Topics in Quantum Mechanics pp 317-320 | Cite as

# Zeta Regularization

## 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.

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