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Dorje C. Brody

Publications and source records attributed to Dorje C. Brody.

At least 91 records · Page 5Linked to original sources
arXiv2007

Unitarity, ergodicity, and quantum thermodynamics

This paper is concerned with the ergodic subspaces of the state spaces of isolated quantum systems. We prove a new ergodic theorem for closed quantum systems which shows that the equilibrium state of the system takes the form of a grand canonical density matrix involving a complete commuting set of observables including the Hamiltonian. The result obtained, which is derived for a generic finite-dimensional quantum system, shows that the equilibrium state arising from unitary evolution is always expressible in the canonical form, without the consideration of a system-bath decomposition.

quant-ph↗
arXiv2007

Note on exponential families of distributions

We show that an arbitrary probability distribution can be represented in exponential form. In physical contexts, this implies that the equilibrium distribution of any classical or quantum dynamical system is expressible in grand canonical form.

cond-mat.stat-mech↗
arXiv2007

Geometry of PT-symmetric quantum mechanics

Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue problem associated with such Hamiltonians have shown that in many cases the entire energy spectrum is real and positive and that the eigenfunctions form an orthogonal and complete basis. Furthermore, the quantum theories determined by such Hamiltonians have been shown to be consistent in the sense that the probabilities are positive and the dynamical trajectories are unitary. However, the geometrical structures that underlie quantum theories formulated in terms of such Hamiltonians have hitherto not been fully understood. This paper studies in detail the geometric properties of a Hilbert space endowed with a parity structure and analyses the characteristics of a PT-symmetric Hamiltonian and its eigenstates. A canonical relationship between a PT-symmetric operator and a Hermitian operator is established. It is shown that the quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative PT norm. The indefiniteness of the norm can be circumvented by introducing a symmetry operator C that defines a positive definite inner product by means of a CPT conjugation operation.

hep-th↗
arXiv2007

Information-Based Asset Pricing

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such "X-factor" we associate a market information process, the values of which are accessible to market agents. Each information process is a sum of two terms; one contains true information about the value of the market factor; the other represents "noise". The noise term is modelled by an independent Brownian bridge. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk-neutral measure, conditional on the information provided by the market filtration. When the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price, and the prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European call option is of the Black-Scholes-Merton type. The information-based framework also generates a natural explanation for the origin of stochastic volatility.

q-fin.PR↗
arXiv2006

On quantum microcanonical equilibrium

A quantum microcanonical postulate is proposed as a basis for the equilibrium properties of small quantum systems. Expressions for the corresponding density of states are derived, and are used to establish the existence of phase transitions for finite quantum systems. A grand microcanonical ensemble is introduced, which can be used to obtain new rigorous results in quantum statistical mechanics.

quant-ph↗
arXiv2006

Entanglement of three-qubit geometry

Geometric quantum mechanics aims to express the physical properties of quantum systems in terms of geometrical features preferentially selected in the space of pure states. Geometric characterisations are given here for systems of one, two, and three spin-1/2 particles, drawing attention to the classification of quantum states into entanglement types.

quant-ph↗
arXiv2006

PT-Symmetric Extension of the Korteweg-de Vries Equation

The Korteweg-de Vries equation u_t+uu_x+u_{xxx}=0 is PT symmetric (invariant under space-time reflection). Therefore, it can be generalized and extended into the complex domain in such a way as to preserve the PT symmetry. The result is the family of complex nonlinear wave equations u_t-iu(i u_x)^epsilon+u_{xxx}=0, where epsilon is real. The features of these equations are discussed. Special attention is given to the epsilon=3 equation, for which conservation laws are derived and solitary waves are investigated.

math-ph↗
arXiv2006

Faster than Hermitian Quantum Mechanics

Given an initial quantum state |psi_I> and a final quantum state |psi_F> in a Hilbert space, there exist Hamiltonians H under which |psi_I> evolves into |psi_F>. Consider the following quantum brachistochrone problem: Subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time tau? For Hermitian Hamiltonians tau has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, tau can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from |psi_I> to |psi_F> can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.

