Euclidean Spacetime Research Articles

The Green's causal functions of bilocal fields

On concrete examples it is demonstrated that in quantum field theory with three universal constants c, h, and k suggested in [1], the limiting value k–1 = 0 corresponding to local theory is a bifurcation point: the Green's causal function for finite k bifurcates into \(\tilde D_L^c {\text{ }}and{\text{ }}\tilde D_E^c \). The natural carrier of the latter is the Euclidean space-time R4, and its continuation from R4 to R3,1 is regular. The former is singular at zero point and on the light cone and hence is rejected.

A class of nonlocal truncations in quantum Einstein gravity and its renormalization group behavior

Motivated by the conjecture that the cosmological constant problem could be solved by strong quantum effects in the infrared we use the exact flow equation of Quantum Einstein Gravity to determine the renormalization group behavior of a class of nonlocal effective actions. They consist of the Einstein-Hilbert term and a general nonlinear function F(k, V) of the Euclidean space-time volume V. A partial differential equation governing its dependence on the scale k is derived and its fixed point is analyzed. For the more restrictive truncation of theory space where F(k, V) is of the form V+V ln V, V+V^2, and V+\sqrt{V}, respectively, the renormalization group equations for the running couplings are solved numerically. The results are used in order to determine the k-dependent curvature of the S^4-type Euclidean space-times which are solutions to the effective Einstein equations, i.e. stationary points of the scale dependent effective action. For the V+V ln V-invariant (discussed earlier by Taylor and Veneziano) we find that the renormalization group running enormously suppresses the value of the renormalized curvature which results from Planck-size bare parameters specified at the Planck scale. Hence one can obtain very large, almost flat universes without fine-tuning the cosmological constant.

Are there really any AdS2 branes in the Euclidean (or not) AdS3?

We do not find any AdS2 branes, neither in the H3+ WZNW model nor in the SL(2,R) WZNW model. We then reexamine the case of the branes that possess a su(2) symmetry: we speculate that they would have to live on the boundary of AdS3. This cannot be realized in an Euclidean spacetime, but in the SL(2,R) WZNW model by analytical continuation.

Comments on the Equivalence between Chern-Simons Theory and Topological Massive Yang-Mills Theory in 3D

The classical formal equivalence upon a redefinition of the gauge connection between Chern-Simons theory and topological massive Yang-Mills theory in three-dimensional Euclidean space-time is analyzed at the quantum level within the BRST formulation of the Equivalence Theorem. The parameter controlling the change in the gauge connection is the inverse $\lambda$ of the topological mass. The BRST differential associated with the gauge connection redefinition is derived and the corresponding Slavnov-Taylor (ST) identities are proven to be anomaly-free. The Green functions of local operators constructed only from the ($\lambda$-dependent) transformed gauge connection, as well as those of BRST invariant operators, are shown to be independent of the parameter $\lambda$, as a consequence of the validity of the ST identities. The relevance of the antighost-ghost fields, needed to take into account at the quantum level the Jacobian of the change in the gauge connection, is analyzed. Their role in the identification of the physical states of the model within conventional perturbative gauge theory is discussed.

Quantum behavior of FRW radiation-filled universes

We study the quantum vacuum fluctuations around closed Friedmann-Robertson-Walker (FRW) radiation-filled universes with nonvanishing cosmological constant. These vacuum fluctuations are represented by a conformally coupled massive scalar field and are treated in the lowest order of perturbation theory. In the semiclassical approximation, the perturbations are governed by differential equations which, properly linearized, become generalized Lame equations. The wave function thus obtained must satisfy appropriate regularity conditions which ensure its finiteness for every field configuration. We apply these results to asymptotically anti de-Sitter Euclidean wormhole spacetimes and show that there is no catastrophic particle creation in the Euclidean region, which would lead to divergences of the wave function.

Fermionic signature of the lattice monopoles

We consider fermions in the field of static monopole-like configurations in the Euclidean space-time. In all the cases considered there exists an infinite number of zero modes, labeled by frequency i\omega. The existence of such modes is a manifestation of instability of the vacuum in the presence of the monopoles and massless fermions. In the Minkowski space the corresponding phenomenon is well known and is a cornerstone of the theory of the magnetic catalysis. Moreover, the well known zero mode of Jackiw and Rebbi corresponds to the limiting case, \omega = 0. We provide arguments why the chiral condensate could be linked to the density of the monopoles in the infrared cluster. A mechanism which can naturally explain the equivalence of the critical temperatures for the deconfinement and chiral transitions, is proposed. We discuss possible implications for the phenomenology of the lattice monopoles.

