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Zeta Regularization

  • Floyd Williams
Chapter
Part of the Progress in Mathematical Physics book series (PMP, volume 27)

Abstract

Zeta functions play a very useful role in physics, especially as they provide for a natural means of regularizing certain quantities which a priori are infinite, or meaningless. In Chapter 14 we presented a zeta function representation of the free energy of a harmonic oscillator. In Chapter 16 we shall employ a zeta function, for example, to provide for a clear mathematical meaning of a certain partition function related to the Hawking path integral [45]. The Vacuum energy of certain quantum fields over various space-times is given by a special value of a suitable zeta function—the value at s = -1/2, for example; cf. [4,11,12,14, 88]; also compare the term in (16.1.47), where the special value s = -p/2 determines the vacuum or Casimir energy of a certain Kaluza-Klein space-time. One uses zeta functions to define the determinant in certain cases of infinite-dimensional operators. This is done by a simple procedure called zeta regularization. If \(D = {{y}^{2}}(\tfrac{{{{\partial }^{2}}}}{{\partial {{x}^{2}}}} + \tfrac{{{{\partial }^{2}}}}{{\partial {{y}^{2}}}})\) is the non-Euclidean Laplace operator on the upper 1/2-plane, for example, then one can define and compute the determinant of the operator −D + s(s − 1). This determinant arises in Polyakov string theory [26, 36, 70, 74]. Zeta regularization without doubt has proven to be a powerful tool for dealing with many interesting phenomena in physics including certain anomalies (more on this in Chapter 17), and ambiguities which arise in one-loop or external field approximations in quantum field theory (so-called ultraviolet divergences), to name a few. Before plunging into examples that involve more sophisticated zeta functions (such will appear in later chapters) we use this chapter to illustrate the regularization procedure in a few simple cases. The following example illustrates the basic idea.

Keywords

Zeta Function Vacuum Energy Riemann Zeta Function Casimir Energy Ultraviolet Divergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Floyd Williams
    • 1
  1. 1.Department of MathematicsUniversity of MassachusettsAmherstUSA

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