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Reference International Centre for C), Theoretical Physics IC/88/346 REFERENCE INTERNATIONAL CENTRE FOR C), THEORETICAL PHYSICS ZETA FUNCTION REGULARIZATION OF QUANTUM FIELD THEORY A.Y. Shiekh INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/88/346 §1 Introduction International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization In generalizing quantum mechanics to quantum field theory one is generally INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS confronted with a scheme plagued by infinities. If one is to make sense of such an ailment, one must find a way to deal with those infinities. In the usual interpretation this difficulty is viewed as stemming from not having yet accounted for the self- interactions which modify the masses and coupling constants to their observed values. A sensible strategy, in order to be useful, must result in only a finite ZETA FUNCTION REGULARIZATION number of physical constants being left, undetermined. Not all theories can be dealt OF QUANTUM FIELD THEORY * with in this way; which has the beneficial consequence of eliminating certain, otherwise prospective, theories of nature. This follows the line of reasoning that physics is more than descriptive, but predictive by virtue of being limited by the A.Y. Shiekh "" requirement of consistency. International Centra foe Theoretical Physics, Tri.fste, Ltaly, In order to proceed with such a theory, the infinities must be restrained (regulated) while the self-interactions are accounted for. Visiting a different space- time dimension is a popular way to regulate (dimensional regularization) as it explicitly maintains the theory's covariance, which may otherwise be difficult to hold on to during reckoning for the self-interactions (renormalization). However, ABSTRACT for string theories, which hinge upon the very special properties of two dimensions, dimensional regularization can be an undesirable procedure leading to regulator- Analytic continuation leads to the finite dependent, and so meaningless, results [Mann, 1988; Leblanc, Mann and Shad wick, re.norma 1 ization of a quantum field theory. 1988]. Another symmetry-maintaining approach is Zeta function regularization, where the divergent integrals are re-interpreted through analytic continuation. In its original form, this technique only extended to treating the one-loop approximation [Salam and Strathdee, 1975; Dowker and Critchley, 1976; Hawking, 1977], but has now been generalized [McKeon and Sherry, 1987]. A common shortcoming of these procedures is that they are perturbatively specified, and one might hope for a non- perturbative formulation, even if one must resort to a perturbative analysis for HIRAHARE - TRIESTE explicit calculation. This motivates the work that follows, which, although leading October 1988 to a finite interpretation of a quantum field theory, does not avoid the ambiguities induced by the self-interactions or the resulting need to renormalize (albeit by a finite amount). " Submitted for publication. "" Present address: Department of Physics, University of Waterloo, Waterloo, Ontario, Canada. §11 Divergent Integrals 8-1 and may be formally confirmed by letting T a f(x) t: Consider the integral: which, if divergent, conceals the u dependence. Divergent integrals are notoriously badly behaved under formal manipulations. For example: Q.E.D. In this way one may achieve 'taming' by consecutively performing the x integral, J then the t integral, and finally the s-» -1 limit. Hence the analytic continuation: Let x-*x/a: H "fa. dx e* dxj dt This is itself ill-defined, unlike the discrete version, as will be seen by taming an example. which is inconsistent. This is important if one intends to work with such quantities. Bearing this in mind, analytic continuation to a convergent object might be contemplated in order to extract the sought-after behaviour. The Zeta function: is used in just such a manner to interpret certain infinite sums and products. One might then analytically continue using the equivalence: CUM T rw where: • fV'e'dt This is formally confirmed by letting: Contemplate the move to the continuum: f°° ' I f °° S-l f f ™ ft >f \ y*i =J f ») dx = —=- I t e dx dt which leads to the claim: -i- §111 Taming of Divergent Integrals The m dependence may then be taken into a to yield: Consider the taming of: ("fa xm dx ~ f^oi j"fwxmdx Such divergent integrals will be referred to as 'pure'. Interpreting as before; Consider separately the special cases of m=0 and -1. For m=0 the only finite -- b+llfV1 (f%-«"»">dxldt interpretation is zero, while the m=-l case is particular in containing both an l_ •'o Vo / Js-l infrared and ultraviolet divergence. This leads to an ambiguity from each regulator, but: and the interpretation: e^'du = — rf-l RC([J) > 0 (otherwise analytically continue) [Gradshteyn and Ryzhik; P3O7, equation 