Euclidean Region Research Articles

On the formula of Mathews and Salam

A direct proof of the Mathews-Salam formula in the Euclidean region is given for a Dirac field interacting with a classical time-dependent field and with a scalar Boson field. Cutoffs are not removed. The result is cast in the setting of a probability gage space.

Some inequalities for correlation functions dominated by repulsive interactions

For certain interactions of a boson or fermion particle in quantum statistical mechanics any correlation function is bounded in absolute value by the corresponding free-boson correlation function (in the Euclidean region).

Two and three body equations in quantum field models

In each pure phase of a ℘(φ)2 quantum field model, we establish local regularity of the Green's functions and exponential decay for noncritical models. We establish the existence of two-particle and three-particle Bethe-Salpeter kernels in the Euclidean region.

Asymptotic behavior and symmetry breaking

The asymptotic behavior of Green's functions for large spacelike momenta in the presence of symmetry breaking is considered using the renormalization-group approach. The disappearance of symmetry-breaking effects due to a generalized mass term in the deep Euclidean region allows us to derive a certain inequality satisfied by a function simply related to the Callan-Symanzik function of a "dimensional" coupling constant of the theory. The inequality holds at a fixed point of the renormalization group.

Asymptotic symmetry

We discuss the deep Euclidean behavior of Green's functions in models in which symmetries are broken spontaneously or by "soft" operators. Our proof that the Green's functions approach their symmetric values in the deep Euclidean region uses the Callan-Symanzik equation in a way which avoids relying on order-by-order power counting for the scalar tadpole insertions. The vacuum expectation value of the scalar field appears in the Callan-Symanzik equation as an additional coupling constant; the solution of the equation introduces a momentum-dependent effective vacuum expectation value. We discuss theories with or without gauge invariance of the second kind.

Deviation from power law at a Callan-Symanzik eigenvalue and the non-negligibility of mass-insertion terms

The mass-insertion term in the Callan-Symanzik equation may not be negligible, even in the deep Euclidean region, because the asymptotic behavior of vertex functions may be different from that of individual dominant graphs. When contributions from different orders of perturbation are related by a constraint, such as the condition that the physical coupling constant is at a nontrivial Callan-Symanzik eigenvalue, cancellations between dominant graphs may occur. This and other circumstances under which vertex functions may fail to satisfy homogeneous Callan-Symanzik equations asymptotically are illustrated with a very simple example, thus emphasizing that the question of asymptotic scale invariance may not be always decided by studying the properties of the $\ensuremath{\beta}$ function alone.

Quasi-bounded and singular functions

A general formulation is given for the concepts of quasi-bounded and singular functions, thereby extending to a much broader class of functions the concepts initially formulated by Parreau in the harmonic case. Let Ω \Omega be a bounded Euclidean region. With the underlying space taken as the class M \mathcal {M} of all nonnegative functions u on Ω \Omega admitting superharmonic majorants, an operator S is introduced by setting Su equal to the regularization of the infimum over λ ≥ 0 \lambda \geq 0 of the regularized reduced functions for ( u − λ ) + {(u - \lambda )^ + } . Quasi-bounded and singular functions are then defined as those u for which S u = 0 Su = 0 and S u = u Su = u , respectively. A development based on properties of the operator S leads to a unified theory of quasi-bounded and singular functions, correlating earlier work of Parreau (1951), Brelot (1967), Yamashita(1968), Heins (1969), and others. It is shown, for example, that a nonnegative function u on Ω \Omega is quasi-bounded if and only if there exists a nonnegative, increasing, convex function φ \varphi on [ 0 , ∞ ] [0,\infty ] such that φ ( x ) / x → + ∞ \varphi (x)/x \to + \infty as x → ∞ x \to \infty and φ ∘ u \varphi \circ u admits a superharmonic majorant. Extensions of the theory are made to the vector lattice generated by the positive cone of functions u in M \mathcal {M} satisfying S u ≤ u Su \leq u .

E+e−Annihilation in Gauge Theories of Strong Interactions

We discuss the behavior of the ${e}^{+}{e}^{\ensuremath{-}}$ total annihilation cross section into hadrons in a class of non-Abelian vector gluon theories. Using the quasi-free-field behavior of these theories in the deep Euclidean region, we show that the asymptotic behavior of ${\ensuremath{\sigma}}_{\mathrm{tot}}^{{e}^{+}{e}^{\ensuremath{-}}}$ is that of the parton model. Nonleading corrections fall off relative to this leading term like inverse powers of logarithms with calculable coefficients. The problem of making more detailed statements about the final state is discussed.

On the asymptotic behavior of Feynman diagrams

Theorem generalizing Weinberg's theorem on asymptotic behavior of Feynman diagrams is proved. Not only the leading asymptotic terms but also the subsequent ones are taken into account. It is shown that if the momenta go to infinity in the Euclidean region, regularized matrix elements of Feynman diagrams can be represented in the form of a polynomial over momenta and their logarithms, plus the terms decreasing faster than any negative power of momenta. (auth)

