Aspects of the functional renormalisation group

We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being NPI quantities. We discuss optimisation and derive a functional optimisation criterion. Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation.     

On the symmetry of real-space renormalisation

A geometric structure, arising from the embedding into a Hilbert space of the parametrised probability measure for a given lattice model, is applied here to study the symmetry properties of real-space renormalisation group (RG) flow. In the projective state space this flow is shown to have two contributions: a gradient term, which generates a projective automorphism of the state space for each given length scale; and a correction term due to the scale change. We argue that this structure implies the absence of any symmetry of a geodesic type for the RG flow when restricted to the parameter space submanifold of the state space. This is demonstrated explicitly via a study of the one-dimensional Ising model in an external field. In this example we construct exact expressions for the beta functions associated with the flow induced by infinitesimal rescaling. These constitute a generating vector field for RG diffeomorphisms on the parameter space manifold, and we analyse the symmetry properties of this transformation. The results indicate an approximate conformal Killing symmetry near the critical point, but no generic symmetry of the RG flow globally on the parameter space.     

Series expansion solution of the Wegner-Houghton renormalisation group equation

The momentum independent projection of the Wegner-Houghton renormalisation group equation is solved with power series expansion. Convergence rate is analyzed for then-vector model. Further evidence is presented for the first order nature of the chiral symmetry restoration at finite temperature in QCD with 3 light flavors.     

Real-space renormalisation approach to the Anderson localisation

A real-space renormalisation method is proposed for random systems. The equation for the Green function in real space is reduced to that for the Green function in renormalised space after the nth decimation transformation to obtain the renormalised Hamiltonian.     

On the reality of the eigenvalues of the linearized renormalisation group transformations on a lattice

We show that the eigenvalues of the T-matrix of the renormalisation group equations on a lattice linearized around a fixed point, are always real although the matrix is non-symmetric.     

Form factors in finite volume I: Form factor bootstrap and truncated conformal space

We describe the volume dependence of matrix elements of local fields to all orders in inverse powers of the volume (i.e., only neglecting contributions that decay exponentially with volume). Using the scaling Lee–Yang model and the Ising model in a magnetic field as testing ground, we compare them to matrix elements extracted in finite volume using truncated conformal space approach to exact form factors obtained using the bootstrap method. We obtain solid confirmation for the form factor bootstrap, which is different from all previously available tests in that it is a non-perturbative and direct comparison of exact form factors to multi-particle matrix elements of local operators, computed from the Hamiltonian formulation of the quantum field theory. We also demonstrate that combining form factor bootstrap and truncated conformal space is an effective method for evaluating finite volume form factors in integrable field theories over the whole range in volume.     

Gradient properties of the renormalisation group equations in multicomponent systems

The existence of a metric, which enables the renormalisation group β functions of a multicomponent field theory to be written as a gradient, has very important implications for the asymptotic behavior of the renormalisation group equations. It is shown that a very simple metric exists in a field theory with n-component Bose fields and arbitrary φ⁴ interaction, when the β functions are calculated perturbatively up to and including the 2-loop diagrams. This same metric is shown to work for all irreducible diagrams, but it must and can be modified to accommodate reducible 3-loop contributions. The prospects and outlook of this aspect of the renormalisation group are discussed.     

Gauging the conformal group

It is demonstrated that Kibble’s method of gauging the Poincaré group can be applied to the gauging of the conformal group. The action of the gauge transformations is the action of general spacetime diffeomorphisms (or coordinate transformations) combined with a local action of an 11-parameter subgroup of SO(4,2). Because the translational subgroup is not an invariant subgroup of the conformal group the appropriate generalisation of the derivative of a physical field is not a covariant derivative in the usual sense, but this does not lead to any inconsistencies.     

Functional representation for the two-dimensional conformal group

A functional projective representation for the two-dimensional conformal group is explicitly constructed. The representation functionals are propagation kernels for self-dual fields.     

Epsilon expansion for multicritical fixed points and exact renormalisation group equations

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the nonlinear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order ε and also for the field anomalous dimension to order ε². An exact marginal operator for the full RG equations is also constructed.     

On the four euclidean conformal group structure of the Sturmian operator

The Sturmian method consists in replacing the eigenvalue problem for the interacting particle Hamiltonian (Schrödinger equation) by the spectral problem for the so-called Sturmian operator (coupling constant quantization). The latter is almost always compact. The main result of this work lies in the disclosure of the algebraic nature of the Sturmian operator as a linear superposition of group representation operators. This group theoretical understanding leads to a large field of physical applications.     

Distributions on Minkowski space and their connection with analytic representations of the conformal group

Unitary analytic representations of the conformal group are relized on Hilbert spaces of holomoprhic or antiholomorphic functions over a tube domain in complex Minkowski space. The distributional boundary values of these functions are tempered distributions on real Minkowski space. The representations are characterized by an integral scale dimension labeln and two spin labelsj ₁ andj ₂. The connection between the dimensionn and the degree of singularity of the tempered distribution is investigated. We propose an application to inclusive reactions of elementary particles.     

Percolation renormalisation group method and its application to the Potts model

A percolation renormalisation group method is proposed to calculate the free energy and other physical quantities for the q-state Potts model (QPM). The method gives the exact free energy and the critical exponent for the one-dimensional QPM and may be extended to other interacting systems.     

A “renormalisation group improved” study of the gauge hierachy problem

In grand unified theories higher order contributions to the effective potential contain terms of the type $\text{(λ}\text{log}\text{υ}{}_1{}^2\text{υ}{}_2{}^2\text{)}{}^{\mathrm{n}}$ where λ is some Higgs coupling and υ₁, υ₂ are the vacuum expectation values of certain scalar fields whose ratio we require phenomenologically to be extremely large ($\text{υ}{}_1{}^2\text{υ}{}_2{}^2\text{≈ 10}{}^{24}$ or so). We study these terms in a toy model and find taht they can be absorbed into suitably defined coupling constant whose evolution equations (in this leading logarithm approximation) seem to indicate that no natural large hierachy is possible. For consistency of the calculation, however, certain “non-leading” terms must also be included and then it is found that the evolution equations have to be reinterpreted, suggesting a very natural explanation of the large ratio of vacuum expectation values.     

On the phase space approach to complexity

We elaborate in some detail on a new phase space approach to complexity, due to Y.-C. Zhang. We show in particular that the connection between maximal complexity and power law noise or correlations can be derived from a simple variational principle. For a 1D signal we find 1/f noise, in accordance with Zhang.     

Representing functionally the two-dimensional conformal group

Projective representations of the two-dimensional conformal group are explicitly constructed in terms of propagation kernels. The representation functionals are then used to study the effect of conformal transformations on the states of a free massless scalar field theory.     

Phase boundary of the square-lattice Ising antiferromagnet by real space renormalization group methods

The transition temperature of the square lattice Ising antiferromagnet at finite magnetic field is calculated by three different approximations within the real space renormalization group approach. The most refined approximation is an extension of Kadanoff's potential moving method to a larger cell-size. The results of this approximation are in good agreement with recent Monte Carlo simulations and the Müller-Hartmann/Zittartz conjecture for the phase boundary.