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Connection of Path Integral to Classical Statistical Mechanics Consider �exible string held under constant tension and con�ned to �gutter� potential...

The lectures provide a pedagogical introduction to the methods of CFT as applied to two-dimensional critical behaviour.

Consistent coupling of effective field theories with a quantum theory of gravity appears to require bounds on the the rank of the gauge group and the amount of matter. We consider landscapes of field theories subject to such to boundedness constraints. We argue that appropriately "coarse-grained" aspects of the randomly chosen field theory in such landscapes, such as the fraction of gauge groups with ranks in a given range, can be statistically predictable. To illustrate our point we show how the uniform measures on simple classes of N=1 quiver gauge theories localize in the vicinity of theories with certain typical structures. Generically, this approach would predict a high energy theory with very many gauge factors, with the high rank factors largely decoupled from the low rank factors if we require asymptotic freedom for the latter.

We present an expression for the shot noise power spectral density in quasi-one dimensional conductors electrostatically controlled by a gate electrode, that includes the effects of Coulomb interaction and of Pauli exclusion among charge carriers. In this sense, our expression extends the well known Landauer-Buttiker noise formula to include the effect of Coulomb interaction through induced fluctuations in the device potential. Our approach is based on the introduction of statistical properties of the scattering matrix and on a second-quantization many-body description. From a quantitative point of view, statistical properties are obtained by means of Monte Carlo simulations on a ensemble of different configurations of injected states, requiring the solution of the Poisson-Schrodinger equation on a three-dimensional grid, with the non-equilibrium Green functions formalism. In a series of example, we show that failure to consider the effects of Coulomb interaction on noise leads to a gross overestimation of the noise spectrum of quasi-one dimensional devices.

A class of models with applications to swarm behavior as well as many other types of spatially extended complex biological and physical systems is studied. Internal fluctuations can play an active role in the organization of the phase structure of such systems. Consequently, it is not possible to fully understand the behavior of these systems without explicitly incorporating the fluctuations. In particular, for the class of models studied here the effect of internal fluctuations due to finite size is a renormalized decrease in the temperature near the point of spontaneous symmetry breaking. We briefly outline how these models can be applied to the behavior of an ant swarm.

Cooperativity effects have been proposed to explain the non-local rheology in the dynamics of soft jammed systems. Based on the analysis of the free-energy model proposed by L. Bocquet, A. Colin \& A. Ajdari ({\em Phys. Rev. Lett.} {\bf 103}, 036001 (2009)), we show that cooperativity effects resulting from the non-local nature of the fluidity (inverse viscosity), are intimately related to the emergence of shear-banding configurations. This connection materializes through the onset of inhomogeneous compact solutions (compactons), wherein the fluidity is confined to finite-support subregions of the flow and strictly zero elsewhere. Compactons coexistence with regions of zero fluidity ("non-flowing vacuum") is shown to be stabilized by the presence of mechanical noise, which ultimately shapes up the equilibrium distribution of the fluidity field, the latter acting as an order parameter for the flow-noflow transitions occurring in the material.

We obtain the Ward identities and the gauge-dependence of Green's functions in non-Abelian gauge theories by using only the canonical commutation relations and the equations of motion for the Heisenberg operators. The consideration is applicable to theories both with and without spontaneous symmetry breaking. We present a definition of a generalized statistical average which ensures that the Fourier images of temperature Green's functions of the Fermionic fields have only even-valued frequencies. This makes it possible to set up a procedure of gauge-invariant statistical averaging in terms of the Hamiltonian and the field operators.

A statistical model of self-organization in a generic class of one-dimensional nonlinear Schrodinger (NLS) equations on a bounded interval is developed. The main prediction of this model is that the statistically preferred state for such equations consists of a deterministic coherent structure coupled with fine-scale, random fluctuations, or radiation. The model is derived from equilibrium statistical mechanics by using a mean-field approximation of the conserved Hamiltonian and particle number for finite-dimensional spectral truncations of the NLS dynamics. The continuum limits of these approximated statistical equilibrium ensembles on finite-dimensional phase spaces are analyzed, holding the energy and particle number at fixed, finite values. The analysis shows that the coherent structure minimizes total energy for a given value of particle number and hence is a solution to the NLS ground state equation, and that the remaining energy resides in Gaussian fluctuations equipartitioned over wavenumbers. Some results of direct numerical integration of the NLS equation are included to validate empirically these properties of the most probable states for the statistical model. Moreover, a theoretical justification of the mean-field approximation is given, in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit.

