Wightman Functions Research Articles

Mode-sum construction of the two-point functions for the Stueckelberg vector fields in the Poincaré patch of de Sitter space

We perform canonical quantization of the Stueckelberg Lagrangian for massive vector fields in the conformally flat patch of de Sitter space in the Bunch-Davies vacuum and find their Wightman two-point functions by the mode-sum method. We discuss the zero-mass limit of these two-point functions and their limits where the Stueckelberg parameter ξ tends to zero or infinity. It is shown that our results reproduce the standard flat-space propagator in the appropriate limit. We also point out that the classic work of Allen and Jacobson [“Vector two-point functions in maximally symmetric spaces,” Commun. Math. Phys. 103, 669 (1986)] for the two-point function of the Proca field and a recent work by Tsamis and Woodard [“Maximally symmetric vector propagator,” J. Math. Phys. 48, 052306 (2007)] for that of the transverse vector field are two limits of our two-point function, one for ξ → ∞ and the other for ξ → 0. Thus, these two works are consistent with each other, contrary to the claim by the latter authors.

Casimir densities from coexisting vacua

The Wightman function, the vacuum expectation values (VEVs) of the field squared, and the energy-momentum tensor are investigated for a massive scalar field with general curvature coupling in a spherically symmetric static background geometry described by two distinct metric tensors inside and outside a spherical boundary. The exterior and interior geometries can correspond to different vacuum states of the same theory. In the region outside the sphere, the contributions in the VEVs, induced by the interior geometry, are explicitly separated. For the special case of the Minkowskian exterior geometry, the asymptotics of the VEVs near the boundary and at large distances are discussed in detail. In particular, it has been shown that the divergences on the boundary are weaker than in the problem of a spherical boundary in Minkowski spacetime with Dirichlet or Neumann boundary conditions. As an application of general results, de Sitter (dS) and anti--de Sitter (AdS) spaces are considered as examples of the interior geometry. For AdS interiors there are no bound states. In the case of dS geometry and for nonminimally coupled fields, bound states appear for a radius of the separating boundary sufficiently close to the dS horizon. Starting from a critical value of the radius, the Minkowskian vacuum in the exterior region becomes unstable. For small values of the AdS curvature radius, to the leading order, the VEVs in the exterior region coincide with those for a spherical boundary in Minkowski spacetime with a Dirichlet boundary condition. The exceptions are the cases of minimal and conformal couplings: for a minimal coupling, the VEVs are reduced to the case with a Neumann boundary condition, whereas for a conformally coupled field there is no reduction to Dirichlet or Neumann results.

On conformal field theories with extremal $ \frac{a}{c} $ values

Unitary conformal field theories (CFTs) are believed to have positive (non-negative) energy correlators. Energy correlators are universal observables in higher-dimensional CFTs built out of integrated Wightman functions of the stress-energy tensor. We analyze energy correlators in parity invariant four-dimensional CFTs. The goal is to use the positivity of energy correlators to further constrain unitary CFTs. It is known that the positivity of the simplest one-point energy correlator implies that $ \frac{1}{3}\leq \frac{a}{c}\leq \frac{31 }{18 } $ where a and c are the Weyl anomaly coefficients. We use the positivity of higher point energy correlators to show that CFTs with extremal values of $ \frac{a}{c} $ have trivial scattering observables. More precisely, for $ \frac{a}{c}=\frac{1}{3} $ and $ \frac{a}{c}=\frac{31 }{18 } $ all energy correlators are fixed to be the ones of the free boson and the free vector theory correspondingly. Similarly, we show that the positivity and finiteness of energy correlators together imply that the three-point function of the stress tensor in a CFT cannot be proportional to the one in the theory of free boson, free fermion or free vector field.

Aspects of the derivative coupling model in four dimensions

A concise discussion of a \(3+1\)-dimensional derivative coupling model, in which a massive Dirac field couples to the four-gradient of a massless scalar field, is given in order to elucidate the role of different concepts in quantum field theory like the regularization of quantum fields as operator-valued distributions, correlation distributions, locality, causality, and field operator gauge transformations.

Wightman function and vacuum polarization in models with non-trivial topology in the presence of boundaries

Wightman function and the vacuum expectation value of the scalar field squared are studied in models with a compact subspace in the presence of parallel plane boundaries, on which the field obeys Robin boundary conditions. By means of the generalized Abel-Plana summation formula we have explicitly extracted from the vacuum expectation values the parts induced by the boundaries. As a simple example, a one-dimensional internal space is considered compactified on a circle.

Scale and Lorentz Transformations at the Light-Front

The light-front (LF) quantization is applied for the model of massive scalar field with self-interaction. We check some of the LF postulates by considering the Wightman function for this model. The scale symmetry imposed only on the LF quantization hypersurface and the Lorentz symmetry assumed for all points in Minkowski’s space-time lead to a strong constraint for the Wightman functions, which is satisfied only by a free and massless scalar field. This result agrees with the recent Weinberg’s result for a scale-symmetric theory. This means that one cannot expect the unitary equivalence of the Fock space for scalar fields with different masses at the LF hypersurface.

