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Response to a poll can be manipulated by means of a series of leading questions. We show that such phenomena cannot be explained by use of classical probability theory, whereas quantum probability theory admits a possibility of offering an explanation. Admissible transformation rules in quantum probability, however, do impose some constraints on the modelling of cognitive behaviour, which are highlighted here. Focusing on a recent poll conducted by Ipsos on a set of questions posed by Sir Humphrey Appleby in an episode of the British political satire \textit{Yes, Prime Minister}, we show that the resulting data cannot be explained quite so simply using quantum rules, although it seems not impossible., Comment: 12 pages, 1 figure

The emergence of classical behaviour in quantum theory is often ascribed to the interaction of a quantum system with its environment, which can be interpreted as environmental monitoring of the system. As a result, off-diagonal elements of the density matrix of the system are damped in the basis of a preferred observable, often taken to be the position, leading to the phenomenon of decoherence. This effect can be modelled dynamically in terms of a Lindblad equation driven by the position operator. Here the question of decoherence resulting from a monitoring of position and momentum, i.e. a phase-space measurement, by the environment is addressed. There is no standard quantum observable corresponding to the detection of phase-space points, which is forbidden by Heisenberg's uncertainty principle. This issue is addressed by use of a coherent-state-based positive operator-valued measure (POVM) for modelling phase-space monitoring by the environment. In this scheme, decoherence in phase space implies the diagonalisation of the density matrix in both position and momentum representations. This is shown to be linked to a Lindblad dynamics where position and momentum appear as two independent Lindblad operators., Comment: 5 pages, 2 figures

It is shown that if the wave function of a quantum system undergoes an arbitrary random transformation such that the diagonal elements of the density matrix in the decoherence basis associated with a preferred observable remain constant, then (i) the off-diagonal elements of the density matrix become smaller in magnitude, and (ii) the state of the system gains information about the preferred observable from its environment in the sense that the uncertainty of the observable is reduced in the transformed state. These results do not depend on the details of how the system-environment interaction generates the random state transformation, and together imply that decoherence leads in general to information gain, not information loss., Comment: 5 pages, version to appear in Physical Review Research

The probability of a given candidate winning a future election is worked out in closed form as a function of (i) the current support rates for each candidate, (ii) the relative positioning of the candidates within the political spectrum, (iii) the time left to the election, and (iv) the rate at which noisy information is revealed to the electorate from now to the election day, when there are three or more candidates. It is shown, in particular, that the optimal strategy for controlling information can be intricate and nontrivial, in contrast to a two-candidate race. A surprising finding is that for a candidate taking the centre ground in an electoral competition among a polarised electorate, certain strategies are fatal in that the resulting winning probability for that candidate vanishes identically., Comment: 17 pages, 6 figures, version to appear in the Royal Society Open Science

The cognitive state of mind concerning a range of choices to be made can effectively be modelled in terms of an element of a high-dimensional Hilbert space. The dynamics of the state of mind resulting form information acquisition is characterised by the von Neumann-L\"uders projection postulate of quantum theory. This is shown to give rise to an uncertainty-minimising dynamical behaviour equivalent to the Bayesian updating, hence providing an alternative approach to characterising the dynamics of cognitive state that is consistent with the free energy principle in brain science. The quantum formalism however goes beyond the range of applicability of classical reasoning in explaining cognitive behaviours, thus opens up new and intriguing possibilities., Comment: 15 pages

Stochastic Schr\"odinger equations that govern the dynamics of open quantum systems are given by the equations for signal processing. In particular, the Brownian motion that drives the wave function of the system does not represent noise, but provides purely the arrival of new information. Thus the wave function is guided by the optimal signal detection about the conditions of the environments under noisy observations. This behaviour is similar to biological systems that detect environmental cues, process this information, and adapt to them optimally by minimising uncertainties about the conditions of their environments. It is postulated that information-processing capability is a fundamental law of nature, and hence that models describing open quantum systems can equally be applied to biological systems to model their dynamics. For illustration, simple stochastic models are considered to capture heliotropic and gravitropic motions of plants. The advantage of such dynamical models is that it allows for the quantification of information processed by the plants. By considering the consequence of information erasure, it is argued that biological systems can process environmental signals relatively close to the Landauer limit of computation, and that loss of information must lie at the heart of ageing in biological systems., Comment: 15 pages

