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The influence of short-range Coulomb correlations on the Mott transition in the single-band Hubbard model at half-filling is studied within cellular dynamical mean field theory for square and triangular lattices. Finite-temperature exact diagonalization is used to investigate correlations within two-, three-, and four-site clusters. Transforming the non-local self-energy from a site basis to a molecular orbital basis, we focus on the inter-orbital charge transfer between these cluster molecular orbitals in the vicinity of the Mott transition. In all cases studied, the charge transfer is found to be small, indicating weak Coulomb induced orbital polarization despite sizable level splitting between orbitals. These results demonstrate that all cluster molecular orbitals take part in the Mott transition and that the insulating gap opens simultaneously across the entire Fermi surface. Thus, at half-filling we do not find orbital-selective Mott transitions, nor a combination of band filling and Mott transition in different orbitals. Nevertheless, the approach towards the transition differs greatly between cluster orbitals, giving rise to a pronounced momentum variation along the Fermi surface, in agreement with previous works. The near absence of Coulomb induced orbital polarization in these clusters differs qualitatively from single-site multi-orbital studies of several transition metal oxides, where the Mott phase exhibits nearly complete orbital polarization as a result of a correlation driven enhancement of the crystal field splitting. The strong single-particle coupling among cluster orbitals in the single-band case is identified as the source of this difference.

We analyse 4-dimensional massive $\vp^4$ theory at finite temperature T in the imaginary-time formalism. We present a rigorous proof that this quantum field theory is renormalizable, to all orders of the loop expansion. Our main point is to show that the counterterms can be chosen temperature independent, so that the temperature flow of the relevant parameters as a function of $T$ can be followed. Our result confirms the experience from explicit calculations to the leading orders. The proof is based on flow equations, i.e. on the (perturbative) Wilson renormalization group. In fact we will show that the difference between the theories at T>0 and at T=0 contains no relevant terms. Contrary to BPHZ type formalisms our approach permits to lay hand on renormalization conditions and counterterms at the same time, since both appear as boundary terms of the renormalization group flow. This is crucial for the proof.

A light front field theory for finite temperature and density is currently being developed. It will be used here to describe the transition region from quark matter to nuclear matter relevant in heavy ion collisions and in the early universe. The energy regime addressed is extremely challenging, both theoretically and experimentally. This is because of the confinement of quarks, the appearance of bound states and correlations, special relativity, and nonlinear phenomena that lead to a change of the vacuum structure of quantum chromodynamics. In the region of the phase transition it eventually leads to a change of the relevant degrees of freedom. We aim at describing this transition from quarks to hadronic degrees of freedom in a unified microscopic approach.

We treat the model which describes "extreme black holes" moving slowly. We derive an effective lagrangian in the low energy for this model and then investigate a statistical behavior of "extreme black holes" in the finite temperature.

The trace of the heat kernel in a (D+1)-dimensional Euclidean spacetime (integer D > 1) is used to derive the free energy in finite temperature field theory. The spacetime presents a D-dimensional compact space (domain) with a (D-1)-dimensional boundary, and a closed dimension, whose volume is proportional the Planck's inverse temperature. The thermal sum appears due to topology of the closed Euclidean time. The obtained free energy in (3+1) and (2+1) dimensions contain two contributions defined by the volume of a domain and by the volume of the domain's boundary. This functional is finite and valid for arbitrary values of the Planck's inverse temperature. The absolute zero of thermodynamic temperature is forbidden topologically, and no universal low temperature asymptotics of the free energy can exist.

We reassess the problem of renormalization in finite temperature field theory (FTFT). A new point of view elucidates the relation between the ultraviolet divergences for T=0 and $T \not= 0$ theories and makes clear the reason why the ultraviolet behavior keeps unaffected when we consider the FTFT version associated to a given quantum field theory (QFT). The strength of the derivation one lies on the H\"ormander's criterion for the existence of products of distributions in terms of the wavefront sets of the respective distributions. The approach allows us to regard the FTFT both imaginary and real time formalism at once in a unified way in the contour ordered formalism.

