Free Quantum Field Research Articles

A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation

We give a microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation (NLS) from many-body quantum theory. The starting point of our proof is [11] on the time-independent problem and [16] on the corresponding problem on a finite lattice. An important new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaussian structure of the free quantum field and prevents the use of the Wick theorem. We overcome it by means of complex analytic methods. Our methods apply to the nonlocal NLS with bounded convolution potential. In the periodic setting, we also consider the local NLS, arising from short-range interactions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in Xs,b spaces.

Split Property for Free Massless Finite Helicity Fields

We prove the split property for any finite helicity free quantum fields. Finite helicity Poincar\'e representations extend to the conformal group and the conformal covariance plays an essential role in the argument. The split property is ensured by the trace class condition: Tr (exp(-s L_0)) is finite for all s>0 where L_0 is the conformal Hamiltonian of the M\"obius covariant restriction of the net on the time axis. We extend the argument for the scalar case presented in [7]. We provide the direct sum decomposition into irreducible representations of the conformal extension of any helicity-h representation to the subgroup of transformations fixing the time axis. Our analysis provides new relations among finite helicity representations and suggests a new construction for representations and free quantum fields with non-zero helicity.

Instantons in asymptotically safe and free quantum field theories

We investigate the instanton dynamics of asymptotically safe and free quantum field theories featuring respectively controllable ultraviolet and infrared fixed points. We start by briefly reviewing the salient points about the instanton calculus for pure Yang Mills (YM) and QCD. We then move on to determine the role of instantons within the controllable regime of the QCD conformal window. In this region we add a fermion-mass operator and determine the density of instantons per unit volume as function of the fermion mass. Finally, for the first time, we extend the instanton calculus to asymptotically safe theories.

Analyticity domain of a quantum field theory and accelero-summation

From 't Hooft's argument, one expects that the analyticity domain of an asymptotically free quantum field theory is horned shaped. In the usual Borel summation, the function is obtained through a Laplace transform and thus has a much larger analyticity domain. However, if the summation process goes through the process called acceleration by Ecalle, one obtains such a horn shaped analyticity domain. We therefore argue that acceleration, which allows to go beyond standard Borel summation, must be an integral part of the toolkit for the study of exactly renormalisable quantum field theories. We sketch how this procedure is working and what are its consequences.

Scalar Models of Formally Interacting Non-Standard Quantum Fields in Minkowski Space-Time

For decades, a lot of work has been devoted to the problem of constructing a non-trivial quantum field theory in four-dimensional space time. This letter addresses the attempts to construct an algebraic quantum field theory in the framework of non-standard theories like hyperfunction or ultra-hyperfunction quantum field theory. For this purpose model theories of formally interacting neutral scalar fields are constructed and some of their characteristic properties like two-point functions are discussed. The formal self-couplings are obtained from local normally-ordered analytic redefinitions of the free scalar quantum field, mimicking a non-trivial structure of the resulting Lagrangians and equations of motion.

Complexity and entanglement for thermofield double states

Motivated by holographic complexity proposals as novel probes of black hole spacetimes, we explore circuit complexity for thermofield double (TFD) states in free scalar quantum field theories using the Nielsen approach. For TFD states at t = 0t=0, we show that the complexity of formation is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals. For TFD states at t>0t>0, we demonstrate that the complexity evolves in time and saturates after a time of the order of the inverse temperature. The latter feature, which is in contrast with the results of holographic proposals, is due to the Gaussian nature of the TFD state of the free bosonic QFT. A novel technical aspect of our work is framing complexity calculations in the language of covariance matrices and the associated symplectic transformations, which provide a natural language for dealing with Gaussian states. Furthermore, for free QFTs in 1+1 dimension, we compare the dynamics of circuit complexity with the time dependence of the entanglement entropy for simple bipartitions of TFDs. We relate our results for the entanglement entropy to previous studies on non-equilibrium entanglement evolution following quenches. We also present a new analytic derivation of a logarithmic contribution due to the zero momentum mode in the limit of vanishing mass for a subsystem containing a single degree of freedom on each side of the TFD and argue why a similar logarithmic growth should be present for larger subsystems.

Momentum-space entanglement after smooth quenches

We compute the total amount of entanglement produced between momentum modes at late times after a smooth mass quench in free bosonic and fermionic quantum field theories. The entanglement and Rényi entropies are obtained in closed form as a function of the parameters characterizing the quench protocol. For bosons, we show that the entanglement production is more significant for light modes and for fast quenches. In particular, infinitely slow or adiabatic quenches do not produce any entanglement. Depending on the quench profile, the decrease as a function of the quench rate delta t can be either monotonic or oscillating. In the fermionic case the situation is more subtle and there is a critical value for the quench amplitude above which this behavior is changed and the entropies become peaked at intermediate values of momentum and of the quench rate. We also show that the results agree with the predictions of a Generalized Gibbs Ensemble and obtain explicitly its parameters in terms of the quench data.