In this work we present the effective field theory of primordial statistical anisotropies generated during anisotropic inflation involving a background $U(1)$ gauge field. Besides the usual Goldstone boson associated with the breaking of time diffeomorphism we have two additional Goldstone bosons associated with the breaking of spatial diffeomorphisms. We further identify these two new Goldstone bosons with the expected two transverse degrees of the $U(1)$ gauge field fluctuations. Upon defining the appropriate unitary gauge, we present the most general quadratic action which respects the remnant symmetry in the unitary gauge. The interactions between various Goldstone bosons leads to statistical anisotropy in curvature perturbation power spectrum. Calculating the general results for power spectrum anisotropy, we recover the previously known results in specific models of anisotropic inflation. In addition, we present novel results for statistical anisotropy in models with non-trivial sound speed for inflaton fluctuations. Also we identify the interaction which leads to birefringence-like effects in anisotropic power spectrum in which the speed of gauge field fluctuations depends on the direction of the mode propagation and the two polarization of gauge field fluctuations contribute differently in statistical anisotropy. As another interesting application, our EFT approach naturally captures interactions generating parity violating statistical anisotropies.

In this work we present the effective field theory of primordial statistical anisotropies generated during anisotropic inflation involving a background $U(1)$ gauge field. Besides the usual Goldstone boson associated with the breaking of time diffeomorphism we have two additional Goldstone bosons associated with the breaking of spatial diffeomorphisms. We further identify these two new Goldstone bosons with the expected two transverse degrees of the $U(1)$ gauge field fluctuations. Upon defining the appropriate unitary gauge, we present the most general quadratic action which respects the remnant symmetry in the unitary gauge. The interactions between various Goldstone bosons leads to statistical anisotropy in curvature perturbation power spectrum. Calculating the general results for power spectrum anisotropy, we recover the previously known results in specific models of anisotropic inflation. In addition, we present novel results for statistical anisotropy in models with non-trivial sound speed for inflaton fluctuations. Also we identify the interaction which leads to birefringence-like effects in anisotropic power spectrum in which the speed of gauge field fluctuations depends on the direction of the mode propagation and the two polarization of gauge field fluctuations contribute differently in statistical anisotropy. As another interesting application, our EFT approach naturally captures interactions generating parity violating statistical anisotropies.

'In July-August 1971, a summer school, a Europhysics Conference and an informal workshop on Statistical Mechanics and Field Theory were held at the Technion-Israel Institute of Technology.' - foreword

'In July-August 1971, a summer school, a Europhysics Conference and an informal workshop on Statistical Mechanics and Field Theory were held at the Technion-Israel Institute of Technology.' - foreword

This paper is devoted to such a fundamental problem of quantum computing as quantum parallelism. It is well known that quantum parallelism is the basis of the ability of quantum computer to perform in polynomial time computations performed by classical computers for exponential time. Therefore better understanding of quantum parallelism is important both for theoretical and applied research, cf. e.g. David Deutsch \cite{DD}. We present a realistic interpretation based on recently developed prequantum classical statistical field theory (PCSFT). In the PCSFT-approach to QM quantum states (mixed as well as pure) are labels of special ensembles of classical fields. Thus e.g. a single (!) ``electron in the pure state'' $\psi$ can be identified with a special `` electron random field,'' say $\Phi_\psi(\phi).$ Quantum computer operates with such random fields. By one computational step for e.g. a Boolean function $f(x_1,...,x_n)$ the initial random field $\Phi_{\psi_0}(\phi)$ is transformed into the final random field $\Phi_{\psi_f}(\phi)$ ``containing all values'' of $f.$ This is the objective of quantum computer's ability to operate quickly with huge amounts of information -- in fact, with classical random fields.

This paper is devoted to such a fundamental problem of quantum computing as quantum parallelism. It is well known that quantum parallelism is the basis of the ability of quantum computer to perform in polynomial time computations performed by classical computers for exponential time. Therefore better understanding of quantum parallelism is important both for theoretical and applied research, cf. e.g. David Deutsch \cite{DD}. We present a realistic interpretation based on recently developed prequantum classical statistical field theory (PCSFT). In the PCSFT-approach to QM quantum states (mixed as well as pure) are labels of special ensembles of classical fields. Thus e.g. a single (!) ``electron in the pure state'' $\psi$ can be identified with a special `` electron random field,'' say $\Phi_\psi(\phi).$ Quantum computer operates with such random fields. By one computational step for e.g. a Boolean function $f(x_1,...,x_n)$ the initial random field $\Phi_{\psi_0}(\phi)$ is transformed into the final random field $\Phi_{\psi_f}(\phi)$ ``containing all values'' of $f.$ This is the objective of quantum computer's ability to operate quickly with huge amounts of information -- in fact, with classical random fields.

