Euclidean Spacetime Research Articles

On the Dynamics of Euclidean Space–Time at the Planck Scale

In this study, a radical hypothesis concerning the wave-particle duality exhibited by extremely small objects in nature is explored by developing a Planck lattice model of space–time that grapples with the uncertain dynamic quantum structure of space–time at the Planck scale. Upon applying the Planck lattice model to two notable experiments that most clearly demonstrate the essence of wave-particle duality, one immediately finds that it successfully shows in a relatively straightforward and physically consistent manner how parcels of matter and energy, i.e., electrons and photons, respectively, can behave like waves. The heuristic concepts regarding the underlying structure of space–time contained herein are intended to show the classical particle description of matter and energy as fundamental, while at the same time doing away with the widely held notion of a continuous space–time.

Comments on wormholes, ensembles, and cosmology

Certain closed-universe big-bang/big-crunch cosmological spacetimes may be obtained by analytic continuation from asymptotically AdS Euclidean wormholes, as emphasized by Maldacena and Maoz. We investigate how these Euclidean wormhole spacetimes and their associated cosmological physics might be described within the context of AdS/CFT. We point out that a holographic model for cosmology proposed recently in arXiv:1810.10601 can be understood as a specific example of this picture. Based on this example, we suggest key features that should be present in more general examples of this approach to cosmology. The basic picture is that we start with two non-interacting copies of a Euclidean holographic CFT associated with the asymptotic regions of the Euclidean wormhole and couple these to auxiliary degrees of freedom such that the original theories interact strongly in the IR but softly in the UV. The partition function for the full theory with the auxiliary degrees of freedom can be viewed as a product of partition functions for the original theories averaged over an ensemble of possible sources. The Lorentzian cosmological spacetime is encoded in a wavefunction of the universe that lives in the Hilbert space of the auxiliary degrees of freedom.

Notes on massless scalar field partition functions, modular invariance and Eisenstein series

The partition function of a massless scalar field on a Euclidean spacetime manifold ℝd−1 × \U0001d54b2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, ℤ) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL(2, ℤ) invariance. When the spacetime manifold is ℝp × \U0001d54bq+1, the partition function is given in terms of a SL(q + 1, ℤ) Eisenstein series again with s = (d + 1)/2. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On \U0001d54bd+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly.

Towards perturbative renormalization ofϕ2(iϕ)ϵquantum field theory

In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in $D$-dimensional Euclidean spacetime, where $\phi$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $\varepsilon$ and were calculated to first order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= \frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\mu^2\phi^2+\frac{1}{2} g\mu_0^2\phi^2\big(i\mu_0^{1-D/2}\phi\big)^\varepsilon-iv\phi$, which now includes renormalization counterterms that are linear and quadratic in the field $\phi$. The parameter $g$ is a dimensionless coupling strength and $\mu_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq D\leq2$). Renormalization is performed perturbatively to second order in $\varepsilon$ and the structure of the Green's functions is analyzed in the limit $D\to 2$. A sum of the most divergent terms is performed to {\it all} orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.

A modified Villain formulation of fractons and other exotic theories

We reformulate known exotic theories (including theories of fractons) on a Euclidean spacetime lattice. We write them using the Villain approach, and then we modify them to a convenient range of parameters. The new lattice models are closer to the continuum limit than the original lattice versions. In particular, they exhibit many of the recently found properties of the continuum theories, including emergent global symmetries and surprising dualities. In addition, these new models provide a clear and rigorous formulation to the continuum models and their singularities. In Appendixes A–C, we use this approach to review the well-studied lattice models and their continuum limits. These include the XY-model, the ZN clock-model, and various gauge theories in diverse dimensions. This paper clarifies the relation between the condensed-matter and the high-energy views of these systems. It emphasizes the role of symmetries associated with the topology of field space, duality, and various anomalies.

