a topos foundation for theories of physics iv categories of systems this paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics the motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos the previous papers in this series are concerned with implementing this program for a single system in the present paper we turn to considering a collection of systems in particular we are interested in the relation between the topos representation for a composite system and the representations for its constituents we also study this problem for the disjoint sum of two systems our approach to these matters is to construct a category of systems and to find a topos representation of the entire category

weak values as context dependent values of observables and born s rule we characterize a value of an observable by a sum rule for generally non commuting observables and a product rule when restricted to a maximal commuting subalgebra of observables together with the requirement that the value is unity for the projection operator of the prepared state and the values are zero for the projection operators of the states which are orthogonal to the prepared state the crucial requirement is that the expectation value and the variance of an observable should be independent of the way of measurement i e the choice of the maximal commuting subalgebra of observables we shall call the value a contextual value we show that the contextual value of an observable coincides with the weak value advocated by aharonov and his colleagues by demanding the consistency of quantum mechanics with kolmogorov s measure theory of probability this also gives a derivation of born s rule which is one of the axioms of conventional quantum mechanics

various pointer states approaches to polar modular values we theoretically analyze the polar decomposition for quantum modular values under various pointer states approaches we consider both the finite dimensional discrete pointer state and continuous pointer state cases in contrast to that a weak value of an observable is usually divided into its real and imaginary parts here we show that separation from the modulus and phase is necessary to a modular value we show that the modulus of the modular value is related to the pointer post selection conditional probability and the phase of the modular value is connected to the summation of a geometric phase and an intrinsic phase we also discuss a relationship between the modulus and phase and therein the derivative of the phase is related to the derivative of the logarithm of the modulus via a berry simon like connection which is in the form of a weak value as a consequence the modulus phase relation allows us to obtain these polar components whenever the connection is specified one of the possible appli

development of a boundary element method for time dependent planar thermoelasticity abstract a new boundary element method is developed for two dimensional quasistatic thermoelasticity this time domain formulation involves only surface quantities consequently volume discretization is completely eliminated and the method becomes a viable alternative to the usual finite element approaches after presenting a brief overview of the governing equations boundary integral equations for coupled quasistatic thermoelasticity are derived by starting with existing fundamental solutions along with an appropriate reciprocal theorem details of a general purpose numerical implementation are then discussed next boundary element methods for the two more practical theories uncoupled quasistatic and steady state thermoelasticity are developed directly from limiting forms of the coupled formulation several numerical examples are provided to illustrate the validity and attractiveness of the boundary element approach for this entire class of problems

inviscid incompressible limits for rotating fluids abstract we consider the inviscid incompressible limits of the rotating compressible navier stokes system for a barotropic fluid we show that the limit system is represented by the rotating incompressible euler equation on the whole space

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

two dimensional elastodynamic displacement discontinuity method a time domain elastodynamic displacement discontinuity method is presented for analysing two dimensional problems in solid mechanics linear continuous in time and piecewise linear in space interpolation functions are assumed for the displacement discontinuities causal and analytical integrations are used throughout the resulting numerical algorithms are time marching and implicit the displacement discontinuity method can be used to model a class of traction boundary value problems where the unknowns are the transient shear and normal discontinuities in displacement across the surface of a mathematical crack in an infinite isotropic and homogeneous linear elastic medium

non oscillating solutions to uncoupled ermakov systems and the semi classical limit the amplitude phase formulation of the schrodinger equation is investigated within the context of uncoupled ermakov systems whereby the amplitude function is given by the auxiliary non linear equation the classical limit of the amplitude and phase functions is analysed by setting up a semi classical ermakov system in this limit it is shown that classical quantities such as the classical probability amplitude and the reduced action are obtained only when the semi classical amplitude and the accumulated phase are non oscillating functions respectively of the space and energy variables conversely among the infinitely many arbitrary exact quantum amplitude and phase functions corresponding to a given wavefunction only the non oscillating ones yield classical quantities in the limit 0

