Euclidean Green Functions Research Articles

Complex conjugate poles and parton distributions

We calculate parton and generalized parton distributions in Minkowski space using a scalar propagator with a pair of complex conjugate poles. Correct spectral and support properties are obtained only after careful analytic continuation from Euclidean space. Alternately the quark distribution function can be calculated from modified cutting rules, which put the intermediate state on its complex mass shells. Distribution functions agree with those resulting from the model's Euclidean space double distribution which we calculate via non-diagonal matrix elements of twist-two operators. Thus one can use a wide class of analytic parameterizations of the quark propagator to connect Euclidean space Green functions to light-cone dominated amplitudes.

Vacuum polarization of a charged massless scalar field on cosmic string spacetime in the presence of a magnetic field

In this paper, we consider a charged massless scalar quantum field operator in the spacetime of an idealized cosmic string, i.e., an infinitely long, straight and static cosmic string, which presents a magnetic field confined in a cylindrical tube of finite radius. Three distinct situations are taken into account in this analysis: (i) a homogeneous field inside the tube, (ii) a magnetic field proportional to 1/r and (iii) a cylindrical shell with δ-function. In these three cases, the axis of the infinitely long tube of radius R coincides with the cosmic string. In order to study the vacuum polarization effects outside the tube, we explicitly calculate the Euclidean Green function associated with this system for the three above situations, considering points in the region outside the tube. Having these Green functions we calculate the renormalized vacuum expectation values, ⟨Φ̂*(x)Φ̂(x)⟩Ren and ⟨T̂νμ(x)⟩Ren, associated with the charged field. In the evaluation of these vacuum polarization effects, two contributions appear for the three models. The first are the standard ones due to the conical geometry of the spacetime and the magnetic flux. The second contributions appear as extra terms. They are corrections due to the finite thickness of the radius of the tube. These extra terms provide relevant contributions, even for points very far away from the system, like a long-range effect.

Euclidean scalar Green function in a higher dimensional global monopole space–time

We construct the explicit Euclidean scalar Green function associated with a massless field in a higher dimensional global monopole space–time, i.e., a (1+d)-space–time with d⩾3 which presents a solid angle deficit. Our result is expressed in terms of an infinite sum of products of Legendre functions with Gegenbauer polynomials. Although this Green function cannot be expressed in a closed form, for the specific case where the solid angle deficit is very small, it is possible to develop the sum and obtain the Green function in a more workable expression. Having this expression it is possible to calculate the vacuum expectation value of some relevant operators. As an application of this formalism, we calculate the renormalized vacuum expectation value of the square of the scalar field, 〈Φ2(x)〉Ren, and the energy-momentum tensor, 〈Tμν(x)〉Ren, for the global monopole space–time with spatial dimensions d=4 and d=5.

Vacuum polarization for a massless spin-1/2 field in the global monopole spacetime at nonzero temperature

In this paper we present the effects produced by temperature in the renormalized vacuum expectation value of the zero–zero component of the energy–momentum tensor associated with a massless left-handed spinor field in the point-like global monopole spacetime. In order to develop this calculation we had to obtain the Euclidean thermal Green function in this background. Because the expression obtained for the thermal energy density cannot be expressed in a closed form, its explicit dependence on temperature is not completely evident. So, in order to obtain concrete information about its thermal behaviour, we develop a numerical analysis of our result in the high-temperature limit for specific values of the parameter α which codify the presence of the monopole.

Nucleon in the chiral quark model: An alternative computation scheme

We formulate the quark sea contributions to the energy and various physical observables in terms of Euclidean Green functions and their K-spin partial wave reduction. In this framework it is not necessary to discretize the continuous spectrum by introducing a finite boundary. Using this formulation we perform a new self-consistent computation of the nucleon state in the Nambu--Jona-Lasinio model. In addition to this technical advantage, such an alternative computation scheme makes it possible to obtain the numerical predictions of the model in an entirely independent way. We use the Pauli-Villars cutoff to define the model. Our results for the nucleon energy, the mesonic profile, and various observables essentially confirm results obtained by other groups using this regularization. The results for ${g}_{\mathrm{A}},$ obtained on the basis of quark currents, are close to the experimental value.

Gauge-invariant charged, monopole and dyon fields in gauge theories

We propose explicit recipes to construct the Euclidean Green functions of gauge-invariant charged, monopole and dyon fields in four-dimensional gauge theories whose phase diagram contains phases with deconfined electric and/or magnetic charges. In theories with only either abelian electric or magnetic charges, our construction is an Euclidean version of Dirac's original proposal, the magnetic dual of his proposal, respectively. Rigorous mathematical control is achieved for a class of abelian lattice theories. In theories where electric and magnetic charges coexist, our construction of Green functions of electrically or magnetically charged fields involves taking an average over Mandelstam strings or the dual magnetic flux tubes, in accordance with Dirac's flux quantization condition. We apply our construction to 't Hooft-Polyakov monopoles and Julia-Zee dyons. Connections between our construction and the semiclassical approach are discussed.

