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Is there a method to renormalize a theory without using perturbative expansions for the divergences? For example, is there a method to get masses and other renormalized quantities without using expansions and counterterms?

O have heard about Lattice Gauge theory

but what other solutions of examples of non perturbative renormalization (numerical or analytical ) are there?

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up vote 4 down vote accepted

The standard nonperturbative way (that provided rigorous constructions in 1+1D and 1+2D QFTs) is constructing the Euclidean (imaginary time) field theory as a limit of lattice theories, and then using analytic continuation to real time via the Osterwalder--Schrader theorem.

In 1+3D, there is so far no rigorous construction of an interacting QFT, but neither is there a corresponding no-go theorem.

In 1+1D, there are also lots of exactly solvable QFTs, where the nonperturbative solution is obtained by the quantum inverse scattering method.

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Is there a rigorous interacting QFT in euclidean 4 dimensions? – drake Aug 14 '12 at 19:26
No known one with $O(4)$ invariance and satisfying reflection positivity, which are necessary conditions for a Euclidean theory corresponding to a physical QFT. – Arnold Neumaier Aug 15 '12 at 9:35

Actually, Wilson (who received the Nobel for his work on renormalisation -- though that's not to say he was the first one to think of it) worked on numerical renormalisation. The starting point is to formulate (in an abstract and somewhat formal (i.e. not very computable) way) the Exact Renormalisation Group Equation.

There are various ways to formulate this, one choice is to either fix bare action and integrating out the higher frequencies to get an effective action, or to fix the desired action and take the cut-off to infinity; the other choice is in the form of the cut-off --- one might want nice properties such as the regulator obeying the symmetries of the action (Poincaré, gauge, etc.).

Up to this point, everything is exact, and in the words of a lecturer of mine, "therefore useless". The other problem is then to solve said equation, which is actually a set of uncountably infinitely many coupled non-linear differential equations (or even differentio-integral equations). The solution, numerically or otherwise, is then a separate problem of approximating suitably. On this front there is not a clear answer, and is the main focus of research. One may easily implement the kinds of analytic approximations that one finds in textbooks (e.g. $n$-th loop, $n$-th order in perturbation, etc.) but there is a lack of real understanding of how the various choices affect the outcome, and when they are appropriate for a given problem.

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Non-perturbative RG methods exist and are useful. They have been used to understand several phenomena in (nonequilibrium) statistical mechanics.

  1. NPRG of the KPZ equation
  2. Reaction-Diffusion processes
  3. Critical dynamics
  4. Nonequilibrium steady-states
  5. Voter models
  6. Branching and annihilating random walks
  7. 3D Ising model

There are also nice reviews and books on the subject.

EDIT (11 Apr 2012)

Another recent review.

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The problem with renormalizations in QFT is that, while the conter-terms are known, they cannot be joined with the bare Lagrangian, but are supposed to be treated perturbatively. A quote of Lagrangian from A. Zee (Edition of 2003, page 175):$$L= \left [ \frac{1}{2}[(\partial \phi)^2 -m_P ^2\phi^2]-\frac{\lambda_P}{4!}\phi^4 \right ] + A(\partial \phi)^2 +B\phi^2+C\phi^4\quad (1)$$ Lagrangian in the square brackets gives itself wrong equations with wrong solutions. The wrongness can be represented as "corrections" to the mass, charge and the fields strength. The rest is OK. In order to subtract these unnecessary and harmful corrections in a systematic way, one introduces the counter-terms (everything that is out of big square brackets in (1)) and joins them to the perturbation $\frac{\lambda_P}{4!}\phi ^4$ in the perturbation theory. Then, with appropriate choice of $A$, $B$, and $C$, one can cancel or subtract the harmful corrections in each order appearing due to the "initial interaction" $\frac{\lambda_P}{4!}\phi ^4$. These heavy techniques ("know how") are offered nowadays because of rejecting recognition of the wrongness of the original Lagrangian which is based on the wrong understanding of physics and on mathematical errors.

An example of exact renormalization is presented in my toy model here: and here. The subtraction of counter-terms is done exactly in the wrong (model) Lagrangian, so the reminder is a physical Lagrangian with no problem.

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