In this work we study in a formal way the density dependent hadron field theory at finite temperature for nuclear matter. The thermodynamical potential and related quantities, as energy density and pressure are derived in two different ways. We first obtain the thermodynamical potential from the grand partition function, where the Hamiltonian depends on the density operator and is truncated at first order. We then reobtain the thermodynamical potential by calculating explicitly the energy density in a Thomas-Fermi approximation and considering the entropy of a fermi gas. The distribution functions for particles and antiparticles are the output of the minimization of the thermodynamical potential. It is shown that in the mean field theory the thermodynamical consistency is achieved. The connection with effective chiral lagrangians with Brown-Rho scaling is discussed.

We study thermal effects for a noncommutative real scalar field in 2+1 dimensions including a Grosse-Wulkenhaar term. Using a perturbative expansion for the free energy, we deduce some general properties of the corresponding contributions, in the thermodynamic limit. We show that the model can be consistently interpreted as defined on a finite volume, which is naturally determined by the noncommutativity scale.

We quantize a scalar field at finite temperature T in the background of a classical black hole, adopting 't Hooft's ``brick wall'' model with generic mixed boundary conditions at the brick wall boundary. We first focus on the exactly solvable case of two dimensional space-time. As expected, the energy density is integrable in the limit of vanishing brick wall thickness only for T=T_H - the Hawking temperature. Consistently with the most general stress energy tensor allowed in this background, the energy density shows a surface contribution localized on the horizon. We point out that the usual divergences occurring in the entropy of the thermal atmosphere are due to the assumption that the third law of thermodynamics holds for the quantum field in the black hole background. Such divergences can be avoided if we abandon this assumption. The entropy density also has a surface term localized on the horizon, which is open to various interpretations. The extension of these results to higher space-time dimensions is briefly discussed.

I discuss three subjects in thermal field theory: why in \sun gauge theories the \zn symmetry is broken at high (instead of low) temperature, the possible singularity structure of gauge variant propagators, and the problem of how to compute the viscosity from the Kubo formula.

I discuss three subjects in thermal field theory: why in \sun gauge theories the \zn symmetry is broken at high (instead of low) temperature, the possible singularity structure of gauge variant propagators, and the problem of how to compute the viscosity from the Kubo formula.

The critical exponents and the critical amplitude ratio of the scalar model are determined using finite-temperature field theory with auxiliary mass. A new numerical method is developed to solve an evolution equation. The results are discussed in comparison with values obtained from the other methods.

The effective field theory approach to high temperature field theory can be used to study the phase transition in theories with spontaneously broken symmetry. I construct a sequence of two effective three--dimensional field theories which are valid on successively longer distance scales for a specific model: $N$ charged scalars coupled to a U(1) gauge field. The resulting effective Lagrangian can be used to investigate the phase transition, in particular the order of the phase transition as a function of $N$, using lattice simulations.

We study symmetry restoration at finite temperature in the theory of a charged scalar field interacting with a constant, external magnetic field. We compute the finite temperature effective potential including the contribution from ring diagrams. We show that in the weak field case, the presence of the field produces a stronger first order phase transition and that the temperature for the onset of the transition is lower, as compared to the case without magnetic field.

In this paper, a real-time formulation of light-cone pp-wave string field theory at finite temperature is presented. This is achieved by developing the thermo field dynamics (TFD) formalism in a second quantized string scenario. The equilibrirum thermodynamic quantities for a pp-wave ideal string gas are derived directly from expectation values on the second quantized string thermal vacuum. Also, we derive the real-time thermal pp-wave closed string propagator. In the flat space limit it is shown that this propagator can be written in terms of Theta functions, exactly as the zero temperature one. At the end, we show how supestrings interactions can be introduced, making this approach suitable to study the BMN dictionary at finite temperature.

Finite Temperature Field Theory - Ashok Das.

