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Dorje C. Brody

Publications and source records attributed to Dorje C. Brody.

107 records · Page 6Linked to original sources
arXiv2005

Twistor cosmology and quantum space-time

The purpose of this paper is to present a model of a quantum space-time in which the global symmetries of space-time are unified in a coherent manner with the internal symmetries associated with the state space of quantum-mechanics. If we take into account the fact that these distinct families of symmetries should in some sense merge and become essentially indistinguishable in the unified regime, our framework may provide an approximate description of or elementary model for the structure of the universe at early times.

hep-th↗
arXiv2004

Theory of Quantum Space-Time

A generalised equivalence principle is put forward according to which space-time symmetries and internal quantum symmetries are indistinguishable before symmetry breaking. Based on this principle, a higher-dimensional extension of Minkowski space is proposed and its properties examined. In this scheme the structure of space-time is intrinsically quantum mechanical. It is shown that the causal geometry of such a quantum space-time possesses a rich hierarchical structure. The natural extension of the Poincare group to quantum space-time is investigated. In particular, we prove that the symmetry group of a quantum space-time is generated in general by a system of irreducible Killing tensors. When the symmetries of a quantum space-time are spontaneously broken, then the points of the quantum space-time can be interpreted as space-time valued operators. The generic point of a quantum space-time in the broken symmetry phase thus becomes a Minkowski space-time valued operator. Classical space-time emerges as a map from quantum space-time to Minkowski space. It is shown that the general such map satisfying appropriate causality-preserving conditions ensuring linearity and Poincare invariance is necessarily a density matrix.

gr-qc↗
arXiv2004

Scalar Quantum Field Theory with Cubic Interaction

In this paper it is shown that an i phi^3 field theory is a physically acceptable field theory model (the spectrum is positive and the theory is unitary). The demonstration rests on the perturbative construction of a linear operator C, which is needed to define the Hilbert space inner product. The C operator is a new, time-independent observable in PT-symmetric quantum field theory.

hep-th↗
arXiv2004

Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction

It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate C is cumbersome in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method is used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory.

hep-th↗
arXiv2003

Shapes of Quantum States

The shape space of k labelled points on a plane can be identified with the space of pure quantum states of dimension k-2. Hence, the machinery of quantum mechanics can be applied to the statistical analysis of planar configurations of points. Various correspondences between point configurations and quantum states, such as linear superposition as well as unitary and stochastic evolution of shapes, are illustrated. In particular, a complete characterisation of shape eigenstates for an arbitrary number of points is given in terms of cyclotomic equations.

quant-ph↗
arXiv2003

Must a Hamiltonian be Hermitian?

A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data. One would think that a quantum theory based on a non-Hermitian Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is possible to use a previously unnoticed physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite. This construction is general and works for any PT-symmetric Hamiltonian. The dynamics is governed by unitary time evolution. This formulation does not conflict with the requirements of conventional quantum mechanics. There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.

hep-th↗
arXiv2003

Quantised Three-Pillar Problem

This paper examines the quantum mechanical system that arises when one quantises a classical mechanical configuration described by an underdetermined system of equations. Specifically, we consider the well-known problem in classical mechanics in which a beam is supported by three identical rigid pillars. For this problem it is not possible to calculate uniquely the forces supplied by each pillar. However, if the pillars are replaced by springs, then the forces are uniquely determined. The three-pillar problem and its associated indeterminacy is recovered in the limit as the spring constant tends to infinity. In this paper the spring version of the problem is quantised as a constrained dynamical system. It is then shown that as the spring constant becomes large, the quantum analog of the ambiguity reemerges as a kind of quantum anomaly.

quant-ph↗
arXiv2003

Elementary Derivation for Passage Times

When a quantum system undergoes unitary evolution in accordance with a prescribed Hamiltonian, there is a class of states |psi> such that, after the passage of a certain time, |psi> is transformed into a state orthogonal to itself. The shortest time for which this can occur, for a given system, is called the passage time. We provide an elementary derivation of the passage time, and demonstrate that the known lower bound, due to Fleming, is typically attained, except for special cases in which the energy spectra have particularly simple structures. It is also shown, using a geodesic argument, that the passage times for these exceptional cases are necessarily larger than the Fleming bound. The analysis is extended to passage times for initially mixed states.

quant-ph↗
arXiv2002

Complex Extension of Quantum Mechanics

It is shown that the standard formulation of quantum mechanics in terms of Hermitian Hamiltonians is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct to explain experimental data. One might expect that a quantum theory based on a non-Hermitian Hamiltonian would violate unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed physical symmetry C of the Hamiltonian. Using C, an inner product is constructed whose associated norm is positive definite. This construction is completely general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalisation of it.

quant-ph↗
arXiv2002

Efficient Simulation of Quantum State Reduction

The energy-based stochastic extension of the Schrodinger equation is a rather special nonlinear stochastic differential equation on Hilbert space, involving a single free parameter, that has been shown to be very useful for modelling the phenomenon of quantum state reduction. Here we construct a general closed form solution to this equation, for any given initial condition, in terms of a random variable representing the terminal value of the energy and an independent Brownian motion. The solution is essentially algebraic in character, involving no integration, and is thus suitable as a basis for efficient simulation studies of state reduction in complex systems.