We review the use of an exact renormalization group equation in quantum field theory and statistical physics. It describes the dependence of the free energy on an infrared cutoff for the quantum or thermal fluctuations. Non-perturbative solutions follow from approximations to the general form of the coarse-grained free energy or effective average action. They interpolate between the microphysical laws and the complex macroscopic phenomena. Our approach yields a simple unified description for O(N)-symmetric scalar models in two, three or four dimensions, covering in particular the critical phenomena for the second-order phase transitions, including the Kosterlitz-Thouless transition and the critical behavior of polymer chains. We compute the aspects of the critical equation of state which are universal for a large variety of physical systems and establish a direct connection between microphysical and critical quantities for a liquid-gas transition. Universal features of first-order phase transitions are studied in the context of scalar matrix models. We show that the quantitative treatment of coarse graining is essential for a detailed estimate of the nucleation rate. We discuss quantum statistics in thermal equilibrium or thermal quantum field theory with fermions and bosons and we describe the high temperature symmetry restoration in quantum field theories with spontaneous symmetry breaking. In particular, we explore chiral symmetry breaking and the high temperature or high density chiral phase transition in quantum chromodynamics using models with effective four-fermion interactions.

We use the non-equlibrium statistical field theory for classical particles, recently developed by Mazenko and Das and Mazenko, together with the free generating functional we have previously derived for point sets initially correlated in phase space, to calculate the time evolution of power spectra in the free theory, i.e. neglecting particle interactions. We provide expressions taking linear and quadratic momentum correlations into account. Up to this point, the expressions are general with respect to the free propagator of the microscopic degrees of freedom. We then specialise the propagator to that expected for particles in cosmology treated within the Zel'dovich approximation and show that, to linear order in the momentum correlations, the linear growth of the cosmological power spectrum is reproduced. Quadratic momentum correlations return a first contribution to the non-linear evolution of the power spectrum, for which we derive a simple closed expression valid for arbitrary wave numbers. This expression is a convolution of the initial density power spectrum with itself, multiplied by a mode-coupling kernel. We also derive the bispectrum expected in this theory within these approximations and show that its connected part reproduces almost, but not quite, the bispectrum expected in Eulerian perturbation theory of the density contrast.

We use the non-equlibrium statistical field theory for classical particles, recently developed by Mazenko and Das and Mazenko, together with the free generating functional we have previously derived for point sets initially correlated in phase space, to calculate the time evolution of power spectra in the free theory, i.e. neglecting particle interactions. We provide expressions taking linear and quadratic momentum correlations into account. Up to this point, the expressions are general with respect to the free propagator of the microscopic degrees of freedom. We then specialise the propagator to that expected for particles in cosmology treated within the Zel'dovich approximation and show that, to linear order in the momentum correlations, the linear growth of the cosmological power spectrum is reproduced. Quadratic momentum correlations return a first contribution to the non-linear evolution of the power spectrum, for which we derive a simple closed expression valid for arbitrary wave numbers. This expression is a convolution of the initial density power spectrum with itself, multiplied by a mode-coupling kernel. We also derive the bispectrum expected in this theory within these approximations and show that its connected part reproduces almost, but not quite, the bispectrum expected in Eulerian perturbation theory of the density contrast.

We develop a reformulation of the functional integral for bosons in terms of bilocal fields. Correlation functions correspond to quantum probabilities instead of probability amplitudes. Discrete and continuous global symmetries can be treated similar to the usual formalism. Situations where the formalism can be interpreted in terms of a statistical field theory in Minkowski space are characterized by violations of unitarity at very large momentum scales. Renormalization group equations suggest that unitarity can be essentially restored by strong fluctuation effects.

Classical $\phi^4$ theory in weak and strong thermal gradients is studied on the lattice in (1+1) dimensions. Classical $\phi^4$ theory in weak and strong thermal gradients is studied on the lattice in (1+1) dimensions. The steady state physics of the theory is investigated from first principles and classified into dynamical regimes. We derive the bulk properties associated with thermal transport, and explore in detail the non-equilibrium statistical mechanics of the theory as well as connections to equilibrium and irreversible thermodynamics. Linear response predictions are found to be valid for systems quite far from equilibrium and are seen to eventually break down simultaneously with local equilibrium.