Signal transduction, or signal-processing capability, is a fundamental property of nature that manifests universally across systems of different scales -- from quantum behaviour to the biological. This includes the detection of environmental cues, particularly relevant to behaviours of both quantum systems and green plants, where there is neither an agent purposely transmitting the signal nor a purposefully built communication channel. To characterise the dynamical behaviours of such systems driven by signal detection followed by transduction, and thus to predict future statistics, it suffices to model the flow of information. This, in turn, provides estimates for the quantity of information processed by the system. The efficiency of biological computation can then be inferred by measuring energy consumption and subsequent heat production., Comment: 13 pages

Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schr\"odinger equation. The present paper considers energy-based stochastic extensions of the Schr\"odinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the L\'evy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for L\'evy models the decoherence rate depends on the overall scale of the energy. Thus, in L\'evy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small., Comment: 29 pages, to appear in Journal of Physics A: Mathematical and Theoretical

A modelling framework, based on the theory of signal processing, for characterising the dynamics of systems driven by the unravelling of information is outlined, and is applied to describe the process of decision making. The model input of this approach is the specification of the flow of information. This enables the representation of (i) reliable information, (ii) noise, and (iii) disinformation, in a unified framework. Because the approach is designed to characterise the dynamics of the behaviour of people, it is possible to quantify the impact of information control, including those resulting from the dissemination of disinformation. It is shown that if a decision maker assigns an exceptionally high weight on one of the alternative realities, then under the Bayesian logic their perception hardly changes in time even if evidences presented indicate that this alternative corresponds to a false reality. Thus confirmation bias need not be incompatible with Bayesian updating. By observing the role played by noise in other areas of natural sciences, where noise is used to excite the system away from false attractors, a new approach to tackle the dark forces of fake news is proposed., Comment: 23 pages, 6 figures, version to appear in Frontiers in Psychology

The dynamics of an open quantum system with balanced gain and loss is not described by a PT-symmetric Hamiltonian but rather by Lindblad operators. Nevertheless the phenomenon of PT-symmetry breaking and the impact of exceptional points can be observed in the Lindbladean dynamics. Here we briefly review the development of PT symmetry in quantum mechanics, and the characterisation of PT-symmetry breaking in open quantum systems in terms of the behaviour of the speed of evolution of the state.

Uncertainty lower bounds for parameter estimations associated with a unitary family of mixed-state density matrices are obtained by embedding the space of density matrices in the Hilbert space of square-root density matrices. In the Hilbert-space setup the measure of uncertainty is given by the skew information of the second kind, while the uncertainty lower bound is given by the Wigner-Yanase skew information associated with the conjugate observable. Higher-order corrections to the uncertainty lower bound are determined by higher-order quantum skew moments; expressions for these moments are worked out in closed form., Comment: 14 pages

The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength., Comment: 20 pages

The space of density matrices is embedded in a Euclidean space to deduce the dynamical equation satisfied by the state of an open quantum system. The Euclidean norm is used to obtain an explicit expression for the speed of the evolution of the state. The unitary contribution to the evolution speed is given by the modified skew information of the Hamiltonian, while the radial component of the evolution speed, connected to the rate at which the purity of the state changes, is shown to be determined by the modified skew information of the Lindblad operators. An open-system analogue of the quantum navigation problem is posed, and a perturbative analysis is presented to identify the amount of change on the speed. Properties of the evolution speed are examined further through example systems, showing that the evolution speed need not be a decreasing function of time., Comment: 5 pages

In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials this paper explains how to use the elementary procedure of Euler summation to sum these divergent series and thereby to make sense of the formal statement of the completeness of the formal sum that represents the reconstruction of the Hamiltonian., Comment: 5 pages, version to appear in American Journal of Physics

Complex dynamical systems driven by the unravelling of information can be modelled effectively by treating the underlying flow of information as the model input. Complicated dynamical behaviour of the system is then derived as an output. Such an information-based approach is in sharp contrast to the conventional mathematical modelling of information-driven systems whereby one attempts to come up with essentially {\it ad hoc} models for the outputs. Here, dynamics of electoral competition is modelled by the specification of the flow of information relevant to election. The seemingly random evolution of the election poll statistics are then derived as model outputs, which in turn are used to study election prediction, impact of disinformation, and the optimal strategy for information management in an election campaign., Comment: 20 pages, 5 figures

A term structure model in which the short rate is zero is developed as a candidate for a theory of cryptocurrency interest rates. The price processes of crypto discount bonds are worked out, along with expressions for the instantaneous forward rates and the prices of interest-rate derivatives. The model admits functional degrees of freedom that can be calibrated to the initial yield curve and other market data. Our analysis suggests that strict local martingales can be used for modelling the pricing kernels associated with virtual currencies based on distributed ledger technologies., Comment: 21 pages, 3 figures, version to appear in SIAM Journal on Financial Mathematics

