Wick Rotation Research Articles

Applying the Euclidean-signature semi-classical method to the quantum Taub models with a cosmological constant and aligned electromagnetic field

We prove the existence of a countably infinite number of “excited” states for the Lorentzian-signature Taub–Wheeler–DeWitt (WDW) equation when a cosmological constant is present using the Euclidean-signature semi-classical method. We also find a “ground” state solution when both an aligned electromagnetic field and cosmological constant are present; as a result, conjecture that the Euclidean-signature semi-classical method can be used to prove the existence of a countably “infinite” number of “excited” states when the two aforementioned matter sources are present. Afterward, we prove the existence of asymptotic solutions to the vacuum Taub–WDW equation using the “no boundary” and “wormhole” solutions of the Taub Euclidean-signature Hamilton–Jacobi equation and compare their mathematical properties. We then discuss the fascinating qualitative properties of the wave functions we have computed. By utilizing the Euclidean-signature semi-classical method in the above manner, we further show its ability to prove the existence of solutions to Lorentzian-signature equations without having to invoke a Wick rotation. This feature of not needing to apply a Wick rotation makes this method potentially very useful for tackling a variety of problems in bosonic relativistic field theory and quantum gravity.

Wick rotations in deformation quantization

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from [Formula: see text] with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space [Formula: see text] and the complex hyperbolic disc [Formula: see text]. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on [Formula: see text] and [Formula: see text]. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the [Formula: see text]-involution of pointwise complex conjugation.

The Hartle–Hawking–Israel state on spacetimes with stationary bifurcate Killing horizons

We consider a free massive quantized Klein–Gordon field in a spacetime [Formula: see text] containing a stationary bifurcate Killing horizon, i.e. a bifurcate Killing horizon whose Killing vector field is globally time-like in the right wedge [Formula: see text] associated to the horizon. We prove the existence of the Hartle–Hawking–Israel (HHI) vacuum state, which is a pure state on the whole spacetime whose restriction to [Formula: see text] is a thermal state [Formula: see text] for the time-like Killing field at Hawking temperature [Formula: see text], where [Formula: see text] is the surface gravity of the horizon. We show that the HHI state is a Hadamard state and is the unique Hadamard state which is equal to the double [Formula: see text]-KMS state in the double wedge [Formula: see text]. We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderón projector for elliptic boundary value problems.

Wick Rotation and the Positivity of Energy in Quantum Field Theory

Abstract We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian metrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.

Accretion Flow onto Ellis–Bronnikov Wormhole

Study of accretion onto wormholes is rather rare compared to that onto black holes. In this paper, we consider accretion flow of cosmological dark energy modeled by barotropic fluid onto the celebrated Ellis–Bronnikov wormhole (EBWH) built by Einstein minimally coupled scalar field ϕ, violating the null energy condition. The accreting fluid is assumed to be phantom, quintessence, dust and stiff matter. We begin by first pointing out a mathematical novelty showing how the EBWH can lead to the Schwarzschild black hole under a complex Wick rotation. Then, we analyze the profiles of fluid radial velocity, density and the rate of mass variation of the EBWH due to accretion and compare the profiles with those of the Schwarzschild black hole. We also analyze accretion to the massless EBWH that has zero ADM mass but has what we call nonzero Wheelerian mass (“mass without mass”), composed of the non-trivial scalar field, that shows gravitational effects. Our conclusion is that the mass of SBH due to phantom accretion decreases consistently with known results, while, in contrast, the mass of EBWH increases. Exactly an opposite behavior emerges for non-phantom accretion to these two objects. Accretion to massless EBWH (i.e., to nonzero Wheelerian mass) shares the same patterns as those of the massive EBWH; hence there is no way to distinguish massive and massless cases by means of accretion flow. The contrasting mass variations due to phantom accretion could be a reflection of the distinct topology of the central objects.

Black holes and large N complex saddles in 3D-3D correspondence

We study the large N sign oscillation of the twisted indices for 3D theories of class ℛ obtained from M5-branes wrapped on a hyperbolic 3-manifold. Holographically, the oscillatory behavior can be understood from the imaginary part of on-shell actions for the two Euclidean supergravity solutions, Bolt± with p = 0, which are Wick rotation of magnetically charged AdS4 black holes. The two solutions have the same imaginary part with opposite sign. The imaginary part comes from the F ∧ F-term in the supergravity and the coefficient is proportional to the Chern-Simons invariant of 3-manifold. Combining the holographic computation with 3D-3D relation for twisted indices, we propose a non-trivial mathematical conjecture regarding the phase factor of a twisted Reidemeister-Ray-Singer torsion on hyperbolic 3-manifold.