We use a ten dimensional dual string background to aspects of the physics large N four dimensional SU(N) gauge theory, where its fundamental quarks are charged under a background electric field. The theory is N=2 supersymmetric for vanishing temperature and electric field. At zero temperature, we observe that the electric field induces a phase transition associated with the dissociation of the mesons into their constituent quarks. This is an analogue of an insulator-metal transition, since the system goes from being an insulator with zero current (in the applied field) to a conductor with free charge carriers (the quarks). At finite temperature this phenomenon persists, with the dissociation transition become subsumed into the more familiar meson melting transition. Here, the dissociation phenomenon reduces the critical melting temperature.

We consider the finite temperature metal-insulator transition in the half filled paramagnetic Hubbard model on the infinite dimensional Bethe lattice. A new method for calculating the Dynamical Mean Field Theory fixpoint surface in the phase diagram is presented and shown to be free from the convergence problems of standard forward recursion. The fixpoint equation is then analyzed using dynamical systems methods. On the fixpoint surface the eigenspectra of its Jacobian is used to characterize the hysteresis boundaries of the first order transition line and its second order critical end point. The critical point is shown to be a cusp catastrophe in the parameter space, opening a pitchfork bifurcation along the first order transition line, while the hysteresis boundaries are shown to be saddle-node bifurcations of two merging fixpoints. Using Landau theory the properties of the critical end point is determined and related to the critical eigenmode of the Jacobian. Our findings provide new insights into basic properties of this intensively studied transition.

We use relative zeta functions technique of W. Muller \cite{Mul} to extend the classical decomposition of the zeta regularized partition function of a finite temperature quantum field theory on a ultrastatic space-time with compact spatial section to the case of non compact spatial section. As an application, we study the case of Schr\"odinger operators with delta like potential, as described by Albeverio & alt. in \cite{AGHH}.

Convenient Cutkosky-like diagrammatic rules for computing the spectral densities of arbitrary two-point correlation functions in finite temperature field theory are derived. The approach is based on an explicit analytic continuation of imaginary-time Feynman diagrams and avoids the complications of real-time finite temperature perturbation theory. The application of this method to the perturbative evaluation of transport coefficients is briefly discussed.

We derive the effective action for a moving magnetic-field-induced spin-density wave (FISDW) in quasi-one-dimensional conductors at zero and nonzero temperatures by taking the functional integral over the electron field. The effective action consists of the (2+1)D Chern-Simons term and the (1+1)D chiral anomaly term, both written for a sum of the electromagnetic field and the chiral field associated with the FISDW phase. The calculated frequency dependence of Hall conductivity interpolates between the quantum Hall effect at low frequencies and zero Hall effect at high frequencies, where the counterflow of FISDW cancels the Hall current. The calculated temperature dependence of the Hall conductivity is interpreted within the two-fluid picture, by analogy with the BCS theory of superconductivity.

Nuclear matter at finite temperature and barion density exhibits several phase transitions that could happen at the early stages of the Universe evolution and could be realized in heavy-ion or hadron-hadron collisions. Microscopic description of phase transitions is notoriously difficult because of the absence of small parameters. Here we present a general approach allowing to treat situations, when there are no small parameters. The approach is based on optimized perturbation theory and self-similar approximation theory. It allows, starting with divergent perturbation series in powers of an asymptotically small parameter, to construct expressions extrapolating asymptotic series to arbitrary values of the parameter, including its infinite limit. Examples of such approximants are: right root approximants, left root approximants, continued root approximants, exponential approximants, and factor approximants. The approach is illustrated by the phase transition of gauge symmetry breaking in a multicomponent field theory. The found critical indices are in very good agreement with Monte Carlo simulations as well as with complicated methods of Pade-Borel summation, while our approach is much simpler. The nice feature of the approach is that it gives exact values for the cases where exact solutions are known.

I show that the novel dual path integral representation for finite temperature quantum field theory proposed in [Phys. Rev. D 77, 105030 (2008), arXiv:0803.1667 ] is a well-known representation of quantum mechanics in terms of symbols of operators.