Method for calculating the imaginary part of the Hadamard elementary functionG(1)in static, spherically symmetric spacetimes

Whenever real particle production occurs in quantum field theory, the imaginary part of the Hadamard Elementary function $G^{(1)}$ is non-vanishing. A method is presented whereby the imaginary part of $G^{(1)}$ may be calculated for a charged scalar field in a static spherically symmetric spacetime with arbitrary curvature coupling and a classical electromagnetic field $A^{\mu}$. The calculations are performed in Euclidean space where the Hadamard Elementary function and the Euclidean Green function are related by $(1/2)G^{(1)}=G_{E}$. This method uses a $4^{th}$ order WKB approximation for the Euclideanized mode functions for the quantum field. The mode sums and integrals that appear in the vacuum expectation values may be evaluated analytically by taking the large mass limit of the quantum field. This results in an asymptotic expansion for $G^{(1)}$ in inverse powers of the mass $m$ of the quantum field. Renormalization is achieved by subtracting off the terms in the expansion proportional to nonnegative powers of $m$, leaving a finite remainder known as the ``DeWitt-Schwinger approximation.'' The DeWitt-Schwinger approximation for $G^{(1)}$ presented here has terms proportional to both $m^{-1}$ and $m^{-2}$. The term proportional to $m^{-2}$ will be shown to be identical to the expression obtained from the $m^{-2}$ term in the generalized DeWitt-Schwinger point-splitting expansion for $G^{(1)}$. The new information obtained with the present method is the DeWitt-Schwinger approximation for the imaginary part of $G^{(1)}$, which is proportional to $m^{-1}$ in the DeWitt-Schwinger approximation for $G^{(1)}$ derived in this paper.

Euclidean Thermal Green Functions of Photons in Generalized Euclidean Rindler Spaces for Any Feynman-Like Gauge

The thermal Euclidean Green functions for Photons propagating in the Rindler wedge are computed by employing an Euclidean approach within any covariant Feynman-like gauge. This is done by generalizing a formula which holds in the Minkowskian case. The coincidence of the found (β = 2π)-Green functions and the corresponding Minkowskian vacuum Green functions is discussed in relation to the remaining static gauge ambiguity already found in previous works. Further generalizations to more complicated manifolds are discussed. Ward identities are verified in the general case.

Magnetic Monopoles in U (1) 4 Lattice Gauge Theory with Wilson Action

We construct the Euclidean Green functions for the soliton (magnetic monopole) field in the U(1)_4 Lattice Gauge Theory with Wilson action. We show that in the strong coupling regime there is monopole condensation while in the QED phase the physical Hilbert space splits into orthogonal soliton sectors labelled by integer magnetic charge.

Canonical quantization of photons in a Rindler wedge

Photons and thermal photons are studied in the Rindler wedge employing Feynman’s gauge and canonical quantization. A Gupta–Bleuler-like formalism is explicitly implemented. Nonthermal Wightman functions and related (Euclidean and Lorentzian) Green functions are explicitly calculated and their complex time analytic structure is carefully analyzed using the Fulling–Ruijsenaars master function. The invariance of the advanced minus retarded fundamental solution is checked and a Ward identity discussed. It is suggested that the KMS condition can be implemented to define thermal states also dealing with unphysical photons. Following this way, thermal Wightman functions and related (Euclidean and Lorentzian) Green functions are built up. Their analytic structure is carefully examined employing a thermal master function as in the nonthermal case and other corresponding properties are discussed. Some subtleties arising dealing with unphysical photons in the presence of the Rindler conical singularity are pointed out. In particular, a one-parameter family of thermal Wightman and Schwinger functions with the same physical content is proved to exist due to a remaining (nontrivial) static gauge ambiguity. A photon version of the Bisognano–Wichmann theorem is investigated in the case of photons propagating in the Rindler Wedge employing Wightman functions. In spite of the found ambiguity in defining Rindler Green functions, the coincidence of (β=2π)-Rindler Wightman functions and Minkowski Wightman functions is proved dealing with test functions related to physical photons and Lorentz photons.

Covariant quantization of the skyrmion

We obtain the internal degrees of freedom of the skyrmion (spin and isospin) within a manifestly Lorentz covariant quantization framework based on defining Green functions for skyrmions and, then, the S-matrix via LSZ reduction. Our method follows Fröhlich and Marchetti's definition of Euclidean soliton Green functions, supplemented with a careful treatment of the boundary conditions around the singularities. The covariant two-point function obtained propagates a tower of spin equal to isospin particles. Our treatment contains the usual method of collective coordinates, as a non-relativistic limit and, because of the new topology introduced, it leads, in a natural way, to the inequivalent (boson/fermion) quantizations of the SU(2) skyrmion.

First-order variational calculation of form factor in a scalar nucleon-meson theory

We investigate a relativistic quantum field theory in the particle representation using a non-perturbative variational technique. The theory is that of two massive scalar particles, `nucleons' and `mesons', interacting via a Yukawa coupling. We calculate the general Euclidean Green function involving two external nucleons and an arbitrary number of external mesons in the quenched approximation for the nucleons. The non-perturbative renormalization and truncation is done in a consistent manner and results in the same variational functional independent of the number of external mesons. We check that the calculation agrees with one-loop perturbation theory for small couplings. As an illustration the special case of meson absorption on the nucleon is considered in detail. We derive the radius of the dressed particle and numerically investigate the vertex function after analytic continuation to Minkowski space.

