Dynamical renormalization group approach to relaxation in quantum field theory

The real time evolution and relaxation of expectation values of quantum fields and of quantum states are computed as initial value problems by implementing the dynamical renormalization group (DRG). Linear response is invoked to set up the renormalized initial value problem to study the dynamics of the expectation value of quantum fields. The perturbative solution of the equations of motion for the field expectation values of quantum fields as well as the evolution of quantum states features secular terms, namely terms that grow in time and invalidate the perturbative expansion for late times. The DRG provides a consistent framework to resum these secular terms and yields a uniform asymptotic expansion at long times. Several relevant cases are studied in detail, including those of threshold infrared divergences which appear in gauge theories at finite temperature and lead to anomalous relaxation. In these cases the DRG is shown to provide a resummation akin to Bloch-Nordsieck but directly in real time and that goes beyond the scope of Bloch-Nordsieck and Dyson resummations. The nature of the resummation program is discussed in several examples. The DRG provides a framework that is consistent, systematic, and easy to implement to study the non-equilibrium relaxational dynamics directly in real time that does not rely on the concept of quasiparticle widths.     

quantum field theory as group algebra

A group theoretical approach to dynamical quantization in general, and quantum field theory in particular, is developed. This approach opens possibilities of new quantization schemes. Some of these schemes are discussed in detail. They offer certain advantages such as relaxation of the conventional principles of unitarity and causality on the one hand and the possibility to attach some meaning to the formal solutions of the equations of unitarity and causality in terms of functional integrals on the other.     

Axioms for renormalization in Euclidean quantum field theory

A set of axioms which fix Euclidean renormalizations up to a finite renormalization is proposed. There exists a one to one correspondence between Euclidean renormalizations and renormalizations in Minkowski space-time satisfying Hepp's axioms. No restrictions on masses are imposed.     

The renormalization group and fractional Brownian motion

We find that in generic field theories the combined effect of fluctuations and interactions leads to a probability distribution function which describes fractional Brownian motion (fBM) and “complex behavior”. To show this we use the renormalization group as a tool to improve perturbative calculations, and check that beyond the classical regime of the field theory (i.e., when no fluctuations are present) the non-linearities drive the probability distribution function of the system away from classical Brownian motion and into a regime which to the lowest order is that of fBM. Our results can be applied to systems away from equilibrium and to dynamical critical phenomena. We illustrate our results with two selected examples: a particle in a heat bath, and the KPZ equation.     

Charge renormalization in quantum field theory by the unitary clothing transformation method

The clothing procedure in several lowest orders in the coupling constant has been implemented using the unitary clothing transformation method. Within a simple field theory model, including interaction of charged spinless nucleons and scalar mesons, an expression for the charge shift is obtained, which is determined by the operators beyond the energy shell.     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

The renormalization group of the model ofA 4-coupling in the abstract approach of quantum field theory

For the model ofA ⁴-interaction the postulates of the renormalization group are stated within the abstract approach of quantum field theory. In the massive case these postulates follow if an on-shell formulation of the model is assumed to exist. For the massless model the postulates of the renormalization group imply that the propagator has a pole at momentum zero. Consequently there is no dynamic mass generation and the propagator is normalizable on the mass shell. It is shown that theS-matrix elements scale with canonical dimensions. A general method of rescaling parameter values is developed which takes into account the possibility of propagator zeros and stationary points of the effective coupling.     

BRST string field theory from Wilson's renormalization group

Wilson “renormalization group” fixed point equations are derived for BRST symmetry in two-dimensional field theory. Their usefulness as equations of motions for string field theory is discussed.     

A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics

We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoir-system coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique [1-4] but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis [5] which has several technical advantages. Besides the formal set-up of the RG equations for generic problems of dissipative quantum mechanics, we demonstrate the method by applying it to the nonequilibrium isotropic Kondo model. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.     

Gauge fixing and renormalization in topological quantum field theory

We show how Witten's topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero.     

Two-loop renormalization group equations in a general quantum field theory: (III). Scalar quartic couplings

The two-loop β-functions for the scalar quartic couplings are computed in a general renormalizable quantum field theory with scalar, $\text{spin}\text{-}\text{1}\text{2}$, and (vector) gauge fields associated with a general gauge group G, using dimensional regularization and modified minimal subtraction (^?MS). A more explicit form is given for the two-loop β-function of the quartic coupling of the Higgs doublet in the minimal QCD electroweak theory based on SU(3) × SU(2) × U(1).     

