How a space-time singularity helps remove the ultraviolet divergence problem

Interior-boundary conditions (IBCs) have been suggested as a possibility to circumvent the problem of ultraviolet divergences in quantum field theories. In the IBC approach, particle creation and annihilation is described with the help of linear conditions that relate the wave functions of two sectors of Fock space: [Formula: see text] at an interior point [Formula: see text] and [Formula: see text] at a boundary point [Formula: see text], typically a collision configuration. Here, we extend IBCs to the relativistic case. To do this, we make use of Dirac’s concept of multi-time wave functions, i.e. wave functions [Formula: see text] depending on [Formula: see text] space-time coordinates [Formula: see text] for [Formula: see text] particles. This provides the manifestly covariant particle-position representation that is required in the IBC approach. In order to obtain rigorous results, we construct a model for Dirac particles in 1+1 dimensions that can create or annihilate each other when they meet. Our main results are an existence and uniqueness theorem for that model, and the identification of a class of IBCs ensuring local probability conservation on all Cauchy surfaces. Furthermore, we explain how these IBCs relate to the usual formulation with creation and annihilation operators. The Lorentz invariance is discussed and it is found that, apart from a constant matrix (which is required to transform in a certain way), the model is manifestly Lorentz invariant. This makes it clear that the IBC approach can be made compatible with relativity.

In quantum field theory, Hamiltonians contain particle creation and annihilation terms that are usually ultraviolet (UV) divergent. It is well known that these divergences can sometimes be removed by adding counter-terms and by taking limits in which a UV cutoff tends toward infinity. Here, I review a novel way of removing UV divergences: by imposing a type of boundary condition on the wave function. These conditions, called interior-boundary conditions (IBCs), relate the values of the wave function at two configurations linked by the creation or annihilation of a particle. They allow for a direct definition of the Hamiltonian without renormalization or limiting procedures. In the last section, I review another boundary condition that serves to determine the probability distribution of detection times and places on a time-like 3-surface.

We consider a way of defining quantum Hamiltonians involving particle creation and annihilation based on an interior-boundary condition (IBC) on the wave function, where the wave function is the particle-position representation of a vector in Fock space, and the IBC relates (essentially) the values of the wave function at any two configurations that differ only by the creation of a particle. Here we prove, for a model of particle creation at one or more point sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian can indeed be rigorously defined in this way without the need for any ultraviolet regularization, and that it is self-adjoint. We prove further that introducing an ultraviolet cut-off (thus smearing out particles over a positive radius) and applying a certain known renormalization procedure (taking the limit of removing the cut-off while subtracting a constant that tends to infinity) yields, up to addition of a finite constant, the Hamiltonian defined by the IBC.

The creation of massless scalar particles by naked singularities in asymptotically flat spacetimes is investigated within the geometrical-optics approximation. To avoid the need to impose boundary conditions on the singularity, we consider models in which a curvature singularity arises at a finite time in the past. We consider two particular types of models. One is a shell-crossing singularity formed in the gravitational collapse of a dust cloud. The energy flux of the created particles remains finite up to the time of formation of the singularity. In the particular case when the singularity forms on the event horizon, geometrical optics yields the exact flux, in spite of the high curvature of spacetime. The radiation obtained is identical to the thermal Hawking radiation emitted by black holes. The other models considered are those of charged shells for which the charge exceeds the mass. If these shells collapse to form naked singularities (which is possible if the proper mass is negative or if Einstein's equations are not imposed), an infinite flux of created particles results. In the cases examined here, the flux is negative for two-dimensional models and for the minimally coupled scalar field in four-dimensional models, whereas it is positive for the conformally coupled scalar field in four-dimensional models. In either case, the back reaction from particle creation will be large and may prevent formation of a naked singularity.

The creation of particles (bosons and fermions) in a hot Friedmann universe from an initial state described by a thermal density matrix is considered in the case when the effective mass of the particles depends on the temperature. It is shown that for scalar particles allowance for the temperature effects leads to a very strong suppression of the creation process. For spinor particles, allowance for the temperature effects leads to a strong suppression of the process of creation from an initial vacuum state. However if the creation of the spinor particles takes place from a state described by a thermal density matrix, then the temperature effect on the particle creation process is relatively weak.