weak values with decoherence the weak value of an observable is experimentally accessible by weak measurements as theoretically analyzed by aharonov et al and recently experimentally demonstrated we introduce a w operator associated with the weak values and give a general framework of quantum operations to the w operator in parallel with the kraus representation of the completely positive map for the density operator the decoherence effect is also investigated in terms of the weak measurement by a shift of a probe wavefunction of continuous variable as an application we demonstrate how the geometric phase is affected by the bit flip noise

pre and post selection weak values and contextuality by analysing the concept of contextuality bell kochen specker in terms of pre and post selection it is possible to assign definite values to observables in a new and surprising way physical reasons are presented for restrictions on these assignments when measurements are performed which do not disturb the pre and post selection i e weak measurements then novel experimental aspects of contextuality can be demonstrated we also prove that every pps paradox with definite predictions directly implies quantum contextuality which is introduced as the analogue of contextuality at the level of quantum mechanics rather than at the level of hidden variable theories finally we argue that certain results of these measurements e g eccentric weak values outside the eigenvalue spectrum cannot be explained by a classical like hidden variable theory

complete description of a quantum system at a given time a generalization of the description of a quantum system in the time interval between two measurements is presented a new concept of a generalized state is introduced generalized states yield a complete description of a quantum system when information about the system is available both from the past snd from the future the formalism of generalized states provides a natural language for describing many peculiar situations in particular situations in which one can ascertain the result of a measurement of any one of several non commuting variables are analysed weak measurements on quantum systems described by generalized states are discussed the relation belween weak and strong measurements is investigated

geometric phase and modulus relations for probability amplitudes as functions on complex parameter spaces we investigate general differential relations connecting the respective behaviors of the phase and modulus of probability amplitudes of the form f where f is a fixed state in hilbert space and is a variable state treated as a section of a u 1 bundle over a complex subspace of the corresponding ray space r cpn amplitude functions on such holomorphic line bundles while not strictly holomorphic nevertheless satisfy generalized cauchy riemann conditions involving the u 1 berry simon connection on the parameter space these conditions entail invertible relations between the gradients of the phase and modulus therefore allowing for the reconstruction of the phase from the modulus or vice versa and other conditions on the behavior of either polar component of the amplitude as a special case we consider amplitude functions valued on the space of pure states the ray space r cpn where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with t

cumulants and the moment algebra tools for analyzing weak measurements recently it has been shown that cumulants significantly simplify the analysis of multipartite weak measurements here we consider the mathematical structure that underlies this and find that it can be formulated in terms of what we call the moment algebra apart from resulting in simpler proofs the flexibility of this structure allows generalizations of the original results to a number of weak measurement scenarios including one where the weakly interacting pointers reach thermal equilibrium with the probed system

two and three dimensional transient heat conduction and thermoelastic analyses by bem via efficient time convolution abstract efficient re integration based convolution algorithms are presented for a large class of problems involving transient two and three dimensional heat conduction and uncoupled thermoelastic analyses based on the earlier works of dargush and banerjee g f dargush p k banerjee application of boundary element method to transient heat conduction int j numer methods engrg 31 1991 1231 1248 g f dargush p k banerjee development of a boundary element method for time dependent planar thermoelasticity int j solids struct 25 1989 999 1021 the main motivation behind the current development is the fact that the storage based convolution approach used in earlier developments was not found to be very suitable for solving large scale practical problems because of substantial computer disk space requirement and the inability to handle time dependent boundary conditions in this paper a re integration based full convolution as well as an efficient and simplified re integration based fast convolution algorithms are presented in which all of the time convolution steps have been calculated by preserving the energy balance as expressed in the convolution while assuming equivalent values of variables to represent the total effect the accuracy and efficiency of the developed algorithms are demonstrated via numerical examples

ermakov systems velocity dependent potentials and nonlinear superposition we derive new ermakov systems with velocity dependent potentials the extended ermakov system presented contains all known one dimensional cases and many new systems these ermakov systems lead to a nonlinear superposition law for the solutions

