------------------------------------- Renormalization without infinities II ------------------------------------- In bare (divergent) QFT, infinities arise because integrals taken over unbounded momenta don't exist; so doing it 'formally' leads to nonsense. Instead, proper QFT takes regularized integrals, for example by adding an explicit cutoff Lambda. This simply means that everything is calculated with an action that depends on Lambda as an additional parameter. Once this is done, everything is finite, but Lambda-dependent. The only problem with that is that the cutoff destroys Lorentz covariance - apart from that it would be a completely respectable field theory in itself. Now Lorentz invariance is violated only at energies > O(Lambda); here O(...) is the Landau symbol that denotes an unspecified factor that might itself depend on Lambda. Hence to have the theory conform to physics that can be checked it suffices to take Lambda large. But for aesthetic reasons and/or since we believe that symmetries are fundamental, we want to have fully invariant theories. This requires that we let Lambda go to infinity. But in order that the results have a finite limit we must at the same time make the bare coupling constants g dependent on Lambda. If this is done in a correct way (and the textbooks on QFT teach one or more of the known correct ways under the heading of 'renormalization'), one encounters no infinities at all in the whole process. Thus renormalized quantities are (typically) never infinite. The essentials of the renormalization process, namely the need for Lambda-dependent coupling constants for sufficiently singular Hamiltonians, can be understood nonperturbatively on the nonrelativistic level. What happens is that one has a family of Hamiltonians H(Lambda,g) that depend on a scale parameter Lambda and and a coupling constant g (or several). H(Lambda,g) has a good limit H(g) as Lambda to inf, with g fixed, but the corresponding limit of the resolvent G(Lambda,g) does not exist; hence if one tries to do calculations with H(g) directly (the 1930 way of doing things, which was a dead end), one gets infinities all over the place. On the other hand, if one chooses a good parameterization g(Lambda,mu) then, although H(Lambda,g(Lambda,mu)) has no longer a good limit as Lambda to inf, its resolvent G(Lambda,g(Lambda,mu)) has a well-defined limit G(mu). (This holds at least in quantum mechanics, which is simply 1D quantum field theory, and in 2D field theory, where this can be proved in certain cases. In 3D and 4D, where things are not well understood rigorously, one probably needs also a Lambda-dependent inner product defining the Hilbert space to ensure that one ends up in the right representation, and Lambda-dependent wave functions to ensure that the limiting renormalized wave functions remain bounded in the limiting renormalized inner product.) Since all dynamical information, including all scattering information, is in the resolvent, G(mu) defines a good physical model for a scattering process. In some simple cases, renormalization can be done nonperturbatively. For example, standard perturbation theory for a Hamiltonian p^2/2m + g delta(x) produces infinities. The renormalization of this particular example is treated nonperturbatively in hep-th/9305052. Thus, infinities only appear if one takes the limit in a way it cannot be taken consistently. The relativistic case is a bit more involved and is at present not understood nonperturbatively in the physically most important case of 4 space-time dimensions, but there is no difference in principle. The local interaction of the formal Lorentz invariant action is replaced by a nonlocal interaction depending on the UV cutoff Lambda. Thus one has V(g,Lambda) in place of V(g), where g are the coupling constants (including masses). To do so, one writes the (Euclidean = Wick rotated) field as Phi(x) = integral dp exp(-i p dot x) Phihat(p) and substitutes it into the action. This gives an action in the momentum representation. Then one regularizes the interaction term by throwing away (or damping more gracefully) the momenta above some cutoff Lambda. Introducing the cutoff makes the interaction nonlocal, as one can see by going from the momentum representation of the regularized interaction term back to the position representation by substituting Phihat(p) = const * integral dx exp(i p dot x) Phi(x). Instead of the delta functions which would appear without the cutoff there are now explicit nonlocal potential terms. (Note that the Coulomb interaction in nonrelativistic QFT is nonlocal. Thus nonlocalities are nothing particularly strange. See H. Ekstein, Phys. Rev. 117, 1590-1595 (1960) for more on nonlocal interactions and relations to the S-matrix. But actually one does not need to care about locality or not, since the regularized interaction in the momentum representation is mathematically ok and one can do everything else in this representation without interpreting it in space.) More precisely, one starts with the smeared Lagrangian interaction defined by the cutoff, uses the representation of the S-matrix as a time-ordered exponential to work out the corresponding Hamiltonian interaction in the interaction picture, and takes this as definition of the regularized dynamics. (Note that Haag's theorem, which asserts that a nontrivial Lorentz-invariant theory satisfying microlocality cannot have an interaction picture, does not apply since the theory with cutoff is neither Lorentz invariant nor microlocal.) From here on, one can do standard perturbation theory without encountering any infinity at all; one gets meaningful formulas throughout the whole renormalization procedure, but everything is dependent on the cutoff Lambda. All contributions to the S-matrix elements of this regularized theory are finite, and give (after analytic continuation back to real time) the S-matrix of the regularized interaction. The result is an asymptotic series S(g,Lambda) for the S-matrix of the regularized interaction, with finite, computable coefficients. This S-matrix is unitary and has all properties one would like to have, except that, because of the cutoff, it is only approximately Lorentz invariant. Of course, for a nonlocal theory in position representation, one gets more complicated Feynman rules than those traditionally written down in the (nonexisting) formal limit Lambda-->inf. In momentum space, the formulas become the standard formulas, but with explicit cutoff included. Thus it is much simpler to always work in the momentum representation. To restore Lorentz invariance, one uses a running coupling constant g=g(Lambda,mu) which, for fixed renormalization point mu (a vector of the same dimension of g containing the free constants in the matching of the renormalization conditions), is uniquely determined (for any fixed renormalization scheme) as the solution of a renormalization group equation whose coefficients are also defined as a (presumably even convergent) asymptotic expansion. Having this, one can take the limit S(mu) = lim_{Lambda to inf} S(g(Lambda,mu),Lambda) which is an asymptotic series in hbar with finite, computable coefficients when the theory is renormalizable, and turns out to be Lorentz invariant and microlocal. Thus one gets the desired Lorentz invariant, microlocal theory as a perturbatively well-defined limit of perturbatively well-defined but not Lorentz invariant or microlocal theories. At the very end one can pass to the limit, but not earlier. The only infinity encountered is not worse than the infinity encountered in defining Riemann integrals over the real line, where one also gets a finite limit by letting a finite cutoff go to infinity. The real mathematical difficulties in QFT are not in the renormalization procedure but in giving a nonperturbative construction of the S-matrix S(mu), i.e., in proving that the above limits exist not only order by order in the coupling constant g but as a function defined for finite values of g.