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.
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.
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.
It often goes unnoticed that, even for a finite number of degrees of freedom, the canonical commutation relations have many inequivalent irreducible unitary representations; the free particle and a particle in a box provide examples that are both simple and well-known. The representations are unitarily inequivalent because the spectra of the position and momentum operators are different, and spectra are invariant under unitary transformations. The existence of these representations can have consequences that run from the merely unexpected to the barely conceivable. To start with, states of a single particle that belong to inequivalent representations will always be mutually orthogonal; they will never interfere with each other. This property, called superseparability elsewhere, is well-defined mathematically, but has not yet been observed. This article suggests two single-particle interference experiments that may reveal its existence. The existence of inequivalent irreducibile representations may be traced to the existence of different self-adjoint extensions of symmetric operators on infinite-dimensional Hilbert spaces. Analysis of the underlying mathematics reveals that some of these extensions can be interpreted in terms of topological, geometrical and physical quantities that can be controlled in the laboratory. The tests suggested are based on these interpretations. In conclusion, it is pointed out that mathematically rigorous many-worlds interpretations of quantum mechanics may be possible in a framework that admits superseparability.
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.
1. This paper, although mathematical in content, is motivated by quantumtheoretical considerations. The states of a quantum-mechanical system are usually described by vectors f of norm 1 in some Hilbert space A, and we assume explicitly that to every unit vector f corresponds a state of the system. This correspondence, however, is not one-to-one. In fact, the vectors which describe the same state form a ray f (in Weyl's terminology, cf. [13], p. 4 and p. 20),1 i.e. a set consisting of all vectors f = Tfo where fo is a fixed unit vector in & and r any complex number of modulus 1. (Every vector f in f will be called a representative of the ray f.) We have therefore a one-to-one correspondence between quantum states and rays, and every significant statement in Quantum Theory is a statement about rays. The transition probability from a state f to a state g equals (f, I)'2 where f, g are representatives of the rays f, g respectively. This suggests the introduction of the inner product of two rays by the definition
This article develops an expanded critical-propositional analysis of S. Dürr, T. Nonn, and G. Rempe’s 1998 experiment, “Origin of Quantum-Mechanical Complementarity Probed by a ‘Which-Way’ Experiment in an Atom Interferometer,” published in Nature, in systematic dialogue with the Theory of Objectivity (TO) developed by Vidamor Cabannas and Denivaldo Silva. The Dürr–Nonn–Rempe experiment constitutes an especially significant empirical case because it addresses a foundational question of quantum mechanics: whether the disappearance of interference following the acquisition of which-way information must be explained through mechanical momentum disturbance, as elementary presentations of the uncertainty principle sometimes suggest, or whether complementarity emerges more fundamentally from correlations between distinguishable degrees of freedom. Using an atom interferometer based on rubidium atoms, optical Bragg diffraction, microwave manipulation of hyperfine states, and path marking through internal atomic degrees of freedom, Dürr, Nonn, and Rempe showed that interference can disappear even when the mechanical back action associated with path detection is too small to account for the observed loss of visibility (Dürr, Nonn, and Rempe 1998b). The phenomenon is instead explained through correlations between atomic motion and the internal which-way marker. Their companion quantitative experiment further examined the relation between path distinguishability D and interference visibility V , providing empirical support for the duality relation associated with Englert’s formulation, D 2 + V 2 ≤ 1. The present article examines these results under the modal and methodological discipline proposed by the Theory of Objectivity. It distinguishes three epistemological levels: (1) empirical confirmation of a physical phenomenon; (2) structural compatibility between that phenomenon and categories of the TO; and (3) specific empirical confir- mation of propositions uniquely derived from the TO. This distinction prevents the mistaken inference that an experiment compatible with an axiom thereby demonstrates the axiom’s claimed modal necessity. Particular attention is given to VA4, concerning boundary and distinction; VA5, concerning observation and relational structure; VA6, concerning composition from prior elements and relations; and VA7, concerning the transcendent element understood within the TO as knowledge or information generated in atomic relations and considered equivalent, within the theory, to atomic radiation. The experiment provides a strong empirical demonstration that physically instantiated information can be generated, stored in atomic internal states, correlated with motional alternatives, and associated with measurable changes in interference. This constitutes a potentially important indirect empirical bridge toward the informational dimensions of VA7. Nevertheless, the experiment does not by itself establish the stronger TO proposition that information is ontologically equivalent to atomic radiation, nor does it prove the necessary truth of VA1–VA7. The analysis also considers the phenomenic elements of light, electromagnetic radiation, atomic transitions, information, boundary formation, and observation; the TO concepts of Expansive and Reductive Inducing Effects (EIE/EIR); the cosmogonic theorem; and the theory’s cosmological eras. It argues that the most productive contribution of the Dürr–Nonn–Rempe experiment to the TO is not cosmological but microphysical and methodological. It suggests a possible operational program in which TO concepts of boundary, observation, and informational differentiation are mapped onto measurable quantities such as path distinguishability, coherence, visibility, mutual information, entropy, and state overlap. The resulting assessment is that the Dürr–Nonn–Rempe experiment has a high degree of dialogue with the Theory of Objectivity, particularly through VA4–VA7, while simultaneously exposing an important burden on the TO: to transform philosophical and modal correspondences into mathematically defined, quantitatively testable, and potentially discriminating empirical propositions. Keywords: Theory of Objectivity; quantum complementarity; which-way information; atom interferometry; distinguishability; interference visibility; physical information; quantum correlations; modal ontology; VA4; VA5; VA6; VA7; information and radiation; operational bridges.
We study quantum behavior from a constructive "finite" point of view, since the introduction of continuum or other actual infinities into physics poses serious conceptual and technical difficulties without any need for these concepts in physics as an empirical science. Taking this approach, we can show that the quantum-mechanical problems can be formulated in the invariant subspaces of permutation representations of finite groups, while the quantum interferences occur as phenomena that are observable in these subspaces. The scalar products in the invariant subspaces (which are needed for formulating the Born rule – the main postulate of quantum mechanics that links mathematical description with experiment) are linear combinations of independent bilinear invariant forms of the permutation representation. A complete set of such forms for any permutation group can be easily calculated by a simple algorithm. Slightly more sophisticated algorithms are required for expressing quantum observables in terms of these forms.