time dependent axisymmetric thermoelastic boundary element analysis a boundary element method is developed for problems of quasistatic axisymmetric thermoelasticity unlike previous approaches this new time domain formulation is written exclusively in terms of surface quantities thereby eliminating the need for volume discretization furthermore since the exact three dimensional infinite space fundamental solutions are employed very accurate solutions are obtainable including the determination of surface stresses in the integral formulation the fundamental solutions are separated into steady state and transient components the steady state portion which contains all of the singularities is integrated analytically in the circumferential direction yielding the familiar axisymmetric kernels the remaining non singular transient integrands are treated by a combination of analytical and numerical quadrature the method is implemented in a general purpose boundary element code which includes multiregion capability along with higher order conforming surface elements several numerical examples are provided to illustrate the validity of the formulation and the attractiveness of this approach for practical engineenring analysis

solution of the schrodinger equation for time dependent 1d harmonic oscillators using the orthogonal functions invariant an extension of the classical orthogonal functions invariant to the quantum domain is presented this invariant is expressed in terms of the hamiltonian unitary transformations which involve the auxiliary function of this quantum invariant are used to solve the time dependent schrodinger equation for a harmonic oscillator with time dependent parameter the solution thus obtained is in agreement with the results derived using other methods which invoke the lewis invariant in their procedures

boundary element methods in three dimensional thermoelasticity abstract a boundary element method is developed for the solution of time dependent problems in three dimensional thermoelasticity the time domain boundary only formulation represents the first of its kind for quasistatic analysis which by definition considers transient heat conduction but ignores the effects of inertia by eliminating the need for volume discretization the method becomes an attractive alternative to finite element analysis for this class of problems additionally because an exact green s function is used in the interior steep thermal gradients can be captured much more readily than with standard domain based methods the presentation includes details of the fundamental solution a derivation of the boundary integral equations and an overview of a general purpose numerical implementation this implementation permits the solution of large multiregion problems with arbitrary geometry and boundary conditions several examples are included to validate the proposed method as well as to highlight its usefulness

on the energy inequality for weak solutions to the navier stokes equations of compressible fluids on unbounded domains abstract we consider the navier stokes equations of compressible isentropic viscous fluids on an unbounded three dimensional domain with a compact lipschitz boundary under the condition that the total mass of the fluid is finite we show the existence of globally defined weak solutions satisfying the energy inequality in differential form

quantum physics and human language human languages employ constructions that tacitly assume specific properties of the limited range of phenomena they evolved to describe these assumed properties are true features of that limited context but may not be general or precise properties of all the physical situations allowed by fundamental physics in brief human languages contain excess baggage that must be qualified discarded or otherwise reformed to give a clear account in the context of fundamental physics of even the everyday phenomena that the languages evolved to describe the surest route to clarity is to express the constructions of human languages in the language of fundamental physical theory not the other way around these ideas are illustrated by an analysis of the verb to happen and the word reality in special relativity and the modern quantum mechanics of closed systems

2d and 3d transient heat conduction analysis by bem via particular integrals abstract particular integral formulations are presented for 2d and 3d transient potential flow heat conduction analysis the results of the analysis are compared with an alternative formulation developed using the volume integral conversion approach although the mathematical foundation of the two methods are different both formulations are shown to produce almost identical results for the particular integral formulation the steady state heat conduction equation is used as the complementary solution and two global shape functions gsfs are considered to approximate the transient term of the heat conduction equation the numerical results for three example problems are given and compared with their analytical solutions

classical three box paradox a simple classical probabilistic system a simple card game classically exemplifies aharonov and vaidman s three box paradox 1991 j phys a math gen 24 2315 implying that the three box example is neither quantal nor a paradox and leaving one with less difficulty to busy the interpreters of quantum mechanics an ambiguity in the usual expression of the retrodiction formula is shown to have misled albert et al 1985 phys rev lett 54 5 to a result not in fact curious the discussion illustrates how to avoid this ambiguity

