An alternate treatment of the results of paper I is given. As in that paper, the Unruh boundary c... more An alternate treatment of the results of paper I is given. As in that paper, the Unruh boundary condition is formulated, the Unruh vacuum is defined as a state satisfying this boundary condition and the thermal character of the state is exhibited. The present work differs in that it uses the double-wedge region of the Kruskal manifold and defines and uses a precise notion of distinguished modes.
On introduit le concept de structure KMS a une particule a partir duquel la construction d'un... more On introduit le concept de structure KMS a une particule a partir duquel la construction d'une classe d'etats thermiques pour des systemes de Bose lineaires peut se reduire a la seconde quantification. Pour de telles structures, on demontre un resultat d'unicite
We define a switch function to be a function from an interval to {1,-1} with a finite number of s... more We define a switch function to be a function from an interval to {1,-1} with a finite number of sign changes. (Special cases are the Walsh functions.) By a topological argument, we prove that, given n real-valued functions, f_1, ..., f_n, in L^1[0,1], there exists a switch function, σ, with at most n sign changes that is simultaneously orthogonal to all of them in the sense that ∫_0^1 σ(t)f_i(t)dt=0, for all i = 1, ... , n. Moreover, we prove that, for each λ∈ (-1,1), there exists a unique switch function, σ, with n switches such that ∫_0^1 σ(t) p(t) dt = λ∫_0^1 p(t)dt for every real polynomial p of degree at most n-1. We also prove the same statement holds for every real even polynomial of degree at most 2n-2. Furthermore, for each of these latter results, we write down, in terms of λ and n, a degree n polynomial whose roots are the switch points of σ; we are thereby able to compute these switch functions.
The authors develop the scattering theory for classical relativistic massive scalar fields propag... more The authors develop the scattering theory for classical relativistic massive scalar fields propagating in the presence of long-range forces. More specifically they consider (a) the Klein-Gordon equation on Minkowski space with a Coulomb potential and (b) the Klein-Gordon equation for the Schwarzschild metric. In either case the results show the existence of certain wave operators which compare the full dynamics
We present a short account of our work to provide quantum electrodynamics (QED) with a product pi... more We present a short account of our work to provide quantum electrodynamics (QED) with a product picture. We aim to complement the longer exposition in a recent paper in Foundations of Physics and to help to make that work more accessible. The product picture is a formulation of QED, equivalent to standard Coulomb gauge QED, in which the Hilbert space arises as (a certain physical subspace of) a product of a Hilbert space for the electromagnetic field and a Hilbert space for charged matter (i.e., the Dirac field) and the Hamiltonian arises as the sum of an electromagnetic Hamiltonian, a charged matter Hamiltonian, and an interaction term. (The Coulomb gauge formulation of QED is not a product picture because, in it, the longitudinal part of the electromagnetic field is made out of charged matter operators.) We also recall a “Contradictory Commutator Theorem” for QED, which exposes flaws in previous attempts at temporal gauge quantization of QED, and we explain how our product picture ...
We introduce a suitable notion of quantum coherent state to describe the electrostatic field of a... more We introduce a suitable notion of quantum coherent state to describe the electrostatic field of a static classical charge distribution, thereby underpinning the author's 1998 formulae for the inner product of a pair of such states. (We also correct an incorrect factor of 4π.) Contrary to what one might expect, this is non-zero whenever the two total charges are equal, even if the charge distributions themselves are different. We then address the problem of furnishing QED with a "product structure", i.e. a formulation in which there is a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a total Hilbert space which is the tensor product of an electromagnetic Hilbert space and a charged-matter Hilbert space. (The traditional Coulomb-gauge formulation of QED doesn't have a product structure in this sense because, in it, the longitudinal part of the electric field is a function...
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entro... more In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_d+2 is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge ...
We combine and further develop ideas and techniques of Allen & Ottewill, Phys. Rev.D, 42, 2669 (1... more We combine and further develop ideas and techniques of Allen & Ottewill, Phys. Rev.D, 42, 2669 (1990) and Kay & Studer Commun. Math. Phys., 139, 103 (1991) for calculating the long range effects of cosmic string cores on classical and quantum field quantities far from an (infinitely long, straight) cosmic string. We find analytical approximations for (a) the gravity-induced ground state renormalized expectation values of φ̂^2 and T̂_μ^ν for a non-minimally coupled quantum scalar field far from a cosmic string (b) the classical electrostatic self force on a test charge far from a superconducting cosmic string. Surprisingly -- even at cosmologically large distances -- all these quantities would be very badly approximated by idealizing the string as having zero thickness and imposing regular boundary conditions; instead they are well approximated by suitably fitted strengths of logarithmic divergence at the string core. Our formula for 〈φ̂^2 〉 reproduces (with much less effort and much...