3.326) where the splitting point is arbitrary and taken at unity. First perform the x-integration: ("Wdx^OM Along the way a possible inconsistency has been ignored, namely one expects that J ,JV-U ° f-<yl under any reasonable interpretation: then the t-integration: f<y>j°°Xmdx - ftyi Now, f(y) might formally be dragged into either the integrand or measure. Consider pulling into the measure: This needs an infrared regulator for m<-l, m>0: f"xmd(fly)x) m r Let e = f(y) x: -i-i e de •f f <y> f<y> which, when interpreted as above, goes correctly like f(y). From this it is realized Only one family of regulators leads to a finite and non-zero result, namely: that one cannot consistently carry past the integral sign, but only through the measure1 (excepting the m=-l case). This rule must therefore be strictly adhered to where a is an arbitrary constant (different for each m). This correctly-* 0 as s->-l, when manipulating such untamed pure divergences. However, once the pure for m<-l, m>0. (The -l<m<0 range may be dealt with by a similar treatment of the divergence is tamed with the correct choice of regulator, the object is well behaved ultraviolet divergence). Hence one is led to an arbitrary, but finite, result which and the restrictions lifted, for then: exposes any functional dependence. Such 'to be determined' factors are typical of fwx dx flm U any renormalization scheme. This yields the strange behaviour: (m<-l and m>0), where it is now clear that f(y) cannot be directly extracted. () fun di • m+1 ' In general formal manipulations with divergent integrals are invalid and each step must be separately justified. -5- -6- Let: §IV One loop Amplitudes in X<|>4 otte+I)-1 So: Divergences begin at the one loop level and one might apply the former analysis for comparison with dimensional regularizatton. f TV"1 f"e«y'I"a(8+"p"Td3[ dr Tadpole diagram: Further let: But, since this depends only on the length of L, it is convertible to a one- which yields the well behaved analytic continuation for m<-l and m>0: dimensional integral using: f -2 f ... [ ^L'ld-L-^-f^TFc '0 [Ramond, 1981] So: This may be evaluated, as before, to yield the same result: .. *- -JL_ ( —^~ dx 2 4 + 2 m+1 (2TT) / * m rty) Extract out the pure divergences, taking care to justify any formal manipulations on divergent integrals: - f "iy» The -l<m<0 case has an analogous treatment leading to the same behaviour. _^fj-dx.f JlLdx) 2 + m So it is found that Zeta function continuation gives a consistent, even though 32* ^> Jo * / strange, interpretation for the 'pure' divergences: Shift x-»x-m2, which may be validated by visiting less than four dimensions, f"V<y)Xmdx~f \y> where the object is finite: For a generally divergent integral, one might consider extracting and so interpreting each divergence consecutively. The associated ambiguities are to be 32*' fixed by 'renormalizing'. A renormalizable theory is then one having only a finite 1 amount of arbitrariness. The scheme is intrinsically non-perturbative , but might 32B ' best be illustrated upon a perturbative approach to X(J>* theory in four dimensions. hence: r i 1 A Hamilton-Jacobi transformation reducing the path integral to a finite number of ordinary integrations [Shiekh, 19*7]. -7- Fish diagram: 1O<=^ f * Adopt the Feynman parameterization which turns the product structure to a sum: So: having shifted L-»L+p(l-u), which may be justified by visiting less than four dimensions, where the object is finite. Converting to a single integral, as before: Then extract out the pure divergences, taking care to justify any formal manipulations on divergent integrals: du 32* but: f 'ln(' ti-ii-«*)du - -2 + a>0 hence: -10- §V Comparison with Dimensional Regularization since: Renormalization then proceeds, as for dimensional regularization, excepting that Tadpole diagram: with Zeta function regularization the process is finite. §VI Summary by dimensional regulari?.atlon Zeta function regularization, as used here, does not avoid the ambiguities usual to (whereiu-+ 2 is the four dimensional limit and \i is arbitrary) quantum field theories, or the resulting need to renormalize. The renormalization, however, becomes finite. I jdx-lnW 327T' J, I by Zeta function regularization Fish diagram: ACKNOWLEDGMENTS The author would like to thank Professor Abdus Salam, the International Atomic Energy Agenuy and UNESCO for hospitality at the fnternational Centre A x for Theoretical Physics, Trieste. He would also like to thank Robert. Mann 2-w 2 ^ p KR' for leading him to these ideas, and John Strathdee for looking at them. by dimensional regularization (whereto-* 2 is the four dimensional limit and u is arbitrary) 1 -lnW- ^- In 32* by Zeta function regularization Zeta function and dimensional regularization are seen to agree if one associatesthe two equally arbitrary quantities: n 2-u -11- -12- REFERENCES 1.
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