist in an interaction between two parallel conducting plates playing the role of boundaries for the electromagnetic field in vacuum state. This effect was predicted theoretically in [26] and measured experimentally 10 years later. Sometimes, all the problems concerning the calculation of vacuum energy under external conditions are being called ”Casimir energy”, including also the problems of vacuum polarization by classical backgrounds. Actually, the vacuum energy of quantum spinor field polarized by a special homogeneous magnetic background was calculated by W.Heisenberg and H.Euler in 1936, 11 years be- fore the discovery of the Casimir force. 6 1.2 Motivation The investigation of vacuum effects in general and vacuum energy in particular has a number of applications to wide variety of physical problems (see recent reviews [23, 65, 62]). Indeed, in a lot of problems concerning the experiments in particle physics, certain effects caused by interaction with vacuum fluctuations must be taken into account as radiation corrections. For example in the bag model of hadrons in Quantum Chromodynamics the vacuum energy of quark and gluon fields makes essential contributions in the total nucleon energy. As further examples of such measurements one can refer to the experiments for the exact mass, isotopic charge, anomalous magnetic moment and so on. Generally speaking, the vacuum corrections are more or less relevant in all problems of particle and high energy physics. Moreover, one can imagine a situation that only the sole possibility to establish the existence of a quantum field in the nature, (since it is present in the model) is the interaction of its vacuum with an observable classical background (and mea- surable effects reasoned thereby). Actually, the interaction of the classical field configuration with quantum fields being in vacuum state leads to modification of macroscopic physical features of this classical object.
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