An alternate treatment of the results of paper I is given. As in that paper, the Unruh boundary c... more An alternate treatment of the results of paper I is given. As in that paper, the Unruh boundary condition is formulated, the Unruh vacuum is defined as a state satisfying this boundary condition and the thermal character of the state is exhibited. The present work differs in that it uses the double-wedge region of the Kruskal manifold and defines and uses a precise notion of distinguished modes.
On introduit le concept de structure KMS a une particule a partir duquel la construction d'un... more On introduit le concept de structure KMS a une particule a partir duquel la construction d'une classe d'etats thermiques pour des systemes de Bose lineaires peut se reduire a la seconde quantification. Pour de telles structures, on demontre un resultat d'unicite
We define a switch function to be a function from an interval to {1,-1} with a finite number of s... more We define a switch function to be a function from an interval to {1,-1} with a finite number of sign changes. (Special cases are the Walsh functions.) By a topological argument, we prove that, given n real-valued functions, f_1, ..., f_n, in L^1[0,1], there exists a switch function, σ, with at most n sign changes that is simultaneously orthogonal to all of them in the sense that ∫_0^1 σ(t)f_i(t)dt=0, for all i = 1, ... , n. Moreover, we prove that, for each λ∈ (-1,1), there exists a unique switch function, σ, with n switches such that ∫_0^1 σ(t) p(t) dt = λ∫_0^1 p(t)dt for every real polynomial p of degree at most n-1. We also prove the same statement holds for every real even polynomial of degree at most 2n-2. Furthermore, for each of these latter results, we write down, in terms of λ and n, a degree n polynomial whose roots are the switch points of σ; we are thereby able to compute these switch functions.
The authors develop the scattering theory for classical relativistic massive scalar fields propag... more The authors develop the scattering theory for classical relativistic massive scalar fields propagating in the presence of long-range forces. More specifically they consider (a) the Klein-Gordon equation on Minkowski space with a Coulomb potential and (b) the Klein-Gordon equation for the Schwarzschild metric. In either case the results show the existence of certain wave operators which compare the full dynamics
We present a short account of our work to provide quantum electrodynamics (QED) with a product pi... more We present a short account of our work to provide quantum electrodynamics (QED) with a product picture. We aim to complement the longer exposition in a recent paper in Foundations of Physics and to help to make that work more accessible. The product picture is a formulation of QED, equivalent to standard Coulomb gauge QED, in which the Hilbert space arises as (a certain physical subspace of) a product of a Hilbert space for the electromagnetic field and a Hilbert space for charged matter (i.e., the Dirac field) and the Hamiltonian arises as the sum of an electromagnetic Hamiltonian, a charged matter Hamiltonian, and an interaction term. (The Coulomb gauge formulation of QED is not a product picture because, in it, the longitudinal part of the electromagnetic field is made out of charged matter operators.) We also recall a “Contradictory Commutator Theorem” for QED, which exposes flaws in previous attempts at temporal gauge quantization of QED, and we explain how our product picture ...
We introduce a suitable notion of quantum coherent state to describe the electrostatic field of a... more We introduce a suitable notion of quantum coherent state to describe the electrostatic field of a static classical charge distribution, thereby underpinning the author's 1998 formulae for the inner product of a pair of such states. (We also correct an incorrect factor of 4π.) Contrary to what one might expect, this is non-zero whenever the two total charges are equal, even if the charge distributions themselves are different. We then address the problem of furnishing QED with a "product structure", i.e. a formulation in which there is a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a total Hilbert space which is the tensor product of an electromagnetic Hilbert space and a charged-matter Hilbert space. (The traditional Coulomb-gauge formulation of QED doesn't have a product structure in this sense because, in it, the longitudinal part of the electric field is a function...
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entro... more In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_d+2 is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge ...
We combine and further develop ideas and techniques of Allen & Ottewill, Phys. Rev.D, 42, 2669 (1... more We combine and further develop ideas and techniques of Allen & Ottewill, Phys. Rev.D, 42, 2669 (1990) and Kay & Studer Commun. Math. Phys., 139, 103 (1991) for calculating the long range effects of cosmic string cores on classical and quantum field quantities far from an (infinitely long, straight) cosmic string. We find analytical approximations for (a) the gravity-induced ground state renormalized expectation values of φ̂^2 and T̂_μ^ν for a non-minimally coupled quantum scalar field far from a cosmic string (b) the classical electrostatic self force on a test charge far from a superconducting cosmic string. Surprisingly -- even at cosmologically large distances -- all these quantities would be very badly approximated by idealizing the string as having zero thickness and imposing regular boundary conditions; instead they are well approximated by suitably fitted strengths of logarithmic divergence at the string core. Our formula for 〈φ̂^2 〉 reproduces (with much less effort and much...
Uploads
Papers by Bernard Kay