The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s= \frac{1}{2}$. Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex $s$ for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region ${\rm Re}\,s<1$ by allowing operator-valued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by $\zeta(s)$., Comment: 8 pages, version to appear in J. Pays. A

Over the past decade it has become evident that intentional disinformation in the political context -- so-called fake news -- is a danger to democracy. However, until now there has been no clear understanding of how to define fake news, much less how to model it. This paper addresses both of these issues. A definition of fake news is given, and two approaches for the modelling of fake news and its impact in elections and referendums are introduced. The first approach, based on the idea of a representative voter, is shown to be suitable for obtaining a qualitative understanding of phenomena associated with fake news at a macroscopic level. The second approach, based on the idea of an election microstructure, describes the collective behaviour of the electorate by modelling the preferences of individual voters. It is shown through a simulation study that the mere knowledge that fake news may be in circulation goes a long way towards mitigating the impact of fake news., Comment: Version to appear as Chapter 18 in Financial Informatics: An Information-Based Approach to Asset Pricing. D. C. Brody, L. P. Hughston & A. Macrina (editors). Singapore: World Scientific Publishing Company (2022)

An elementary 'quantum-mechanical' derivation of the conditions for a system of functions to form a Reisz basis of a Hilbert space on a finite interval is presented., Comment: 4 pages, 1 figure

The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads to a challenging open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives the exact asymptotic behavior of the eigenfunction.

This paper is in response to a recent comment by Bellissard [arXiv:1704.02644] on the paper [Phys. Rev. Lett. 118, 130201 (2017)]. It is explained that the issues raised in the comment have already been discussed in the paper and do not affect the conclusions of the paper., Comment: 2 pages

We propose an energy-driven stochastic master equation for the density matrix as a dynamical model for quantum state reduction. In contrast, most previous studies of state reduction have considered stochastic extensions of the Schr\"odinger equation, and have introduced the density matrix as the expectation of the random pure projection operator associated with the evolving state vector. After working out properties of the reduction process we construct a general solution to the energy-driven stochastic master equation. The solution is obtained by the use of nonlinear filtering theory and takes the form of a completely positive stochastic map., Comment: To appear in: Collapse of the Wave Function (S. Gao, editor), Cambridge University Press (24 pages)

The classical derivation of the well-known Vasicek model for interest rates is reformulated in terms of the associated pricing kernel. An advantage of the pricing kernel method is that it allows one to generalize the construction to the L\'evy-Vasicek case, avoiding issues of market incompleteness. In the L\'evy-Vasicek model the short rate is taken in the real-world measure to be a mean-reverting process with a general one-dimensional L\'evy driver admitting exponential moments. Expressions are obtained for the L\'evy-Vasicek bond prices and interest rates, along with a formula for the return on a unit investment in the long bond, defined by $L_t = \lim_{T \rightarrow \infty} P_{tT} / P_{0T}$, where $P_{tT}$ is the price at time $t$ of a $T$-maturity discount bond. We show that the pricing kernel of a L\'evy-Vasicek model is uniformly integrable if and only if the long rate of interest is strictly positive., Comment: 23 pages

A Hamiltonian operator $\hat H$ is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of $\hat H$ is $2xp$, which is consistent with the Berry-Keating conjecture. While $\hat H$ is not Hermitian in the conventional sense, ${\rm i}{\hat H}$ is ${\cal PT}$ symmetric with a broken ${\cal PT}$ symmetry, thus allowing for the possibility that all eigenvalues of $\hat H$ are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that ${\hat H}$ is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true., Comment: 5 pages, version to appear in Phys. Rev. Lett

In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully consistent with standard quantum mechanics. This follows from the surprising fact that the much-discussed metric operator on Hilbert space is not physically observable. In particular, for closed quantum systems in finite dimensions there is no statistical test that one can perform on the outcomes of measurements to determine whether the Hamiltonian is Hermitian in the conventional sense, or PT-symmetric---the two theories are indistinguishable. Nontrivial physical effects arising as a consequence of PT symmetry are expected to be observed, nevertheless, for open quantum systems with balanced gain and loss., Comment: 6 pages, final version to appear in J Phys A

In certain circumstances tools of Riemannian geometry are sufficient to address questions arising in the more general Finslerian context. We show that one such instance presents itself in the characterisation of geodesics in Randers spaces of constant flag curvature. To achieve a simple, Riemannian derivation of this special family of curves, we exploit the connection between Randers spaces and the Zermelo problem of time-optimal navigation in the presence of background fields. The characterisation of geodesics is then proven by generalising an intuitive argument developed recently for the solution of the quantum Zermelo problem., Comment: 9 pages, published version