Black hole remnants may exist if Starobinsky inflation occurred

Zero temperature black hole solutions to the semiclassical backreaction equations are investigated. Evidence is provided that certain components of the stress-energy tensors for free quantum fields at the horizon only depend on the local geometry near the horizon. This allows the semiclassical backreaction equations to be solved near the horizon. It is found that macroscopic uncharged zero temperature black hole solutions to the equations may exist if the coefficient of one of the higher derivative terms in the gravitational Lagrangian is large enough and of the right sign for Starobinsky inflation to have occurred in the early Universe.

Circuit complexity in fermionic field theory

We define and calculate versions of complexity for free fermionic quantum field theories in 1+1 and 3+1 dimensions, adopting Nielsen's geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for fermions: viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cut-off dependence are discussed.

Was Polchinski wrong? Colombeau distributional Rindler space-time with distributional Levi-Cività connection induced vacuum dominance. Unruh effect revisited

The vacuum energy density of free scalar quantum field ϕ in a Rindler distributional space-time with distributional Levi-Cività connection is considered. It has been widely believed that, except in very extreme situations, the influence of acceleration on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is wrong by showing that in a Rindler distributional background space-time with distributional Levi-Cività connection the vacuum energy of free quantum fields is forced, by the very same background distributional space-time such the Rindler distributional background space-time, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on a Rindler distributional space-times with distributional Levi-Cività connection. In particular we obtain that the vacuum fluctuations 〈ϕ2〉 has a singular behavior on a Rindler horizon. Therefore sufficiently strongly accelerated observer burns up near the Rindler horizon. Thus Polchinski’s account doesn’t violate of the Einstein equivalence principle.

Hamiltonian renormalisation II. Renormalisation flow of 1+1 dimensional free scalar fields: derivation

In the companion paper Lang et al (2017 arXiv:1711.05685) we motivated a renormalisation flow on Osterwalder–Schrader data (OS-data) consisting of (1.) a Hilbert space, (2.) a cyclic vacuum and (3.) a Hamiltonian annihilating that vacuum. As the name suggests, the motivation was via the OS reconstruction theorem which allows to reconstruct the OS data from an OS measure satisfying (a subset of) the OS axioms, in particular reflection positivity. The guiding principle was to map the usual Wilsonian path integral renormalisation flow onto a flow of the corresponding OS data.We showed that this induced flow on the OS data has an unwanted feature which disqualifies the associated coarse grained Hamiltonians from being the projections of a continuum Hamiltonian onto vectors in the coarse grained Hilbert space. This motivated the definition of a direct Hamiltonian renormalisation flow which follows the guiding principle but does not suffer from the afore mentioned caveat.In order to test our proposal, we apply it to the only known completely solvable model, namely the case of free scalar quantum fields. In this paper we focus on the Klein Gordon field in two spacetime dimensions and illustrate the difference between the path integral induced and direct Hamiltonian flow. Generalisations to more general models in higher dimensions will be discussed in our companion papers.

Dynamical Casimir effect in a double tunable superconducting circuit

We present an analytical and numerical analysis of the particle creation in a cavity ended with two SQUIDs, both subjected to time dependent magnetic fields. In the linear and lossless regime, the problem can be modeled by a free quantum field in $1+1$ dimensions, in the presence of boundary conditions that involve a time dependent linear combination of the field and its spatial and time derivatives. We consider a situation in which the boundary conditions at both ends are periodic functions of time, focusing on interesting features as the dependence of the rate of particle creation with the characteristics of the spectrum of the cavity, the conditions needed for parametric resonance, and interference phenomena due to simultaneous time dependence of the boundary conditions. We point out several concrete effects that could be tested experimentally

Conformal data of fundamental gauge-Yukawa theories

We determine central charges, critical exponents and appropriate gradient flow relations for nonsupersymmetric vector-like and chiral Gauge-Yukawa theories that are fundamental according to Wilson and that feature calculable UV or IR interacting fixed points. We further uncover relations and identities among the various local and global conformal data. This information is used to provide the first extensive characterisation of general classes of free and safe quantum field theories of either chiral or vector-like nature via their conformal data. Using large $N_f$ techniques we also provide examples in which the safe fixed point is nonperturbative but for which conformal perturbation theory can be used to determine the global variation of the $a$ central charge.

