Free Scalar Field Research Articles

Momentum distribution and correlation for a free scalar field in the Tsallis nonextensive statistics based on density operator

We derived the expression of the normalized $q$-expectation value based on the density operator to the order $1-q$ with the physical temperature in the Tsallis nonextensive statistics of entropic parameter $q$. With the derived expression of the normalized $q$-expectation value, we calculated the momentum distribution and the correlation to the order $1-q$ as functions of the inverse physical temperature for a free scalar field. To the order $1-q$, the momentum distribution derived by using the density operator coincides with the momentum distribution derived from the entropic measure described with the distribution, when the physical temperature equals the temperature in the distribution derived from the entropic measure. The correlation depends on the momentums for $q \neq 1$. The factor two appears in the correlation for the same momentums, and indicates that the effects of boson at $q \neq 1$ and those at $q=1$ are similar for the correlation.

Vacuum state in de Sitter spacetime with static charts

We study the free massive scalar field in de Sitter spacetime with static charts. In particular, we find positive-frequency modes for the Bunch-Davies vacuum state natural to the static charts as superpositions of the well-known positive-frequency modes in the conformally-flat chart. We discuss in detail how these modes are defined globally in the two static charts and the region in their future. The global structure of these solutions leads to the well-known description of the Bunch-Davies vacuum state as an entangled state. Our results are expected to be useful not only for studying the thermal properties in the vacuum fluctuations in de Sitter spacetime but also for understanding the nonlocal properties of the vacuum state.

New perspective on the Unruh effect

In this work, based on the worldline path integral representation of the vacuum energy in spacetime with a Lorentzian metric, we provide a new but complementary interpretation of the Unruh effect. We perform the quantization of the massless free scalar field in Rindler space specifying initial and final conditions. After quantization, the final outcome for the vacuum energy is interpreted as world line path integrals. In this picture we find that the Unruh radiation is made of real particles as well as real antiparticles. The prediction regarding the presence of antiparticles in the radiation might open new lines for experimental detection of the effect. We present a thought experiment which offers a clear picture and supports the new interpretation.

Pauli-Jordan function and scalar field quantization in κ -Minkowski noncommutative spacetime

We study a complex free scalar field theory on a noncommutative background spacetime called $\kappa$-Minkowski. In particular we address the problem of second quantization. We obtain the algebra of creation and annihilation operators in an explicitly covariant way. Our procedure does not use canonical/Hamiltonian formulations, which turn out to be ill-defined in our context. Instead we work in a spacetime covariant way by introducing a noncommutative Pauli-Jordan function. This function is obtained as a generalization of the ordinary, commutative, one by taking into account the constraints imposed by the symmetries of our noncommutative spacetime. The Pauli-Jordan function is later employed to study the structure of the light cone in $\kappa$-Minkowski spacetime, and to draw conclusions on the superluminal propagation of signals.

Free field primaries in general dimensions: counting and construction with rings and modules

We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as elements of the polynomial ring over d × n variables obeying a system of first and second order partial differential equations. LWPs invariant under Sn correspond to primary fields in free scalar field theory in d dimensions, constructed from n fields. The LWPs are in one-to-one correspondence with a quotient of the polynomial ring in d × (n − 1) variables by an ideal generated by n quadratic polynomials. The implications of this description for the counting and construction of primary fields are described: an interesting binomial identity underlies one of the construction algorithms. The product on the ring of LWPs can be described as a commutative star product. The quadratic algebra of lowest weight polynomials has a dual quadratic algebra which is non-commutative. We discuss the possible physical implications of this dual algebra.

The art of the state

Quantum field theory (QFT) on curved spacetimes lacks an obvious distinguished vacuum state. We review a recent no-go theorem that establishes the impossibility of finding a preferred state in each globally hyperbolic spacetime, subject to certain natural conditions. The result applies in particular to the free scalar field, but the proof is model-independent and therefore of wider applicability. In addition, we critically examine the recently proposed “SJ states”, that are determined by the spacetime geometry alone, but which fail to be Hadamard in general. We describe a modified construction that can yield an infinite family of Hadamard states, and also explain recent results that motivate the Hadamard condition without direct reference to ultra-high energies or ultra-short distance structure.