We introduce a family of operations in quantum mechanics that one can regard as "universal quantum measurements" (UQMs). These measurements are applicable to all finite-dimensional quantum systems and entail the specification of only a minimal amount of structure. The first class of UQM that we consider involves the specification of the initial state of the system---no further structure is brought into play. We call operations of this type "tomographic measurements", since given the statistics of the outcomes one can deduce the original state of the system. Next, we construct a disentangling operation, the outcome of which, when the procedure is applied to a general mixed state of an entangled composite system, is a disentangled product of pure constituent states. This operation exists whenever the dimension of the Hilbert space is not a prime, and can be used to model the decay of a composite system. As another example, we show how one can make a measurement of the direction along which the spin of a particle of spin s is oriented (s = 1/2, 1, ...). The required additional structure in this case involves the embedding of CP(1) as a rational curve of degree 2s in CP(2s)., Comment: 13 pages

If X and Y are independent, Y and Z are independent, and so are X and Z, one might be tempted to conclude that X, Y, and Z are independent. But it has long been known in classical probability theory that, intuitive as it may seem, this is not true in general. In quantum mechanics one can ask whether analogous statistics can emerge for configurations of particles in certain types of entangled states. The explicit construction of such states, along with the specification of suitable sets of observables that have the purported statistical properties, is not entirely straightforward. We show that an example of such a configuration arises in the case of an N-particle GHZ state, and we are able to identify a family of observables with the property that the associated measurement outcomes are independent for any choice of 2, 3, ..., N-1 of the particles, even though the measurement outcomes for all N particles are not independent. Although such states are highly entangled, the entanglement turns out to be 'fragile', i.e. the associated density matrix has the property that if one traces out the freedom associated with even a single particle, the resulting reduced density matrix is separable., Comment: 17 pages, version to appear in J. Phys. A

The quantum navigation problem of finding the time-optimal control Hamiltonian that transports a given initial state to a target state through quantum wind, that is, under the influence of external fields or potentials, is analysed. By lifting the problem from the state space to the space of unitary gates realising the required task, we are able to deduce the form of the solution to the problem by deriving a universal quantum speed limit. The expression thus obtained indicates that further simplifications of this apparently difficult problem are possible if we switch to the interaction picture of quantum mechanics. A complete solution to the navigation problem for an arbitrary quantum system is then obtained, and the behaviour of the solution is illustrated in the case of a two-level system., Comment: 7 pages, 4 figures; version to appear in New Journal of Physics

A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realises a required quantum process or task, under the influence of a prevailing `background' Hamiltonian that cannot be manipulated. When the task is to transform one quantum state into another, finding the solution in closed form to the problem is nontrivial even in the case of time-independent Hamiltonians. An elementary solution, based on trigonometric analysis, is found here when the Hilbert space dimension is two. Difficulties arising from generalisations to higher-dimensional systems are discussed., Comment: 5 pages

The solution to the problem of finding a time-optimal control Hamiltonian to generate a given unitary gate, in an environment in which there exists an uncontrollable ambient Hamiltonian (e.g., a background field), is obtained. In the classical context, finding the time-optimal way to steer a ship in the presence of a background wind or current is known as the Zermelo navigation problem, whose solution can be obtained by working out geodesic curves on a space equipped with a Randers metric. The solution to the quantum Zermelo problem, which is shown here to take a remarkably simple form, is likewise obtained by finding explicit solutions to the geodesic equations of motion associated with a Randers metric on the space of unitary operators. The result reveals that the optimal control in a sense `goes along with the wind'., Comment: 5 pages, 1 figure; version to appear in Phys. Rev. Lett

A model for the thermodynamics of a quantum heat bath is introduced. Under the assumption that the bath molecules have finitely many degrees of freedom and are weakly interacting, we present a general derivation of the equation of state of the bath in the thermodynamic limit. The relation between the temperature and the specific energy of the bath depends on (i) the spectral properties of the molecules, and (ii) the choice of probability measure on the state space of a representative molecule. The results obtained illustrate how the microscopic features of the molecular constituents determine the macroscopic thermodynamic properties of the bath. Our findings can thus be used to compare the merits of different hypotheses for the equilibrium states of quantum systems. Two examples of plausible choices for the probability measure are considered in detail., Comment: 26 pages; version to appear in J. Phys. A

