Wick Rotation Research Articles

A spacetime derivation of the Lorentzian OPE inversion formula

Caron-Huot has recently given an interesting formula that determines OPE data in a conformal field theory in terms of a weighted integral of the four-point function over a Lorentzian region of cross-ratio space. We give a new derivation of this formula based on Wick rotation in spacetime rather than cross-ratio space. The derivation is simple in two dimensions but more involved in higher dimensions. We also derive a Lorentzian inversion formula in one dimension that sheds light on previous observations about the chaos regime in the SYK model.

Families of spectral triples and foliations of space(time)

We study a noncommutative analog of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a “Lorentzian spectral triple,” which can also be viewed as the “reverse Wick rotation” of a product spectral triple. This construction of “Lorentzian spectral triples” fits well into the Krein space approach to noncommutative Lorentzian geometry.

Symplectic Wick rotations between moduli spaces of 3-manifolds

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric $m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and $E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic.

Lefschetz thimbles in fermionic effective models with repulsive vector-field

We discuss two problems in complexified auxiliary fields in fermionic effective models, the auxiliary sign problem associated with the repulsive vector-field and the choice of the cut for the scalar field appearing from the logarithmic function. In the fermionic effective models with attractive scalar and repulsive vector-type interaction, the auxiliary scalar and vector fields appear in the path integral after the bosonization of fermion bilinears. When we make the path integral well-defined by the Wick rotation of the vector field, the oscillating Boltzmann weight appears in the partition function. This “auxiliary” sign problem can be solved by using the Lefschetz-thimble path-integral method, where the integration path is constructed in the complex plane. Another serious obstacle in the numerical construction of Lefschetz thimbles is caused by singular points and cuts induced by multivalued functions of the complexified scalar field in the momentum integration. We propose a new prescription which fixes gradient flow trajectories on the same Riemann sheet in the flow evolution by performing the momentum integration in the complex domain.

Wick–Tzitzeica solitons and their Monge–Ampér equation

We apply the Wick rotation to the Monge–Ampère equation of Tzitzeica graphs and we introduce the Wick–Tzitzeica solitons as complex functions solving the new equation. Some known Tzitzeica surfaces yields examples of these new solitons and we analyze them. To a second Wick–Tzitzeica soliton, we associate a homogeneous ODE system of gradient type which is Nambu–Poisson.

The Ising model coupled to 2d orders

In this article we make first steps in coupling matter to causal set theory in the path integral. We explore the case of the Ising model coupled to the 2d discrete Einstein Hilbert action, restricted to the 2d orders. We probe the phase diagram in terms of the Wick rotation parameter β and the Ising coupling j and find that the matter and the causal sets together give rise to an interesting phase structure. The couplings give rise to five different phases. The causal sets take on random or crystalline characteristics as described in Surya (2012 Class. Quantum Grav. 29 132001) and the Ising model can be correlated or uncorrelated on the random orders and correlated, uncorrelated or anti-correlated on the crystalline orders. We find that at least one new phase transition arises, in which the Ising spins push the causal set into the crystalline phase.

Lorentz signature and twisted spectral triples

We show how twisting the spectral triple of the Standard Model of elementary particles naturally yields the Krein space associated with the Lorentzian signature of spacetime. We discuss the associated spectral action, both for fermions and bosons. What emerges is a tight link between twists and Wick rotation.

Signs and stability in higher-derivative gravity

Perturbatively renormalizable higher-derivative gravity in four space–time dimensions with arbitrary signs of couplings has been considered. Systematic analysis of the action with arbitrary signs of couplings in Lorentzian flat space–time for no-tachyons, fixes the signs. Feynman [Formula: see text] prescription for these signs further grants necessary convergence in path-integral, suppressing the field modes with large action. This also leads to a sensible wick rotation where quantum computation can be performed. Running couplings for these sign of parameters make the massive tensor ghost innocuous leading to a stable and ghost-free renormalizable theory in four space–time dimensions. The theory has a transition point arising from renormalization group (RG) equations, where the coefficient of [Formula: see text] diverges without affecting the perturbative quantum field theory (QFT). Redefining this coefficient gives a better handle over the theory around the transition point. The flow equations push the flow of parameters across the transition point. The flow beyond the transition point is analyzed using the one-loop RG equations which shows that the regime beyond the transition point has unphysical properties: there are tachyons, the path-integral loses positive definiteness, Newton’s constant [Formula: see text] becomes negative and large, and perturbative parameters become large. These shortcomings indicate a lack of completeness beyond the transition point and need of a nonperturbative treatment of the theory beyond the transition point.

