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A theory of large distance physics

1978–
Daniel Friedan

In my 1980 PhD thesis, I showed that the renormalization group flow of 2d quantum field theory produces the solutions to Einstein's equation for the gravitational metric of space-time. The coupling constants of the general 2d nonlinear field theory comprise a Riemannian metric on a manifold. The 2d renormalization group drives the Riemannian metric to a solution of Einstein's equation.

I pursued the idea that the 2d renormalization group might produce the laws of physics. This led into string theory in the early 1980s. In 2002 I formulated a theory of the quantum string background produced by a “quantization” of the 2d renormalization group.

Unfortunately, I have been unable to derive anything from this theory that can be checked against experiment. It does not solve the basic problem that string theory has too many classical backgrounds (there are too many 2d conformal field theories). I hoped that the characterization of the quantum string background would predict some subtle observable effects. But I could not make the idea work.

One idea, expressed in the paper Cosmology from the two-dimensional renormalization group acting as the Ricci flow, might still work. Solutions of the 2d renormalization group fixed point equation are in principle more general than solutions of the Einstein equation. This might show up in the cosmological solution. To test the idea, one needs a first principles theory of cosmology that does not quite work with the standard Einstein equation, but does work when the fixed point equation of the 2d renormalization group is substituted. This led me into the first principles cosmology project.

My thesis was partially incorporated into string theory in the early 1980s. The 2d quantum field theory is the string world-sheet. The fixed point equation of the 2d rg — 2d scale invariance — is the consistency condition for the perturbative string S-matrix recipe. String theory provided more elaborate 2d nonlinear models whose couplings included space-time gauge and matter fields in addition to a Riemannian metric. The 2d couplings parametrize the string backgrounds. The 2d rg drives the string backgrounds to the classical solutions, the solutions of the classical space-time field equations.

Missing was a notion of a quantum string background related to a quantum field theory in space-time. There was no mechanism in string theory that produced quantum field theory — only a correspondence between the perturbative string S-matrix at low momenta and the perturbative S-matrix derived from the classical space-time field theory corresponding to the 2d rg fixed point.

In 2002, I proposed a mechanism that produces a space-time quantum field theory at large distances (in Planck units) and a quantum string background for string scattering at shorter distances, in such a way that they agree at the intermediate scale. This seems to me a perfectly acceptable form for a fundamental theory of physics to take. An S-matrix is not adequate as a complete theory of physics. Everything we know about the real physical world is described by quantum mechanics and its classical approximation. On the other hand, we have no reason to expect or need a quantum mechanical description of short distance physics, at distances many orders of magnitude smaller than we have any chance of observing experimentally. An S-matrix would be a quite suitable description of small distance physics.

The proposed mechanism is a 2d quantum field theory, the λ-model, a natural 2d nonlinear model whose target space is the space of classical space-time fields. The a priori measure of the 2d model is a functional integral on the classical fields — a quantum field theory in space-time. Perturbatively, this quantum field theory is the canonically quantized field theory. Semi-classical effects in the λ-model might produce non-canonical effects in the space-time quantum field theory. Winding modes would produce non-canonical degrees of freedom. Two-dimensional instantons would produce non-canonical couplings. Winding modes will be present in 4d space-time when there is an SU(2) gauge symmetry group. Two-dimensional instantons will be present when there is an SU(N) gauge symmetry, N ≥ 2. Predictions of non-canonical effects in the standard model might give a way to test the proposed mechanism.

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