Light-Cone Operator Expansions in Perturbation Theory. II

A systematic investigation, in perturbation theory, is presented of the light-cone behavior of multiparticle matrix elements of time-ordered products of local fields: $\ensuremath{\int}{d}^{4}x{e}^{\mathrm{idx}}〈\ensuremath{\beta}|T{\ensuremath{\psi}}_{1}(x){\ensuremath{\psi}}_{2}(0)|\ensuremath{\alpha}〉$. In the limit ${q}_{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\infty}$, the contribution of any single Feynman graph is of the form ${{q}_{\ensuremath{-}}}^{\ensuremath{\beta}}{(\mathrm{ln}{q}_{\ensuremath{-}})}^{\ensuremath{\gamma}}$. The main result here is a rule by means of which the integers $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ can be read off from the topology of the graph. The implications of this investigation for local field theories are organized and discussed in operator language in a companion paper. A by-product of the special methods here developed to obtain asymptotic estimates in perturbation theory is a refinement of Weinberg's theorem for the Euclidean region: the determination of the logarithmic factors in the asymptotic form of Feynman amplitudes when a set of external momenta ${q}_{1}, {q}_{2}, \dots{}$ is allowed to approach infinity according to ${q}_{i}=\ensuremath{\eta}{{q}^{\ensuremath{'}}}_{i}, \ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\infty}$.

Statistical mechanics results in the P( φ) 2 quantum field theory

We apply ideas and methods from classical statistical mechanics to study the P( φ) 2 self-coupled two-dimensional Boson field theory in the Euclidean region. In particular, we consider correlation inequalities of Griffiths type; the thermodynamic limit for the pressure, the average interaction and the entropy; and the equilibrium equations for states associated with a given interaction.

Consistency in the Reconstruction of Patterns from Sample Data

Let A be a k-dimensional Euclidean region having unit volume. An m-colors pattern is a partition PA of A into m sets Ai, i—1,…, m with positive volume. PA is called a random pattern if in addition the partition of A is a realization of a random process with the following stationarity and isotropy properties:(i) for all points s ∊ A, P(s ∊ Ai) =pi9, i = 1,..., m(ii) for all pair of points s, s'∊ A with distance |s—s|=d between them, P(s' ∊ Ai | s ∊ Aj)Pij(d) i,j= 1,..., m.

Generalized Transformation Functional of a Continuum Model of the Dual Amplitude

The aim of this paper is to clarify the space-time picture of the functional integral formu­ lation of the duality theory. It is shown that the functional underlying in this formulation can be considered as Dirac's generalized transformation functional (g.t.f.) of the position operator of an elastic string. The g.t.f. can be introduced only in the Euclidean metric theory. Several postulates are imposed on it. A functional differential equation of the g.t.f. is obtained. A manifestly crossing symmetric coupling scheme of the string with external sources is set up on the basis of the g.t.f. in the Euclidean framework; physical amplitude is obtained by analytic continuation with respect to the external momenta from the Euclidean region to the Minkowskian region. A few elementary consequences for the scattering amplitude in this scheme are obtained.

Construction of Unitary, CovariantSMatrices Defined by Convergent Perturbation Series

A method is given for the construction of $S$ matrices which are unitary and covariant. Moreover, the $S$ matrix elements are the analytic continuations to timelike momenta of quantities which are defined by convergent perturbation series for Euclidean momenta. It is shown that the $S$ matrix in the Euclidean region has the same form as the correlation functions of classical gases which interact through stable, regular pair potentials. The known results on the existence of a thermodynamic limit and convergence of activity (fugacity) series for such gases are exploited in the $S$-matrix theory. Approximation schemes suggested by the analogy are noted.

On the zero-mass superpropagator

A rigorous construction for the zero-mass exponential superpropagator is given. It is based on an explicit regularization of the propagator in coordinate space and a suitable choice of the coupling constant is taken in order to damp down the regularized superpropagator to an L 2 function. By applying a Sommerfeld-Watson transformation and a Fourier transformation we are able to show that the regularized superpropagator tends, in an appropriate sense, both in momentum and coordinate space, to some localizable generalized function. Analytic continuation in the coupling constant shows the appearance of a logarithmic cut. By means of the regularization of the propagator and the special choice of the coupling constant we have neither to cut divergent integrals nor to pass to Euclidean region. Extensions to superpropagators which are formally written as entire functions of the propagator are given.

On the Renormalization of Feynman Integrals

AbstractAn arbitrary Feynman integral is considered for external momenta in the Euclidean region, the usual rotation of energy contours having been used to write the integral as an integral over Euclidean internal momenta. A compactification of the space of internal momenta is defined, and the Feynman integral is written as the integral of a current on this compact manifold. This presentation of the integral is used to give a proof of the convergence criterion for Feynman integrals, and to show that a well‐defined renormalized integral may be obtained by a subtraction operation or by analytic renormalization.

Reconstructing Patterns from Sample Data

Abstract : A Euclidean region is partitioned in an unknown way into a known number of subregions. For a discrete set of points in the region we may observe into which subregion they each fall. The problem is to obtain an estimated reconstruction of the partition based on the observations. Section 1 outlines a probabilistic approach to the problem and defines criteria of goodness for an estimated reconstruction. Section 2 evaluates the performance of nearest-point rules. Section 3 examines the effect of sample-size and compares several arrangements of the data points. Section 4 proposes a modification of nearest-point procedures having a certain optimality property. Section 5 is more or less independent of the earlier sections; it explores several probabilistic partitioning models. (Author)

Formulation of a scalar quantum field theory with an essentially non-linear

In scalar quantum field theory with a local, essentially non-linear, interaction a method is suggested of constructing an S-matrix which is finite in all orders of perturbation theory. The construction is made for the amplitudes of physical processes in the Euclidean region of spacelike external momenta. It is shown that there is a principle non-uniqueness when formulating the problem.