Pushing the limits of time beyond the Big Bang singularity: Scenarios for the branch cut universe

AbstractIn this contribution, we identify two scenarios for the evolutionary branch cut universe. In the first scenario, the universe evolves continuously from the negative complex cosmological time sector, prior to a primordial singularity, to the positive one, circumventing continuously a branch cut, and no primordial singularity occurs in the imaginary sector, only branch points. In the second scenario, the branch cut and branch point disappear after the realization of the imaginary component of the complex time by means of a Wick rotation, which is replaced by the thermal time. In the second scenario, the universe has its origin in the Big Bang, but the model contemplates simultaneously a mirrored parallel evolutionary universe going backwards in the cosmological thermal time negative sector. A quantum formulation based on the Wheeler–DeWitt equation is sketched and preliminary conclusions are drawn.

Quantized noncommutative Riemann manifolds and stochastic processes: The theoretical foundations of the square root of Brownian motion

We lay the theoretical and mathematical foundations of the square root of Brownian motion and we prove the existence of such a process. In doing so, we consider Brownian motion on quantized noncommutative Riemannian manifolds and show how a set of stochastic processes on sets of complex numbers can be devised. This class of stochastic processes are shown to yield at the outset a Chapman–Kolmogorov equation with a complex diffusion coefficient that can be straightforwardly reduced to the Schrödinger equation. The existence of these processes has been recently shown numerically. In this work we provide an analogous support for the existence of the Chapman–Kolmogorov–Schrödinger equation for them, performing a Monte Carlo study. It is numerically seen as a Wick rotation can turn the heat kernel into the Schrödinger one, mapping such kernels through the corresponding stochastic processes. In this way, we introduce a new kind of improper complex stochastic process. This permits a reformulation of quantum mechanics using purely geometrical concepts that are strongly linked to stochastic processes. Applications to economics are also entailed.

Non-Hermiticity-Governed Active Photonic Resonances.

Photonic resonances play an essential role in the generation and propagation of light in optical and photonic devices, as well as in light-matter interaction, including nonlinear optical responses. Previous studies in lasers and other open systems have shown exotic roles played by non-Hermiticity on modifying passive resonances, defined in the absence of optical gain and loss. Here we report a new type of resonances in non-Hermitian photonic systems that does not originate from a passive resonance, identified by analyzing a unique quantization condition in the non-Hermitian extension of the Wentzel-Kramers-Brillouin method. Termed active photonic resonances, these unique resonances are found in non-Hermitian systems with a spatially correlated complex dielectric function, which is related to supersymmetry quantum mechanics after a Wick rotation. Remarkably, such an active photonic resonance shifts continuously on the real frequency axis as optical gain increases, suggesting the possibility of a tunable on-chip laser that can span a wavelength range over 100nm.

Heat kernel for the quantum Rabi model: II. Propagators and spectral determinants

The quantum Rabi model (QRM) is widely recognized as an important model in quantum systems, particularly in quantum optics. The Hamiltonian H Rabi is known to have a parity decomposition H Rabi = H + ⊕ H −. In this paper, we give the explicit formulas for the propagator of the Schrödinger equation (integral kernel of the time evolution operator) for the Hamiltonian H Rabi and H ± by the Wick rotation (meromorphic continuation) of the corresponding heat kernels. In addition, as in the case of the full Hamiltonian of the QRM, we show that for the Hamiltonians H ±, the spectral determinant is, up to a non-vanishing entire function, equal to the Braak G-function (for each parity) used to prove the integrability of the QRM. To do this, we show the meromorphic continuation of the spectral zeta function of the Hamiltonians H ± and give some of its basic properties.

Eigenstates of quasi-Keplerian self-gravitating particle discs

ABSTRACT Although quasi-Keplerian discs are among the most common astrophysical structures, computation of secular angular momentum transport within them routinely presents a considerable practical challenge. In this work, we investigate the secular small-inclination dynamics of a razor-thin particle disc as the continuum limit of a discrete Lagrange–Laplace secular perturbative theory and explore the analogy between the ensuing secular evolution – including non-local couplings of self-gravitating discs – and quantum mechanics. We find the ‘quantum’ Hamiltonian that describes the time evolution of the system and demonstrate the existence of a conserved inner product. The lowest-frequency normal modes are numerically approximated by performing a Wick rotation on the equations of motion. These modes are used to quantify the accuracy of a much simpler local-coupling model, revealing that it predicts the shape of the normal modes to a high degree of accuracy, especially in narrow annuli, even though it fails to predict their eigenfrequencies.

