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Superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics and in many fields of science and technology such as optics, signal processing and antenna theory. In this paper we introduce a new method to generate superoscillatory functions that allows us to construct explicitly a very large class of superoscillatory functions.

In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schr\"odinger equation and other equations., Comment: to appear in Memoirs of the American Mathematical Society

The pigeonhole principle: "If you put three pigeons in two pigeonholes at least two of the pigeons end up in the same hole" is an obvious yet fundamental principle of Nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above "quantum pigeonhole principle" is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoticed. Our results shed new light on the very notions of separability and correlations in quantum mechanics and on the nature of interactions. It also presents a new role for entanglement, complementary to the usual one. Finally, interferometric experiments that illustrate our effects are proposed.

A "strongly" interacting, and entangling, heavy, non recoiling, external particle effects a significant change of the environment. Described locally, the corresponding entanglement event is a generalized electric Aharonov Bohm effect, that differs from the original one in a crucial way. We propose a gedanken interference experiment. The predicted shift of the interference pattern is due to a self induced or "private" potential difference experienced while the particle is in vacuum. We show that all non trivial Born Oppenheimer potentials are "private" potentials. We apply the Born Oppenheimer approximation to interference states. Using our approach we calculate the relative phase of the external heavy particle as well as its uncertainty throughout an interference experiment /entanglement event. We thus complement the Born Oppenheimer approximation for interference states., Comment: Accepted for publication in The Physical Review A

This paper has been withdrawn by the author., Comment: This paper has been withdrawn by the author. 10 pages, 3 figures

This paper has been withdrawn by the author., Comment: This paper has been withdrawn by the author. 11 pages, 4 figures

We discuss experimental situations that consist of multiple preparation and measurement stages. This leads us to a new approach to quantum mechanics. In particular, we introduce the idea of multi-time quantum states which are the appropriate tools for describing these experimental situations. We also describe multi-time measurements and discuss their relation to multi-time states. A consequence of our new formalism is to put states and operators on an equal footing. Finally we discuss the implications of our new approach to quantum mechanics for the problem of the flow of time., Comment: 17 pages, 8 figures

We discuss the consequences of the Aharonov-Bohm effect in setups involving several charged particles, wherein none of the charged particles encloses a closed loop around the magnetic flux. We show that in such setups, the AB phase is encoded either in the relative phase of a bi-partite or multi-partite entangled photons states, or alternatively, gives rise to an overall AB phase that can be measured relative to another reference system. These setups involve processes of annihilation or creation of electron/hole pairs. We discuss the relevance of such effects in "vacuum Birefringence" in QED, and comment on their connection to other known effects., Comment: 4 pages, 3 figures

The nonloclal exchange of the conserved, gauge invariant quantity $e^{\frac{i}{\hbar} (p_{k}-\frac{e}{c}A_{k})L^{k}}, L^{k}=const., k=1,2$ between the charged particle and the magnetic flux line (in the $k=3$ direction), is responsible for the Aharonov-Bohm effect. This exchange occurs at a definite time, before the wavepackets are brought together to interfere, and can be verified experimentally., Comment: LaTeX, 13 pages with 3 figures

The two main features of the Aharonov-Bohm effect are the topological dependence of accumulated phase on the winding number around the magnetic fluxon, and non-locality -- local observations at any intermediate point along the trajectories are not affected by the fluxon. The latter property is usually regarded as exclusive to quantum mechanics. Here we show that both the topological and non-local features of the Aharonov-Bohm effect can be manifested in a classical model that incorporates random noise. The model also suggests new types of multi-particle topological non-local effects which have no quantum analog., Comment: 4 pages, to be published in Phys. Rev. Lett

It is shown that a quantum shutter, pre- and post-selected in particular quantum states, can close simultaneously arbitrary number of slits preventing the passage of a single photon in an arbitrary state. A set of K pre- and post-selected shutters can close the slits preventing the passage of K or less photons. This result indicates that the surprising properties of pre- and post-selected quantum systems are even more robust than previously expected., Comment: 3 pages RevTex4

Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine what is the Hamiltonian. We show that in general this requires a time $\Delta t$ which obeys the uncertainty relation $\Delta t \Delta H \gtrsim 1$ where $\Delta H$ is a measure of how accurately the unknown Hamiltonian must be estimated. We then apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can in principle be measured in an arbitrarily short time. On the other hand we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time $\Delta t$ which obeys the uncertainty relation $\Delta t \Delta E \gtrsim 1$ where $\Delta E$ is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail., Comment: 12pages, revised version with small corrections

Two recent works suggest a possibility of sending signals to a space-like separated region, contrary to the spirit of special relativity. In the first work [J. Grunhaus, S. Popescu, and D. Rohrlich, Phys. Rev. A 53, 3781 (1996)] it has been shown that sending signals to particular union of space-like separated region cannot cause causality paradoxes. Another work [Y. Aharonov and L. Vaidman, Phys. Rev. A 61, 052108 (2000)] showed that the relative phase of quantum superposition of a particle in two separate locations can be measured locally. Together with the possibility of changing the relative phase in a nonlocal way using potential effect we, apparently, have a method of sending signals to space-like separated regions. These arguments are critically analyzed in this paper., Comment: 6 pages. Contribution to the "Mysteries and Paradoxes in Quantum Mechanics", Garda Lake 2000

Various aspects of nonlocality of a quantum wave are discussed. In particular, the question of the possibility of extracting information about the relative phase in a quantum wave is analyzed. It is argued that there is a profound difference in the nonlocal properties of the quantum wave between fermion and boson particles. The phase of the boson quantum state can be found from correlations between results of measurements in separate regions. These correlations are identical to the Einstein-Podolsky-Rosen (EPR) correlations between two entangled systems. An ensemble of results of measurements performed on fermion quantum waves does not exhibit the EPR correlations and the relative phase of fermion quantum waves cannot be found from these results. The existence of a physical variable (the relative phase) which cannot be measured locally is the nonlocality aspect of the quantum wave of a fermion., Comment: 12 pages

Protective measurement, which was proposed as a method of observing the wavefunction of a single system, is extended to the observation of the density matrix of a single system. d'Espagnat's definition of `proper mixture' is shown to be improper because it does not allow for appropriate fluctuations. His claim that there could be different mixtures corresponding to the same density matrix is critically examined. These results provide a new meaning to the density matrix, which gives it the same ontological status as the wavefunction describing a pure state. This also enables quantum entropy to be associated with a single system., Comment: 12 pages, latex, no figures, paper presented at the International Symposium on UNCERTAIN REALITY, New Delhi, India, Jan. 5-9, 1998

It is argued that the time-of-arrival cannot be precisely defined and measured in quantum mechanics. By constructing explicit toy models of a measurement, we show that for a free particle it cannot be measured more accurately then $\Delta t_A \sim 1/E_k$, where $E_k$ is the initial kinetic energy of the particle. With a better accuracy, particles reflect off the measuring device, and the resulting probability distribution becomes distorted. It is shown that a time-of-arrival operator cannot exist, and that approximate time-of-arrival operators do not correspond to the measurements considered here., Comment: References added. To appear in Phys. Rev. A

It is shown that a topological vector-potential (Berry phase) is induced by the act of measuring angular momentum in a direction defined by a reference particle. This vector potential appears as a consequence of the back-reaction due to the quantum measurement., Comment: 7 pages, Latex. Revised article. A detailed description of the regime of validity has been added. To appear in Phys. Rev. Lett

The measurability by means of continuous measurements, of an observable $\A(t_0)$, at an instant, and of a time averaged observable, $\bar \A=1/T\int \A(t')dt'$, is examined for linear and in particular for non-linear quantum mechanical systems. We argue that only when the exact (non-perturbative) solution is known, an exact measurement may be possible. A perturbative approach is shown to fail in the non-linear case for measurements with accuracy $\Delta \bar \A < \Delta \bar \A_{min}(T)$, giving rise to a restriction on the accuracy. Thus, in order to prepare an initial pure state of a non-linear system, by means of a continuous measurement, the exact non-perturbative solution must be known., Comment: 16 pages, Revtex

Alter and Yamamoto [Phys. Rev. A 53, R2911 (1996)] claimed to consider ``protective measurements'' [Phys. Lett. A 178, 38 (1993)] which we have recently introduced. We show that the measurements discussed by Alter and Yamamoto ``are not'' the protective measurements we proposed. Therefore, their results are irrelevant to the nature of protective measurements., Comment: 2 pages LaTeX

