Loop expansion in light-cone phi4 field theory.

We apply Kleinert's variational method to field theory and calculate the effective potential of the $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The potential is regularized in the standard way using dimensional regularization, but we use a nonconventional renormalization scheme to subtract the divergences. This scheme emerges in a natural way when one tries to renormalize the potential before calculating the optimum value of the variational parameter. In order to do this we also have to renormalize the variational parameter. We have demonstrated that this alternative renormalization scheme eliminates the divergences to all orders in the cumulant expansion of the effective potential. We have also tested its compatibility with critical phenomena by calculating critical exponents and the equation of state to order $\ensuremath{\epsilon}.$ We have shown how to obtain an exact implicit form of the equation of state, which can be solved after performing an $\ensuremath{\epsilon}$ expansion. The equation of state and the critical exponents coincide to order $\ensuremath{\epsilon}$ with the usual ones calculated through the loop expansion. The renormalization scheme proposed is also applied to the one-loop effective potential. Even though we obtain the same critical behavior in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions as in more conventional approaches, our renormalization scheme yields a $\ensuremath{\beta}$ function that goes to zero in the limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}0.$ This is an indication that variational techniques and the loop expansion, depending on the renormalization method employed, may be sensitive to the triviality of the $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory in four dimensions.

It is well-known that non-commutative (NC) field theories at theta = infinity are ``equivalent'' to large N matrix field theories to all orders in perturbation theory, due to the dominance of planar diagrams. By formulating a NC field theory on the lattice non-perturbatively and mapping it onto a twisted reduced model, we point out that the above equivalence does not hold if the translational symmetry of the NC field theory is broken spontaneously. As an example we discuss NC scalar field theory, where such a spontaneous symmetry breakdown has been confirmed by Monte Carlo simulations.

Accumulation of Discrete Eigenvalues of the Radial Dirac Operator

A consistent equation for the quark condensate in the chiral limit provides a determination of the renormalization-group-invariant quark condensate. A critical point at which the strong coupling constant is big enough for chiral symmetry breaking to take place is found. They are analyzed at the one- and two-loop levels respectively. An intuitive picture of the condensation above the critical coupling constant is discussed.

Dilatation invariance and spontaneous symmetry breakdown in dimensional renormalization

The non-perturbative ultraviolet divergence of the sine-Gordon model is used to study the k{sup +} = 0 region of light-cone perturbation theory. The light-cone vacuum is shown to be unstable at the non- perturbative {beta}{sup 2} = 8{pi} critical point by a light-cone version of Coleman's variational method. Vacuum bubbles, which are k{sup +} = 0 diagram in light-cone field theory and are individually finite and non-vanishing for all {beta}, conspire to generate ultraviolet divergences of the light-cone energy density. The k{sup +} = 0 region of momentum also contributed to connected Green's functions: the connected two point function will not diverge, as it should, at the critical point unless diagrams which contribute only at k {sup +} = 0 are properly included. This analysis shows in a simple way how the k {sup +} = 0 region cannot be ignored even for connected diagrams. This phenomenon is expected to occur in higher dimensional gauge theories starting at two loop order in light-cone perturbation theory.

The localization properties of particles in field theory are studied with the help of a convenient mathematical description of counters. It is shown that field theory is capable of explaining the observed localization patterns. Apart from the usual axioms of field theory we have to assume some smoothness properties of the Green's functions in momentum space.

Non-differentiability of the effective potential

Quantum field theories become relevant to statistical physics for systems at or near their critical points. Under such circumstances physical observables obey simple scaling relations and the parameters or critical exponents describing the power law relations are essentially independent of the microscopic details of the interactions and depend only on general features of the system such as the overall symmetry invariance. This crucial property of universality allows critical exponents to be calculated by using continuum quantum field theories with the same symmetry content. The couplings of the quantum field theory must be tuned to a critical point, or surface, when the theory becomes scale invariant and correlation functions of various operators have a power law dependence on their spatial separations determined by the dimensions of the operators which are directly related to the statistical mechanical critical exponents. Thus the Ising model, describing simple magnetic systems on a lattice, is related to the field theory with a single scalar field ∅. For the renormalisable ∅ 4 theory there is one coupling g and at the critical point, determined by the vanishing of the β function, β(g *) = 0, the theory becomes scale invariant.

