The quantized Dirac field is known, by a result of Fewster and Verch, to satisfy a Quantum Weak E... more The quantized Dirac field is known, by a result of Fewster and Verch, to satisfy a Quantum Weak Energy Inequality (QWEI) on its averaged energy density along time-like curves in arbitrary four-dimensional globally hyperbolic spacetimes. However, this result does not provide an explicit form for the bound. By adapting ideas from the earlier work, we give a simplified derivation of a QWEI for the Dirac field leading to an explicit bound. The bound simplifies further in the case of static curves in static spacetimes, and, in particular, coincides with a result of Fewster and Mistry in four-dimensional Minkowski spacetime. We also show that our QWEI is compatible with local covariance and derive a simple consequence.
Cosmic strings arising from GUTs can catalyse baryon decay processes with strong interaction cros... more Cosmic strings arising from GUTs can catalyse baryon decay processes with strong interaction cross sections. We examine the mechanism by which the cross section is enhanced and find that it depends strongly on the details of the distribution of gauge fields within the string core. We propose a calculational scheme for estimating wavefunction amplification factors and also a physical understanding of the nature of the enhancement process.
We present an inverse scattering construction of generalised point interactions (GPI) – point-lik... more We present an inverse scattering construction of generalised point interactions (GPI) – point-like objects with non-trivial scattering behaviour. The construction is developed for single centre S-wave GPI models with rational S-matrices, and starts from an integral transform suggested by the scattering data. The theory of unitary dilations is then applied to construct a unitary mapping between Pontryagin spaces which extend the usual position and momentum Hilbert spaces. The GPI Hamiltonian is defined as a multiplication operator on the momentum Pontryagin space and its free parameters are fixed by a physical locality requirement. We determine the spectral properties and domain of the Hamiltonian in general, and construct the resolvent and Møller wave operators thus verifying that the Hamiltonian exhibits the required scattering behaviour. The physical Hilbert space is identified. The construction is illustrated by GPI models representing the effective range approximation. For negat...
We study the classical and quantum theory of a class of nonlinear differential equations on model... more We study the classical and quantum theory of a class of nonlinear differential equations on models of chronology violating spacetimes in which space consists of only finitely many discrete points. Classically, we study the initial value problem for data specified before the dischronal region. In the linear and weakly coupled nonlinear regimes, we show (for generic choices of parameters) that solutions always exist and are unique; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution is shown to preserve the symplectic structure. The quantum theory is approached via the quantum initial value problem (QIVP); that is, by seeking operator-valued solutions to the equation of motion whose initial data forms a representation of the canonical (anti)commutation relations. Using normal operator ordering, we construct solutions to the QIVP for both Bose and Fermi statistics (again for generic choice of parameters) and prove that these solutions are unique....
We study the classical and quantum theory of a class of nonlinear differential equations on chron... more We study the classical and quantum theory of a class of nonlinear differential equations on chronology violating spacetime models in which space consists of finitely many discrete points. Classically, in the linear and weakly nonlinear regimes (for generic choices of parameters) we prove existence and uniqueness of solutions corresponding to initial data specified before the dischronal region; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution preserves the symplectic structure. The quantum theory is approached via the quantum initial value problem (QIVP); that is, by seeking operator-valued solutions to the equation of motion with initial data representing the canonical (anti)commutation relations. Using normal operator ordering, we construct solutions to the QIVP for both Bose and Fermi statistics (again for generic choice of parameters) and prove that these solutions are unique. For models with two spatial points, the resulting evolution i...
We consider 2-dimensional cylinder spacetimes whose metrics differ from the flat Minkowskian metr... more We consider 2-dimensional cylinder spacetimes whose metrics differ from the flat Minkowskian metric within a compact region K. By choice of time orientation, these spacetimes may be regarded as either globally hyperbolic timelike cylinders or nonglobally hyperbolic spacelike cylinders. For generic metrics in our class, we classify all possible candidate quantum field algebras for massive Klein–Gordon theory which obey the F-locality condition introduced by Kay. This condition requires each point of spacetime to have an intrinsically globally hyperbolic neighbourhood N such that the commutator (in the candidate algebra) of fields smeared with test functions supported in N agrees with the value obtained in the usual construction of Klein–Gordon theory on N. By considering bisolutions to the Klein–Gordon equation, we prove that generic timelike cylinders admit a unique F-local algebra – namely the algebra obtained by the usual construction – and that generic spacelike cylinders do not ...