Journal of Research of the National Bureau of Standards

I propose a new version of the Hierarchical Reference Theory of liquids. Two formalisms, one in the grand canonical ensemble, the other in the framework of statistical field theory are given in parallel. In the latter the theory is an avatar of a new version of the non perturbative renormalization group (J. Phys. A : Math. Gen. \textbf{42}, 225004 (2009)). The flow of the Wilsonian action as well as that of the effective average action of Wetterich are derived and a simple relation between the two functionals is established. The standard Hierarchical Reference Theory for liquids (\textit{Adv. Phys.} \textbf{44}, 211 (1995)) is recovered for a sharp infra-red cut-off of the propagator

I propose a new version of the Hierarchical Reference Theory of liquids. Two formalisms, one in the grand canonical ensemble, the other in the framework of statistical field theory are given in parallel. In the latter the theory is an avatar of a new version of the non perturbative renormalization group (J. Phys. A : Math. Gen. \textbf{42}, 225004 (2009)). The flow of the Wilsonian action as well as that of the effective average action of Wetterich are derived and a simple relation between the two functionals is established. The standard Hierarchical Reference Theory for liquids (\textit{Adv. Phys.} \textbf{44}, 211 (1995)) is recovered for a sharp infra-red cut-off of the propagator

The mean spherical approximation (MSA) is a closure relation for pair correlation functions (two-point functions) in statistical physics. It can be applied to a wide range of systems, is computationally fairly inexpensive, and when properly applied and interpreted lead to rather good results. In this paper we promote its applicability to euclidean quantum field theories formulated on a lattice, by demonstrating how it can be used to locate the critical lines of a class of multi-component bosonic models. The MSA has the potential to handle models lacking a positive definite integration measure, which therefore are difficult to investigate by Monte-Carlo simulations.

In this paper, we show how classical statistical field theory techniques can be used to efficiently perform the numerical evaluation of the non-perturbative Schwinger mechanism of particle production by quantum tunneling. In some approximation, we also consider the back-reaction of the produced particles on the external field, as well as the self-interactions of the produced particles.

-elementary excitations. the energy spectrum and properties of liquid He 4 at low temperatures; the fermi liquid; second quantization; the dilute bose gas; the dilute fermi gas; the interaction representation; the green's function; basic principles of the diagram technique; rules for constructing diagrams for interactions of various types; dyson's equation. the vertex part. many-particle green's functions; temperature green's functions; perturbation theory; the diagram technique in coordinate space; the diagram technique in momentum space; the perturbation series for the thermodynamic potential Ω ; dyson's equation. many-particle green's functions; time-dependent green's function for T ≠ 0. analytic properties of the green's functions; properties of the vertex part for small momentum transfer. zero sound; effective mass. relation between the fermi momentum and the particle number. excitation of the bose type. heat capacity; singularities of the vertex part when the total momentum of the colliding particles is small; electron-photon interactions; some properties of a degenerate plasma; application of field theory methods to a system of interacting bosons; the green's functions; the dilute nonideal bose gas; properties of one-particle excitations near the cutoff point; application of field theory methods to a system of interacting bosons for T ≠ 0; the green's functions of radiation in an absorbing medium; calculation of the dielectric constant; van der waals forces in an inhomogeneous dielectric; molecular interaction forces; the phenomenon of superconductivity. the model. the interaction hamiltonian; the cooper phenomenon. instability of the ground state of a system of noninteracting fermions with respect to arbitrarily weak attraction between the particles; the basic system of equations for a superconductor; derivation of the equations of the theory of superconductivity in the phonon model; the thermodynamics of superconductors; a superconductor in a weak electromagnetic field; properties of a superconductor in an arbitrary magnetic field near the critical temperature; theory of superconducting alloys-  