Infrared exponents and running coupling of SU( N) Yang–Mills theories

We present approximate solutions for the gluon and ghost propagators as well as the running coupling in Landau gauge Yang–Mills theories. We solve the corresponding Dyson–Schwinger equations in flat Euclidean spacetime without any angular approximation. This supplements recently obtained results employing a four-torus, i.e., a compact spacetime manifold, as infrared regulator. We confirm previous findings deduced from an extrapolation with tori of different volumes: the gluon propagator is weakly vanishing in the infrared and the ghost propagator is highly singular. For non-vanishing momenta our propagators are in remarkable agreement with recent lattice calculations.

The dual of non-Abelian Lattice Gauge Theory

Non-Abelian Lattice Gauge Theory in Euclidean space-time of dimension d ≥ 2 whose gauge group is any compact Lie group is related to a Spin Foam Model by an exact strong-weak duality transformation. The group degrees of freedom are integrated out and replaced by combinatorial expressions involving irreducible representations and intertwiners of the gauge group. This transformation is available for the partition function, for the expectation value of observables (spin networks), and for the correlator of centre monopoles which is a ratio of partition functions in the original model and an ordinary expectation value in the dual formulation.

Curvature Vacuum Correlations in N -Dimensional Einstein Gravity

Under the flat Euclidean space–time background, the expressions for the leading terms of several two-point curvature vacuum correlation functions in N-dimensional Einstein gravity are calculated by using the perturbative expansion of the metric. It is shown that the contributions of the leading terms of such two-point curvature vacuum correlation functions are all vanishing.

Notes on noncommutative instantons

We study in detail the ADHM construction of U( N) instantons on noncommutative Euclidean space–time R NC 4 and noncommutative space R NC 2× R 2 . We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U( N) instanton solutions on R NC 4 and on R NC 2× R 2 . We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons is always an integer.

Four-dimensional lattice gauge theory with ribbon categories and the Crane–Yetter state sum

Lattice Gauge Theory in four-dimensional Euclidean space–time is generalized to ribbon categories which replace the category of representations of the gauge group. This provides a framework in which the gauge group becomes a quantum group while space–time is still given by the “classical” lattice. At the technical level, this construction generalizes the spin foam model dual to lattice gauge theory and defines the partition function for a given triangulation of a closed and oriented piecewise-linear four-manifold. This definition encompasses both the standard formulation of d=4 pure Yang–Mills theory on a lattice and the Crane–Yetter invariant of four-manifolds. The construction also implies that certain classes of spin foam models formulated using ribbon categories are well-defined even if they do not correspond to a topological quantum field theory.

Examining the Origin of Cosmic Rays: A New Conceptual Approach

A new conceptual approach for examining the origin of cosmic rays is developed by considering the characteristics of particle trajectory distributions in four-dimensional Euclidean space-time. Transformation of an isotropic distribution into a velocity distribution in ordinary three-dimensional space results in a spatially isotropic flux of highly relativistic particles. This particle flux exhibits a power-law energy spectrum that decreases with increasing energy approximately inversely as the square of the energy and peaks at much lower energies. This model also provides additional results resembling the observed characteristics of cosmic radiation.

Hybrid Monte Carlo algorithm for the double exchange model

The Hybrid Monte Carlo algorithm is adapted to the simulation of a system of classical degrees of freedom coupled to non self-interacting lattices fermions. The diagonalization of the Hamiltonian matrix is avoided by introducing a path-integral formulation of the problem, in $d+1$ Euclidean space-time. A perfect action formulation allows to work on the continuum euclidean time, without need for a Trotter-Suzuki extrapolation. To demonstrate the feasibility of the method we study the Double Exchange Model in three dimensions. The complexity of the algorithm grows only as the system volume, allowing to simulate in lattices as large as $16^3$ on a personal computer. We conclude that the second order paramagnetic-ferromagnetic phase transition of Double Exchange Materials close to half-filling belongs to the Universality Class of the three-dimensional classical Heisenberg model.

Polyakov's spin factor and new algorithms for efficient perturbative computations in QCD

Polyakov's spin factor enters as a weight in the path-integral description of particle-like modes propagating in Euclidean space–times, accounting for particle spin. The Fock–Feynman–Schwinger path integral applied to QCD accommodates Polyakov's spin factor in a natural manner while, at the same time, it identifies Wilson line (loop) operators as sole agents of interaction dynamics among matter and gauge field quanta. A direct application of such a separation between spin content and dynamics is the emergence of master expressions for the perturbative series involving either open or closed fermionic lines which provide new, comprehensive approaches to perturbative QCD.

Signature Change and Clifford Algebras

Given the real Clifford algebra of a quadratic space with a given signature, we define a new product in this structure such that it simulates the Clifford product of a quadratic space with another signature different from the original one. Among the possible applications of this new product, we use it in order to write the Minkowskian Dirac equation over the Euclidean spacetime and to define a new duality operation in terms of which one can find self-dual and anti-self-dual solutions of gauge fields over Minkowski spacetime analogous to the ones over Euclidean spacetime and without needing to complexify the original real algebra.