Emergent universe with wormholes in massive gravity

An emergent universe (EU) scenario is proposed to obtain a universe free from big-bang singularity. In this framework the present universe emerged from a static Einstein universe phase in the infinite past. A flat EU scenario is found to exist in Einstein’s gravity with a non-linear equation of state (EoS). It has been shown subsequently that a physically realistic EU model can be obtained considering cosmic fluid composed of interacting fluids with a non-linear equation of state. It results a viable cosmological model accommodating both early inflation and present accelerating phases. In the present paper, the origin of an initial static Einstein universe needed in the EU model is explored in a massive gravity theory which subsequently emerged to be a dynamically evolving universe. A new gravitational instanton solution in a flat universe is obtained in the massive gravity theory which is a dynamical wormhole that might play an important role in realizing the origin of the initial state of the emergent universe. The emergence of a Lorentzian universe from a Euclidean gravity is understood by a Wick rotation . A universe with radiation at the beginning finally transits into the present observed universe with a non-linear EoS as the interactions among the fluids set in. Thus a viable flat EU scenario where the universe stretches back into time infinitely, with no big bang is permitted in a massive gravity.

Matrix quantum mechanics on [formula omitted

We study Matrix Quantum Mechanics on the Euclidean time orbifold S1/Z2. Upon Wick rotation to Lorentzian time and taking the double-scaling limit this theory provides a toy model for a big-bang/big crunch universe in two dimensional non-critical string theory where the orbifold fixed points become cosmological singularities. We derive the MQM partition function both in the canonical and grand canonical ensemble in two different formulations and demonstrate agreement between them. We pinpoint the contribution of twisted states in both of these formulations either in terms of bi-local operators acting at the end-points of time or branch-cuts on the complex plane. We calculate, in the matrix model, the contribution of the twisted states to the torus level partition function explicitly and show that it precisely matches the world-sheet result, providing a non-trivial test of the proposed duality. Finally we discuss some interesting features of the partition function and the possibility of realising it as a τ-function of an integrable hierarchy.

Fakeons and Lee-Wick models

The “fakeon” is a fake degree of freedom, i.e. a degree of freedom that does not belong to the physical spectrum, but propagates inside the Feynman diagrams. Fakeons can be used to make higher-derivative theories unitary. Moreover, they help us clarify how the Lee-Wick models work. In this paper we study the fakeon models, that is to say the theories that contain fake and physical degrees of freedom. We formulate them by (nonanalytically) Wick rotating their Euclidean versions. We investigate the properties of arbitrary Feynman diagrams and, among other things, prove that the fakeon models are perturbatively unitary to all orders. If standard power counting constraints are fulfilled, the models are also renormalizable. The S matrix is regionwise analytic. The amplitudes can be continued from the Euclidean region to the other regions by means of an unambiguous, but nonanalytic, operation, called average continuation. We compute the average continuation of typical amplitudes in four, three and two dimensions and show that its predictions agree with those of the nonanalytic Wick rotation. By reconciling renormalizability and unitarity in higher-derivative theories, the fakeon models are good candidates to explain quantum gravity.

Action and observer dependence in Euclidean quantum gravity

Given a Lorentzian spacetime and a non-vanishing timelike vector field with level surfaces Σ, one can construct on a Euclidean metric (Hawking and Ellis 1973 The Large Scale Structure of Space-Time (Cambridge: Cambridge University Press)). Motivated by this, we consider a class of metrics with an arbitrary function Θ that interpolates between the Euclidean () and Lorentzian () regimes, separated by the codimension one hypersurface defined by . Since can not, in general, be obtained from by a diffeomorphism, its Euclidean regime is in general different from that obtained from Wick rotation . For example, if is the k = 0 Lorentzian de Sitter metric corresponding to , the Euclidean regime of is the k = 0 Euclidean anti-de Sitter space with .We analyze the curvature tensors associated with for arbitrary Lorentzian metrics and timelike geodesic fields , and show that they have interesting and remarkable mathematical structures: (i) Additional terms arise in the Euclidean regime of . (ii) For the simplest choice of a step-profile for Θ, the Ricci scalar Ric of reduces, in the Lorentzian regime , to the complete Einstein–Hilbert lagrangian with the correct Gibbons–Hawking–York boundary term; the latter arises as a delta-function of strength 2K supported on . (iii) In the Euclidean regime , Ric also has an extra term of the -foliation. We highlight similar foliation dependent terms in the full Riemann tensor.We present some explicit examples for FLRW spacetimes in standard foliation and spherically symmetric spacetimes in the Painleve–Gullstrand foliation. We briefly discuss implications of the results for Euclidean quantum gravity and quantum cosmology.

Quasitriangular structure and twisting of the 3D bicrossproduct model

We show that the bicrossproduct model C[SU2∗] ▶ ◁ U (su2) quantum Poincaré group in 2+1 dimensions acting on the quantum spacetime [xi, t] = ıλxi is related by a Drinfeld and module-algebra twist to the quantum double U (su2)⊲<C[SU2] acting on the quantum spacetime [xμ, xν] = ıλϵμνρxρ. We obtain this twist by taking a scaling limit as q → 1 of the q-deformed version of the above, where it corresponds to a previous theory of q-deformed Wick rotation from q-Euclidean to q-Minkowski space. We also recover the twist result at the Lie bialgebra level.