Weyl–Wigner representation of canonical equilibrium states

The Weyl–Wigner representations for canonical thermal equilibrium quantum states are obtained for the whole class of quadratic Hamiltonians through a Wick rotation of the Weyl–Wigner symbols of Heisenberg and metaplectic operators. The behavior of classical structures inherently associated to these unitaries is described under the Wick mapping, unveiling that a thermal equilibrium state is fully determined by a complex symplectic matrix, which sets all of its thermodynamical properties. The four categories of Hamiltonian dynamics (parabolic, elliptic, hyperbolic and loxodromic) are analyzed. Semiclassical and high temperature approximations are derived and compared to the classical and/or quadratic behavior.

Twistors and the Amplitudes in the Integrable Sector of Superstring Theory

Twistors can represent the spin in superstring theory and provide a theoretical explanation of the two-dimensional spherical model of elementary particles together with the surface distributions of the charges. The integrability of certain matter sectors of the strong interactions also may be traced to the action of the conformal group on collinear trajectories that occur in a twistor formulation. The amplitudes for these scattering processes in four dimensions have a representation with quantum states in a nonsupersymmetric theory defined by the low-energy limit of a solution to the N = 2 string equations with a Wick rotation of the metric with (2, 2) signature. The twistor form of an N = 2 scattering amplitude is given.

О термодинамических параметрах адиабатически изолированного тела

The definition of the main thermodynamic functions for an adiabatically isolated body with constant internal energy is proposed in the framework of the formalism of the covariant quantum theory with reparametrization invariance of proper time. The modification does not change the dynamic content of the theory at the classical level, but it allows one to define the unitary evolution operator in quantum theory. In this operator, proper time is a measure of the internal movement of the body. The transition to statistical mechanics is carried out by the Wick rotation of proper time in the complex plane. As a result, a representation of the partition function of an isolated body in the form of a Euclidean functional integral on the space of closed trajectories in the configuration space is obtained. For a given internal energy, the average return temperature and free energy are determined, which under lie the thermomechanics of an adiabatically isolated body.

Codimension-two holography for wedges

We propose a codimension two holography between a gravitational theory on a $d+1$ dimensional wedge spacetime and a $d-1$ dimensional CFT which lives on the corner of the wedge. Formulating this as a generalization of AdS/CFT, we explain how to compute the free energy, entanglement entropy and correlation functions of the dual CFTs from gravity. In this wedge holography, the holographic entanglement entropy is computed by a double minimization procedure. Especially, for a four dimensional gravity ($d=3$), we obtain a two dimensional CFT and the holographic entanglement entropy perfectly reproduces the known result expected from the holographic conformal anomaly. We also discuss a lower dimensional example ($d=2$) and find that a universal quantity naturally arises from gravity, which is analogous to the boundary entropy. Moreover, we consider a gravity on a wedge region in Lorentzian AdS, which is expected to be dual to a CFT with a space-like boundary. We formulate this new holography and compute the holographic entanglement entropy via a Wick rotation of the AdS/BCFT construction. Via a conformal map, this wedge spacetime is mapped into a geometry where a bubble-of-nothing expands under time evolution. We reproduce the holographic entanglement entropy for this gravity dual via CFT calculations.

Can accretion properties distinguish between a naked singularity, wormhole and black hole?

We first advance a mathematical novelty that the three geometrically and topologically distinct objects mentioned in the title can be exactly obtained from the Jordan frame vacuum Brans I solution by a combination of coordinate transformations, trigonometric identities and complex Wick rotation. Next, we study their respective accretion properties using the Page–Thorne model which studies accretion properties exclusively for rge r_{text {ms}} (the minimally stable radius of particle orbits), while the radii of singularity/throat/horizon r<r_{text {ms}}. Also, its Page–Thorne efficiency epsilon is found to increase with decreasing r_{text {ms}} and also yields epsilon =0.0572 for Schwarzschild black hole (SBH). But in the singular limit rrightarrow r_{s} (radius of singularity), we have epsilon rightarrow 1 giving rise to 100 % efficiency in agreement with the efficiency of the naked singularity constructed in [10]. We show that the differential accretion luminosity frac{d{mathcal {L}}_{infty }}{dln {r}} of Buchdahl naked singularity (BNS) is always substantially larger than that of SBH, while Eddington luminosity at infinity L_{text {Edd}}^{infty } for BNS could be arbitrarily large at rrightarrow r_{s} due to the scalar field phi that is defined in (r_{s}, infty ). It is concluded that BNS accretion profiles can still be higher than those of regular objects in the universe.