All order renormalizable refined Gribov-Zwanziger model with BRST invariant fermionic horizon function in linear covariant gauges

We introduce, within the refined Gribov-Zwanziger setup, a composite Becchi-Rouet-Stora-Tyutin (BRST) invariant fermionic operator coupled to the inverse of the Faddeev-Popov operator. As a result, an effective BRST invariant action in Euclidean space-time is constructed, enabling us to pave the first step towards the study of the behavior of the fermion propagator in the infrared region in the class of the linear covariant gauges. The aforementioned action is proven to be renormalizable to all orders by means of the algebraic renormalization procedure.

Thermal Casimir effect for the scalar field in flat spacetime under a helix boundary condition

In this work we consider the generalized zeta function method to obtain temperature corrections to the vacuum (Casimir) energy density, at zero temperature, associated with quantum vacuum fluctuations of a scalar field subjected to a helix boundary condition and whose modes propagate in ($3+1$)-dimensional Euclidean spacetime. We find closed and analytical expressions for both the two-point heat kernel function and free energy density in the massive and massless scalar field cases. In particular, for the massless scalar field case, we also calculate the thermodynamics quantities internal energy density and entropy density, with their corresponding high- and low-temperature limits. We show that the temperature correction term in the free energy density must suffer a finite renormalization, by subtracting the scalar thermal blackbody radiation contribution, in order to provide the correct classical limit at high temperatures. We check that, at low temperature, the entropy density vanishes as the temperature goes to zero, in accordance with the third law of thermodynamics. We also point out that, at low temperatures, the dominant term in the free energy and internal energy densities is the vacuum energy density at zero temperature. Finally, we also show that the pressure obeys an equation of state.

N-point correlators of twist-2 operators in SU(N) Yang-Mills theory to the lowest perturbative order

We compute, to the lowest perturbative order in SU(N) Yang-Mills theory, n-point correlators in the coordinate and momentum representation of the gauge-invariant twist-2 operators with maximal spin along the p+ direction, both in Minkowskian and — by analytic continuation — Euclidean space-time. We also construct the corresponding generating functionals. Remarkably, they have the structure of the logarithm of a functional determinant of the identity plus a term involving the effective propagators that act on the appropriate source fields.

Truncation effects in the charge representation of the O(2) model

The O(2) model in Euclidean space-time is the zero-gauge-coupling limit of the compact scalar quantum electrodynamics. We obtain a dual representation of it called the charge representation. We study the quantum phase transition in the charge representation with a truncation to ``spin $S$," where the quantum numbers have an absolute value less than or equal to $S$. The charge representation preserves the gapless-to-gapped phase transition even for the smallest spin truncation $S = 1$. The phase transition for $S = 1$ is an infinite-order Gaussian transition with the same critical exponents $\delta$ and $\eta$ as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there are true BKT transitions for $S \ge 2$. The essential singularity in the correlation length for $S = 1$ is different from that for $S \ge 2$. The exponential convergence of the phase-transition point is studied in both Lagrangian and Hamiltonian formulations. We discuss the effects of replacing the truncated $\hat{U}^{\pm} = \exp(\pm i \hat{\theta})$ operators by the spin ladder operators $\hat{S}^{\pm}$ in the Hamiltonian. The marginal operators vanish at the Gaussian transition point for $S = 1$, which allows us to extract the $\eta$ exponent with high accuracy.

A possible symmetry and role for Euclidean space-time in cosmology

In this paper, we consider conservation equation in cosmology and propose four possible forms of space-times for universe resulting from dual symmetry between scale factor and energy density in the universe with constant equation of state. We start to describe these four possible types of space-times for universe and it’s possible consequences. Due to the uniformity of the metric signature in Euclidean space-time, for the first time, we introduce a new symmetry for Euclidean space-time and consider it as a transition unstable state between the visible 4D universe and the invisible world with an extra dimension. Finally, we schematically represent these four possible space-times in a unique scenario for the universe.