A new construction is presented for point interactions (PI) and generalised point interactions (G... more A new construction is presented for point interactions (PI) and generalised point interactions (GPI). The construction is an inverse scattering procedure, using integral transforms suggested by the required scattering theory. The usual class of PI in 3 dimensions (i.e. the self adjoint extensions of the Laplacian on the domain of smooth functions compactly supported away from the origin) is reconstructed. In addition a 1-parameter family of GPI models termed resonance point interactions (RPI) is constructed, labelled by M. The case M < 0 coincides with a special case of a known GPI model; the case M> 0 appears to be new. In both cases, the Hilbert space of states must be extended, for M < 0, a larger Hilbert space is required, whilst for M> 0, the Hilbert space is extended to a Pontryagin space. In the latter case, the space of physical states is identified as a positive definite invariant subspace. Complete Møller wave operators are constructed for the models considered...
Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTCs) are ... more Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTCs) are subject to a non-unitary evolution X. Recently, a prescription has been proposed, which restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the composition of two or more non-unitary evolutions. We propose an alternative method by which unitarity of the evolution may be regained, by extending X to a unitary evolution on a larger (possibly indefinite) inner product space. The proposal removes the ambiguity noted by Jacobson in assigning expectation values to observables localised in regions spacelike separated from the CTC region. We comment on the physical significance of the possible indefiniteness of the inner product introduced in our proposal. Typeset using REVTEX
arXiv: General Relativity and Quantum Cosmology, 2004
In a recent preprint, Krasnikov has claimed that to show that quantum energy inequalities (QEIs) ... more In a recent preprint, Krasnikov has claimed that to show that quantum energy inequalities (QEIs) are violated in curved spacetime situations, by considering the example of a free massless scalar field in two-dimensional de Sitter space. We show that this claim is incorrect, and based on misunderstandings of the nature of QEIs. We also prove, in general two-dimensional spacetimes, that flat spacetime QEIs give a good approximation to the curved spacetime results on sampling timescales short in comparison with natural geometric scales.
We study real linear scalar field theory on two simple non-globally hyperbolic spacetimes contain... more We study real linear scalar field theory on two simple non-globally hyperbolic spacetimes containing closed timelike curves within the framework proposed by Kay for algebraic quantum field theory on non-globally hyperbolic spacetimes. In this context, a spacetime (M,g) is said to be ‘F-quantum compatible’ with a field theory if it admits a ∗-algebra of local observables for that theory which satisfies a locality condition known as ‘F-locality’. Kay’s proposal is that, in formulating algebraic quantum field theory on (M,g), F-locality should be imposed as a necessary condition on the ∗-algebra of observables. The spacetimes studied are the 2and 4-dimensional spacelike cylinders (Minkowski space quotiented by a timelike translation). Kay has shown that the 4dimensional spacelike cylinder is F-quantum compatible with massless fields. We prove that it is also F-quantum compatible with massive fields and prove the Fquantum compatibility of the 2-dimensional spacelike cylinder with both m...
arXiv: General Relativity and Quantum Cosmology, 2005
A quantum inequality bound on the expectation value of the null-contracted stress tensor in an ar... more A quantum inequality bound on the expectation value of the null-contracted stress tensor in an arbitrary Hadamard state is used to obtain constraints on the geometries of traversable wormholes. Particular attention is given to the wormhole models of Visser, Kar, and Dadhich (VKD) and to those of Kuhfittig. These are models which use arbitrarily small amounts of exotic matter for wormhole maintenance. It is shown that macroscopic VKD models are either ruled out or severely constrained, while a recent model of Kuhfittig is shown to be, in fact, non-traversable.
arXiv: General Relativity and Quantum Cosmology, 2019
Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), whic... more Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a non-negative effective energy density (EED). The SEC ensures the timelike convergence property. However, for both classical and quantum fields, violations of the SEC can be observed even in the simplest of cases, like the Klein-Gordon field. Therefore there is a need to develop theorems with weaker restrictions, namely energy conditions averaged over an entire geodesic and weighted local averages of energy densities such as quantum energy inequalities (QEIs). We present lower bounds of the EED for both classical and quantum scalar fields allowing nonzero mass and nonminimal coupling to the scalar curvature. In the quantum case these bounds take the form of a set of state-dependent QEIs valid for the class of Hadamard states. We also discuss how these lower bounds are applied to prove Hawking-type singularity theorems asserting that, along with suff...
We re-examine the justification for the imposition of regular boundary conditions on the wavefunc... more We re-examine the justification for the imposition of regular boundary conditions on the wavefunction at the Coulomb singularity in the treatment of the hydrogen atom in non-relativistic quantum mechanics. We show that the issue of the correct boundary conditions is not independent of the physical structure of the proton. Under the physically reasonable assumption that the finite size and structure of the proton can be represented as a positive correction to the Coulomb potential, we give a justification for the regular boundary condition, which, in contrast to the usual treatments, is physically motivated and mathematically rigorous. We also describe how irregular boundary conditions can be used to model non-positive corrections to the Coulomb potential.