-elementary excitations. the energy spectrum and properties of liquid He 4 at low temperatures; the fermi liquid; second quantization; the dilute bose gas; the dilute fermi gas; the interaction representation; the green's function; basic principles of the diagram technique; rules for constructing diagrams for interactions of various types; dyson's equation. the vertex part. many-particle green's functions; temperature green's functions; perturbation theory; the diagram technique in coordinate space; the diagram technique in momentum space; the perturbation series for the thermodynamic potential Ω ; dyson's equation. many-particle green's functions; time-dependent green's function for T ≠ 0. analytic properties of the green's functions; properties of the vertex part for small momentum transfer. zero sound; effective mass. relation between the fermi momentum and the particle number. excitation of the bose type. heat capacity; singularities of the vertex part when the total momentum of the colliding particles is small; electron-photon interactions; some properties of a degenerate plasma; application of field theory methods to a system of interacting bosons; the green's functions; the dilute nonideal bose gas; properties of one-particle excitations near the cutoff point; application of field theory methods to a system of interacting bosons for T ≠ 0; the green's functions of radiation in an absorbing medium; calculation of the dielectric constant; van der waals forces in an inhomogeneous dielectric; molecular interaction forces; the phenomenon of superconductivity. the model. the interaction hamiltonian; the cooper phenomenon. instability of the ground state of a system of noninteracting fermions with respect to arbitrarily weak attraction between the particles; the basic system of equations for a superconductor; derivation of the equations of the theory of superconductivity in the phonon model; the thermodynamics of superconductors; a superconductor in a weak electromagnetic field; properties of a superconductor in an arbitrary magnetic field near the critical temperature; theory of superconducting alloys-  

-elementary excitations. the energy spectrum and properties of liquid He 4 at low temperatures; the fermi liquid; second quantization; the dilute bose gas; the dilute fermi gas; the interaction representation; the green's function; basic principles of the diagram technique; rules for constructing diagrams for interactions of various types; dyson's equation. the vertex part. many-particle green's functions; temperature green's functions; perturbation theory; the diagram technique in coordinate space; the diagram technique in momentum space; the perturbation series for the thermodynamic potential Ω ; dyson's equation. many-particle green's functions; time-dependent green's function for T ≠ 0. analytic properties of the green's functions; properties of the vertex part for small momentum transfer. zero sound; effective mass. relation between the fermi momentum and the particle number. excitation of the bose type. heat capacity; singularities of the vertex part when the total momentum of the colliding particles is small; electron-photon interactions; some properties of a degenerate plasma; application of field theory methods to a system of interacting bosons; the green's functions; the dilute nonideal bose gas; properties of one-particle excitations near the cutoff point; application of field theory methods to a system of interacting bosons for T ≠ 0; the green's functions of radiation in an absorbing medium; calculation of the dielectric constant; van der waals forces in an inhomogeneous dielectric; molecular interaction forces; the phenomenon of superconductivity. the model. the interaction hamiltonian; the cooper phenomenon. instability of the ground state of a system of noninteracting fermions with respect to arbitrarily weak attraction between the particles; the basic system of equations for a superconductor; derivation of the equations of the theory of superconductivity in the phonon model; the thermodynamics of superconductors; a superconductor in a weak electromagnetic field; properties of a superconductor in an arbitrary magnetic field near the critical temperature; theory of superconducting alloys-  

-elementary excitations. the energy spectrum and properties of liquid He 4 at low temperatures; the fermi liquid; second quantization; the dilute bose gas; the dilute fermi gas; the interaction representation; the green's function; basic principles of the diagram technique; rules for constructing diagrams for interactions of various types; dyson's equation. the vertex part. many-particle green's functions; temperature green's functions; perturbation theory; the diagram technique in coordinate space; the diagram technique in momentum space; the perturbation series for the thermodynamic potential Ω ; dyson's equation. many-particle green's functions; time-dependent green's function for T ≠ 0. analytic properties of the green's functions; properties of the vertex part for small momentum transfer. zero sound; effective mass. relation between the fermi momentum and the particle number. excitation of the bose type. heat capacity; singularities of the vertex part when the total momentum of the colliding particles is small; electron-photon interactions; some properties of a degenerate plasma; application of field theory methods to a system of interacting bosons; the green's functions; the dilute nonideal bose gas; properties of one-particle excitations near the cutoff point; application of field theory methods to a system of interacting bosons for T ≠ 0; the green's functions of radiation in an absorbing medium; calculation of the dielectric constant; van der waals forces in an inhomogeneous dielectric; molecular interaction forces; the phenomenon of superconductivity. the model. the interaction hamiltonian; the cooper phenomenon. instability of the ground state of a system of noninteracting fermions with respect to arbitrarily weak attraction between the particles; the basic system of equations for a superconductor; derivation of the equations of the theory of superconductivity in the phonon model; the thermodynamics of superconductors; a superconductor in a weak electromagnetic field; properties of a superconductor in an arbitrary magnetic field near the critical temperature; theory of superconducting alloys-  

Starting from an extension of the Poisson bracket structure and Kubo-Martin-Schwinger-property of classical statistical mechanics of continuous systems to spin systems, defined on a lattice, we derive a series of, as we think, new and interesting bounds on correlation functions for general lattice systems. Our method is expected to yield also useful results in Euclidean Field Theory. Furthermore the approach is applicable in situations where other techniques fail, e.g. in the study of phase transitions without breaking of a {\bf continuous} symmetry like $P(\phi)$-theories with $\phi (x)$ scalar.