I show that the novel dual path integral representation for finite temperature quantum field theory proposed in [Phys. Rev. D 77, 105030 (2008), arXiv:0803.1667 ] is a well-known representation of quantum mechanics in terms of symbols of operators.

We present a relativistic three-body equation to investigate the properties of nucleons in hot and dense nuclear/quark matter. Within the light front approach we utilize a zero-range interaction to study the three-body dynamics. The relativistic in-medium equation is derived within a systematic Dyson equation approach that includes the dominant medium effects due to Pauli blocking and self energy corrections. We present the in-medium nucleon mass and calculate the dissociation of the three-body system.

We use the non-perturbative renormalization group to clarify some features of perturbation theory in thermal field theory. For the specific case of the scalar field theory with O(N) symmetry, we solve the flow equations within the local potential approximation. This approximation reproduces the perturbative results for the screening mass and the pressure up to order g^3, and starts to differ at order g^4. The method allows a smooth extrapolation to the regime where the coupling is not small, very similar to that obtained from a simple self-consistent approximation.

We derive finite temperature time-dependent effective actions for the phase of the pairing field, which are appropriate for a 2D electron system with both non-retarded d- and s-wave attraction. As for s-wave pairing the d-wave effective action contains terms with Landau damping, but their structure appears to be different from the s-wave case due to the fact that the Landau damping is determined by the quasiparticle group velocity v_{g}, which for d-wave pairing does not have the same direction as the non-interacting Fermi velocity v_{F}. We show that for d-wave pairing the Landau term has a linear low temperature dependence and in contrast to the s-wave case are important for all finite temperatures. A possible experimental observation of the phase excitations is discussed.

We extend to finite temperature the time-dependent effective theory for the Goldstone field (the phase of the pair field) $ \theta $ which is appropriate for a superfluid containing one species of fermions with s-wave interactions, described by the BCS Lagrangian. We show that, when Landau damping is neglected, the effective theory can be written as a local time-dependent non-linear Schr\"{o}dinger Lagrangian (TDNLSL) which preserves the Galilean invariance of the zero temperature effective theory and is identified with the superfluid component. We then calculate the relevant Landau terms which are non-local and which destroy the Galilean invariance. We show that the retarded $\theta$-propagator (in momentum space) can be well represented by two poles in the lower-half frequency plane, describing damping with a predicted temperature, frequency and momentum dependence. It is argued that the real parts of the Landau terms can be approximately interpreted as contributing to the normal fluid component.

We discuss the derivation of Newtonian potentials in the framework of quantum field theory. We focus on two particular points: on long range forces i.e. forces which fall off as $1/r^n$ being mediated by light quanta like neutrinos or Goldstone bosons and on possible temperature dependence of such forces arising in situations when the exchanged quanta are in a thermal heat bath. Examples of the latter are cosmic relic photons and relic neutrinos. Among other things, we will show that the existence of cosmic relic neutrinos modifies the long tail of the two-neutrino exchange Feinberg-Sucher force drastically. Results concerning the potential mediated by two Goldstone bosons are also presented.

This book develops the basic formalism and theoretical techniques for studying relativistic quantum field theory at high temperature and density. Specific physical theories treated include QED, QCD, electroweak theory, and effective nuclear field theories of hadronic and nuclear matter. Topics covered include: functional integral representation of the partition function, diagrammatic expansions, linear response theory, screening and plasma oscillations, spontaneous symmetry breaking, Goldstone theorem, resummation and hard thermal loops, lattice gauge theory, phase transitions, nucleation theory, quark-gluon plasma, and color superconductivity. Applications to astrophysics and cosmology cover white dwarf and neutron stars, neutrino emissivity, baryon number violation in the early universe, and cosmological phase transitions. Applications to relativistic nucleus-nucleus collisions are also included. The book is written for theorists in elementary particle physics, nuclear physics, astrophysics, and cosmology.

