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Wilson Renormalization Group formulation of Real Time thermal field theories

Massimo Pietroni

Published Jan 1, 1996• 15 citations
OpenAlex

Abstract

We apply Renormalization Group techniques to the Real Time formulation of thermal field theory. Due to the separation between the T = 0 and the T = 0 parts of the propagator in this formalism, one can derive exact evolution equations for the Green functions describing the effect of integrating out thermal fluctuations of increasing wavelengths, the initial conditions being the renormalized Green functions of the T = 0 theory. As a first application, we study the phase transition for the real scalar theory, computing the order of the transition, the critical temperature, and critical exponents, in different approximations to the evolution equations for the scalar potential. CERN-TH/96-23 The dynamics of a second order, or weakly first order, phase transition is governed by long wavelength fluctuations of the order parameter. These fluctuations, whose length scale is much larger than the inverse temperature of the system, are essentially classical, since the

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The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].

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To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.

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The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is the outgrowth of one hundred and fifty years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different behavior separated by abrupt phase transitions. These qualitative differences seem to be present in experimentally observed condensed-matter systems. However, the "extended singularity theorem" in statistical mechanics shows that sharp changes can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters. Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales. The renormalization (semi-)group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson's theory is the first approach to phase transitions to agree with the extended singularity theorem. Some of the history of the study of these correlations and singularities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications are summarized.

arXiv:1102.3705•Feb 17, 2011•5 citations
We apply Renormalization Group techniques to the Real Time formulation of thermal field theory. Due to the separation between the T = 0 and the T = 0 parts of the propagator in this formalism, one can derive exact evolution equations for the Green functions describing the effect of integrating out thermal fluctuations of increasing wavelengths, the initial conditions being the renormalized Green functions of the T = 0 theory. As a first application, we study the phase transition for the real scalar theory, computing the order of the transition, the critical temperature, and critical exponents, in different approximations to the evolution equations for the scalar potential. CERN-TH/96-23 The dynamics of a second order, or weakly first order, phase transition is governed by long wavelength fluctuations of the order parameter. These fluctuations, whose length scale is much larger than the inverse temperature of the system, are essentially classical, since the
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
The perturbative approach to the description of long-wavelength excitations at high temperature breaks down near the critical point of a second order phase transition. We study the dynamics of these excitations in a relativistic scalar field theory at and near the critical point via a renormalization group approach at high temperature and an $\ensuremath{\epsilon}$ expansion in $d=5\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions. The long-wavelength physics is determined by a nontrivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum p are ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{p}^{z}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p},$ respectively, and the group velocity of quasiparticles ${v}_{g}\ensuremath{\sim}{p}^{z\ensuremath{-}1}$ vanishes in the long-wavelength limit at the critical point. Away from the critical point for $T\ensuremath{\gtrsim}{T}_{c}$ we find ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}z}[1+(p\ensuremath{\xi}{)}^{2z}{]}^{1/2}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p}(p\ensuremath{\xi}{)}^{2z}/[1+(p\ensuremath{\xi}{)}^{2z}]$ with $\ensuremath{\xi}$ the finite temperature correlation length $\ensuremath{\xi}\ensuremath{\propto}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}.$ The new dynamical exponent z results from anisotropic renormalization in the spatial and time directions. For a theory with O(N)O(N)O(N) symmetry we find $z=1+\ensuremath{\epsilon}{(N+2)/(N+8)}^{2}+\mathcal{O}({\ensuremath{\epsilon}}^{2}).$ This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.
It is explained how field-theoretic methods and the dynamic renormalisation group (RG) can be applied to study the universal scaling properties of systems that either undergo a continuous phase transition or display generic scale invariance, both near and far from thermal equilibrium. Part 1 introduces the response functional field theory representation of (nonlinear) Langevin equations. The RG is employed to compute the scaling exponents for several universality classes governing the critical dynamics near second-order phase transitions in equilibrium. The effects of reversible mode-coupling terms, quenching from random initial conditions to the critical point, and violating the detailed balance constraints are briefly discussed. It is shown how the same formalism can be applied to nonequilibrium systems such as driven diffusive lattice gases. Part 2 describes how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then used to analyse simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which
The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is the outgrowth of one hundred and fifty years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different behavior separated by abrupt phase transitions. These qualitative differences seem to be present in experimentally observed condensed-matter systems. However, the "extended singularity theorem" in statistical mechanics shows that sharp changes can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters. Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales. The renormalization (semi-)group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson's theory is the first approach to phase transitions to agree with the extended singularity theorem. Some of the history of the study of these correlations and singularities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications are summarized.