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Galilei invariance and the welcher Weg problem

Rathindra Nath Sen

Published Jul 18, 2013• 0 citations
OpenAlex

Abstract

If it is possible to detect the path of an atom in a double-slit atom interferometer, then interference will not be observed even if path detection causes no change in the atom's momentum. This fact was explained by Scully et al and Duerr et al by using a principle ofcomplementarity that could not be formulated mathematically, together with a rather peculiar notion of entanglement. We provide an explanation based on considerations of Galilei invariance, without recourse to any notion of complementarity or entanglement. However, we do have to assume that, under conditions of the experiment, composite systems such as atoms and molecules may be regarded as single particles. This assumption is the basis of modern atom and molecule interferometry, but it lacks justification from the first principles of quantum mechanics. In our world-view the welcher Weg problem merely highlights this fact. The empirical validity of this assumption seems to reveal an aspect of quantum mechanics which can justly be termed anti-reductionist, and we conclude that use of unnecessarily strong hypotheses to explain observed phenomena can hide important physical -- and occasionally philosophical -- problems.

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If it is possible to detect the path of an atom in a double-slit atom interferometer, then interference will not be observed even if path detection causes no change in the atom's momentum. This fact was explained by Scully et al and Duerr et al by using a principle ofcomplementarity that could not be formulated mathematically, together with a rather peculiar notion of entanglement. We provide an explanation based on considerations of Galilei invariance, without recourse to any notion of complementarity or entanglement. However, we do have to assume that, under conditions of the experiment, composite systems such as atoms and molecules may be regarded as single particles. This assumption is the basis of modern atom and molecule interferometry, but it lacks justification from the first principles of quantum mechanics. In our world-view the welcher Weg problem merely highlights this fact. The empirical validity of this assumption seems to reveal an aspect of quantum mechanics which can justly be termed anti-reductionist, and we conclude that use of unnecessarily strong hypotheses to explain observed phenomena can hide important physical -- and occasionally philosophical -- problems.
Results of the welcher Weg experiment of Duerr, Nonn and Rempe are explained by using ray representations of the Galilei group. The key idea is that the state of the incoming atom be regarded as belonging to an irreducible unitary ray representation of this group. If this is the case, interaction with an interferometer with a which-way detector must split this state into the direct sum of two states belonging to representations with different internal energies. (While the zero of internal energy is arbitrary, the difference between two internal energies is well-defined and is invariant under unitary transformations.) The state of the outgoing atom will then be a superposition of two mutually orthogonal states, so that there will be no interference. Neither complementarity nor entanglement plays a role in this explanation. Furthermore, in atom interferometry it is not enough for a quantum eraser to erase the internal energy difference; to restore interference, two copies of a representation have to be collapsed into one. In a direct sum of copies of the same representation, copies of the same state will still be orthogonal. These assertions may be testable, and two new atom interferometry experiments are suggested. One of them is an `own-goal' experiment which may decisively refute the explanation offered here, and restore the aura of mystery that this paper tries to dispel.
Early on, Albert Einstein and Niels Bohr heatedly debated wave-particle duality. To explain duality Bohr postulated his principle of complementarity, which is now widely accepted, but has mystifying metaphysical implications. Albert Einstein disagreed. Richard Feynman called it the “only mystery�? in quantum mechanics. Ingenious experiments including those using entanglement have confirmed Bohr’s complementarity. In this paper a new Duality Theorem is presented with proof which explains duality with particle always remaining particle and wave always remaining wave throughout, without using wave-particle complementarity or “which way�? (welcher-weg) observation. We explain results of reported experiments by showing the equivalence: Coherence and alignment ≡ Interference ≡ No “which way�? observation; No coherence or alignment ≡ No interference ≡ “which way�? observation That is, complementarity is redundant; conventional criteria of alignment and coherence alone suffice for interference. Our explanation does not require that photon somehow “know�? about the test setup or “which way�? observation and change its behavior from particle to wave and vice versa. No new assumptions are made, only new reasoning. In the context of entanglement we also help explain non-local action at a distance, and rephrase Albert Einstein’s unanswered question “Is quantum mechanics complete?�? at a more fundamental level.
An Axiom is presented and justified which (a) Explains duality in interference without complementarity or “which way” (welcher-weg) observation (b) Shows the equivalence: Coherence and alignment ≡ Interference ≡ No “which way” observation; No coherence or alignment ≡ No interference ≡ “which way” observation (c) Explains Wheeler’s delayed choice thought experiment (d) Explains results of experimental implementations of Wheeler’s experiment which show retro-causality with and without entanglement (e) Explains non-local action at a distance, and (f) Rephrases Albert Einstein’s unanswered question “Is quantum mechanics complete?” at a more fundamental level than just duality and non-locality. The explanation given does not require that the particle (photon) somehow “know” about the test setup or “which way” observation or change its behavior from particle to wave and vice versa as required by currently accepted explanation based on Niels Bohr’s complementarity principle. No new assumptions are made, only a new complete interpretation of probability which is already a fundamental assumption of quantum mechanics.
Complementarity lies at the heart of conceptual foundation of orthodox quantum mechanics. The wave-particle duality makes it impossible to tell which slit each particle passes through and still observe an interference pattern in a Young's double-slit experiment. In this paper, this duality is appraised under the standard formulation of quantum mechanics for atom interferometers. It is found that the internal freedoms like electronic states of an atom can be used to detect the which-path information of each atom while uphold the interference pattern of atoms when the center-of-mass motion of atoms is detected.
In the history of quantum mechanics, much has been written about the double-slit experiment, and much debate as to its interpretation has ensued. Indeed, to explain the interference patterns for sub-atomic particles, explanations have been given not only in terms of the principle of complementarity and wave-particle duality but also in terms of quantum consciousness and parallel universes. In this paper, the topic will be discussed from the perspective of spin-coupling in the hope of further clarification. We will also suggest that this explanation allows for a realist interpretation of the Afshar Experiment.
As was demonstrated in the previous chapter, the process of observation and acquisition of information or at least the possibility of ‘knowing’ (whether or not we bother to ‘look’) can profoundly change the outcome of an experiment. For example, in the case of the micromaser which-path detector, we do not need to ‘look at’ or ‘interrogate’ the masers in order to lose the interference cross term; it is enough that we could have known. Experiments along these lines provide a dramatic example of the importance of which-path, or ‘Welcher-Weg’, information. The present chapter treats the Welcher-Weg quantum eraser problem from a different vantage. We first consider the interference of light as it is scattered from simple atomic systems consisting of single atoms located at two neighboring sites. From this simple model, we can gain a wealth of insight into such problems as complementarity, delayed choice, and the quantum eraser via field–field and photon–photon correlation functions, i.e., via G (1) ( r , t ) and G (2) ( r , r ′ t , t ′). The chapter concludes with a demonstration that such considerations can, in principle, even lead to new kinds of high-resolution spectroscopy.