Scalar field theory at finite temperature is investigated via an improved renormalization group prescription which provides an effective resummation over all possible non-overlapping higher loop graphs. Explicit analyses for the lambda phi^4 theory are performed in d=4 Euclidean space for both low and high temperature limits. We generate a set of coupled equations for the mass parameter and the coupling constant from the renormalization group flow equation. Dimensional reduction and symmetry restoration are also explored with our improved approach.

The method of the real time perturbative calculations of nonequilibrium averages is generalised to the case of varying chemical potential. Calculations are performed in the frame of Zubarev's nonequilibrium density matrix approach. In this approach perturbations of temperature and other thermodynamical parameters are taken into account explicitly including nonlinear terms. It differs from the Schwinger-Keldysh approach through the choice of more general initial conditions for the density matrix.

We study the perturbation expansion of the free energy of N=4 supersymmetric SU(N) Yang-Mills at finite temperature in powers of 't Hooft's coupling g^2 N in the large N limit. Infrared divergences are controlled by constructing a hierarchy of two 3 dimensional effective field theories. This procedure is applied to the calculation of the free energy to order (g^2 N)^(3/2), but it can be extended to higher order corrections.

Conventional finite-temperature perturbation theory in which propagators have poles at $k^{2}=m^{2}$ is shown to break down at the two-loop level for self-interacting scalar fields. The breakdown is avoided by using free thermal propagators that have poles at the same energy as the exact thermal propagator. This quasiparticle energy ${\cal E}(\vec{k})$ is temperature-dependent, complex, and gauge invariant. An operator theory containing two self-adjoint scalar fields is presented in which all temperature dependence is incorporated into the Hamiltonian. No thermal traces are required to compute thermal Green functions. Choosing the spectrum of the unperturbed part of the Hamiltonian to contain the exact quasiparticle energy ${\cal E}(\vec{k})$ produces a resummed perturbation theory that has the correct poles and branch cuts. The location of the poles and cuts is explained directly in terms of the spectrum of the Hamiltonian.

The thermodynamics of a scalar field with a quartic interaction is studied within the linear delta expansion (LDE) method. Using the imaginary-time formalism the free energy is evaluated up to second order in the LDE. The method generates nonperturbative results that are then used to obtain thermodynamic quantities like the pressure. The phase transition pattern of the model is fully studied, from the broken to the symmetry restored phase. The results are compared with those obtained with other nonperturbative methods and also with ordinary perturbation theory. The results coming from the two main optimization procedures used in conjunction with the LDE method, the Principle of Minimal Sensitivity (PMS) and the Fastest Apparent Convergence (FAC) are also compared with each other and studied in which cases they are applicable or not. The optimization procedures are applied directly to the free energy.

The values of the presently available truncated perturbative expressions for the pressure of the quark-gluon plasma at finite temperatures and finite chemical potential are trustworthy only at very large energies. When used down to temperatures close to the critical one Tc, they suffer from large uncertainties due to the renormalization scale freedom. In order to reduce these uncertainties, we perform resummations of the pressure by applying Pade-related approximants to the available perturbation series for the short-distance and for the long-distance contributions. In the two contributions, we use two different renormalization scales which reflect different energy regions contributing to the different parts. Application of the obtained expressions at low temperatures is made possible by replacing the usual four-loop barMS beta function for alpha_s by its Borel-Pade resummation, eliminating thus the unphysical Landau singularities of alpha_s. The obtained results are remarkably insensitive to the chosen renormalization scale and can be compared with lattice results -- for the pressure (p), the chemical potential contribution (delta p) to the pressure, and various susceptibilities. A good qualitative agreement with the lattice results is revealed down to temperatures close to Tc.

Symmetry restoration in a theory of a self-interacting charged scalar field at finite temperature and in the presence of an external magnetic field is examined. The effective potential is evaluated nonperturbatively in the context of the optimized perturbation theory method. It is explicitly shown that in all ranges of the magnetic field, from weak to large fields, the phase transition is second order and that the critical temperature increases with the magnetic field. In addition, we present an efficient way to deal with the sum over the Landau levels, which is of interest especially in the case of working with weak magnetic fields.