Reflection positivity in higher derivative scalar theories

Reflection positivity constitutes an integral prerequisite in the Osterwalder–Schrader reconstruction theorem which relates quantum field theories defined on Euclidean space to their Lorentzian signature counterparts. In this work, we rigorously prove the violation of reflection positivity in a large class of free scalar fields with a rational propagator. This covers, in particular, higher-derivative theories where the propagator admits a partial fraction decomposition as well as degenerate cases including, e.g., p4-type propagators.

Entanglement entropy on a fuzzy sphere with a UV cutoff

We introduce a UV cutoff into free scalar field theory on the noncommutative (fuzzy) two-sphere. Due to the IR-UV connection, varying the UV cutoff allows us to control the effective nonlocality scale of the theory. In the resulting fuzzy geometry, we establish which degrees of freedom lie within a specific geometric subregion and compute the associated vacuum entanglement entropy. Entanglement entropy for regions smaller than the effective nonlocality scale is extensive, while entanglement entropy for regions larger than the effective nonlocality scale follows the area law. This reproduces features previously obtained in the strong coupling regime through holography. We also show that mutual information is unaffected by the UV cutoff.

Lifshitz entanglement entropy from holographic cMERA

We study entanglement entropy in free Lifshitz scalar field theories holographically by employing the metrics proposed by Nozaki, Ryu and Takayanagi in [1] obtained from a continuous multi-scale entanglement renormalisation ansatz (cMERA). In these geometries we compute the minimal surface areas governing the entanglement entropy as functions of the dynamical exponent z and we exhibit a transition from an area law to a volume law analytically in the limit of large z. We move on to explore the effects of a massive deformation, obtaining results for any z in arbitrary dimension. We then trigger a renormalisation group flow between a Lifshitz theory and a conformal theory and observe a monotonic decrease in entanglement entropy along this flow. We focus on strip regions but also consider a disc in the undeformed theory.

What Spatial Geometries do (2+1)-Dimensional Quantum Field Theory Vacua Prefer?

We consider relativistic (2+1)-dimensional quantum field theories (QFTs) on a product of time with a two-space and study the vacuum free energy as a functional of the temperature and spatial geometry. We focus on free scalar and Dirac fields on arbitrary perturbations of flat space, finding that the free energy difference from flat space is finite and always negative to leading order in the perturbation. Thus, free (2+1)-dimensional QFTs appear to always energetically favor a crumpled space on all scales; this is true both as a purely quantum effect at zero temperature and as a purely thermal effect at high temperature. Importantly, we show that this quantum effect is non-negligible for the relativistic Dirac degrees of freedom on monolayer graphene even at room temperature, so we argue that this vacuum energy effect should be included for a proper analysis of the equilibrium configuration of graphene or similar materials.

Manifestly covariant canonical quantization of the scalar field and particle localization

Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface [Formula: see text], which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to [Formula: see text]. Position on [Formula: see text] is determined by a 4-vector [Formula: see text]. The corresponding quantum position operator, formed in terms of the operators [Formula: see text], [Formula: see text], that create/annihilate particles on [Formula: see text], has thus well-behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with [Formula: see text], the superposition coefficients forming multiparticle wave packet profiles — wave functions, including a single-particle wave function that satisfies the covariant generalization of the Foldy equation. The covariant center-of-mass operator is introduced and its expectation values in a generic multiparticle state calculated.

Evolution of complexity following a quantum quench in free field theory

Using a recent proposal of circuit complexity in quantum field theories introduced by Jefferson and Myers, we compute the time evolution of the complexity following a smooth mass quench characterized by a time scale δt in a free scalar field theory. We show that the dynamics has two distinct phases, namely an early regime of approximately linear evolution followed by a saturation phase characterized by oscillations around a mean value. The behavior is similar to previous conjectures for the complexity growth in chaotic and holographic systems, although here we have found that the complexity may grow or decrease depending on whether the quench increases or decreases the mass, and also that the time scale for saturation of the complexity is of order δt (not parametrically larger).