Compact space and infrared behavior of the effective QCD

We aim to investigate the infrared regime in the effective theory of the new quadratic gauge in the physical compact space by defining it on 4-sphere. Abelian dominance is characterized by off-diagonal gluons acquiring dynamical masses and it hints at existence of confinement. We do observe Abelian dominance in the ghost condensed vacuum of this theory on 4-sphere. Along with this observation, we find an unusual result that mass of an off-diagonal gluon on 4-sphere is position dependent in this theory as a consequence of the curved geometry. This suggests that the curvature of 4-sphere does not change the infrared behavior of present theory on 4-sphere from that of the theory with the same quadratic gauge in Euclidean spacetime as Abelian dominance is observed in Euclidean spacetime too while it has caused mass of an off-diagonal gluon on 4-sphere to be position dependent. The effective action in the confined phase on 4-sphere and that in the 4-dim Euclidean space are found to be identical in a profound outcome.

Renormalization in Minkowski space–time

The multiplicative and the functional renormalization group methods are applied for the four-dimensional scalar theory in Minkowski space–time. It is argued that the appropriate choice of the subtraction point is more important in Minkowski than in Euclidean space–time. The parameters of the cutoff theory, defined by a subtraction point in the quasi-particle domain, are complex due to the mass-shell contributions and the renormalization group flow becomes much more involved than its Euclidean counterpart.

Schwinger pair production in SL(2,C) topologically nontrivial fields via non-Abelian worldline instantons

Schwinger pair production is analyzed in a BPST instanton background and in its SL$(2,\mathbb{C})$ complex extension for complex scalar particles. A non-Abelian extension of the worldline instanton method is utilized, wherein Wong's equations in a coherent state picture adopted for SL$(2,\mathbb{C})$ are solved in Euclidean spacetime. While pair production is not predicted in the BPST instanton, a complex extension of the BPST instanton, existing as parallel fields in Minkowski spacetime, is shown to decay via the Schwinger effect.

Two-neutrino double- β decay in pionless effective field theory from a Euclidean finite-volume correlation function

Two-neutrino double-beta decay of certain nuclear isotopes is one of the rarest Standard Model processes observed in nature. Its neutrinoless counterpart is an exotic lepton-number nonconserving process that is widely searched for to determine if the neutrinos are Majorana fermions. In order to connect the rate of these processes to the Standard Model and beyond the Standard Model interactions, it is essential that the corresponding nuclear matrix elements are constrained reliably from theory. Lattice quantum chromodynamics (LQCD) and low-energy effective field theories (EFTs) are expected to play an essential role in constraining the matrix element of the two-nucleon subprocess, which could in turn provide the input into ab initio nuclear-structure calculations in larger isotopes. Focusing on the two-neutrino process $nn \to pp \, (ee \bar{\nu}_e\bar{\nu}_e)$, the amplitude is constructed in this work in pionless EFT at next-to-leading order, demonstrating the emergence of a renormalization-scale independent amplitude and the absence of any new low-energy constant at this order beyond those present in the single-weak process. Most importantly, it is shown how a LQCD four-point correlation function in Euclidean and finite-volume spacetime can be used to constrain the Minkowski infinite-volume amplitude in the EFT. The same formalism is provided for the related single-weak process, which is an input to the double-$\beta$ decay formalism. The LQCD-EFT matching procedure outlined for the double-weak amplitude paves the road toward constraining the two-nucleon matrix element entering the neutrinoless double-beta decay amplitude with a light Majorana neutrino.

The cosmological constant of emergent spacetime in the Newtonian approximation

We present a simple quantum-mechanical estimate of the cosmological constant of a Newtonian Universe. We first mimic the dynamics of a Newtonian spacetime by means of a nonrelativistic quantum mechanics for the matter contents of the Universe (baryonic and dark) within a fixed (i.e. nondynamical) Euclidean spacetime. Then we identify an operator that plays, on the matter states, a role analogous to that played by the cosmological constant. Finally, we prove that there exists a quantum state for the matter fields, in which the above-mentioned operator has an expectation value equal to the cosmological constant of the given Newtonian Universe.

A geometric algebraic approach to fluid dynamics

Geometric algebra plays a major role in merging the physical and mathematical ideas in the context of various physical systems. In this paper, we explore certain properties associated with barotropic and non-barotropic fluid flows with the help of geometric algebra over a four-dimensional Euclidean space time manifold. We introduce the concepts of multivectors associated with vorticity, helicity, and parity, which evolve from a four-velocity field. In this context, the fluid dynamical analogs of the Poynting theorem, Lorentz force, and Maxwell’s equations are derived. The fluid Maxwell’s equations can be extracted from a single equation.