Starting from an extension of the Poisson bracket structure and Kubo-Martin-Schwinger-property of classical statistical mechanics of continuous systems to spin systems, defined on a lattice, we derive a series of, as we think, new and interesting bounds on correlation functions for general lattice systems. Our method is expected to yield also useful results in Euclidean Field Theory. Furthermore the approach is applicable in situations where other techniques fail, e.g. in the study of phase transitions without breaking of a {\bf continuous} symmetry like $P(\phi)$-theories with $\phi (x)$ scalar.

We report on a numerical study of real-time dynamics of electromagnetically interacting chirally imbalanced lattice Dirac fermions within the classical statistical field theory approach. Namely, we perform exact simulations of the real-time quantum evolution of fermionic fields coupled to classical electromagnetic fields, which are in turn coupled to the vacuum expectation value of the fermionic electric current. We use Wilson-Dirac Hamiltonian for fermions, and non-compact action for the gauge field. In general, we observe that the backreaction of fermions on the electromagnetic field prevents the system from acquiring chirality imbalance. In the case of chirality pumping in parallel electric and magnetic fields, electric field is screened by the produced on-shell fermions and the accumulation of chirality is hence stopped. In the case of evolution with initially present chirality imbalance, axial charge tends to transform to helicity of electromagnetic field. By performing simulations on large lattices we show that in most cases this decay process is accompanied by the inverse cascade phenomenon which transfers energy from short-wavelength to long-wavelength electromagnetic fields. In some simulations, however, we observe a very clear signature of inverse cascade for the helical magnetic fields which is not accompanied by the axial charge decay. This suggests that the relation between inverse cascade and axial charge decay is not as straightforward as predicted by the simplest form of anomalous Maxwell equations.

A four part series of lectures on the connection of statistical mechanics and quantum field theory. The general principles relating statistical mechanics and the path integral formulation of quantum field theory are presented in the first lecture. These principles are then illustrated in lecture 2 by a presentation of the theory of the Ising model for $H=0$, where both the homogeneous and randomly inhomogeneous models are treated and the scaling theory and the relation with Fredholm determinants and Painlev{\'e} equations is presented. In lecture 3 we consider the Ising model with $H\neq 0$, where the relation with gauge theory is used to discuss the phenomenon of confinement. We conclude in the last lecture with a discussion of quantum spin diffusion in one dimensional chains and a presentation of the chiral Potts model which illustrates the physical effects that can occur when the Euclidean and Minkowski regions are not connected by an analytic continuation. (To be published as part of the Proceedings of the Sixth Annual Theoretical Physics Summer School of the Australian National University which was held in Canberra during Jan. 1994.)

We have written a {\it Mathematica} program that calculates the integrand corresponding to any amplitude in the closed-time-path formulation of real time statistical field theory. The program is designed so that it can be used by someone with no previous experience with {\it Mathematica}. It performs the contractions over the tensor indices that appear in real time statistical field theory and gives the result in the 1-2, Keldysh or RA basis. We have used the program to calculate the ward identity for the QED 3-point function, the QED 4-point function for two photons and two fermions, and the QED 5-point function for three photons and two fermions. In real time statistical field theory, there are seven 3-point functions, 15 4-point functions and 31 5-point functions. We produce a table that gives the results for all of these functions. In addition, we give a simple general expression for the KMS conditions between $n$-point green functions and vertex functions, in both the Keldysh and RA bases

Using the collective variables (CV) method the basic relations of statistical field theory of a multicomponent non-homogeneous fluids are reconsidered. The corresponding CV action depends on two sets of scalar fields - fields $\rho_{\alpha}$ connected to the local density fluctuations of the $\alpha$th species of particles and fields $\omega_{\alpha}$ conjugated to $\rho_{\alpha}$. The explicit expressions for the CV field correlations and their relation to the density correlation functions are found. The perturbation theory is formulated and a mean field level (MF) of the theory is considered in detail.

Using the collective variables (CV) method the basic relations of statistical field theory of a multicomponent non-homogeneous fluids are reconsidered. The corresponding CV action depends on two sets of scalar fields - fields $\rho_{\alpha}$ connected to the local density fluctuations of the $\alpha$th species of particles and fields $\omega_{\alpha}$ conjugated to $\rho_{\alpha}$. The explicit expressions for the CV field correlations and their relation to the density correlation functions are found. The perturbation theory is formulated and a mean field level (MF) of the theory is considered in detail.