quant-ph↗
arXiv2006

Exactly solvable quantum state reduction models with time-dependent coupling

A closed-form solution to the energy-based stochastic Schrodinger equation with a time-dependent coupling is obtained. The solution is algebraic in character, and is expressed directly in terms of independent random data. The data consist of (i) a random variable H which has the distribution P(H=E_i) = pi_i, where pi_i is the transition probability from the initial state to the Luders state with energy E_i; and (ii) an independent P-Brownian motion, where P is the physical probability measure associated with the dynamics of the reduction process. When the coupling is time-independent, it is known that state reduction occurs asymptotically--that is to say, over an infinite time horizon. In the case of a time-dependent coupling, we show that if the magnitude of the coupling decreases sufficiently rapidly, then the energy variance will be reduced under the dynamics, but the state need not reach an energy eigenstate. This situation corresponds to the case of a ``partial'' or ``incomplete'' measurement of the energy. We also construct an example of a model where the opposite situation prevails, in which complete state reduction is achieved after the passage of a finite period of time.

quant-ph↗
arXiv2006

Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly

In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\cP\cT$-symmetric wrong-sign quartic Hamiltonian $H=\half p^2-gx^4$ has the same spectrum as the conventional Hermitian Hamiltonian $\tilde H=\half p^2+4g x^4-\sqrt{2g} x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian $\cP\cT$-symmetric Hamiltonian. This anomaly in the Hermitian form of a $\cP\cT$-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into $H$. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to $-ϕ^4$ quantum field theory in higher-dimensional space-time are discussed.

hep-th↗
arXiv2006

Combinatorics and field theory

For any given sequence of integers there exists a quantum field theory whose Feynman rules produce that sequence. An example is illustrated for the Stirling numbers. The method employed here offers a new direction in combinatorics and graph theory.

quant-ph↗
arXiv2006

Quantum phase transitions without thermodynamic limits

A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterised by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order (continuous) phase transitions at finite temperatures.

quant-ph↗
arXiv2006

On optimum Hamiltonians for state transformations

For a prescribed pair of quantum states |psi_I> and |psi_F> we establish an elementary derivation of the optimum Hamiltonian, under constraints on its eigenvalues, that generates the unitary transformation |psi_I> --> |psi_F> in the shortest duration. The derivation is geometric in character and does not rely on variational calculus.

quant-ph↗
arXiv2006

Quantum states and space-time causality

Space-time symmetries and internal quantum symmetries can be placed on equal footing in a hyperspin geometry. Four-dimensional classical space-time emerges as a result of a decoherence that disentangles the quantum and the space-time degrees of freedom. A map from the quantum space-time to classical space-time that preserves the causality relations of space-time events is necessarily a density matrix.

quant-ph↗
arXiv2005

Semiclassical analysis of a complex quartic Hamiltonian

It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\half p^2+\halfμ^2x^2-λx^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half μ^2x^2+igx^3-λx^4, whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter λ.

quant-ph↗
arXiv2005

Quantum noise and stochastic reduction

In standard nonrelativistic quantum mechanics the expectation of the energy is a conserved quantity. It is possible to extend the dynamical law associated with the evolution of a quantum state consistently to include a nonlinear stochastic component, while respecting the conservation law. According to the dynamics thus obtained, referred to as the energy-based stochastic Schrodinger equation, an arbitrary initial state collapses spontaneously to one of the energy eigenstates, thus describing the phenomenon of quantum state reduction. In this article, two such models are investigated: one that achieves state reduction in infinite time, and the other in finite time. The properties of the associated energy expectation process and the energy variance process are worked out in detail. By use of a novel application of a nonlinear filtering method, closed-form solutions--algebraic in character and involving no integration--are obtained for both these models. In each case, the solution is expressed in terms of a random variable representing the terminal energy of the system, and an independent noise process. With these solutions at hand it is possible to simulate explicitly the dynamics of the quantum states of complicated physical systems.

quant-ph↗
arXiv2005

Finite-time stochastic reduction models

A new energy-based stochastic extension of the Schrodinger equation for which the wave function collapses after the passage of a finite amount of time is proposed. An exact closed-form solution to the dynamical equation, valid for all finite-dimensional quantum systems, is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs. A time-change technique is introduced to construct a `clock' variable that relates the asymptotic and the finite-time collapse models by means of a nonlinear transformation.

quant-ph↗