The Game of Triangles

We present STR (Star-Triangle Relations), a Mathematica package designed to solve Feynman integrals by means of the method of uniqueness in any Euclidean spacetime dimension. We provide a set of tools to draw Feynman diagrams and interact with them only by the use of the mouse. Throughout the use of a graphic interface, the package should be easily accessible to users with little or no previous experience on diagrams computation.

Long-range electroweak amplitudes of single hadrons from Euclidean finite-volume correlation functions

A relation is presented between single-hadron long-range matrix elements defined in a finite Euclidean spacetime, and the corresponding infinite-volume Minkowski amplitudes. This relation is valid in the kinematic region where any number of two-hadron states can simultaneously go on shell, so that the effects of strongly-coupled intermediate channels are included. These channels can consist of non-identical particles with arbitrary intrinsic spins. The result accommodates general Lorentz structures as well as non-zero momentum transfer for the two external currents inserted between the single-hadron states. The formalism, therefore, generalizes the work by Christ et al.~[Phys.Rev. D91 114510 (2015)], and extends the reach of lattice quantum chromodynamics (QCD) to a wide class of new observables beyond meson mixing and rare decays. Applications include Compton scattering of the pion ($\pi \gamma^\star \to [\pi \pi, K \overline K] \to \pi \gamma^\star$), kaon ($K \gamma^\star \to [\pi K, \eta K] \to K \gamma^\star$) and nucleon ($N \gamma^\star \to N \pi \to N \gamma^\star$), as well as double-$\beta$ decays, and radiative corrections to the single-$\beta$ decay, of QCD-stable hadrons. The framework presented will further facilitate generalization of the result to studies of nuclear amplitudes involving two currents from lattice QCD.

A comparative study of quaternionic rotational Dirac equation and its interpretation

In this study, we develop the generalized Dirac-like four-momentum equation for rotating spin-1/2 particles in four-dimensional quaternionic algebra. The generalized quaternionic Dirac equation consists of the rotational energy and angular momentum of particle and antiparticle. Accordingly, we also discuss the four-vector form of quaternionic relativistic mass, moment of inertia and rotational energy-momentum in Euclidean space-time. The quaternionic four-angular momentum, (i.e. the rotational analogy of four-linear momentum) predicts the dual energy (rest mass energy and pure rotational energy) and dual momentum (linear-like momentum and pure rotational momentum). Further, the solutions of quaternionic rotational Dirac energy-momentum are obtained by using one-, two- and four-component of quaternionic spinor. We also demonstrate the solutions of quaternionic plane wave equation which gives the rotational frequency and wave propagation vector of Dirac particles and antiparticles in terms of quaternionic form.

Large-N \u2102\u2119N \u22121 sigma model on a Euclidean torus: uniqueness and stability of the vacuum

In this paper we examine analytically the large-N gap equation and its solution for the 2D ℂℙN −1 sigma model defined on a Euclidean spacetime torus of arbitrary shape and size (L, β), β being the inverse temperature. We find that the system has a unique homogeneous phase, with the ℂℙN −1 fields ni acquiring a dynamically generated mass (λ) ≥ Λ2 (analogous to the mass gap of SU(N ) Yang-Mills theory in 4D), for any β and L. Several related topics in the recent literature are discussed. One concerns the possibility, which turns out to be excluded according to our analysis, of a “Higgs-like” — or deconfinement — phase at small L and at zero temperature. Another topics involves “soliton-like” (inhomogeneous) solutions of the generalized gap equation, which we do not find. A related question concerns a possible instability of the standard ℂℙN −1 vacuum on R2, which is shown not to occur. In all cases, the difference in the conclusions can be traced to the existence of certain zeromodes and their proper treatment. The ℂℙN −1 model with twisted boundary conditions is also analyzed. The θ dependence and different limits involving N , β and L are briefly discussed.