UNIFORMLY ACCELERATED DETECTOR IN THE κ-DEFORMED DIRAC VACUUM

In this paper, we investigate how a uniformly accelerated detector responds to vacuum state of a Dirac field in the κ-Minkowski spacetime. Starting from κ-deformed Dirac theory, which is invariant under κ-Poincaré algebra, we derive κ-deformed Wightman function for Dirac field, which is valid up to first-order in the deformation parameter a. Using this, we calculate the response function of the uniformly accelerated detector, which is coupled to massless Dirac field in κ-spacetime. From this, we obtain the modification to Unruh effect for the κ-deformed Dirac field, valid up to first-order in the deformation parameter.

On the response of particle detectors in Vaidya spacetimes

Using the formalism of the interaction picture we calculate an expression for the Wightman function for only the spherically symmetric modes of a quantum Klein–Gordon scalar field in a general Vaidya spacetime with ingoing null dust. It is demonstrated that particle detectors following time-like trajectories that are in the ground state at some instant of time outside of the collapsing shell will have a subsequent evolution that is independent of the configuration of the ingoing null dust. For detectors that are taken to be in the ground state at a time interior to the shell it is shown that their response will depend on the configuration of the ingoing null dust. Relevance to the information loss paradox is discussed.

A ground state for the causal diamond in 2 dimensions

We apply a recent proposal for a distinguished ground state of a quantum field in a globally hyperbolic spacetime to the free massless scalar field in a causal diamond in two-dimensional Minkowski space. We investigate the two limits in which the Wightman function is evaluated (i) for pairs of points that lie in the centre of the diamond (i.e. far from the boundaries), and (ii) for pairs of points that are close to the left or right corner. We find that in the centre, the Minkowski vacuum state is recovered, with a definite value of the infrared cutoff. Interestingly, the ground state is not the Rindler vacuum in the corner of the diamond, as might have been expected, but is instead the vacuum of a flat space in the presence of a static mirror on that corner. We confirm these results by numerically evaluating the Wightman function of a massless scalar field on a causal set corresponding to the continuum causal diamond.

THERMAL RADIATION FROM A FLUCTUATING EVENT HORIZON

We consider a pointlike two-level system undergoing uniformly accelerated motion. We evaluate the transition probability rate of this system coupled to a massless scalar field near a fluctuating event horizon. Such horizon fluctuations are generated using random noise. We study the case of centered, stationary and Gaussian random processes. The transition probability of the two-level system is obtained from the positive-frequency Wightman function calculated to one-loop order in the noise averaging process. Our results show how this fluctuating-horizon model modifies the response function of the two-level system.

Summing planar diagrams by an integrable bootstrap. II

We continue our investigation of correlation functions of the large-N (planar) limit of the (1+1)-dimensional principal chiral sigma model, whose bare field U(x) lies in the fundamental matrix representation of SU(N). We find all the form factors of the renormalized field. An exact formula for Wightman and time-ordered correlation functions is found.

Uniformly accelerating observer inκ-deformed space-time

In this paper, we study the effect of $\kappa$-deformation of the space-time on the response function of a uniformly accelerating detector coupled to a scalar field. Starting with $\kappa$-deformed Klein-Gordon theory, which is invariant under a $\kappa$-Poincar\'e algebra and written in commutative space-time, we derive $\kappa$-deformed Wightman functions, valid up to second order in the deformation parameter $a$. Using this, we show that the first non-vanishing correction to the Unruh thermal distribution is only in the second order in $a$. We also discuss various other possible sources of $a$-dependent corrections to this thermal distribution.

Vacuum polarization induced by a cosmic string in anti-de Sitter spacetime

In this paper, we investigate the vacuum expectation values (VEVs) of the field squared and the energy–momentum tensor associated with a massive scalar quantum field induced by a generalized cosmic string in D-dimensional anti-de Sitter (AdS) spacetime. In order to develop this analysis we evaluate the corresponding Wightman function. As we shall observe, this function is expressed as the sum of two terms: the first one corresponds to the Wightman function in pure AdS bulk and the second one is induced by the presence of the string. The second contribution is finite at the coincidence limit and is used to provide closed expressions for the parts in the VEVs of the field squared and the energy–momentum tensor induced by the presence of the string. Because the analysis of vacuum polarization effects in pure AdS spacetime has been developed in the literature, here we are mainly interested in the investigation of string-induced effects. We show that the curvature of the background spacetime has an essential influence on the VEVs at distances larger than the curvature radius. In particular, at large distances the decay of the string-induced VEVs is power law for both massless and massive fields. The string-induced parts vanish on the AdS boundary and they dominate the pure AdS part for points near the AdS horizon.

Flavour-coherent propagators and Feynman rules: covariant cQPA formulation

We present a simplified and generalized derivation of the flavour-coherent propagators and Feynman rules for the fermionic kinetic theory based on coherent quasiparticle approximation (cQPA) [1–7]. The new formulation immediately reveals the composite nature of the cQPA Wightman function as a product of two spectral functions and an effective two-point interaction vertex, which contains all quantum statistical and coherence information. We extend our previous work to the case of nonzero dispersive self-energy, which leads to a broader range of applications. By this scheme, we derive flavoured kinetic equations for local 2-point functions $$ S_{\text{k}}^{{ <, > }} $$ (t, t), which are reminiscent of the equations of motion for the density matrix. We emphasize that in our approach all the interaction terms are derived from first principles of nonequilibrium quantum field theory.