Quasinormal modes and black hole hairs in AdS

Holography relates the quasinormal modes frequencies of AdS black holes to the pole structure of the dual field theory propagator. These modes thus provide the timescale for the approach to thermal equilibrium in the CFT. Here, we study how such pole structure and, in particular, the time to equilibrium can get modified in the presence of a black hole hair. More precisely, we consider in AdS a set of relaxed boundary conditions that allow for a low decaying graviton mode near the boundary, which triggers an additional degree of freedom. We solve the scalar field response on such background analytically and non-perturbatively in the hair parameter, and we obtain how the pole structure gets affected by the presence of a black hole hair, relative to that of the usual AdS black hole geometry. The setup we consider is a massive 3D gravity theory, which admits a one-parameter family deformation of BTZ solution and enables us to solve the problem analytically. The theory also admits an AdS$_3$ soliton which gives a family of vacua that can be constructed from the hairy black hole by means of a double Wick rotation. The spectrum of normal modes on the latter geometry can also be solved analytically; we study its properties in relation to those of the AdS$_3$ vacuum.

Higher spin partition functions via the quasinormal mode method in de Sitter quantum gravity

In this note we compute the 1-loop partition function of spin-ss fields on Euclidean de Sitter space S^{2n+1}S2n+1 using the quasinormal mode method. Instead of computing the quasinormal mode frequencies from scratch, we use the analytic continuation prescription L_{\text{AdS}}\to iL_{\text{dS}}LAdS→iLdS, appearing in the dS/CFT correspondence, and Wick rotate the normal mode frequencies of fields on thermal \text{AdS}_{2n+1}AdS2n+1 into the quasinormal mode frequencies of fields on de Sitter space. We compare the quasinormal mode and heat kernel methods of calculating 1-loop determinants, finding exact agreement, and furthermore explicitly relate these methods via a sum over the conformal dimension. We discuss how the Wick rotation of normal modes on thermal \text{AdS}_{2n+1}AdS2n+1 can be generalized to calculating 1-loop partition functions on the thermal spherical quotients S^{2n+1}/\mathbb{Z}_{p}S2n+1/ℤp. We further show that the quasinormal mode frequencies encode the group theoretic structure of the spherical spacetimes in question, analogous to the analysis made for thermal AdS in [1-3] .

Spinning black holes for generalized scalar-tensor theories in three dimensions

We consider a general class of scalar tensor theories in three dimensions whose action contains up to second-order derivatives of the scalar field with coupling functions that only depend on the standard kinetic term of the scalar field, thus ensuring the invariance under the constant shift of the scalar field. For this model, we show that the field equations for a stationary metric ansatz together with a purely radial scalar field can be fully integrated. The kinetic term of the scalar field solution is shown to satisfy an algebraic relation depending only on the coupling functions, and hence is constant while the metric solution is nothing but the BTZ metric with an effective cosmological constant fixed in terms of the coupling functions. As a direct consequence the thermodynamics of the solution is shown to be identical to the BTZ one with an effective cosmological constant, despite the presence of a scalar field. Finally, the expression of the semi-classical entropy of this solution is also confirmed through a generalized Cardy-like formula involving the mass of the scalar soliton obtained from the black hole by means of a double Wick rotation.

Beating the Big Bang – Version 2

In “The Science Show” on Australian Broadcasting Corporation’s “Radio National” (July 4, 2020 - Robyn Williams says “Let's continue this quest into the weird and wonderful with Geraint Lewis's latest book, The Cosmic Revolutionary's Handbook (Or: How to Beat the Big Bang).” Then Professor Lewis says, “What we wanted to lay out in the book that we have is essentially what is the observational evidence that astronomers used to build up the picture of the Big Bang theory, and outline that if you want your theory to supplant the Big Bang theory, you're going to have to explain this observational evidence as well as the Big Bang picture does, if not better, before scientists will take note. Number one fact; the night sky is dark. It's Olbers' paradox, it's one of the nice famous philosophical, scientific ideas. If your scientific theory can't explain that simple fact, you are already in trouble.” This article will address several points in its refutation of the Big Bang – Olbers’ paradox, Edwin Hubble, redshift, blueshift, the cosmic microwave background, gravitational waves, quantum entanglement, the quantizing of E=mc2, the First Law of Thermodynamics, the initially plausible (though, I conclude, ultimately incorrect) multiverse hypothesis, vector-tensor-scalar geometry, dark matter, dark energy, mass, quantum spin, the Higgs boson and Higgs field, bosons of the strong and weak nuclear forces, as well as advanced waves and retarded waves. It concludes with the Mobius strip, figure-8 Klein bottle, Wick rotation, Albert Einstein’s time dilation, and Rene Descartes’ space-matter relation. Following the advice of Dr. Luke Barnes (coauthor of The Cosmic Revolutionary's Handbook), I’ve refined my ideas - by adding about 1,500 words I previously wrote to explain the ideas better. This new section is called 'Interstellar, Intergalactic and Time Travel plus Simply-Connected and Nonorientable Topology' and all additions, such as these extra lines in the abstract and the new paragraph about dark matter and dark energy, are highlighted in yellow.