Time, classical and quantum

Nambu quantum mechanics: a nonlinear generalization of geometric quantum mechanics

Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially un-distinguishables.

Quantum mechanics is like a pot: it is almost indestructible and extremely rigid, but also very flexible because you can use any ingredients for your soup. Goran Lindblad Quantum mechanics Although our first four chapters have been very mathematical, quantum mechanics has never been very far away. Let us recall how the mathematical structure of quantum mechanics is usually summarized at the end of a first course in the subject. First of all the pure states are given by vectors in a Hilbert space. If that Hilbert space is finite dimensional it is simply the vector space equipped with a scalar product of the particular kind that we called a Hermitian form in Eq. (3.82). Actually a pure state corresponds to an entire equivalence class of vectors; this is usually treated in such a way that the vectors are normalized to have length one and afterwards vectors differing by an overall phase e io are regarded as physically equivalent. In effect then the space of pure states is the complex projective space; as always in this book n = N – 1. The notation used in quantum mechanics differs from what we have used so far. We have denoted vectors in by Z α , while in quantum mechanics they are usually denoted by a ket vector. We can think of the index α as just a label telling us that Z α is a vector, and then these two notations are in fact exactly equivalent.

Anticipating the realization of quantum computers, we propose the most reader-friendly exposition of quantum information and qubits theory. Although the latter lies within framework of linear algebra, it has some fl avor of quantum mechanics and it would be easier to get used to special symbols and terminologies. Quantum mechanics is described in the language of functional analysis: the state space (the totality of all states) of a quantum system is a Hilbert space over the complex numbers and all mechanical quantities are taken as Hermite operators. Hence some basics of functional analysis is necessary. We make a smooth transition from linear algebra to functional analysis by comparing the elements in these theories: Hilbert space vs. fi nite dimensional vector space, Hermite operator vs. linear map given by a Hermite matrix. Then from Newtonian mechanics to quantum mechanics and then to the theory of qubits. We elucidate qubits theory a bit by accommodating it into linear algebra framework under these precursors.

Finding ‘‘real’’ time in quantum mechanics

Time: And time again

We review the postulates of quantum mechanics with respect to the representation of physical states and measurable quantities, their time evolution, and the interpretation of measurements. We first formulate the postulates in terms of wave functions and differential operators, and then reformulate them in the abstract Hilbert space of state vectors, using Dirac’s notations. Improper states subject to Dirac’s delta normalization are introduced, and the space of physical states is extended to include them. The postulates are rationalized by associating each Hermitian linear operator with a complete orthonormal system of its eigenvectors, where measurement probabilities depend on the projections of these eigenvectors on the system’s state vector. Particularly, wave functions are identified as projections of state vectors on the position operator eigenstates. State vectors representing multidimensional systems are formulated as tensor products of vectors in their subspaces. Finally, we address the general uncertainty relations in simultaneous measurements of different observables.

The state of a concrete system (from physics, chemistry, ecology, or other sciences) is described using (finitely many, say n) observable quantities (e.g., positions and velocities for mechanical systems, population densities for echological systems, etc.). Hence, the state of a system may be represented as a point $x$ in a geometrical space $\mathbb R^n$. In many cases, the quantities describing the state are related, so that the phase space (space of all possible states) is a submanifold $M\subset \mathbb R^n$. The time evolution of the system is represented by a curve $x_t$, $t \in\mathbb R$ drawn on the phase space $M$, or by a sequence $x_n \in M$, $n \in\mathbb Z$, if we consider discrete time.

Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be jointly measurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton–Isham–Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry à la Connes et al. To each unital C*- algebra {mathcal{A}} we associate a geometric object—a diagram of topological spaces collating quotient spaces of the noncommutative space underlying {mathcal{A}}—that performs the role of a generalised Gel'fand spectrum. We show how any functor F from compact Hausdorff spaces to a suitable target category {mathsf{C}} can be applied directly to these geometric objects to automatically yield an extension {tilde{F}} acting on all unital C*-algebras. This procedure is used to give a novel formulation of the operator K0-functor via a finitary variant {tilde{K}_{ f}} of the extension {tilde{K}} of the topological K-functor. We then delineate a C*-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital C*-algebra the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.

Time has always been a fundamental yet enigmatic concept in physics. From the arrow of time in thermodynamics to the timeless nature of quantum mechanics, understanding time's essence has challenged the foundations of modern physics. In this paper, we propose a novel interpretation of time in quantum mechanics, distinguishing between real time as observable events (wave function collapse or address changes) and imaginary time as the continuous evolution of the quantum state. By connecting this framework with the holographic principle, we aim to provide insights into the nature of time, its arrow, and its relationship with quantum mechanics. Keywords Thermodynamics; Wave function; Quantum mechanics; Holographic principle

Notes from the lectures by the author at the 7th Jerusalem Winter School 1990 on Quantum Cosmology and Baby Universes. The lectures covered quantum mechanics for closed systems like the universe, generalized quantum mechanics, time in quantum mechanics, the quantum mechanics spacetime, and practical quantum cosmology. References have not been updated.

(in the sense of Lyapunov) should be considered in the physical state space, not in the state space of another system which is a lower dimensional realization of some input-output relation. The dimension of the stat.e space depends on t,he stability property one is interested in. It. is, hence, not correct to state that the dimension of the st.ate space is (an - 1) for a nonuniformly damped power system, and only (2n - 2) for a uniformly damped or undamped power system; eveh this is a priori' expect,ed for the power system model, since t.he physical variables are t.he speeds and the load angle differences, but not the load angles themselves. However, I usel Anderson’s t.heorem only to derive a Lyapunov funct.ion for the power syst,em stability problem and not, to conclude its sign dehiteness; the validity and the usefulness of the Lyapunov function are checked a posteriori. Because of t.he reasons pointed out by Sastry and Murthy, it would indeed be incorrect to conclude that for the power syst,em case a positive-definite mat.rix P is obtained from Anderson’s technique. My purpose is to develop a systematic procedure to derive a useful Lyapunov function. Actually, once the Lyapunov function is obtained, one can forget about Anderson’s theorem and investigate the sign definiteness of the tentative Lyapunov function. This function should be considered in the (2n - l>dimensional state space with

Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on [Formula: see text] which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.

In this paper we develop a picture of Quantum Mechanics based on the description of physical observables in terms of expectation value functions, generalizing thus the so called Ehrenfest theorems for quantum dynamics. Our basic technical ingredient is the set of tools which has been developed in the last years for the geometrical formulation of Quantum Mechanics. In the new picture, we analyze the problem of the dynamical equations, the uncertainty relations and interference and illustrate the construction with the simple case of a two-level system.

We present a method of constructing a space of quantum states for a field theory: given phase space of a theory, we define a family of physical systems each possessing a finite number of degrees of freedom, next we define a space of quantum states for each finite system, finally using projective techniques we organize all these spaces into a space of quantum states which corresponds to the original phase space. This construction is kinematic in this sense that it bases merely on the structure of the phase space of a theory and does not take into account possible constraints on the space. The construction is a generalization of a construction by Kijowski—the latter one is limited to theories of linear phase spaces, while the former one is free of this limitation. The method presented in this paper enables to construct a space of quantum states for the teleparallel equivalent of general relativity.