The desirability of evaluating the effective potential in field theories near a phase transition has been recognized in a number of different areas. We show that recent Monte Carlo simulations for the probability distribution for the order parameter in an equilibrium Ising system, when combined with low-order renormalization group results for an ordinary ${\ensuremath{\varphi}}^{4}$ system, can be used to extract the effective potential. All scaling features are included in the process.

One of the early concerns in quantum field theory was causality — can one ensure that measurements at spacelike separation do not interfere? General issues of this type were important in the days when little was understood about the underlying dynamics of particle interactions; the hope was to place constraints on the types of allowed dynamics and interactions on the basis of cherished tenets of kinematics. There were in fact two ways to approach the problem. First, one could take the point of view that only the results of scattering experiments had physical content; then causal behavior means that the scattered wave cannot reach the detector before the incident wave strikes the target. We might describe this as global causality, and it is obeyed by tachyon-free string theory S-matrices. However there is a second, local version of causality, which asks whether local measurements commute at spacelike separation. Local causality ensures global causality in ordinary field theory, and typically one can construct global violations from local ones, so it would seem that the two are equivalent. But in quantum field theory the local question is more fundamental, and can be resolved without solving any complicated global problem such as long-distance signal propagation. Local causality rests on the universal properties of field theory, e.g. that any manifold locally looks the same, rather than their implementation in any particular spacetime background. For instance we can be sure that an interacting scalar field propagating in a Schwarzschild geometry obeys the axioms of local field theory without having to solve for the full S-matrix of the problem.

An integral representation for the f-dimensional nonrelativistic Coulomb Green's function in momentum space and an expansion of this function in a series of Gegenbauer polynomials are obtained. It is shown that the momentum space representatives of the f-dimensional Coulomb Green's function and the related reduced Green's functions can be obtained by differentiation with respect to the momentum transfer of the corresponding functions in the one-dimensional (f odd) or two-dimensional (f even) case. Expressions in closed form are then obtained for the momentum space representatives in the one-dimensional case of the full Coulomb Green's function and of the general nth excited state reduced Coulomb Green's function.

Correlation inequalities are used to show that the two component λ(φ2)2 model (with HD, D, HP, P boundary conditions) has a unique vacuum if the field does not develop a non-zero expectation value. It follows by a generalized Coleman theorem that in two space-time dimensions the vacuum is unique for all values of the coupling constant. In three space-time dimensions the vacuum is unique below the critical coupling constant. For then-componentP(|φ|2)2+μφ1 model, absence of continuous symmetry breaking, as μ goes to zero, is proven for all states which are translation invariant, satisfy the spectral condition, and are weak* limit points of finite volume states satisfyingN loc τ and higher order estimates.

The nonlinear $\ensuremath{\sigma}$ model is renormalizable and asymptotically free in two dimensions. We show here how to construct this model in $2+\ensuremath{\epsilon}$ dimensions. Renormalization-group equations follow and exhibit a nontrivial uv stable fixed point, which corresponds in the language of Heisenberg ferromagnets to a critical point. The existence of systematic expansions in powers of $d\ensuremath{-}2$ follows from this analysis. In the presence of a source information about long-distance behavior above the critical coupling constant is obtainable.

In this paper we compare the construction of equilibrium states at finite temperature for self-interacting massive scalar quantum field theories on Minkowski spacetime proposed by Fredenhagen and Lindner with results obtained in ordinary thermal field theory, by means of real time and Matsubara formalisms. In the construction of this state, even if the adiabatic limit is considered, the interaction Lagrangian is multiplied by a smooth time cut-off. In this way the interaction starts adiabatically and the correlation functions are free from divergences. The corresponding interaction Hamiltonian is a local interacting field smeared over the interval of time where the chosen cut-off is not constant. In order to cope with this smearing, the Matsubara propagator needs to be modified. We obtain an expansion of the correlation functions of the interacting equilibrium state as a sum over certain type of graphs with mixed edges, some of them correspond to modified Matsubara propagators and others to propagators of the real time formalism. An integration over the adiabatic time cut-off is present in every vertex. However, at every order in perturbation theory, the final result does not depend on the particular form of the cut-off function. The obtained graphical expansion contains in it both the real time formalism and the Matsubara formalism as particular cases. Finally, we show that a particular factorisation which is used to derive the ordinary real time formalism holds only in special cases and we present a counterexample. We conclude with the analysis of certain correlation functions and we notice that corrections to the self-energy in a $\lambda\phi^4$ at finite temperature theory are expected.