three box paradox and cheshire cat grin the case of spin 1 atoms in this work we propose a definite theoretical implementation of the three box paradox a scheme in which a single quantum particle appears to be present with certainty in two separate boxes with spin 1 atoms we further show how our setup can give rise to a cheshire cat grin type of situation in which an atom can apparently be found with certainty in one of the boxes while one of its properties the angular momentum projection along a specifically chosen axis appears to be in a different box the significance of our findings is discussed relative to the status of the properties of a system obtained from weak measurements

quantum measuring processes of continuous observables the purpose of this paper is to provide a basis of theory of measurements of continuous observables we generalize von neumann s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables and show how every such measuring process determines the state change caused by the measurement we establish a one to one correspondence between completely positive instruments in the sense of davies and lewis and the state changes determined by the measuring processes we also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis a proof of the wigner araki yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework we also discuss the implication of these results for the recent results due to srinivas and due to mercer on measurements of continuous observables

weak measurements of trajectories in quantum systems classical bohmian and sum over paths weak values obtained from weak measurements attempt to describe the properties of a quantum system as it evolves from an initial to a final state without practically altering this evolution trajectories can be defined from weak measurements of the position or inferred from weak values of the momentum operator the former can be connected to the asymptotic form of the feynman propagator and the latter to bohmian trajectories employing a time dependent oscillator as a model this work analyzes to what extent weak measurements can shed light on the underlying dynamics of a quantum system expressed in terms of trajectories in particular by comparing the two approaches

time domain transient elastodynamic analysis of 3 d solids by bem the numerical implementation of the direct boundary element formulation for time domain transient analysis of three dimensional solids is presented in a most general and complete manner the present formulation employs the space and time dependent fundamental solution stokes solution and graffi s dynamic reciprocal theorem to derive the boundary integral equations in the time domain a time stepping scheme is then used to solve the boundary initial value problem by marching forward in time higher order shape functions are used to approximate the field quantities in space as well as in time and a combination of analytical time integration and numerical spatial integration integration is carried out to form a system of linear equations at the end of each time step these equations are solved to obtain the unknown field quantities at that time finally the accuracy and reliability of this algorithm is demonstrated by solving a number of example problems and comparing the results against the available analytical and numerical solution

two and three dimensional transient thermoelastic analysis by bem via particular integrals abstract particular integral formulations are presented for two and three dimensional transient uncoupled thermoelastic analysis these formulations differ from previous particular integral formulations in that the equation of uncoupled thermoelasticity including the heat conduction equation is fully satisfied to obtain particular integrals the equation of the steady state thermoelasticity is now used as the complementary solution and two global shape functions are considered to approximate the transient term of the heat conduction equation so that two sets of particular integrals could be derived the numerical results for both sets of particular integrals are given for three example problems and compared with their analytical solutions

analysis of symmetric domains in advanced applications with the boundary element method the strategy in the boundary element method for the analysis of symmetric domains that does not require the modelling of contours corresponding to the axes of symmetry is investigated for a number of advanced applications these applications include treatment of domain loadings two dimensional time domain transient elastodynamics and the analysis of probabilistic problems in elastostatics with a random geometric configuration both symmetric as well as arbitrary loadings acting on the symmetric objects are considered a number of case studies are presented to provide comparisons of computer memory and cpu time requirements for the analyses of the entire object versus the analyses of only the symmetric portion of the object the numerical data presented clearly demonstrates the advantages of developing efficient symmetric formulations for advanced applications

low mach number limit of viscous compressible flows in the whole space this paper is devoted to the low mach number limit of weak solutions to the compressible navier stokes equations for isentropic fluids in the whole space r d d 2 or 3 this problem was investigated by p l lions and n masmoudi we present here a different approach based upon strichartz s estimates for the linear wave equation in the inviscid case which improves the convergence result and simplifies the proof we prove that the velocity field is strongly compact and converges to a global weak solution of the incompressible navier stokes equations

