Concepts and Paradoxes in a Quantum Universe

In classical mechanics, an action is defined only modulo additive terms which do not modify the equations of motion; in certain cases, these terms are topological quantities. We construct an infinite sequence of higher order topological actions and argue that they play a role in quantum mechanics, and hence can be accessed experimentally.

Phase space methods are ubiquitous in quantum mechanics. From the Weyl-Wigner Moyal formalism to coherent states and discrete phase spaces we see the imprints of the classical world again and again. In this presentation, we address one of two major developments introduced by Aharonov and his collaborators: The concept of weak values that stems from a time-symmetric view of quantum physics. We look at the weak measurement through two distinct geometric frames: The geometry of the measuring apparatus and the geometry of the measured system. We …nalize with some brief comments on the second major conceptual due to Aharonov and collaborators: the theory of modular variables and how Schwinger´s discrete phase space structure helps to shed light on it. We conclude then with the following mantra: It´ s all about phase space.

In this brief talk we will show how weak values appear in a wide range of physical contexts beyond the usual context of weak measurements. Among others, we will discuss how weak values appear in: the physics of classical parameters in a quantum evolution; the statistics of strong measurements; formulas for probability amplitudes in quantum mechanics; and finally, in the classical correspondence of quantum mechanics.

The state vector describing the physical situation of the magnetic A-B effect should depend upon all three quantizeable entities in the problem, the electron orbiting the solenoid, the moving charged particles in the solenoid and the vector potential. One may imagine three approximate solutions to the exact dynamics, where two of the three entities do not interact at all, and the third, quantized, entity interacts with a classical approximation. Thus, fifty-five years ago, A-B showed that, if the interaction is between the quantized electron current and the classical approximation to the solenoid’s vector potential, the state vector acquires a measurable phase shift. Four years ago Vaidman showed that, if the interaction is between the quantized solenoid current and the classical approximation to the electron’s vector potential, the state vector acquires the A-B phase shift. I shall first show why these two results have to be the same. Then, I shall show that, if the interaction is between the quantized vector potential and the classical approximation to the electron and solenoid currents, the state vector acquires the A-B phase shift. Lastly, I shall show how to reconcile these three mathematically and conceptually different calculations.

I discuss the outcome statistics of sequential weak measurement of general observables. In sequential weak measurement of canonical variables, without post-selection, correlations yield the corresponding correlations of the Wigner function. Outcome correlations in spin-1/2 sequential weak measurements without post-selection coincide with those in strong measurements, they are constrained kinematically so that they yield as much information as single measurements. In sequential weak measurements with post-selection, a new anomaly occurs, different from the weak value anomaly in single weak measurements. I consider trivial post-selection, i.e.: re-selection |f>=|i>, which should intuitively not differ from no post-selection since weak measurements are considered non-invasive. Indeed, re-selection does not matter, compared with no-selection, for single weak measurement. It does so, however, for sequential ones. I illustrate it in spin-1/2 weak measurement.

The products of weak values of quantum observables have interesting implications in deriving quantum uncertainty and complementarity relations for both weak and strong measurement statistics. We show that a product representation formula allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables. This formula also leads to a strong uncertainty relation for unitary operators which displays a new preparation uncertainty relation for quantum systems. Furthermore, the two system observables that are weakly and strongly measured in a weak measurement context are shown to obey a complementarity relation under the interchange of these observables, in the form of an upper bound on the product of the corresponding weak values. Moreover, we derive general tradeoff relations, between weak purity, quantum purity and quantum incompatibility using the weak value formalism. Our results may open up new ways of thinking about uncertainty and complementarity relations using products of weak values.

The weak value, as an expectation value, requires an ensemble to be found. Nevertheless, we argue that the physical meaning of the weak value is much more close to the physical meaning of an eigenvalue than to the physical meaning of an expectation value. Theoretical analysis and experimental results performed in the MPQ laboratory of Harald Weinfurter are presented. Quantum systems described by numerically equal eigenvalue, weak value and expectation value cause identical average shift of an external system interacting with them during an infinitesimal time. However, there are differences between the final states of the external system. In the case of an eigenvalue, the shift is the only change in the wavefunction of the external system. In case of the expectation value, there is an additional change in the quantum state of the same order, while in the case of the weak value the additional distortion is negligible. The understanding of weak value as a property of a single system refutes recent claims that there exist classical statistical analogue to the weak value.