The quantized Dirac field is known, by a result of Fewster and Verch, to satisfy a Quantum Weak E... more The quantized Dirac field is known, by a result of Fewster and Verch, to satisfy a Quantum Weak Energy Inequality (QWEI) on its averaged energy density along time-like curves in arbitrary four-dimensional globally hyperbolic spacetimes. However, this result does not provide an explicit form for the bound. By adapting ideas from the earlier work, we give a simplified derivation of a QWEI for the Dirac field leading to an explicit bound. The bound simplifies further in the case of static curves in static spacetimes, and, in particular, coincides with a result of Fewster and Mistry in four-dimensional Minkowski spacetime. We also show that our QWEI is compatible with local covariance and derive a simple consequence.
Cosmic strings arising from GUTs can catalyse baryon decay processes with strong interaction cros... more Cosmic strings arising from GUTs can catalyse baryon decay processes with strong interaction cross sections. We examine the mechanism by which the cross section is enhanced and find that it depends strongly on the details of the distribution of gauge fields within the string core. We propose a calculational scheme for estimating wavefunction amplification factors and also a physical understanding of the nature of the enhancement process.
We present an inverse scattering construction of generalised point interactions (GPI) – point-lik... more We present an inverse scattering construction of generalised point interactions (GPI) – point-like objects with non-trivial scattering behaviour. The construction is developed for single centre S-wave GPI models with rational S-matrices, and starts from an integral transform suggested by the scattering data. The theory of unitary dilations is then applied to construct a unitary mapping between Pontryagin spaces which extend the usual position and momentum Hilbert spaces. The GPI Hamiltonian is defined as a multiplication operator on the momentum Pontryagin space and its free parameters are fixed by a physical locality requirement. We determine the spectral properties and domain of the Hamiltonian in general, and construct the resolvent and Møller wave operators thus verifying that the Hamiltonian exhibits the required scattering behaviour. The physical Hilbert space is identified. The construction is illustrated by GPI models representing the effective range approximation. For negat...
We study the classical and quantum theory of a class of nonlinear differential equations on model... more We study the classical and quantum theory of a class of nonlinear differential equations on models of chronology violating spacetimes in which space consists of only finitely many discrete points. Classically, we study the initial value problem for data specified before the dischronal region. In the linear and weakly coupled nonlinear regimes, we show (for generic choices of parameters) that solutions always exist and are unique; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution is shown to preserve the symplectic structure. The quantum theory is approached via the quantum initial value problem (QIVP); that is, by seeking operator-valued solutions to the equation of motion whose initial data forms a representation of the canonical (anti)commutation relations. Using normal operator ordering, we construct solutions to the QIVP for both Bose and Fermi statistics (again for generic choice of parameters) and prove that these solutions are unique....
We study the classical and quantum theory of a class of nonlinear differential equations on chron... more We study the classical and quantum theory of a class of nonlinear differential equations on chronology violating spacetime models in which space consists of finitely many discrete points. Classically, in the linear and weakly nonlinear regimes (for generic choices of parameters) we prove existence and uniqueness of solutions corresponding to initial data specified before the dischronal region; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution preserves the symplectic structure. The quantum theory is approached via the quantum initial value problem (QIVP); that is, by seeking operator-valued solutions to the equation of motion with initial data representing the canonical (anti)commutation relations. Using normal operator ordering, we construct solutions to the QIVP for both Bose and Fermi statistics (again for generic choice of parameters) and prove that these solutions are unique. For models with two spatial points, the resulting evolution i...
We consider 2-dimensional cylinder spacetimes whose metrics differ from the flat Minkowskian metr... more We consider 2-dimensional cylinder spacetimes whose metrics differ from the flat Minkowskian metric within a compact region K. By choice of time orientation, these spacetimes may be regarded as either globally hyperbolic timelike cylinders or nonglobally hyperbolic spacelike cylinders. For generic metrics in our class, we classify all possible candidate quantum field algebras for massive Klein–Gordon theory which obey the F-locality condition introduced by Kay. This condition requires each point of spacetime to have an intrinsically globally hyperbolic neighbourhood N such that the commutator (in the candidate algebra) of fields smeared with test functions supported in N agrees with the value obtained in the usual construction of Klein–Gordon theory on N. By considering bisolutions to the Klein–Gordon equation, we prove that generic timelike cylinders admit a unique F-local algebra – namely the algebra obtained by the usual construction – and that generic spacelike cylinders do not ...
