Spectral Methods in Quantum Field Theory

1. Front Matter

   Pages 1-10

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2. Introduction

   • Noah Graham, Markus Quandt, Herbert Weigel

   Pages 1-13

3. Review of Scattering Theory

   • Noah Graham, Markus Quandt, Herbert Weigel

   Pages 15-32

4. Quantum Field Theory and the Spectral Method

   • Noah Graham, Markus Quandt, Herbert Weigel

   Pages 33-61

5. Applications in One Space Dimension

   • Noah Graham, Markus Quandt, Herbert Weigel

   Pages 63-89

6. Spectral Analysis of Charges

   • Noah Graham, Markus Quandt, Herbert Weigel

   Pages 91-101

7. Hedgehog Configurations in d = 3+1

   • Noah Graham, Markus Quandt, Herbert Weigel

   Pages 103-128

8. Boundary Conditions and Casimir Forces

   • Noah Graham, Markus Quandt, Herbert Weigel

   Pages 129-142

9. String-Type Configurations

   • Noah Graham, Markus Quandt, Herbert Weigel

   Pages 143-169

10. Quantum Corrections to Q-Balls

    • Noah Graham, Markus Quandt, Herbert Weigel

    Pages 171-177

11. Back Matter

    Pages 1-4

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In this monograph we apply scattering theory methods to calculations in quantum ?eld theory, with a particular focus on properties of the quantum vacuum. These methods will provide e?cient and reliable solutions to a - riety of problems in quantum ?eld theory. Our approach will also elucidate in a concrete context many of the subtleties of quantum ?eld theory, such as divergences, regularization, and renormalization, by connecting them to more familiar results in quantum mechanics. We will use tools of scattering theory to characterize the spectrum of energyeigenstatesinapotentialbackground,hencethetermspectralmethods. This mode spectrum comprises both discrete bound states and a continuum of scattering states. We develop a powerful formalism that parameterizes the e?ects of the continuum by the density of states, which we compute from scattering data. Summing the zero-point energies of these modes gives the energy of the quantum vacuum, which is one of the central quantities we study.Althoughthemostcommonlystudiedbackgroundpotentialsarisefrom static soliton solutions to the classical equations of motion, these methods are not limited to such cases.

• calculus
• casimir forces
• mathematical physics
• non-perturbative quantum fluid theory
• quantum mechanics
• scattering theory
• spectral method
• vaccum energy

From the reviews:

“The authors present the mathematics that is foundational to a variety of problems in quantum field theory, in particular the subtleties of calculating the zero-point energy, i.e. the energy of the vacuum. … This monograph … should be used by graduate students as a companion to the standard textbooks of QFT.” (Gert Roepstorff, Zentralblatt MATH, Vol. 1169, 2009)

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