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Volume 60 Issue 6 Dec , pp. Volume 59 Issue 6 Dec , pp. Volume 58 Issue 6 Dec , pp. Volume 57 Issue 6 Dec , pp. Previous Article. Next Article. Quantum measure theory. Abstract We first present some basic properties of a quantum measure space. Citing Articles Here you can find all Crossref-listed publications in which this article is cited. This result, together with others of a similar nature, furnishes to my way of thinking the single best argument in favor of spatio-temporal discreteness, and the most direct indication so far of where the discreteness-scale lies.
If, for example, the kinematic counting of "horizon molecules" in causal set theory were to find a dynamical justification, one would be able to derive the coefficient of proportionality relating the true discreteness scale to the nominal Planck length of 0. Or maybe it would be better to say that it concerns the possibility of re-formulating quantum mechanics entirely as a theory of quantal histories, without ever needing to call on state-vectors, measurements, or external agents as fundamental notions.
But what purpose would such a re-formulation serve, and would it aid in the quest for quantum gravity? Part of the answer which I've already alluded to above is that quantum gravity seems to resist the Hamiltonian framework of "canonical quantization". This statement reflects first of all the so-called problem of time and the need to deal with diffeomorphism-invariant quantities, but when, as with causets, we add to the mix a discreteness which is not only spatial but temporal, the contradiction with the Hamiltonian as a generator of continuous time-evolution becomes severe.
And when we allow for topology change or do away with the continuum altogether any thought of quantum gravity as a theory of operator-valued fields on a fixed manifold also fades out. A second part of the answer harks back directly to the question of the unity of physics. The lack of a theory of quantum gravity is an obvious symptom of the disunity in our present understanding, but it is not the only symptom one can point to.
Another notorious instance is the "cut" which quantum mechanics supposedly forces on us, that severs the macroscopic world of classical physics from the microscopic world of quantal objects and processes. Some authors have taken this cut so seriously as to deny reality to the micro-world altogether.
This issue of the quantal cut might appear to be remote from quantum gravity, but insofar as "quantum vs. Even if such suspicions are mistaken, though, the practical need for a histories formulation remains, and further reasons to desire a more realistic re-formulation of quantum mechanics crop up when one tries to generalize to the quantal case the condition of "Bell Causality" that figures so heavily in connection with classical models of "sequential growth" for causal sets.
How then does the path-integral offer an alternative to the textbook formalism of state-vectors, Hamiltonians, and external observers? This measure is closely related to the so called decoherence functional. In saying this, I am presupposing that the Born rule or rule of thumb! The former obeys a "2-slit" sum-rule that expresses the absence of interference between alternative histories, the latter obeys a weaker, "3-slit" sum-rule that expresses the absence of "higher order" interference beyond pairwise. This 3-slit sum-rule, which reflects the quadratic nature of Bornian probabilities, has now been tested directly in a literal 3-slit experiment with individual photons.
The formal framework I have just sketched rests mathematically on a space of histories and a notion of integration thereon that allows one to compute the quantal measure mu A for any sufficiently regular subset of the history-space. Satisfactory as this framework is in some ways, the interpretation of mu A as a probability still does not take us beyond the realm of "pointer events" that occur in laboratory instruments or other macroscopic objects.
But the laboratory, or even the observatory, is not the place where we expect to encounter quantum gravity. There are no laboratories deep inside black holes or in the very early universe. Not instrument-events, but events like the "big bang" are the natural domain of quantum gravity, and one would like to be able to reason about these events in direct physical terms.
When one ventures into such realms, however, the physical significance of the quantal measure becomes uncertain: the Born rule loses its force and, because of interference, one can no longer interpret mu A as a probability in any ordinary sense. The question then is whether mu admits of a broader interpretation that remains serviceable outside the city limits of Copenhagen. If it does not, then perhaps an entirely different type of dynamical framework will be called for. My current belief is that the quantal-measure meaning in effect the path-integral does have a direct microscopic significance, not as a probability per se, but as an arbiter of whether a given event can occur at all.
In other words, we concede that an event A of vanishing measure mu A will never occur. Drawing out the implications of this "preclusion principle", one soon finds that it conflicts with the classical conception of reality as a single history a single point in the "history space" over which the path-integral integrates. Of course, such a contradiction with classical conceptions of reality is just what one would have expected, but an escape route also opens up.
It turns out that -- at least so far -- the preclusion principle does not seem to conflict with a modified conception of reality that replaces the single classical history with a set of one or more histories.
Measure theory for physicists
Such a set is a special case of what is called an "anhomomorphic coevent", and the new point of view is that reality is best described by such a coevent. In the resulting re-formulation of quantum theory, it is most natural to reason about events using rules of inference that differ from those of classical logic. One could even regard these modified rules as the essence of the new formulation, but one should not confuse them with what has previously been called "quantum logic".
We thus acquire an interpretation of the quantal measure in conjunction with a particular answer to the question, What is quantum mechanics telling us about the nature of reality? As a byproduct of this development, one solves the "measurement problem", or more properly, one obtains a solution of that problem if instrument-events can be proved to obey a certain separability condition. It remains to be seen whether this condition can be established in sufficient generality.
It also remains to be seen whether the new formulation will be be able to accomplish the task for which it was ultimately intended, namely to provide a more precise framework for thinking about quantum gravity, and thereby to help clear away some of the obstacles standing between us and that theory. Jump to Navigation. Rafael Sorkin. Senior Research Affiliate. Quantum Foundations ,. Quantum Gravity. Education Ph. California Institute of Technology, A. Summa cum laude Harvard, Research Interests [Most references still to be added] OVERVIEW OF MY WORK More than anything else, my work in physics has been guided by a desire to overcome the disunity which characterizes our present conception of nature, and which shows itself most obviously in our failure to have reconciled our best theory of spacetime structure general relativity with the best dynamical framework we know of for describing the behavior of matter on small scales quantum field theory.
Quantum Grav. Myers, Eric Poisson, Rafael D. Sorkin, Gravitational action with null boundaries, Phys. Rev D 94, , arXiv: Sorkin, Yasaman K. Sorkin, Expressing entropy globally in terms of 4D field-correlations, J. D 87 6 , , arXiv: Afshordi, M.
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