A new construction is presented for point interactions (PI) and generalised point interactions (G... more A new construction is presented for point interactions (PI) and generalised point interactions (GPI). The construction is an inverse scattering procedure, using integral transforms suggested by the required scattering theory. The usual class of PI in 3 dimensions (i.e. the self adjoint extensions of the Laplacian on the domain of smooth functions compactly supported away from the origin) is reconstructed. In addition a 1-parameter family of GPI models termed resonance point interactions (RPI) is constructed, labelled by M. The case M < 0 coincides with a special case of a known GPI model; the case M> 0 appears to be new. In both cases, the Hilbert space of states must be extended, for M < 0, a larger Hilbert space is required, whilst for M> 0, the Hilbert space is extended to a Pontryagin space. In the latter case, the space of physical states is identified as a positive definite invariant subspace. Complete Møller wave operators are constructed for the models considered...
Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTCs) are ... more Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTCs) are subject to a non-unitary evolution X. Recently, a prescription has been proposed, which restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the composition of two or more non-unitary evolutions. We propose an alternative method by which unitarity of the evolution may be regained, by extending X to a unitary evolution on a larger (possibly indefinite) inner product space. The proposal removes the ambiguity noted by Jacobson in assigning expectation values to observables localised in regions spacelike separated from the CTC region. We comment on the physical significance of the possible indefiniteness of the inner product introduced in our proposal. Typeset using REVTEX
arXiv: General Relativity and Quantum Cosmology, 2004
In a recent preprint, Krasnikov has claimed that to show that quantum energy inequalities (QEIs) ... more In a recent preprint, Krasnikov has claimed that to show that quantum energy inequalities (QEIs) are violated in curved spacetime situations, by considering the example of a free massless scalar field in two-dimensional de Sitter space. We show that this claim is incorrect, and based on misunderstandings of the nature of QEIs. We also prove, in general two-dimensional spacetimes, that flat spacetime QEIs give a good approximation to the curved spacetime results on sampling timescales short in comparison with natural geometric scales.
We study real linear scalar field theory on two simple non-globally hyperbolic spacetimes contain... more We study real linear scalar field theory on two simple non-globally hyperbolic spacetimes containing closed timelike curves within the framework proposed by Kay for algebraic quantum field theory on non-globally hyperbolic spacetimes. In this context, a spacetime (M,g) is said to be ‘F-quantum compatible’ with a field theory if it admits a ∗-algebra of local observables for that theory which satisfies a locality condition known as ‘F-locality’. Kay’s proposal is that, in formulating algebraic quantum field theory on (M,g), F-locality should be imposed as a necessary condition on the ∗-algebra of observables. The spacetimes studied are the 2and 4-dimensional spacelike cylinders (Minkowski space quotiented by a timelike translation). Kay has shown that the 4dimensional spacelike cylinder is F-quantum compatible with massless fields. We prove that it is also F-quantum compatible with massive fields and prove the Fquantum compatibility of the 2-dimensional spacelike cylinder with both m...
arXiv: General Relativity and Quantum Cosmology, 2005
A quantum inequality bound on the expectation value of the null-contracted stress tensor in an ar... more A quantum inequality bound on the expectation value of the null-contracted stress tensor in an arbitrary Hadamard state is used to obtain constraints on the geometries of traversable wormholes. Particular attention is given to the wormhole models of Visser, Kar, and Dadhich (VKD) and to those of Kuhfittig. These are models which use arbitrarily small amounts of exotic matter for wormhole maintenance. It is shown that macroscopic VKD models are either ruled out or severely constrained, while a recent model of Kuhfittig is shown to be, in fact, non-traversable.
arXiv: General Relativity and Quantum Cosmology, 2019
Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), whic... more Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a non-negative effective energy density (EED). The SEC ensures the timelike convergence property. However, for both classical and quantum fields, violations of the SEC can be observed even in the simplest of cases, like the Klein-Gordon field. Therefore there is a need to develop theorems with weaker restrictions, namely energy conditions averaged over an entire geodesic and weighted local averages of energy densities such as quantum energy inequalities (QEIs). We present lower bounds of the EED for both classical and quantum scalar fields allowing nonzero mass and nonminimal coupling to the scalar curvature. In the quantum case these bounds take the form of a set of state-dependent QEIs valid for the class of Hadamard states. We also discuss how these lower bounds are applied to prove Hawking-type singularity theorems asserting that, along with suff...
We re-examine the justification for the imposition of regular boundary conditions on the wavefunc... more We re-examine the justification for the imposition of regular boundary conditions on the wavefunction at the Coulomb singularity in the treatment of the hydrogen atom in non-relativistic quantum mechanics. We show that the issue of the correct boundary conditions is not independent of the physical structure of the proton. Under the physically reasonable assumption that the finite size and structure of the proton can be represented as a positive correction to the Coulomb potential, we give a justification for the regular boundary condition, which, in contrast to the usual treatments, is physically motivated and mathematically rigorous. We also describe how irregular boundary conditions can be used to model non-positive corrections to the Coulomb potential.
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Papers by C. Fewster