Aspects of Quantum Measurement 28 November 2008
Posted by integralscience in Science.add a comment
In the mathematical formalism of quantum theory, there are two levels that are important to keep clearly distinct: 1) observables/operators and 2) outcomes/eigenstates. At the first level, one observable is selected by the experimenter from among various observables. At the second level, the measurement of a selected observable is actually performed and an outcome is randomly selected by nature from among the eigenstates of that observable’s operator.
For example, consider observable such as spin with an associated operator, S. The eigenvectors of this operator S correspond to the possible outcomes of a measurement of that observable, e.g., |up> and |down>. These eigenvectors have corresponding eigenvalues which are the probability amplitudes of each outcome. The probabilities of all the outcomes of the single observable are normalized. For a classical analog to the spin, we can consider an observable S for a coin that corresponds to the measurement of its heads/tails position. This single observable S has two outcomes, |heads> and |tails>, each with probability 50%. We’re still just talking about one observable, though.
We can envision another observable T of the coin, e.g., the measurement of its angular momentum. The operator for this observable will have its own set of orthogonal eigenvectors corresponding to the different possible outcomes of measuring this momentum observable. The corresponding probabilities of these momentum outcomes will be normalized amongst themselves.
The two different operators S and T may be non-commuting, i.e., the two observables may be incompatible, which means that they can not be simultaneously measured with arbitrary precision. What is important to note here is that the different orthogonal eigenvectors |heads> and |tails> of the single observable S are very different from different non-commuting operators T and S. In other words, the two different outcomes of a single observable S are not the same as incompatible observables S and T. For example, commutation relations and Heisenberg’s uncertainty principle apply to two incompatible observables, not to two possible outcomes of one observable. Normalization and the collapse of the state vector apply to the possible outcomes of a single observable, not to incompatible observables.
The selection of an observable (e.g, position or momentum) is a free choice of the experimenter. In effect, it is a choice of a reference frame or coordinate system. Mathematically, it is a choice of the basis according to which the state vector of the system is represented. The choice of an observable is not part of the dynamics of quantum theory. It is not something the theory predicts, but something we must define, just as in relativity we are free to select any particular reference frame for observation. We are, in effect, talking about different ways of describing the world, not different underlying physics. A unique feature of quantum physics, however, is the incompatibility of different observables. Multiple complementary perspectives are needed to see the “whole” of reality; but we can not simultaneously view reality from these different perspectives. This is one unique and mysterious aspect of quantum theory.
Once an observable is selected, the selection of a particular outcome of a measurement of that observable is not a choice of the experimenter. The result of a measurement appears to be a free choice of nature, subject to the probability distributions dictated by quantum theory. If the same observable is measured, each individual outcome is unpredictable, but the ensemble of outcomes will conform to the theoretically predicted probability distribution.
Thus, there are two quite distinct aspects of quantum measurement. The first is the free selection by the experimenter of the observable to be measured. This selection defines the set of possible measurement results and their respective probabilities, but does not select a particular result. Even if the measurement interaction takes place, and the system decoheres, there is still not a selection of one result. All that has taken place is that the coherence between the different possible results is no longer measurable. The selection of one result does not take place until a result has been actualized by the actualization of a unique measurement result. This selection of one result does not obey a physical law (in fact, it violates Schrodinger’s equation). Nor is it, like the selection of the observable, something that the experimenter is free to choose. It is a unique and mysterious feature of quantum theory.
Pseudo-problems in physics 25 November 2008
Posted by integralscience in Philosophy, Science.add a comment
Most problems in physics are genuine problems with interesting solutions. There are a few problems, though, that are described as problems of physics, but are actually pseudo-problems. They arise from a fundamental misunderstanding about the nature of theories. One such problem arises as the explanatory gap between the abstract description of the world by a theory and the concrete and particular present moment, actualized here and now. This can take several forms. For example, there is the problem of explaining how it is that, of all the possible universes consistent with the laws of physics, we happen to live in this particular one, with the fundamental constants of physics having the particular values that they do. Another is the measurement problem of quantum physics, which is the problem of relating the probabilistic description of the world given by the theory to the particular world that is actually experienced. We know from our immediate experience that this particular world exists here and now. The problem is that we don’t have an explanation of how such a specific actual world is selected from the general description of reality that the theory provides. We expect the theory to dictate not only the general laws of many possible worlds, but to explain as well why it is that this particular world is actual rather than various other possibilities.