Scrambling in the quantum Lifshitz model

We study signatures of chaos in the quantum Lifshitz model through out-of-time ordered correlators (OTOC) of current operators. This model is a free scalar field theory with dynamical critical exponent z = 2. It describes the quantum phase transition in 2D systems, such as quantum dimer models, between a phase with a uniform ground state to another one with spontaneously broken translation invariance. At the lowest temperatures the chaotic dynamics are dominated by a marginally irrelevant operator which induces a temperature dependent stiffness term. The numerical computations of OTOC exhibit a non-zero Lyapunov exponent (LE) in a wide range of temperatures and interaction strengths. The LE (in units of temperature) is a weakly temperature-dependent function; it vanishes at weak interaction and saturates for strong interaction. The Butterfly velocity increases monotonically with interaction strength in the studied region while remaining smaller than the interaction-induced velocity/stiffness.

Quantum entanglement in de Sitter space with a wall and the decoherence of bubble universes

We study the effect of a bubble wall on the entanglement entropy of a free massive scalar field between two causally disconnected open charts in de Sitter space. We assume there is a delta-functional wall between the open charts. This can be thought of as a model of pair creation of bubble universes in de Sitter space. We first derive the Euclidean vacuum mode functions of the scalar field in the presence of the wall in the coordinates that respect the open charts. We then derive the Bogoliubov transformation between the Euclidean vacuum and the open chart vacua that makes the reduced density matrix diagonal. We find that larger walls lead to less entanglement. Our result may be regarded as evidence of decoherence of bubble universes from each other. We also note an interesting relationship between our results and discussions of the black hole firewall problem.

Exact ghost-free bigravitational waves

We study the propagation of exact gravitational waves in the ghost-free bimetric theory. Our focus is on type N spacetimes compatible with the cosmological constants provided by the bigravity interaction potential, and particularly in the single class known by allowing at least a Killing symmetry: the AdS waves. They have the advantage of being represented by a generalized Kerr-Schild transformation from AdS spacetime. This means a notorious simplification in bigravity by allowing to straightforwardly compute any power of its interaction square root matrix, opening the door to explore physically meaningful exact configurations. For these exact gravitational waves the complex dynamical structure of bigravity decomposes into elementary exact massless or massive excitations propagating on AdS. We use a complexified formulation of the Euler-Darboux equations to provide for the first time the general solutions to the massive version of the Siklos equation which rules the resulting AdS-waves dynamics, using an integral representation originally due to Poisson. Inspired in this progress we tackle the subtle problem on how matter couples to bigravity and concretely if this occurs through a composite metric, which is hard to handle in a general setting. Surprisingly, the Kerr-Schild ansatz brings again a huge simplification in how the related energy-momentum tensors are calculated. This allows us to explicitly characterize AdS waves supported in one case by a massless free scalar field and by a wavefront-homogeneous Maxwell field in another. Considering the most general allowed Maxwell source instead is a highly nontrivial task, that we accomplish by exploiting again the complexified Euler-Darboux description and taking advantage of the classical Riemann method. In fact, this allow us to find at the end the most general configurations for any matter source.

Inflationary universe in deformed phase space scenario

We consider a noncommutative (NC) inflationary model with a homogeneous scalar field minimally coupled to gravity. The particular NC inflationary setting herein proposed, produces entirely new consequences as summarized in what follows. We first analyze the free field case and subsequently examine the situation where the scalar field is subjected to a polynomial and exponential potentials. We propose to use a canonical deformation between momenta, in a spatially flat Friedmann–Lemaî tre–Robertson–Walker (FLRW) universe, and while the Friedmann equation (Hamiltonian constraint) remains unaffected the Friedmann acceleration equation (and thus the Klein–Gordon equation) is modified by an extra term linear in the NC parameter. This concrete noncommutativity on the momenta allows interesting dynamics that other NC models seem not to allow. Let us be more precise. This extra term behaves as the sole explicit pressure that under the right circumstances implies a period of accelerated expansion of the universe. We find that in the absence of the scalar field potential, and in contrast with the commutative case, in which the scale factor always decelerates, we obtain an inflationary phase for small negative values of the NC parameter. Subsequently, the period of accelerated expansion is smoothly replaced by an appropriate deceleration phase providing an interesting model regarding the graceful exit problem in inflationary models. This last property is present either in the free field case or under the influence of the scalar field potentials considered here. Moreover, in the case of the free scalar field, we show that not only the horizon problem is solved but also there is some resemblance between the evolution equation of the scale factor associated to our model and that for the R2 (Starobinsky) inflationary model. Therefore, our herein NC model not only can be taken as an appropriate scenario to get a successful kinetic inflation, but also is a convenient setting to obtain inflationary universe possessing the graceful exit when scalar field potentials are present.