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

A fully regulated definition of Feynman’s path integral is presented here. The proposed re-formulation of the path integral coincides with the familiar formulation whenever the path integral is well defined. In particular, it is consistent with respect to lattice formulations and Wick rotations, i.e., it can be used in Euclidean and Minkowski space-time. The path integral regularization is introduced through the generalized Kontsevich-Vishik trace, that is, the extension of the classical trace to Fourier integral operators. Physically, we are replacing the time-evolution semi-group by a holomorphic family of operators such that the corresponding path integrals are well defined in some half space of C. The regularized path integral is, thus, defined through analytic continuation. This regularization can be performed by means of stationary phase approximation or computed analytically depending only on the Hamiltonian and the observable (i.e., known a priori). In either case, the computational effort to evaluate path integrals or expectations of observables reduces to the evaluation of integrals over spheres. Furthermore, computations can be performed directly in the continuum and applications (analytic computations and their implementations) to a number of models including the non-trivial cases of the massive Schwinger model and a φ4 theory.

Non-perturbative decay of non-Abelian hair

We construct a solution of Heterotic supergravity which interpolates between two different AdS3 × S3 × T4 geometries corresponding to the near-horizon limits of two 5-dimensional black holes, only one of which has non-Abelian hair. This solution can be used to estimate the amplitude of probability of the non-perturbative decay of the gauge 5-brane responsible for the non-Abelian hair into eight solitonic 5-branes by evaluating its Euclidean action. The Wick rotation of this solution poses several problems which we argue can be overcome by using a non-extremal off-shell (NEOS) deformation of the solution. This NEOS field configuration can be Wick rotated straight away and its Euclidean action can be computed for any value of the deformation parameter. The Euclidean result can then be anti-Wick-rotated and its extremal limit gives the Euclidean action of the original solution, which turns out to be one half of the difference between the entropies of the 5-dimensional black holes.

Wick rotations and real GIT

We define Wick-rotations by considering pseudo-Riemannian manifolds as real slices of a holomorphic Riemannian manifold. From a frame bundle viewpoint Wick-rotations between different pseudo-Riemannian spaces can then be studied through their structure groups which are real forms of the corresponding complexified Lie group (different real forms $O(p,q)$ of the complex Lie group $O(n,\mathbb{C})$). In this way, we can use real GIT (geometric invariant theory) to derive several new results regarding the existence, and non-existence, of such Wick-rotations. As an explicit example, we Wick rotate a known $G_2$-holonomy manifold to a pseudo-Riemannian manifold with split-$G_2$ holonomy.

Wick Rotation for D-modules

We extend the classical Wick rotation to D-modules and higher codimensional submanifolds.

An adaptive FEM with ITP approach for steady Schrödinger equation

ABSTRACTIn this paper, an adaptive numerical method is proposed for solving a 2D Schrödinger equation with an imaginary time propagation approach. The differential equation is first transferred via a Wick rotation to a real time-dependent equation, whose solution corresponds to the ground state of a given system when time approaches infinity. The temporal equation is then discretized spatially via a finite element method, and temporally utilizing a Crank–Nicolson scheme. A moving mesh strategy based on harmonic maps is considered to eliminate possible singular behaviour of the solution. Several linear and nonlinear examples are tested by using our method. The experiments demonstrate clearly that our method provides an effective way to locate the ground state of the equations through underlying eigenvalue problems.

Anisotropic SD2 brane: accelerating cosmology and Kasner-like space-time from compactification

Starting from an anisotropic (in all directions including the time direction of the brane) non-SUSY D2 brane solution of type IIA string theory we construct an anisotropic space-like D2 brane (or SD2 brane, for short) solution by the standard trick of a double Wick rotation. This solution is characterized by five independent parameters. We show that compactification on six-dimensional hyperbolic space (H_6) of a time-dependent volume of this SD2 brane solution leads to accelerating cosmologies (for some time tsim ,t_0, with t_0 some characteristic time) where both the expansions and the accelerations are different in three spatial directions of the resultant four-dimensional universe. On the other hand at early times (t ll t_0) this four-dimensional space, in certain situations, leads to four-dimensional Kasner-like cosmology, with two additional scalars, namely, the dilaton and a volume scalar of H_6. Unlike in the standard four-dimensional Kasner cosmology here all three Kasner exponents could be positive definite, leading to expansions in all three directions.

Cosmological acceleration from a scalar field and classical and quantum gravitational waves (Inflation and dark energy)

We show that on the average, homogeneous and isotropic scalar field and on the average homogeneous and isotropic ensembles of classical and quantum gravitational waves generate the de Sitter expansion of the empty (with no matter) space-time. At the start and by the end of its cosmological evolution the Universe is empty. The contemporary Universe is about 70% empty, so the effect of cosmological acceleration should be very noticeable. One can assume that it manifests itself as dark energy. At the start of the cosmological evolution, before the first matter was born, the Universe is also empty. The cosmological acceleration of such an empty space-time can manifests itself as inflation. To get the de Sitter accelerated expansion of the empty space-time under influence of scalar fields and classical and quantum gravitational waves, one needs to make a mandatory Wick rotation, i.e. one needs to make a transition to the Euclidean space of imaginary time. One can assume that the very existence of inflation and dark energy could be considered as a possible observable evidence of the fact that time by its nature could be a complex value which manifests itself precisely at the start and by the end of the evolution of the Universe, i.e. in those periods when the Universe is empty (or nearly empty).