quant-ph↗
arXiv2001

Entanglement Induced Phase Transitions

Starting from the canonical ensemble over the space of pure quantum states, we obtain an integral representation for the partition function. This is used to calculate the magnetisation of a system of N spin-1/2 particles. The results suggest the existence of a new type of first order phase transition that occurs at zero temperature in the absence of spin-spin interactions. The transition arises as a consequence of quantum entanglement. The effects of internal interactions are analysed and the behaviour of the magnetic susceptibility for a small number of interacting spins is determined.

quant-ph↗
arXiv1999

Geometric Quantum Mechanics

The manifold of pure quantum states is a complex projective space endowed with the unitary-invariant geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given quantum system can be represented by specific geometrical features that are selected and preferentially identified in this complex manifold. Here we construct a number of examples of such geometrical features as they arise in the state spaces for spin-1/2, spin-1, and spin-3/2 systems, and for pairs of spin-1/2 systems. A study is undertaken on the geometry of entangled states, and a natural measure is assigned to the degree of entanglement of a given state for a general multi-particle system. The properties of this measure are analysed for the entangled states of a pair of spin-1/2 particles. With the specification of a quantum Hamiltonian, the resulting Schrodinger trajectory induces a Killing field, which is quasiergodic on a toroidal subspace of the energy surface. When the dynamical trajectory is lifted orthogonally to Hilbert space, it induces a geometric phase shift on the wave function. The uncertainty of an observable in a given state is the length of the gradient vector of the level surface of the expectation of the observable in that state, a fact that allows us to calculate higher order corrections to the Heisenberg relations. A general mixed state is determined by a probability density function on the state space, for which the associated first moment is the density matrix. The advantage of a general state is in its applicability in various attempts to go beyond the standard quantum theory.

quant-ph↗
arXiv1997

Statistical Geometry in Quantum Mechanics

A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the Hilbert space H. By consideration of the square-root density function we can regard M as a submanifold of the unit sphere in H. Therefore, H embodies the `state space' of the probability distributions, and the geometry of M can be described in terms of the embedding of in H. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramer-Rao and Bhattacharyya inequalities. The statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.

gr-qc↗
arXiv1997

Geometrisation of Statistical Mechanics

Classical and quantum statistical mechanics are cast here in the language of projective geometry to provide a unified geometrical framework for statistical physics. After reviewing the Hilbert space formulation of classical statistical thermodynamics, we introduce projective geometry as a basis for analysing probabilistic aspects of statistical physics. In particular, the specification of a canonical polarity on $RP^{n}$ induces a Riemannian metric on the state space of statistical mechanics. In the case of the canonical ensemble, we show that equilibrium thermal states are determined by the Hamiltonian gradient flow with respect to this metric. This flow is concisely characterised by the fact that it induces a projective automorphism on the state manifold. The measurement problem for thermal systems is studied by the introduction of the concept of a random state. The general methodology is then extended to include the quantum mechanical dynamics of equilibrium thermal states. In this case the relevant state space is complex projective space, here regarded as a real manifold endowed with the natural Fubini-Study metric. A distinguishing feature of quantum thermal dynamics is the inherent multiplicity of thermal trajectories in the state space, associated with the nonuniqueness of the infinite temperature state. We are then led to formulate a geometric characterisation of the standard KMS-relation often considered in the context of $C^{*}$-algebras. The example of a quantum spin one-half particle in heat bath is studied in detail.

gr-qc↗
arXiv1997

Generalised Heisenberg Relations

A geometric framework for quantum statistical estimation is used to establish a series of higher order corrections to the Heisenberg uncertainty relations associated with pairs of canonically conjugate variables. These corrections can be expressed in terms of linear combinations of higher order cumulants for the distributions, and thus vanish for Gaussian distributions. Estimates for typical numerical values arising from these corrections are indicated in the case of a gamma distribution.

quant-ph↗
arXiv1997

Geometry of Thermodynamic States

A novel geometric formalism for statistical estimation is applied here to the canonical distribution of classical statistical mechanics. In this scheme thermodynamic states, or equivalently, statistical mechanical states, can be characterised concisely in terms of the geometry of a submanifold ${\cal M}$ of the unit sphere ${\cal S}$ in a real Hilbert space ${\cal H}$. The measurement of a thermodynamic variable then corresponds to the reduction of a state vector in ${\cal H}$ to an eigenstate, where the transition probability is the Boltzmann weight. We derive a set of uncertainty relations for conjugate thermodynamic variables in the equilibrium thermodynamic states. These follow as a consequence of a striking thermodynamic analogue of the Anandan-Aharonov relations in quantum mechanics. As a result we are able to provide a resolution to the controversy surrounding the status of `temperature fluctuations' in the canonical ensemble. By consideration of the curvature of the thermodynamic trajectory in its state space we are then able to derive a series of higher order variance bounds, which we calculate explicitly to second order.

quant-ph↗