Multiple vacua in two-dimensional Yang-Mills theory

Two-dimensional SU( N) Yang-Mills theory is endowed with a non-trivial vacuum structure (κ-sectors). The presence of κ-sectors modifies the energy spectrum of the theory and its instanton content, the (Euclidean) spacetime being compactified on a sphere. For the exact solution, in the limit in which the sphere is decompactified, a κ-sector cane mimicked by the presence of κ-fundamental charges at ∞, according to a Witten's suggestion. However, this property neither holds before decompactification nor for the genuine perturbative solution which corresponds to the zero-instanton contribution on the sphere.

Wilson Loops in the Adjoint Representation and Multiple Vacua in Two-Dimensional Yang–Mills Theory

QCD2 with fermions in the adjoint representation is invariant under SU(N)/ZN and thereby is endowed with a nontrivial vacuum structure (k-sectors). The static potential between adjoint charges, in the limit of infinite mass, can be therefore obtained by computing Wilson loops in the pure Yang–Mills theory with the same nontrivial structure. When the (Euclidean) space-time is compacted on a sphere S2, Wilson loops can be exactly expressed in terms of an infinite series of topological excitations (instantons). The presence of k-sectors modifies the energy spectrum of the theory and its instanton content. For the exact solution, in the limit in which the sphere is decompacted, a k-sector can be mimicked by the presence of k-fundamental charges at ∞, according to Witten's suggestion. However, this property does not hold before decompaction or for the genuine perturbative solution which corresponds to the zero-instanton contribution on S2.

A Newtonian Expanding Universe

1. The phenomenon of the expansion of the universe has usually been discussed by students of relativity by means of the concept of ‘expanding space’. This concept, though mathematically significant, has by itself no physical content; it is merely the choice of a particular mathematical apparatus for describing and analysing phenomena. An alternative procedure is to choose a static space, as in ordinary physics, and analyse the expansion-phenomenon as actual motions in this space. Moving particles in a static space will give the same observable phenomena as stationary particles in ‘expanding’ space. In each case the space is a construct built up by the mathematician out of observations that could in principle be made; it is built up around the matter in motion according to certain rules. The formulation of the relevant laws of nature depends on the rules adopted, and the laws will be quite different if different rules are adopted, as I have elsewhere† explained. The alternative procedures have been tersely described in a recent paper by S. R. Milner.‡ He explained that we can either modify our geometry in order to retain d ∫ ds c 0 as the paths of free particles, or retain Euclidean geometry and Minkowski space-time and modify the variational principle by weighting the elements of path ds with appropriate invariant weighting factors. Einstein’s general relativity adopts the first procedure; in my recent treatment of the cosmological problem I adopted the second procedure. In neither case has the space constructed any physical significance—no content attaches to the phrase ‘the space of nature’—but the second procedure has the advantage that it employs the space commonly used in physics. The equations of motion I obtained for free particles moving in the presence of a certain distribution of matter in motion were derived by the straightforward method of making

Further studies on the smoothed SU(2) gauge configurations

The field variables (FV), the strength tensor (ST) and the topological charge density (TCD) of lattice SU(2) gauge field are studied in the momentum space. The evolution behaviors of their Fourier components are monitored during the smoothing procedure. The high frequency components of the FV can not be eliminated by smoothing, but all the Fourier amplitudes of the ST are uniformly suppressed. The distribution of the TCD in Euclidean spacetime is highly smoothed. Based on these facts, we give a new interpretation of the structure of the smoothed gauge configurations and argue that the ultraviolet and infrared cutoff introduced by lattice discretization account for the suppression of the instanton size distribution at both the large size end and the small size end.

Path integral representation for Wilson loops and the non-Abelian Stokes theorem

We discuss the derivation of the path integral representation over gauge degrees of freedom for Wilson loops in $\mathrm{SU}(N)$ gauge theory and 4-dimensional Euclidean space-time by using well-known properties of group characters. A discretized form of the path integral is naturally provided by the properties of group characters and does not need any artificial regularization. We show that the path integral over gauge degrees of freedom for Wilson loops derived by Diakonov and Petrov [Phys. Lett. B 224 131 (1989)] by using a special regularization is erroneous and predicts zero for the Wilson loop. This property is obtained by direct evaluation of path integrals for Wilson loops defined for pure $\mathrm{SU}(2)$ gauge fields and $Z(2)$ center vortices with spatial azimuthal symmetry. Further we show that both derivations given by Diakonov and Petrov for their regularized path integral, if done correctly, predict also zero for Wilson loops. Therefore, the application of their path integral representation of Wilson loops cannot give a new way to check confinement in lattice as has been declared by Diakonov and Petrov [Phys. Lett. B 242 425 (1990)]. From the path integral representation which we consider we conclude that no new non-Abelian Stokes theorem can exist for Wilson loops except the old-fashioned one derived by means of the path-ordering procedure.