Topological interactions in (2 + 1)-gravity: classical fields

In the framework of classical field theory we consider some new topological effects in (2 + 1)-dimensional Einstein gravity. The investigation is based on the explicit expression for the Euclidean Green function for massless fields (scalar with minimal coupling or electrostatic) on the two-dimensional Riemannian surface. Expressions for the topological self-energy of a point charge and point dipole are obtained. These results are interpreted in electrostatics terms on the Euclidean plane. The applications to the cases of (2 + 1)-dimensional conical and multiconical spaces and to the spacetime of a circular dust string and spherical star in (2 + 1) dimensions are considered. We have found the existence of an additional interaction between charged particles which depends on the topology of the Riemannian surface. The same effect arises when we consider two uncharged gravitating particles in the presence of classical charged matter. Both these effects may be considered as a Casimir-like interaction in classical theory.

Equivalence of Euclidean and Wightman field theories

A new inversion formula for the Laplace transformation of tempered distributions with supports in the closed positive semiaxis is obtained. The inverse Laplace transform of a tempered distribution is defined by means of a limit of a special distribution constructed from this distribution. The weak spectral condition on the Euclidean Green's functions implies that some of the limits needed for the inversion formula exist for any Euclidean Green's function with an even number of variables. We then prove that the initial Osterwalder-Schrader axioms [1] and the weak spectral condition are equivalent with the Wightman axioms.

W*-KMS structure from multi-time Euclidean Green functions

An axiomatic approach to the problem of reconstruction of a dynamics from the Euclidean multi-time Green functions is proposed. The existence of a W*-algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown. A condition under which the reconstruction from the one-time and multi-time Green functions are equivalent is formulated.

Reconstruction of Kubo–Martin–Schwinger structure from Euclidean Green functions

The explicit form of the Poisson kernel for the strip Tβ={z∈C:0<Im z<β} is used to derive some conditions for a state ω on a given C*-algebra dynamical system (𝒜,αt) to be a Kubo–Martin–Schwinger (KMS) state. Reflection positivity and generalized periodicity of the Euclidean Green functions corresponding to ω are established and used for an analytic continuation to real time. Reflection positivity is also used for the construction of a cyclic and faithful representation of (𝒜,αt) on some Hilbert space. A sufficient condition for the identification of the constructed dynamics with the modular group of a von Neumann algebra generated by the representation is formulated in terms of the Euclidean Green functions.

Quantum mechanical tunnelling at finite energy and its equivalent amplitudes in the (vacuum) instanton approximation

The transition amplitude for quantum tunnelling between neighbouring vacua of the sine-Gordon potential at finite energy is estimated with the help of the path-integral method. The tunnelling process is dominated by nonvacuum (periodic) instanton configurations. It is explicitly shown that the amplitude for tunnelling between the nth excited states which is dominated by nonvacuum instantons is equivalent to that of the vacuum instanton approximation calculated with the 2n-point euclidean Green's function, which is the standard calculation of instanton-like transitions in high energy collisions. The equivalence, however, is valid only in the low energy limit, whereas the amplitude for quantum tunnelling with nonvacuum instantons is valid for energies up to the sphaleron mass.

Dirac thermal propagator in a constant external magnetic field.

A real-time fermion propagator in a static uniform magnetic field and at a finite chemical potential is obtained by analytically continuing the corresponding Euclidean Green's function back to the real time. The resulting propagator, which is written explicitly in terms of the $\ensuremath{\gamma}$ matrices, recovers the known result in the chiral representation.

Vacuum fluctuations of twisted fields in the spacetime of a cosmic string-thermal effects

In the case of quantum field theory at finite temperature the author investigates the thermal Euclidean Green function for massive twisted scalar fields in the spacetime of an infinite straight cosmic string. Then taking the limit, when mass goes to zero, the author obtains the integral form of the mean-square field as a function of temperature.

Self-interaction and quantum effects near a point mass in three-dimensional gravitation

The authors determine some gravitational effects in the classical and quantum massive scalar field theories near a point mass in the Einstein theory in three dimensions and near a particular spinning point mass in the topologically massive gravity. Classically, they find the expression, to a first-order approximation in the deficit angle of the conical spacetime, for the self-force acting on a scalar charge at rest. Within quantum field theory, they obtain the vacuum expectation value of the energy-momentum operator for a massive scalar field by determining the Euclidean Green function. They provide an explicit expression for the vacuum energy-momentum tensor to first order in the deficit angle for an arbitrary coupling parameter. Near a point mass in the Einstein theory, they calculate the gravitational force on a massive test particle due to the backreaction of the vacuum energy-momentum tensor. For the minimal and conformal couplings, they show that the gravitational force is always attractive. Near the particular spinning point mass in the topologically massive gravity, they can prove that the gravitational force is always attractive for a massless scalar field.