Thermodynamic equilibrium with acceleration and the Unruh effect

We address the problem of thermodynamic equilibrium with constant acceleration along the velocity field lines in a quantum relativistic statistical mechanics framework. We show that for a free scalar quantum field, after vacuum subtraction, all mean values vanish when the local temperature T is as low as the Unruh temperature T_U = A/2\pi where A is the magnitude of the acceleration four-vector. We argue that the Unruh temperature is an absolute lower bound for the temperature of any accelerated fluid at global thermodynamic equilibrium. We discuss the conditions of this bound to be applicable in a local thermodynamic equilibrium situation.

An inverse mass expansion for entanglement entropy in free massive scalar field theory

We extend the entanglement entropy calculation performed in the seminal paper by Srednicki (Phys Rev Lett 71:666, 1993) for free real massive scalar field theories in 1+1, 2+1 and 3+1 dimensions. We show that the inverse of the scalar field mass can be used as an expansion parameter for a perturbative calculation of the entanglement entropy. We perform the calculation for the ground state of the system and for a spherical entangling surface at third order in the inverse mass expansion. The calculated entanglement entropy contains a leading area law term, as well as subleading terms that depend on the regularization scheme, as expected. Universal terms are non-perturbative effects in this approach. Interestingly, this perturbative expansion can be used to approximate the coefficient of the area law term, even in the case of a massless scalar field in 2+1 and 3+1 dimensions. The presented method provides the spectrum of the reduced density matrix as an intermediate result, which is an important advantage in comparison to the replica trick approach. Our perturbative expansion underlines the relation between the area law and the locality of the underlying field theory.

Entanglement of purification in free scalar field theories

We compute the entanglement of purification (EoP) in a 2d free scalar field theory with various masses. This quantity measures correlations between two subsystems and is reduced to the entanglement entropy when the total system is pure. We obtain explicit numerical values by assuming minimal gaussian wave functionals for the purified states. We find that when the distance between the subsystems is large, the EoP behaves like the mutual information. However, when the distance is small, the EoP shows a characteristic behavior which qualitatively agrees with the conjectured holographic computation and which is different from that of the mutual information. We also study behaviors of mutual information in purified spaces and violations of monogamy/strong superadditivity.

The AdS3 propagator and the fate of locality

We recently used Virasoro symmetry considerations to propose an exact formula for a bulk proto-field ϕ in AdS3. In this paper we study the propagator 〈ϕϕ〉. We show that many techniques from the study of conformal blocks can be generalized to compute it, including the semiclassical monodromy method and both forms of the Zamolodchikov recursion relations. When the results from recursion are expanded at large central charge, they match gravitational perturbation theory for a free scalar field coupled to gravity in our chosen gauge.We find that although the propagator is finite and well-defined at long distances, its perturbative expansion in {G}_N=frac{3}{2c} exhibits UV/IR mixing effects. If we nevertheless interpret 〈ϕϕ〉 as a probe of bulk locality, then when {G}_{N{m}_{phi }}ll 1 locality breaks down at the new short-distance scale {sigma}_{ast}sim sqrt[4]{G_N{R}_{mathrm{AdS}}^3} . For ϕ with very large bulk mass, or at small central charge, bulk locality fails at the AdS length scale. In all cases, locality ‘breakdown’ manifests as singularities or branch cuts at spacelike separation arising from non-perturbative quantum gravitational effects.