Contribution to the proceedings of the NATO workshop on the Electroweak Phase Transition and the Early Universe, Sintra, March 1994. To be published by Plenum Press.

We use a functional approach to evaluate the Casimir free energy for a self-interacting scalar field in $d+1$ dimensions, satisfying Dirichlet boundary conditions on two parallel planes. When the interaction is turned off, exact results for the free energy in some particular cases may be found, as well as low and high temperature expansions based on a duality relation that involves the inverse temperature $\beta$ and the distance between the mirrors, $a$. For the interacting theory, we derive and implement two different approaches. The first one is a perturbative expansion built with a thermal propagator that satisfies Dirichlet boundary conditions on the mirrors. The second approach uses the exact finite-temperature generating functional as a starting point. In this sense, it allows one to include, for example, non-perturbative thermal corrections into the Casimir calculation, in a controlled way. We present results for calculations performed using those two approaches.

We impose the periodicity conditions corresponding to the Matsubara formalism for Thermal Field Theory as constraints in the imaginary time path integral. These constraints are introduced by means of time-independent auxiliary fields which, by integration of the original variables, become dynamical fields in the resulting `dual' representation for the theory. This alternative representation has the appealing property of involving fields which live in one dimension less than the original ones, with a quantum partition function whose integration measure is identical to the one of its classical counterpart, albeit with a different (spatially nonlocal) action.

Using a ten dimensional dual string background, we study aspects of the physics of finite temperature large N four dimensional SU(N) gauge theory, focusing on the dynamics of fundamental quarks in the presence of a background magnetic field. At vanishing temperature and magnetic field, the theory has N=2 supersymmetry, and the quarks are in hypermultiplet representations. In a previous study, similar techniques were used to show that the quark dynamics exhibit spontaneous chiral symmetry breaking. In the present work we begin by establishing the non-trivial phase structure that results from finite temperature. We observe, for example, that above the critical value of the field that generates a chiral condensate spontaneously, the meson melting transition disappears, leaving only a discrete spectrum of mesons at any temperature. We also compute several thermodynamic properties of the plasma.

We investigate the properties of $qq$ and $q\bar q$ states in hot and dense quark matter in the framework of light-front finite temperature field theory. Presently we use the Nambu Jona-Lasinio model of QCD and derive the gap equation at finite temperature and density. We study pionic and scalar diquark dynamics in quark matter and calculate the masses and the Mott dissociation as a function of the temperature $T$ and the chemical potential $\mu$. For the scalar diquark we determine the critical temperature of color superconductivity.

The formulation of statistical physics using light-front quantization, instead of conventional equal-time boundary conditions, has important advantages for describing relativistic statistical systems, such as heavy ion collisions. We develop light-front field theory at finite temperature and density with special attention to quantum chromodynamics. First, we construct the most general form of the statistical operator allowed by the Poincare algebra. In light-front quantization, the Green's functions of a quark in a medium can be defined in terms of just 2-component spinors and does not lead to doublers in the transverse directions. Since the theory is non-local along the light cone, we use causality arguments to construct a solution to the related zero-mode problem. A seminal property of light-front Green's functions is that they are related to parton densities in coordinate space. Namely, the diagonal and off-diagonal parton distributions measured in hard scattering experiments can be interpreted as light-front density matrices.

We consider the logarithmic negativity of a finite interval embedded in an infinite one dimensional system at finite temperature. We focus on conformal invariant systems and we show that the naive approach based on the calculation of a two-point function of twist fields in a cylindrical geometry yields a wrong result. The correct result is obtained through a four-point function of twist fields in which two auxiliary fields are inserted far away from the interval, and they are sent to infinity only after having taken the replica limit. In this way, we find a universal scaling form for the finite temperature negativity which depends on the full operator content of the theory and not only on the central charge. In the limit of low and high temperatures, the expansion of this universal form can be obtained by means of the operator product expansion. We check our results against exact numerical computations for the critical harmonic chain.