Erratum to: General equilibrium second-order hydrodynamic coefficients for free quantum fields

In section 3, the partition function is included in the definition of the statistical operator.

Infrared divergences for free quantum fields in cosmological spacetimes

We investigate the nature of infrared divergences for the free graviton and inflaton two-point functions in flat Friedman–Lemaître–Robertson–Walker spacetime. These divergences arise because the momentum integral for these two-point functions diverges in the infrared. It is straightforward to see that the power of the momentum in the integrand can be increased by 2 in the infrared using large gauge transformations, which are sufficient for rendering these two-point functions infrared finite for slow-roll inflation. In other words, if the integrand of the momentum integral for these two-point functions behaves like , where p is the momentum, in the infrared, then it can be made to behave like by large gauge transformations. On the other hand, it is known that, if one smears these two-point functions in a gauge-invariant manner, the power of the momentum in the integrand is changed from to . This fact suggests that the power of the momentum in the integrand for these two-point functions can be increased by 4 using large gauge transformations. In this paper we show that this is indeed the case. Thus, the two-point functions for the graviton and inflaton fields can be made finite by large gauge transformations for a large class of potentials and states in single-field inflation.

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.

Toward a Definition of Complexity for Quantum Field Theory States.

We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form su(1,1) algebras. On the manifold of Gaussian states generated by these operations, the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and those of holographic complexity proposals.

The solution of the sixth Hilbert problem: the ultimate Galilean revolution

I argue for a full mathematization of the physical theory, including its axioms, which must contain no physical primitives. In provocative words: 'physics from no physics'. Although this may seem an oxymoron, it is the royal road to keep complete logical coherence, hence falsifiability of the theory. For such a purely mathematical theory the physical connotation must pertain only the interpretation of the mathematics, ranging from the axioms to the final theorems. On the contrary, the postulates of the two current major physical theories either do not have physical interpretation (as for von Neumann's axioms for quantum theory), or contain physical primitives as 'clock', 'rigid rod', 'force', 'inertial mass' (as for special relativity and mechanics). A purely mathematical theory as proposed here, though with limited (but relentlessly growing) domain of applicability, will have the eternal validity of mathematical truth. It will be a theory on which natural sciences can firmly rely. Such kind of theory is what I consider to be the solution of the sixth Hilbert problem. I argue that a prototype example of such a mathematical theory is provided by the novel algorithmic paradigm for physics, as in the recent information-theoretical derivation of quantum theory and free quantum field theory.This article is part of the theme issue 'Hilbert's sixth problem'.

Was Polchinski Wrong? Colombeau Distributional Rindler Space-Time with Distributional Levi-Cività Connection Induced Vacuum Dominance. Unruh Effect Revisited

In this paper we leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. Using nonlinear superdistributional geometry and supergeneralized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates. However the Tolman formula for the total energy $E$ of a static and asymptotically flat spacetime,gives $E=mc^2$, as it should be. New class Colombeau solutions to Einstein field equations is obtained.New class Colombeau solutions to Einstein field equations is obtained. The vacuum energy density of free scalar quantum field ${\Phi}$ with a distributional background spacetime also is considered.It has been widely believed that, except in very extreme situations, the influence of acceleration on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is wrong by showing that in a Rindler distributional background spacetime with distributional Levi-Civit\`a connection the vacuum energy of free quantum fields is forced, by the very same background distributional space-time such a Rindler distributional background space-time, to become dominant over any classical energy density component.This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on a Rindler distributional space-times with distributional Levi-Civit\`a connection. In particular we obtain that the vacuum fluctuations $<{\Phi}^2({\delta})>$ have a singular behavior at a Rindler horizon $\delta = 0$.Therefore sufficiently strongly accelerated observer burns up near the Rindler horizon. Thus Polchinski account does not violate of the Einstein equivalence principle.

On the Uniqueness of the Fock Quantization of the Dirac Field in the Closed FRW Cosmology

The Fock quantization of free fields propagating in cosmological backgrounds is in general not unambiguously defined due to the nonstationarity of the space-time. For the case of a scalar field in cosmological scenarios, it is known that the criterion of unitary implementation of the dynamics serves to remove the ambiguity in the choice of Fock representation (up to unitary equivalence). Here, applying the same type of arguments and methods previously used for the scalar field case, we discuss the issue of the uniqueness of the Fock quantization of the Dirac field in the closed FRW space-time proposed by D’Eath and Halliwell.