The Heisenberg ferromagnet as a quantum field theory

We consider a lattice of spin 1/2 ions, described by the discrete form of the current commutation relationsJ _i J _(i) =1/2, [J _i ,J _i ]=i _ij J _i where =1, 2, 3 andi label the lattice sites. The algebra is realized as the Clifford algebra over a Hilbert space. The equations of motion are specified by a formal Hamiltonian of the Heisenberg form: , wheref _ij 0 and only a finite numberQ of ions are linked to any given lattice site. We prove that the Hamiltonian is non-negative in a representation of , and has a ground state exhibiting ferromagnetism. The time displacement group acts continuously on , inducing automorphisms. is asymptotically abelian with respect to the space translations of the lattice.The model is an example of an algebraic quantum field theory and possesses a broken symmetry, the rotation group 0(3). The consequent Goldstone theorem is proved, namely, there is no energy gap in the spectrum ofH.     

Monte Carlo renormalization group study of φ field theory

A previously proposed general method for evaluating block renormalized coupling constants within the framework of the Monte Carlo renormalization group (MCRG) is applied to φ⁴ field theory. The flow diagrams, fixed points, and critical exponents are determined in two, three and four dimensions. Results in four dimensions are consistent with the idea that φ⁴ field theory is trivial (non-interacting) in the continuum limit. The possibility of using MCRG techniques to ascertain whether a general non-asymptotically free theory is trivial or not is also discussed.     

The renormalization group and quantum edge states

The role of edge states in phenomena like the quantum Hall effect is well known, and the basic physics has a wide field-theoretic interest. In this paper we introduce a new model exhibiting quantum Hall-like features. We show how the choice of boundary conditions for a one-particle Schrödinger equation can give rise to states localized at the edge of the system. We consider both the example of a free particle and the more involved example of a particle in a magnetic field. In each case, edge states arise from a non-trivial scaling limit involving the boundary condition, and chirality of the boundary condition plays an essential role. Second quantization of these quantum mechanical systems leads to a multi-particle ground state carrying a persistent current at the edge. We show that the theory quantized with this vacuum displays an “anomaly” at the edge which is the mark of a quantized Hall conductivity in the presence of an external magnetic field. These models therefore possess characteristics which make them indistinguishable from the quantum Hall effect at macroscopic distances. We also offer interpretations for the physics of such boundary conditions which may have a bearing on the nature of the excitations in these models.     

Real-space renormalization group study of a two-dimensional nonlinear massless field

The magnetic susceptibility of CuF₂·2H₂O has been measured as a function of magnetic field from 1.5 to 10 K. The spin-flop transition was observed and its value extrapolated to zero temperature is H_SF(0) = 30.5 kOe. This critical field is in very good agreement with data obtained from zero field measurements.     

Monte Carlo renormalization group study of φ4 field theory

A previously proposed general method for evaluating block renormalized coupling constants within the framework of the Monte Carlo renormalization group (MCRG) is applied to φ⁴ field theory. The flow diagrams, fixed points, and critical exponents are determined in two, three and four dimensions. Results in four dimensions are consistent with the idea that φ⁴ field theory is trivial (non-interacting) in the continuum limit. The possibility of using MCRG techniques to ascertain whether a general non-asymptotically free theory is trivial or not is also discussed.     

Dynamical mean field theory with the density matrix renormalization group

A new numerical method for the solution of the dynamical mean field theory's self-consistent equations is introduced. The method uses the density matrix renormalization group technique to solve the associated impurity problem. The new algorithm makes no a priori approximations and is only limited by the number of sites that can be considered. We obtain accurate estimates of the critical values of the metal-insulator transitions and provide evidence of substructure in the Hubbard bands of the correlated metal. With this algorithm, more complex models having a larger number of degrees of freedom can be considered and finite-size effects can be minimized.     

The density matrix renormalization group for ab initio quantum chemistry

During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. Its underlying wavefunction ansatz, the matrix product state (MPS), is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS, the rank of the decomposition, controls the size of the corner of the many-body Hilbert space that can be reached with the ansatz. This parameter can be systematically increased until numerical convergence is reached. The MPS ansatz naturally captures exponentially decaying correlation functions. Therefore DMRG works extremely well for noncritical one-dimensional systems. The active orbital spaces in quantum chemistry are however often far from one-dimensional, and relatively large virtual dimensions are required to use DMRG for ab initio quantum chemistry (QC-DMRG). The QC-DMRG algorithm, its computational cost, and its properties are discussed. Two important aspects to reduce the computational cost are given special attention: the orbital choice and ordering, and the exploitation of the symmetry group of the Hamiltonian. With these considerations, the QC-DMRG algorithm allows to find numerically exact solutions in active spaces of up to 40 electrons in 40 orbitals.