We review the Extended Mean Field Theory (EMFT) approximation and apply it to complex, scalar $\phi^4$-theory on the lattice. We study the critical properties of the Bose condensation driven by a nonzero chemical potential $\mu$ at both zero and nonzero temperature and determine the $(T,\mu)$ phase diagram. The results are in very good agreement with recent Monte Carlo data for all parameter values considered. EMFT can be formulated directly in the thermodynamic limit which allows us to study lattice spacings for which Monte Carlo studies are not feasible with present techniques. We find that the EMFT approximation accurately reproduces many known phenomena of the exact solution, like the "Silver Blaze" behavior at zero temperature and dimensional reduction at finite temperature.

The finite-temperature one-loop effective potential for a scalar field in the static de Sitter space-time is obtained. Within this framework, by using zeta-function regularization, one can get, in the conformally invariant case, the explicit expression for the stress tensor anomaly. Its value turns out to depend on the thermal state of the system. This conclusion is different from the one derived by other authors, who considered thermal properties of ultraviolet divergences in static spaces ignoring the effects of horizons. The behaviour of the effective potential in the ground state and in de Sitter-invariant state is also studied, showing the role played by the curvature on the minima.

We study the finite temperature properties of the SO(3) lattice gauge theory using mean field theory. The main result is the calculation of the effective action at finite temperature. The form of the effective action is used to explain the behaviour of the adjoint Wilson line in numerical simulations. Numerical simulations of the SO(3) lattice gauge theory show that the adjoint Wilson line has a very small value at low temperatures; at high temperatures, metastable states are observed in which the adjoint Wilson line takes positive or negative values. The effective action is able to explain the origin of these metastable states. A comparison of the effective actions of the SU(2) and the SO(3) lattice gauge theories explains their different behaviour at high temperatures. The mean field theory also predicts a finite temperature phase transition in the SO(3) lattice gauge theory.

We study nonlinear response in weakly coupled hot $\phi^4$ theory. We obtain an expression for a quadratic shear viscous response coefficient using two different formalisms: transport theory and response theory. The transport theory calculation is carried out by assuming a local equilibrium form for the distribution function and expanding in the gradient of the local four dimensional velocity field. By doing a gradient expansion on the Boltzmann equation we obtain a hierarchy of equations for the coefficients of this expansion.To do the response theory calculation we use Zubrave's techniques in nonequilibrium statistical mechanics to derive a generalized Kubo formula. Using this formula allows us to obtain the quadratic shear viscous response from the three-point retarded green function of the viscous shear stress tensor. We use the closed time path formalism of real time finite temperature field theoryto show that this three-point function can be calculated using equilibrium quantum field theory by writing it as an integral equation involving a four-point vertex. This four-point vertex can in turn be obtained from an integral equation which represents the resummation of an infinite series of ladder and extended-ladder diagrams.The connection between transport theory and response theory is made when we show that the integral equation for this four-point vertex has exactly the same form as the equation obtained from the Boltzmann equation for the coefficient of the quadratic term of the gradient expansion of the distribution function.

These lectures review phases and phase transitions of the Standard Model, with emphasis on those aspects which are amenable to a first principle study. Model calculations and theoretical ideas of practical applicability are discussed as well. Contents: 1.Overview; 2. Field Theory at Finite Temperature and Density; 3.Critical Phenomena; 4.Electroweak Interactions at Finite Temperature; 5. Thermodynamics of Four Fermions models; 6.The Phases of QCD; 7.QCD at Finite Temperature, $\mu_B = 0$; 8.QCD at Finite Temperature, $\mu_B \ne 0$.

The use of classical thermal field to approximate real-time quantum thermal field theory is discussed. For a \lambda\phi^4 theory, it is shown that the classical Rayleigh-Jeans divergence can be canceled with the appropriate counterterms, and a comparison is made between the classical and quantum perturbative expansion. It is explained why Hard Thermal Loops prevent the same method to work for gauge theories.