Posted by: Xhyra Graf | 29 January 2007

Quantum Mechanics

Relational quantum mechanics is an interpretation of quantum theory which discards the notions of absolute state of a system, absolute value of its physical quantities, or absolute event. The theory describes only the way systems affect each other in the course of physical interactions. State and physical quantities refer always to the interaction, or the relation, between two systems. Nevertheless, the theory is assumed to be complete.  The physical content of quantum theory is understood as expressing the net of relations connecting all different physical systems.

[Similar to the point I was unsuccessfully trying to make to Harris about what was the underlying cause of the half-life determinancy problem.]

  • 1. Introduction
  • Quantum theory is our current general theory of physical motion.

    But the interpretation of what the theory actually tells us about the physical world raises a lively debate, which has continued with alternating fortunes, from the early days of the theory in the late twenties, to nowadays. The relational interpretations are a number of reflections by different authors, which were independently developed, but converge in indicating an interpretation of the physical content of the theory. The core idea is to read the theory as a theoretical account of the way distinct physical systems affect each other when they interact (and not of the way physical systems “are”), and the idea that this account exhausts all that can be said about the physical world. The physical world is thus seen as a net of interacting components, where there is no meaning to the state of an isolated system. A physical system (or, more precisely, its contingent state) is reduced to the net of relations it entertains with the surrounding systems, and the physical structure of the world is identified as this net of relationships.

    The difficulty in the interpretation of quantum mechanics derives from the fact that the theory was first constructed for describing microscopic systems (atoms, electrons, photons) and the way these interact with macroscopic apparatuses built to measure their properties. Such interactions are denoted as “measurements”. The theory consists in a mathematical formalism, which allows probabilities of alternative outcomes of such measurements to be calculated. If used just for this purpose, the theory raises no difficulty. But we expect the macroscopic apparatuses themselves — in fact, any physical system in the world — to obey quantum theory, and this seems to raise contradictions in the theory.

    In classical mechanics, a system S is described by a certain number of physical variables. For instance, an electron is described by its position and its spin (intrinsic angular momentum). These variables change with time and represent the contingent properties of the system. We say that their values determine, at every moment, the “state” of the system.

    The characteristic feature of quantum mechanics is that it does not allow us to assume that all variables of the system have determined values at every moment (this irrespectively of whether or not we know such values). It was Werner Heisenberg who first realized the need to free ourselves from the belief that, say, an electron has a well determined position at every time. When it is not interacting with an external system that can detect its position, the electron can be “spread out” over different positions.  In the jargon of the theory, one says that the electron is in a “quantum superposition” of two (or many) different positions. It follows that the state of the system cannot be captured by giving the value of its variables. Instead, quantum theory introduces a new notion of “state” of a system, which is different from a list of values of its variables. Such a new notion of state was developed in the work of Erwin Schrödinger in the form of the “wave function” of the system, usually denoted by Ψ. Paul Adrien Maurice Dirac gave a general abstract formulation of the notion of quantum state, in terms of a vector Ψ moving in an abstract vector space. The time evolution of the state Ψ is deterministic and is governed by the Schrödinger equation. From the knowledge of the state Ψ, one can compute the probability of the different measurement outcomes q. That is, the probability of the different ways in which the system S can affect a system O in an interaction with it. The theory then prescribes that at every such ‘measurement’, one must update the value of Ψ, to take into account which of the different outcomes has happened. This sudden change of the state Ψ depends on the specific outcome of the measurement and is therefore probabilistic. It is called the “collapse of the wave function”.

    A better alternative is to take the observed values q, q′, q″, … as the actual elements of reality, and view Ψ just as a bookkeeping device, determined by the actual values q, q′, q″, … that happened in past. From this perspective, the real events of the world are the “realization” (the “coming to reality”, the “actualization”) of the values q, q′, q″, … in the course of the interaction between physical systems. This actualization of a variable q in the course of an interaction can be denoted as the quantum event q. An exemple of a quantum event is the detection of an electron in a certain position. The position variable of the electron assumes a determined value in the course of the interaction between the electron and an external system and the quantum event is the “manifestation” of the electron in a certain position. Quantum events have an intrinsically discrete (“quantized”) granular structure.

    The difficulty of this second option is that if we take the quantum nature of all physical systems into account, the statement that a certain specific event q “has happened” (or, equivalently that a certain variable has or has not taken the value q) can be true and not-true at the same time.

    More precisely, it shows that we cannot disentangle the two: according to the theory an observed quantity (q) can be at the same time determined and not determined. An event may have happened and at the same time may not have happened.

  • 2. Relational view of quantum states
  • The way out from this dilemma suggested by the relational interpretation is that the quantum events, and thus the values of the variables of a physical system S, namely the q‘s, are relational. That is, they do not express properties of the system S alone, but rather refer to the relation between two systems. In particular, the central tenet of relational quantum mechanics (Rovelli 1996, 1997) is that there is no meaning in saying that a certain quantum event has happened or that a variable of the system S has taken the value q: rather, there is meaning in saying that the event q has happened or the variable has taken the value q for O, or with respect to O. The apparent contradiction between the two statements that a variable has or hasn’t a value is resolved by indexing the statements with the different systems with which the system in question interacts. If I observe an electron at a certain position, I cannot conclude that the electron is there: I can only conclude that the electron as seen by me is there. Quantum events only happen in interactions between systems, and the fact that a quantum event has happened is only true with respect to the systems involved in the interaction.