These notes are concerned with the physics of phase transitions: the phenomenon that in particular environments, qualified by particular values of external parameters such as temperature, magnetic field etc., many systems exhibit singularities in the thermodynamic variables which best describe the macroscopic state of the system. For example...

We show that the one-loop effective action at finite temperature for a scalar field with quartic interaction has the same renormalized expression as at zero temperature if written in terms of a certain classical field $\phi_c$, and if we trade free propagators at zero temperature for their finite-temperature counterparts. The result follows if we write the partition function as an integral over field eigenstates (boundary fields) of the density matrix element in the functional Schr\"{o}dinger field-representation, and perform a semiclassical expansion in two steps: first, we integrate around the saddle-point for fixed boundary fields, which is the classical field $\phi_c$, a functional of the boundary fields; then, we perform a saddle-point integration over the boundary fields, whose correlations characterize the thermal properties of the system. This procedure provides a dimensionally-reduced effective theory for the thermal system. We calculate the two-point correlation as an example.

We show that the one-loop effective action at finite temperature for a scalar field with quartic interaction has the same renormalized expression as at zero temperature if written in terms of a certain classical field $\phi_c$, and if we trade free propagators at zero temperature for their finite-temperature counterparts. The result follows if we write the partition function as an integral over field eigenstates (boundary fields) of the density matrix element in the functional Schr\"{o}dinger field-representation, and perform a semiclassical expansion in two steps: first, we integrate around the saddle-point for fixed boundary fields, which is the classical field $\phi_c$, a functional of the boundary fields; then, we perform a saddle-point integration over the boundary fields, whose correlations characterize the thermal properties of the system. This procedure provides a dimensionally-reduced effective theory for the thermal system. We calculate the two-point correlation as an example.

Recently Mazenko and Das and Mazenko introduced a non-equilibrium field theoretical approach to describe the statistical properties of a classical particle ensemble starting from the microscopic equations of motion of each individual particle. We use this theory to investigate the transition from those microscopic degrees of freedom to the evolution equations of the macroscopic observables of the ensemble. For the free theory, we recover the continuity and Jeans equations of a collisionless gas. For a theory containing two-particle interactions in a canonical perturbation series, we find the macroscopic evolution equations to be described by the Born-Bogoliubov-Green-Kirkwood-Yvon hierarchy (BBGKY hierarchy) with a truncation criterion depending on the order in perturbation theory. This establishes a direct link between the classical and the field-theoretical approaches to kinetic theory that might serve as a starting point to investigate kinetic theory beyond the classical limits.

We develop a statistical theory of the mean field. It is based on the proposition that the mean field can be obtained as an energy average. Moreover, it is assumed that the matrix elements of the residual interaction are random with the average value of zero. Explicit expressions for the mean field and the fluctuation away from the average are obtained. The fluctuation is expanded in terms of more and more complex excitations. Using the randomness of the matrix elements one can then obtain formulas for the contribution to the error from each class of complex excitations and a general condition for the convergence of the expansion is derived. Making some simplifying assumptions a schematic model is developed and applied to the problem of nuclear matter. It yields a measure of the strength of the effective interaction. The latter turns out to be three orders of magnitude less than that calculated using a potential which gives a binding energy of about -7 MeV/nucleon demonstrating the strong damping of the interaction strength induced by the averaging process.

We investigate the relativistic mean field theory of nuclear matter at finite temperature and baryon density taking into account of nonlinear statistical effects, characterized by power-law quantum distributions. The analysis is performed by requiring the Gibbs conditions on the global conservation of baryon number and electric charge fraction. We show that such nonlinear statistical effects play a crucial role in the equation of state and in the formation of mixed phase also for small deviations from the standard Boltzmann-Gibbs statistics.

A statistical theory of the mean field is developed. It is based on the proposition that the mean field can be obtained as an energy average. Moreover, it is assumed that the matrix elements of the residual interaction, obtained after the average interaction is removed, are random with the average value of zero. With these two assumptions one obtains explicit expressions for the mean field and the fluctuation away from the average. The fluctuation is expanded in terms of more and more complex excitations. Using the randomness of the matrix elements one can then obtain formulas for the contribution to the error from each class of complex excitations and a general condition for the convergence of the expansion is derived. It is to be emphasized that no conditions on the nature of the system being studied are made. Making some simplifying assumptions a schematic model is developed. This model is applied to the problem of nuclear matter. The model yields a measure of the strength of the effective interaction. It turns out to be three orders of magnitude less than that calculated using a potential which gives a binding energy of about -7 MeV/nucleon demonstrating the strong damping of the interaction strength induced by the averaging process.