In several situations, most notably when describing metastable states, a system can evolve according to an effective non hermitian Hamiltonian. To each eigenvalue of a non hermitian Hamiltonian is associated an eigenstate $\vert\phi\rangle$ which evolves forward in time and an eigenstate $\langle{\psi}\vert$ which evolves backward in time. Quantum measurements on such systems are analyzed in detail with particular emphasis on adiabatic measurements in which the measuring device is coupled weakly to the system. It is shown that in this case the outcome of the measurement of an observable $A$ is the weak value $\langle{\psi}\vert A\vert\phi\rangle / \langle{\psi}\vert{\phi}\rangle $ associated to the two-state vector $\langle{\psi}\vert$ $\vert\phi\rangle$ corresponding to one of the eigenvalues of the non hermitian Hamiltonian. The possibility of performing such measurements in a laboratory is discussed., Comment: RevTeX, 4 pages

A recent result about measurability of a quantum state of a single quantum system is generalized to the case of a single pre- and post-selected quantum system described by a two-state vector. The protection required for such measurement is achieved by coupling the quantum system to a pre- and post-selected protected device yielding a nonhermitian effective Hamiltonian., Comment: TeX, 11 pages

Forces related to A-B phases between fluxons with $\Phi=\alpha\Phi_0\ \ \ $ $\alpha\ne integer$ are discussed. We find a $\alpha^2\ln(r)$ type interaction screened on a scale $\lambda_s$. The forces exist only when the fluxons are actually immersed in the region with non vanishing charge density and are periodic in $\alpha$. We briefly comment on the problem of observing such forces., Comment: 10 pages, latex, no fig

We explore further the suggestion to describe a pre- and post-selected system by a two-state, which is determined by two conditions. Starting with a formal definition of a two-state Hilbert space and basic operations, we systematically recast the basics of quantum mechanics - dynamics, observables, and measurement theory - in terms of two-states as the elementary quantities. We find a simple and suggestive formulation, that ``unifies'' two complementary observables: probabilistic observables and non-probabilistic `weak' observables. Probabilities are relevant for measurements in the `strong coupling regime'. They are given by the absolute square of a two-amplitude (a projection of a two-state). Non-probabilistic observables are observed in sufficiently `weak' measurements, and are given by linear combinations of the two-amplitude. As a sub-class they include the `weak values' of hermitian operators. We show that in the intermediate regime, one may observe a mixing of probabilities and weak values. A consequence of the suggested formalism and measurement theory, is that the problem of non-locality and Lorentz non-covariance, of the usual prescription with a `reduction', may be eliminated. We exemplify this point for the EPR experiment and for a system under successive observations., Comment: LaTex, 44 pages, 4 figures included. Figure captions and related text in sections 3.1, 4.2 are revised. A paragraph in pages 9-10 about non-generic two-states is clarified. Footnotes added

Comment on [R.L. Ingraham, Phys. Rev. A 50, 4502 (1994)]. Ingraham suggested ``a delayed-choice experiment with partial, controllable memory erasing''. It is shown that he cannot be right since his predictions contradict relativistic causality. A subtle quantum effect which was overlooked by Ingraham is explained., Comment: 4 pages, LaTeX

It is shown that the ``retrodiction paradox'' recently introduced by Peres arises not because of the fallacy of the time-symmetric approach as he claimed, but due to an inappropriate usage of retrodiction., Comment: 4 pages

We investigate the phase accumulated by a charged particle in an extended quantum state as it encircles one or more magnetic fluxons, each carrying half a flux unit. A simple, essentially topological analysis reveals an interplay between the Aharonov-Bohm phase and Berry's phase., Comment: 10 pages, TAUP 2110-93. TeX

An analysis of errors in measurement yields new insight into the penetration of quantum particles into classically forbidden regions. In addition to ``physical" values, realistic measurements yield ``unphysical" values which, we show, can form a consistent pattern. An experiment to isolate a particle in a classically forbidden region obtains negative values for its kinetic energy. These values realize the concept of a {\it weak value}, discussed in previous works., Comment: 22 pp, TAUP 1850-90