The Wiener chaos approach to interest rate modelling arises from the observation that the pricing kernel admits a representation in terms of the conditional variance of a square-integrable random variable, which in turn admits a chaos expansion. When the expansion coefficients factorise into multiple copies of a single function, then the resulting interest rate model is called coherent, whereas a generic interest rate model will necessarily be incoherent. Coherent representations are nevertheless of fundamental importance because incoherent ones can always be expressed as a linear superposition of coherent elements. This property is exploited to derive general expressions for the pricing kernel and the associated bond price and short rate processes in the case of an n-th order chaos model for each $n$. The pricing formulae for bond options and swaptions are obtained in closed forms for a number of examples. An explicit representation for the pricing kernel of a generic---incoherent---model is then obtained by use of the underlying coherent elements. Finally, finite-dimensional realisations of the coherent chaos models are investigated in detail. In particular, it is shown that a class of highly tractable models can be constructed having the characteristic feature that the discount bond price is given by a piecewise flat (simple) process., Comment: 26 pages

The Hermiticity condition in quantum mechanics required for the characterisation of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert space dimensionality is finite. Specifically, characterisations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems., Comment: 26 pages, final version to appear in J. Phys. A

This paper introduces an information-based model for the pricing of storable commodities such as crude oil and natural gas. The model uses the concept of market information about future supply and demand as a basis for valuation. Physical ownership of a commodity is taken to provide a stream of convenience dividends equivalent to a continuous cash flow. The market filtration is assumed to be generated jointly by (i) current and past levels of the dividend rate, and (ii) partial information concerning the future of the dividend flow. The price of a commodity is the expectation under a suitable pricing measure of the totality of the discounted risk-adjusted future convenience dividend, conditional on the information provided by the market filtration. In the situation where the dividend rate is modelled by an Ornstein-Uhlenbeck process, the prices of options on commodities can be derived in closed form. The approach that we present can be applied to other assets that yield potentially negative effective cash flows, such as real estate, factories, refineries, mines, and power generating plants., Comment: Version to appear as Chapter 17 in Financial Informatics: An Information-Based Approach to Asset Pricing. D. C. Brody, L. P. Hughston & A. Macrina (editors). Singapore: World Scientific Publishing Company (2022)

Information geometry provides a tool to systematically investigate parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric., Comment: 17 pages, invited contribution for a Special Issue on "Distance in Information and Statistical Physics"

The well-known theorem of Dybvig, Ingersoll and Ross shows that the long zero-coupon rate can never fall. This result, which, although undoubtedly correct, has been regarded by many as surprising, stems from the implicit assumption that the long-term discount function has an exponential tail. We revisit the problem in the setting of modern interest rate theory, and show that if the long "simple" interest rate (or Libor rate) is finite, then this rate (unlike the zero-coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators. New interest rate models are constructed, under this hypothesis and certain generalizations thereof, that illustrate explicitly the good asymptotic behaviour of the resulting discount bond systems. The conditions necessary for the existence of such "hyperbolic" and "generalized hyperbolic" long rates are those of so-called social discounting, which allow for long-term cash flows to be treated as broadly "just as important" as those of the short or medium term. As a consequence, we are able to provide a consistent arbitrage-free valuation framework for the cost-benefit analysis and risk management of long-term social projects, such as those associated with sustainable energy, resource conservation, and climate change., Comment: 30 pages, version to appear in Mathematical Finance

Wigner's theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner's theorem: a holomorphically projective (complex geodesic-curves preserving) transformation on a quantum state space must be generated by a Hamiltonian that is not necessarily Hermitian., Comment: 13 pages, no figure

When investors have heterogeneous attitudes towards risk, it is reasonable to assume that each investor has a pricing kernel, and that these individual pricing kernels are aggregated to form a market pricing kernel. The various investors are then buyers or sellers depending on how their individual pricing kernels compare to that of the market. In Brownian-based models, we can represent such heterogeneous attitudes by letting the market price of risk be a random variable, the distribution of which corresponds to the variability of attitude across the market. If the flow of market information is determined by the movements of prices, then neither the Brownian driver nor the market price of risk are directly visible: the filtration is generated by an "information process" given by a combination of the two. We show that the market pricing kernel is then given by the harmonic mean of the individual pricing kernels associated with the various market participants. Remarkably, with an appropriate definition of L\'evy information one draws the same conclusion in the case when asset prices can jump. As a consequence we are led to a rather general scheme for the management of investments in heterogeneous markets subject to jump risk., Comment: Version to appear in: Proceedings of the Royal Society London A