This technical report revisits the analysis of family-wise error rates in statistical parametric mapping - using random field theory - reported in (Eklund et al., 2015). Contrary to the understandable spin that these sorts of analyses attract, a review of their results suggests that they endorse the use of parametric assumptions - and random field theory - in the analysis of functional neuroimaging data. We briefly rehearse the advantages parametric analyses offer over nonparametric alternatives and then unpack the implications of (Eklund et al., 2015) for parametric procedures.

This technical report revisits the analysis of family-wise error rates in statistical parametric mapping - using random field theory - reported in (Eklund et al., 2015). Contrary to the understandable spin that these sorts of analyses attract, a review of their results suggests that they endorse the use of parametric assumptions - and random field theory - in the analysis of functional neuroimaging data. We briefly rehearse the advantages parametric analyses offer over nonparametric alternatives and then unpack the implications of (Eklund et al., 2015) for parametric procedures.

A statistical field theory of particle production is presented using a gaussian functional in three dimensions. Identifying the field with the particle density fluctuation results in zero correlations of order three and higher, while the second order correlation function is of a Yukawa form. A detailed scheme for projecting the theoretical three-dimensional correlation onto data of three and fewer dimensions illustrates how theoretical predictions are tested against experimental moments in the different dimensions. An example given in terms of NA35 parameters should be testable against future NA35 data.

Strongly correlated electrons with box disorder in high-dimensional lattices are investigated. We apply the statistical dynamical mean-field theory, which treats local correlations non-perturbatively. The incorporation of a finite lattice connectivity allows for the detection of disorder-induced localization via the probability distribution function of the local density of states. We obtain a complete paramagnetic ground state phase diagram and find correlation-induced as well as disorder-induced metal-insulator transitions. Our results qualitatively confirm predictions obtained by typical medium theory. Moreover, we find that the probability distribution function of the local density of states in the metallic phase strongly deviates from a log-normal distribution as found for the non-interacting case.

Strongly correlated electrons with box disorder in high-dimensional lattices are investigated. We apply the statistical dynamical mean-field theory, which treats local correlations non-perturbatively. The incorporation of a finite lattice connectivity allows for the detection of disorder-induced localization via the probability distribution function of the local density of states. We obtain a complete paramagnetic ground state phase diagram and find correlation-induced as well as disorder-induced metal-insulator transitions. Our results qualitatively confirm predictions obtained by typical medium theory. Moreover, we find that the probability distribution function of the local density of states in the metallic phase strongly deviates from a log-normal distribution as found for the non-interacting case.

We describe how the concepts of quantum open systems and the methods of closed-time-path (CTP) effective action and influence functional (IF) can be usefully applied to the analysis of statistical mechanical problems involving quantum fields in gravitation and cosmology. In the first lecture we discuss in general terms the relevance of open system concepts in the description of a variety of physical processes, and outline the basics of the CTP and IF formalisms. In the second lecture we illustrate the IF method with a model of two interacting quantum fields, deriving the influence action via a perturbative expansion involving the closed-time-path Green functions. We show how noise of quantum fields can be defined and derive a general fluctuation- dissipation relation for quantum fields. In the third lecture we discuss the problem of backreaction in semiclassical gravity with the example of a scalar field in a Bianchi Type-I universe. We show that the CTP effective action not only yields a real and causal equation of motion with a dissipative term depicting the effect of particle creation, as was found earlier, it also contains a noise term measuring the fluctuations in particle number and governing the metric fluctuations. The particle creation-backreaction problem can be understood as a manifestation of a fluctuation-dissipation relation for quantum fields in dynamic spacetimes, generalizing Sciama's observation for black hole Hawking radiation. A more complete description of semiclassical gravity is given by way of an Einstein-Langevin equation, the conventional theory based on the expectation value of the energy momentum tensor being its

We survey some results relating noncommutative geometry to the class field theory of number fields. These results appear within the context of quantum statistical mechanics where some arithmetic properties of a given number field can be realized in terms of the structure of equilibrium states of a quantum statistical mechanical system.

This is a set of review notes on combinatorial aspects of Bosonic quantum field theory. We collect together several related issues concerning moments of distributions, moments of stochastic processes and Ito's formula, and Green's functions and cumulant moments in quantum field theory.