    This relativisation of actuality is viable thanks to a remarkable property of the formalism of quantum mechanics. John von Neumann was the first to notice that the formalism of the theory treats the measured system (S ) and the measuring system (O) differently, but the theory is surprisingly flexible on the choice of where to put the boundary between the two. Different choices give different accounts of the state of the world (for instance, the collapse of the wave function happens at different times); but this does not affect the predictions on the final observations. Von Neumann only described a rather special situation, but this flexibility reflects a general structural property of quantum theory, which guarantees the consistency among all the distinct “accounts of the world” of the different observing systems. The manner in which this consistency is realized, however, is subtle.

  • 3. Correlations
  • Mermin points out that a theorem on correlations in Hilbert space quantum mechanics is relevant to the problem of what exactly quantum theory tells us about the physical world. Consider a quantum system S with internal parts s, s′,…, that may be considered as subsystems of S , and define the correlations among subsystems as the expectation values of products of subsystems’ observables. It can be proved that, for any resolution of S into subsystems, the subsystems’ correlations determine uniquely the state of S. According to Mermin, this theorem highlights two major lessons that quantum mechanics teaches us: first, the relevant physics of S is entirely contained in the correlations both among the s, s′,…, themselves (internal correlations) and among the s′,…, and other systems (external correlations); second, correlations may be ascribed physical reality whereas, according to well-known ‘no-go’ theorems, the quantities that are the terms of the correlations cannot (Mermin 1998).

  • 4. Self measurement
  • From a relational point of view, the properties of a system exists only in reference to another system. What about the properties of a system with respect to itself? Can a system measure itself? Is there any meaning of the correlations of a system with itself? Implicit in the relational point of view is the intuition that a complete self-measurement is impossible. It is this impossibility that forces all properties to be referred to another system. The issue of the self-measurement has been analyzed in details in two remarkable works, from very different perspectives, but with similar conclusions, by Marisa Dalla Chiara and by Thomas Breuer.

    Marisa Dalla Chiara (1977) has addressed the logical aspect of the measurement problem. She observes that the problem of self-measurement in quantum mechanics is strictly related to the self-reference problem, which has an old tradition in logic.

    Nevertheless, any apparatus which realizes the reduction of the wave function is necessarily only a metatheoretical object ” (Dalla Chiara 1977, p. 340).

    As is well known, from a purely logical point of view self-reference properties in formal systems impose limitations on the descriptive power of the systems themselves. Thomas Breuer has shown that, from a physical point of view, this feature is expressed by the existence of limitations in the universal validity of physical theories, no matter whether classical or quantum (Breuer 1995).

    He defines a map from the space of all sets of states of the apparatus to the space of all sets of states of the system. Such a map assigns to every set of apparatus states the set of system states that is compatible with the information that — after the measurement interaction — the apparatus is in one of these states. Under reasonable assumptions on this map, Breuer is able to prove a theorem stating that no such map can exist that can distinguish all the states of the system.

  • 5. Other relational views 
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    Relational views on quantum theory have been defended also by Lee Smolin (1995) and by Louis Crane (1995) in a cosmological context. If one is interested in the quantum theory of the entire universe, then, by definition, an external observer is not available. Breuer’s theorem shows then that a quantum state of the universe, containing all correlations between all subsystems, expresses information that is not available, not even in principle, to any observer. In order to write a meaningful quantum state, argue Crane and Smolin, we have to divide the universe in two components and consider the relative quantum state predicting the outcomes of the observations that one component can make on the other.

     

  • 6. Some consequences of the relational point of view
  • Finally, it has been suggested in (Rovelli 1997) that the relationalism at the core of quantum theory pointed out by the relational interpretations might be connected with the spatiotemporal relationalism that characterizes general relativity.

  • 7. Conclusion
  •  Relational interpretations of quantum mechanics propose a solution to the interpretational difficulties of quantum theory based on the idea of weakening the notions of the state of a system, event, and the idea that a system, at a certain time, may just have a certain property. The world is described as an ensemble of events (“the electron is the point x“) which happen only relatively to a given observer. Accordingly, the state and the properties of a system are relative to another system only. There is a wide diversity in style, emphasis, and language in the authors that we have mentioned. Indeed, most of the works mentioned have developed independently from each other. But it is rather clear that there is a common idea underlying all these approaches, and the convergence is remarkable.

    With special relativity, simultaneity of two distant events has been recognized as meaningless, unless referred to a specific state of motion of something. (This something is usually denoted as “the observer” without, of course, any implication that the observer is human or has any other peculiar property besides having a state of motion. Similarly, the “observer system” O in quantum mechanics need not to be human or have any other property beside the possibility of interacting with the “observed” system S.)

    This way of thinking the world has certainly heavy philosophical implications. The claim of the relational interpretations is that it is nature itself that is forcing us to this way of thinking. If we want to understand nature, our task is not to frame nature into our philosophical prejudices, but rather to learn how to adjust our philosophical prejudices to what we learn from nature.

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    Laudisa, Federico, Rovelli, Carlo, “Relational Quantum Mechanics”, The Stanford Encyclopedia of Philosophy (Fall 2005 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/entries/qm-relational/

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