The creation of massless scalar particles in asymptotically flat spacetimes containing shell-focusing naked singularities which evolve from nonsingular initial data is studied. In the case where the singularity is marginally naked, i.e., its Cauchy horizon coincides with the event horizon, we are able to compute the spectrum of created particles by Hawking's method. The spectrum of particles is no longer thermal, but can be expressed as a quasithermal spectrum with a frequency-dependent temperature. In the high-frequency limit the effective temperature approaches a constant value greater than the Hawking temperature. In the more general case where the Cauchy horizon and event horizon do not coincide, we calculate the expectation value of the stress-energy tensor of the scalar field in the two-dimensional spacetimes obtained by suppressing the spherical coordinates. In all cases the energy flux along the Cauchy horizon diverges in a positive sense. This strongly suggests that the metric's back-reaction to the flux of created particles will prevent the formation of naked shell-focusing singularities.

A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, i.e., for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has successfully been done already in one space dimension, and more generally for codimension-1 boundaries, the situation of point sources in three dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3D, which also correspond to a boundary condition. Indeed, we confirm this expectation here by proving that there is no self-adjoint operator on a (truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with an IBC (on the boundary consisting of configurations with a particle at the origin) that are away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential.

Recently, there has been progress in developing interior-boundary conditions (IBCs) as a technique of avoiding the problem of ultraviolet divergence in non-relativistic quantum field theories while treating space as a continuum and electrons as point particles. An IBC can be expressed in the particle-position representation of a Fock vector $\psi$ as a condition on the values of $\psi$ on the set of collision configurations, and the corresponding Hamiltonian is defined on a domain of vectors satisfying this condition. We describe here how Bohmian mechanics can be extended to this type of Hamiltonian. In fact, part of the development of IBCs was inspired by the Bohmian picture. Particle creation and annihilation correspond to jumps in configuration space; the annihilation is deterministic and occurs when two particles (of the appropriate species) meet, whereas the creation is stochastic and occurs at a rate dictated by the demand for the equivariance of the $|\psi|^2$ distribution, time reversal symmetry, and the Markov property. The process is closely related to processes known as Bell-type quantum field theories.

The paper deals with collapse dynamics of a spherically symmetric massive star in the framework of non-equilibrium thermodynamic prescription through particle creation mechanism. The matter content in the star is in the form of perfect fluid with barotropic equation of state, and the dissipative phenomena due to non-equilibrium thermodynamics is in the form of bulk viscosity. For simplicity, the thermodynamic system is chosen to be adiabatic so that the effective bulk viscous pressure is linearly related to the particle creation rate. As a result, the evolution of the collapsing star also depends on the particle creation rate. By proper choice of creation rate as a function of the Hubble parameter, it is found that the end state of the collapse may be either a black hole (BH) or a naked singularity (NS).

We investigate the spherically symmetric gravitational collapse of an incoherent dust cloud by considering a LTB-type spacetime in third-order Lovelock Gravity without cosmological constant, and give three families of LTB-like solutions which separately corresponding to hyperbolic, parabolic and elliptic. Notice that the contribution of high-order curvature corrections have a profound influence on the nature of the singularity, and the global structure of spacetime changes drastically from the analogous general relativistic case. Interestingly, the presence of high order Lovelock terms leads to the formation of massive, naked and timelike singularities in the 7D spacetime, which is disallowed in general relativity. Moveover, we point out that the naked singularities in the 7D case may be gravitational weak therefore may not be a serious threat to the cosmic censorship hypothesis, while the naked singularities in the D ≥ 8 inhomogeneous collapse violate the cosmic censorship hypothesis seriously.