To see how these problems are pseudo-problems, we need only realize that physical theories are abstractions from the here and now. They are conceptual systems that posit and describe a general reality, much of which is forever inaccessible to experiment (e.g., decohered regions of a many worlds theory, or space-like separated regions of space-time). We always begin and end with the present moment. In fact, we never escape it. With our theories, though, we imagine a vast world extended into the future and past, reaching far into the depths of intergalactic space, and including various branches of possibility in a reality that includes many worlds of possibility. The problem begins when we forget that this is all imagined in the present moment, and take these theories to be describing a primary reality.
For example, if we think that the quantum theoretical description of the world is real, we can become perplexed trying to explain how to relate it to our present experience, e.g., how is it that the present experience is “actualized” from the many possibilities? Why is this possible outcome actualized and not another? Or consider the four dimensional space-time of special relativity. Why is the present here-and-now located at this particular point in space-time, and not another? This problem only arises because we have forgotten that we never really left the present here-and-now in the first place. The measurement problem, the actualization of a particular “here-now” in spacetime, is looking at it backwards. The key is to recognize that we do not live in the abstraction but in the here and now, and have never left it. The key is to become aware of how we are abstracting from the present here-now, and what presuppositions of time and space we are making when we do so. This approach to fundamental physics removes the pseudo-problems from physics that have their root in our own misconception of the nature of scientific theory. And it opens up a more fruitful approach to understanding the basis of physical theory and its relationship to the inescapable present moment of experience.
A moving image of eternity 24 November 2008
Posted by integralscience in Philosophy, Science.add a comment
What is the nature of time? In our contemporary culture, this is considered a question for physics to answer. In Einstein’s general relativity, time (as well as space) are relative and dynamic, depending on gravity and mass. Time is part of the nature of the physical world. In the Timaeus, Plato also expresses a view in which the time is created along with the physical world as an image of a deeper eternal unity:
Wherefore he resolved to have a moving image of eternity, and when he set in order the heaven, he made this image eternal but moving according to number, while eternity itself rests in unity; and this image we call time. For there were no days and nights and months and years before the heaven was created, but when he constructed the heaven he created them also. They are all parts of time, and the past and future are created species of time, which we unconsciously but wrongly transfer to the eternal essence.
In other words, time is not fundamental but a derivative image. And the movement of this image is mathematical. The manifest multiplicity in time is the mathematical movement of an underlying eternal unity. This Platonic vision is remarkably similar to the notion of symmetry. Put simply, a symmetry is an imagined change that produces no change. Where there is an invariant amidst variation, there is a symmetry. For example, a rotation of a circle by 90 degrees changes the circle but leaves it unchanged in a deeper, more fundamental sense. The symmetry transformation is a moving image of the eternal invariant structure. And in modern physics, fundamental constants of motion (conserved quantities) are directly related to symmetries through Noether’s theorem. Mathematical symmetry expresses the relationship of time and eternity, the sense in which time is a moving image of eternity, change within the unchanging, variation within an invariant.
A physical law itself is also an invariant that expresses the variations in the physical world. A law of physics provides a mathematical form that describes the invariant relationships between variable quantities. For example, F=ma is (within classical physics) a universal formula that is invariant across all times and places. While the values for m, a, and F can change, their relationship expressed by this equation is an invariant amidst this change. Thus, physical law itself is a symmetry that reveals the physical world as a moving image of eternity.
A paradox of sorts emerges at this point. On the one hand, the very notion of symmetry, of invariance within variation, presupposes a very basic form of change and time. On the other hand, physical time itself is part of the physical world that manifests as a moving image of eternity. Thus, time seems to be at once both cause and effect of manifestation. It is here that we are prompted to distinguish between layers of time. The most basic “proto-time,” the bare possibility of imagined change of the unchanging, is without any elaborate structure. Physical time, on the other hand, is rich with structure. It is entangled with space in a four-dimensional curved space-time whose dynamics depend on gravity and mass. The nature of physical time is part of the solution to equations of physics. But these equations themselves presuppose a more basic proto-time that is embedded in the very notion of law itself. The origin of this proto-time is presupposed by physical law, and is therefore not within the domain of physics per se. It is beyond or outside of physics. One might say it is part of meta-physics. Deeply understanding the foundations of physics necessarily takes one into this realm beyond physics, just as studying the foundations of a building takes one into the Earth. Structural engineering ultimately opens up to geology.