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

strichartz estimates for the euler equations in the rotational framework abstract we consider the initial value problems of the incompressible euler equations in the rotational framework we obtain the optimal range of the strichartz estimate for the linear group associated with the coriolis forces as an application we prove that the lifespan of the solution can be taken arbitrarily large provided that the speed of rotation is sufficiently high

thermal stress analysis of axisymmetric bodies via the boundary element method abstract a time dependent boundary element method bem for evaluating thermal stresses in an axisymmetric thermoelastic body has been developed previous bem work on axisymmetric thermoelasticity is extended to permit the calculation of stresses at both boundary and interior points however the new formulation still requires only surface discretization and is capable of very accurate evaluation of stress the boundary displacement integral equation is successively differentiated to obtain the corresponding stress and strain integral equations explicit expressions for the steady state axisymmetric fundamental solutions are derived in this process the non steady components of the integrands are obtained directly from the corresponding three dimensional counterparts the numerical implementation of this formulation is carried out within a general purpose bem computer code and several numerical examples are presented to illustrate the accuracy and applicability of the method

strange weak values we develop a formal theory of the weak values with emphasis on the consistency conditions and a probabilistic interpretation in the counter factual processes we present the condition for the choice of the post selected state to give a negative weak value of a given projection operator and strange values of an observable in general the general framework is applied to hardy s paradox and the spin 1 2 system to explicitly address the issues of counter factuality and strange weak values the counter factual arguments which characterize the paradox specify the pre selected state and a complete set of the post selected states clarifies how the strange weak values emerge

quantum non markovian processes break conditional past future independence for classical markovian stochastic systems past and future events become statistically independent when conditioned to a given state at the present time memory non markovian effects break this condition inducing a nonvanishing conditional past future correlation here this classical memory indicator is extended to a quantum regime which provides an operational definition of quantum non markovianity based on a minimal set of three time ordered quantum system measurements and postselection the detection of memory effects through the measurement scheme is univocally related to departures from born markov and white noise approximations in quantum and classical environments respectively

decoherent histories and realism we reconsider the decoherent histories approach to quantum mechanics and analyze some problems related to its interpretation which we believe have not been adequately clarified by its proponents we put forward some assumptions which in our opinion are necessary for a realistic interpretation of the probabilities that the formalism attaches to decoherent histories we prove that such assumptions unless one limits the set of the decoherent families which can be taken into account lead to a logical contradiction the line of reasoning we follow is conceptually different from other arguments which have been presented and which have been rejected by the supporters of the decoherent histories approach the conclusion is that the decoherent histories approach to be considered as an interesting realistic alternative to the orthodox interpretation of quantum mechanics requires the identification of a mathematically precise criterion to characterize an appropriate set of decoherent families which does not give rise to any problem

quantum theory of probability and decisions the probabilistic predictions of quantum theory are conventionally obtained from a special probabilistic axiom but that is unnecessary because all the practical consequences of such predictions fo

topological sensitivity for 3d elastodynamic and acoustic inverse scattering in the time domain building on previous work for 3d inverse scattering in the frequency domain this article develops the concept of topological derivative for 3d elastic and acoustic wave imaging of media of arbitrary geometry using data in the time domain the topological derivative which quantifies the sensitivity of the cost functional associated with the inverse scattering problem due to the creation at a specified location of an infinitesimal hole for the elastodynamic case or rigid inclusion for the acoustic case is found to be expressed in terms of the time convolution of the free field and a supplementary adjoint field the derivation of the topological derivative follows the generic pattern proposed in previous studies which is transposable to a variety of other physical problems a numerical example where the featured cost function is defined in terms of synthetic data arising from the scattering of plane acoustic waves by a rigid spherical inclusion illustrates the utility of the topological derivative concept for defect identification using time varying data