Recently, the possibility was pointed out by one of the present authors and his collaborators that an effective naked singularity referred to as "a visible border of spacetime" is generated by high-energy particle collision in the context of large extra dimensions or TeV-scale gravity. In this paper, we investigate the particle creation by a naked singularity in general dimensions, while adopting a model in which a marginally naked singularity forms in the collapse of a homothetic lightlike pressureless fluid. We find that the spectrum deviates from that of Hawking radiation due to scattering near the singularity but can be recast in quasi-thermal form. The temperature is always higher than that of Hawking radiation of a same-mass black hole, and can be arbitrarily high depending on a parameter in the model. This implies that, in principle, the naked singularity may be distinguished from a black hole in collider experiments.

While fundamental physically realistic Hamiltonians should be invariant under time reversal, time asymmetric Hamiltonians can occur as mathematical possibilities or effective Hamiltonians. Here, we study conditions under which non-relativistic Hamiltonians involving particle creation and annihilation, as come up in quantum field theory (QFT), are time asymmetric. It turns out that the time reversal operator T can be more complicated than just complex conjugation, which leads to the question which criteria determine the correct action of time reversal. We use Bohmian trajectories for this purpose and show that time reversal symmetry can be broken when charges are permitted to be complex numbers, where ‘charge’ means the coupling constant in a QFT that governs the strength with which a fermion emits and absorbs bosons. We pay particular attention to the technique for defining Hamiltonians with particle creation based on interior-boundary conditions, and we find them to generically be time asymmetric. Specifically, we show that time asymmetry for complex charges occurs whenever not all charges have equal or opposite phase. We further show that, in this case, the corresponding ground states can have non-zero probability currents, and we determine the effective potential between fermions of complex charge.

There are many solutions to the Einstein field equations that demonstrate naked singularity (NS) formation after regular evolution. It is possible, however, that such a quantum effect as particle creation prevents NSs from forming. We investigate the relation between the curvature strength and the quantum effects of NSs in a very wide class of spherical dust collapse. Through a perturbative calculation, we find that if the NS is very strong, the quantum particle creation diverges as the Cauchy horizon is approached, while if the NS is very weak, the creation should be finite. In the context of cosmic censorship, strong NSs will be subjected to the backreaction of quantum effects and may disappear or be hidden behind horizons, while weak NSs will not.

We investigate matter creation processes during the reheating period of the early Universe, by using the thermodynamic of open systems. The Universe is assumed to consist of the inflationary scalar field, which, through its decay, generates relativistic matter and pressureless dark matter. The inflationary scalar field transfers its energy to the newly created matter particles, with the field energy decreasing to near zero. The equations governing the irreversible matter creation are obtained by combining the thermodynamics description of the matter creation and the gravitational field equations. The role of the different inflationary scalar field potentials is analyzed by using analytical and numerical methods. The values of the energy densities of relativistic matter and dark matter reach their maximum when the Universe is reheated up to the reheating temperature, which is obtained as a function of the scalar field decay width, the scalar field particle mass, and the cosmological parameters. Particle production leads to the acceleration of the Universe during the reheating phase, with the deceleration parameter showing complex dynamics. Once the energy density of the scalar field becomes negligible with respect to the matter densities, the expansion of the Universe decelerates, and inflation has a graceful exit.

We investigate gravitational effects of extreme, non-extreme and ultra- extreme domain walls in the presence of a dilaton field. The dilaton is a scalar field without self-interaction that couples to the matter po- tential that is responsible for the formation of the wall. Motivated by superstring and supergravity theories, we consider both an exponential dilaton coupling (parametrized with the coupling constant alpha and the case where the coupling is self-dual, i.e. it has an extremum for a fi- nite value of the dilaton. For an exponential dilaton coupling, extreme walls (which are static planar configurations with surface energy density sigma_ext saturating the corresponding Bogomol'nyi bound) have a naked (planar) singularity outside the wall for alpha>1, while for alpha smaller or equal to 1 the singularity is null. On the other hand, non-extreme walls (bubbles with two insides and sigma_non > sigma_ext and ultra-extreme walls bubbles of false vacuum decay with sigma_ultra < sigma_ext always have naked singularities. There are solutions with self-dual couplings, which reduce to singularity-free vacuum domain wall space--times. However, only non- and ultra-extreme walls of such a type are dynamically stable.