Flavoured quantum Boltzmann equations from cQPA

We develop a Boltzmann-type quantum transport theory for interacting fermion and scalar fields including both flavour and particle-antiparticle mixing. Our formalism is based on the coherent quasiparticle approximation (cQPA) for the 2-point correlation functions, whose extended phase-space structure contains new spectral shells for flavour- and particle-antiparticle coherence. We derive explicit cQPA propagators and Feynman rules for the transport theory. In particular the nontrivial Wightman functions can be written as composite operators $\sim {\cal A} F {\cal A}$, which generalize the usual Kadanoff-Baym ansatz. Our numerical results show that particle-antiparticle coherence can strongly influence CP-violating flavour mixing even for relatively slowly-varying backgrounds. Thus, unlike recently suggested, these correlations cannot be neglected when studying asymmetry generation due to time-varying mass transition, for example in electroweak-type baryogenesis models. Finally, we show that the cQPA coherence solutions are directly related to squeezed states in the more familiar operator formalism.

Topological Casimir effect in compactified cosmic string spacetime

We investigate the Wightman function, the vacuum expectation values of the field squared and the energy–momentum tensor for a massive scalar field with general curvature coupling in the generalized cosmic string geometry with a compact dimension along its axis. The boundary condition along the compactified dimension is taken in general form with an arbitrary phase. The vacuum expectation values are decomposed into two parts. The first one corresponds to the uncompactified cosmic string geometry and the second one is the correction induced by the compactification. The asymptotic behavior of the vacuum expectation values of the field squared, energy density and stresses is investigated near the string and at large distances. We show that the nontrivial topology due to the cosmic string enhances the vacuum polarization effects induced by the compactness of spatial dimension for both the field squared and the vacuum energy density. A simple formula is given for the part of the integrated topological Casimir energy induced by the planar angle deficit. The results are generalized for a charged scalar field in the presence of a constant gauge field. In this case, the vacuum expectation values are periodic functions of the component of the vector potential along the compact dimension.

WIGHTMAN FUNCTIONS IN DE SITTER AND ANTI-DE SITTER SPACETIMES IN THE PRESENCE OF COSMIC STRINGS

In this paper we evaluate the Wightman functions associated with a massive quantum scalar field in de Sitter and anti-de Sitter spacetimes in the presence of a cosmic string. Having these functions we calculate the corresponding renormalized vacuum expectation values of the field squared and present the behavior of the contributions induced by the cosmic string as function of the proper distance to it for different values of the parameter which codify the presence of this linear topological defect.

Gauging away physics

We consider the recent argument by Higuchi et al(2011 Class. Quantum Grav. 28 245012) that a nonlocal gauge transformation can be used to eliminate the infrared divergence of the graviton propagator, when evaluated in Bunch–Davies vacuum on the open coordinate submanifold of de Sitter space in the transverse-traceless-synchronous gauge. Because the transformation is not local, the equal time commutator of undifferentiated fields no longer vanishes. From explicit examination of the Wightman function we demonstrate that the transformation adds anti-sources in the far future which cancel the bad infrared behavior but also change the propagator equation. The same problem exists in the localized version of the recent argument. Adding such anti-sources does not seem to be legitimate and could be used to eliminate the infrared divergence of the massless, minimally coupled scalar. The addition of such anti-sources in flat space QED could be effected by an almost identical gauge transformation, and would seem to eliminate the well-known infrared divergences which occur in loop corrections to exclusive amplitudes.

Summing planar diagrams by an integrable bootstrap

Correlation functions of matrix-valued fields are not generally known for massive renormalized field theories. We find the large-N limit of form factors of the (1+1)-dimensional sigma model with SU(N) X SU(N) symmetry. These form factors give a correction to the free-field approximation for the N=infinity Wightman function. The method is a combination of the 1/N-expansion of the S-matrix and Smirnov's form-factor axioms. We expand the renormalized field in terms of a free massive Bosonic field as N goes to infinity.

Finite width in out-of-equilibrium propagators and kinetic theory

We derive solutions to the Schwinger–Dyson equations on the Closed-Time-Path for a scalar field in the limit where backreaction is neglected. In Wigner space, the two-point Wightman functions have the curious property that the equilibrium component has a finite width, while the out-of equilibrium component has zero width. This feature is confirmed in a numerical simulation for scalar field theory with quartic interactions. When substituting these solutions into the collision term, we observe that an expansion including terms of all orders in gradients leads to an effective finite-width. Besides, we observe no breakdown of perturbation theory, that is sometimes associated with pinch singularities. The effective width is identical with the width of the equilibrium component. Therefore, reconciliation between the zero-width behaviour and the usual notion in kinetic theory, that the out-of-equilibrium contributions have a finite width as well, is achieved. This result may also be viewed as a generalisation of the fluctuation–dissipation relation to out-of-equilibrium systems with negligible backreaction.