The Kochen-Specker theorem is an important and subtle topic in thefoundations of quantum mechanics (QM). The theorem demonstrates theimpossibility of a certain type of interpretation of QM in terms ofhidden variables (HV) that naturally suggests itself when one beginsto consider the project of interpretating QM.We here present thetheorem/argument and the foundational discussion surrounding it atdifferent levels. The reader looking for a quick overview should readthe following sections and subsections: 1, 2, 3.1, 3.2, 4, and6. Those who read the whole entry will find proofs of some non-trivialclaims in supplementary documents.
Putting Paradoxes to Work: Contextuality in Measurement-Based Quantum Computation
We describe a joint cohomological framework for measurement-based quantum computation (MBQC) and the corresponding contextuality proofs. The central object in this framework is an element $$[\beta _\Psi ]$$[βΨ]in the second cohomology group of the chain complex describing a given MBQC. $$[\beta _\Psi ]$$[βΨ]contains the function computed therein up to gauge equivalence, and at the same time is a contextuality witness. The present cohomological description only applies to temporally flat MBQCs, and we outline an approach for extending it to the temporally ordered case.
QBism: Quantum Theory as a Hero's Handbook
This paper represents an elaboration of the lectures delivered by one of us (CAF) during "Course 197 -- Foundations of Quantum Physics" at the International School of Physics "Enrico Fermi" in Varenna, Italy, July 2016. Much of the material for it is drawn from arXiv:1003.5209, arXiv:1401.7254, and arXiv:1405.2390. However there are substantial additions of original material in Sections 4, 7, 8 and 9, along with clarifications and expansions of the older content throughout. Topics include the meaning of subjective probability; no-cloning, teleportation, and quantum tomography from the subjectivist Bayesian perspective; the message QBism receives from Bell inequality violations (namely, that nature is creative); the import of symmetric informationally complete (SIC) quantum measurements for the technical side of QBism; quantum cosmology QBist-style; and a potential meaning for the holographic principle within QBism.
Collapse Theories
Quantum mechanics, with its revolutionary implications, has posedinnumerable problems to philosophers of science. In particular, it hassuggested reconsidering basic concepts such as the existence of aworld that is, at least to some extent, independent of the observer,the possibility of getting reliable and objective knowledge about it,and the possibility of taking (under appropriate circumstances) atleast some properties to be objectively possessed by physical systems.It has also raised many others questions which are well known to thoseinvolved in the debate on the interpretation of this pillar of modernscience. One can argue that most of the problems are not only due tothe intrinsic revolutionary nature of the phenomena which have led tothe development of the theory. They are also related to the fact that,in its standard formulation and interpretation, quantum mechanics is atheory which is excellent (in fact it has an unprecedented success inthe history of science) in telling us everything about what weobserve, but it meets with serious difficulties in telling uswhat there is. We are making here specific reference to thecentral problem of the theory, usually referred to as themeasurement problem, which is accompanying quantum theory sinceits birth. It is just one of the many attempts to overcome thedifficulties posed by this problem that has led to the development ofCollapse Theories, i.e., to the Dynamical ReductionProgram (DRP). As we shall see, this approach consists inaccepting that the dynamical equation of the standard theory should bemodified by the addition of stochastic and nonlinear terms. The nicefact is that the resulting theory is capable, on the basis of a singledynamics which is assumed to govern all natural processes, to accountat the same time for all well-established facts about microscopicsystems as described by the standard theory, as well as for theso-called postulate of wave packet reduction (WPR), which accompaniesthe interaction of a microscopic system with a measuring device. As iswell known, such a postulate is assumed in the standard scheme just inorder to guarantee that measurements have outcomes but, as weshall discuss below, it meets with insurmountable difficulties if onetries to derive it by assuming the measurement itself to be a processgoverned by the linear laws of the theory. Finally, the collapsetheories account in a completely satisfactory way for the classicalbehavior of macroscopic systems.
The Causal Axioms of Algebraic Quantum Field Theory: A Diagnostic
This paper examines the axioms of algebraic quantum field theory (AQFT) that aim to characterize the theory as one that implements relativistic causation. I suggest that the spectrum condition (SC), microcausality (MC), and primitive causality axioms (PC), taken individually, fall short of fulfilling this goal against what some philosophers have claimed. Instead, I will show that the “local primitive causality” (LPC) condition captures each axiom’s advantages. However, this is only the case because SC, MC, and PC, taken together, imply LPC, as I will show from a construction by Haag and Schroer (1962).
The Consistent Histories Approach to Quantum Mechanics
The consistent histories, also known as decoherent histories, approachto quantum interpretation is broadly compatible with standard quantummechanics as found in textbooks. However, the concept ofmeasurement by which probabilities are introduced in standardquantum theory no longer plays a fundamental role. Instead,all quantum time dependence is probabilistic (stochastic),with probabilities given by the Born rule or its extensions. Byrequiring that the description of a quantum system be carried outusing a well-defined probabilistic sample space (called a“framework”) this approach resolves many well-knownquantum paradoxes of quantum foundations. In particular, quantummechanics is local and consistent with special relativity. Classicalmechanics emerges as a useful approximation to the more fundamentalquantum mechanics under suitable conditions. The price to be paid forthis is a set of rules for reasoning resembling, but very much simplerthan, those of quantum logic. An important philosophical implicationis the absence of a single universally-true state of affairs at eachinstant of time. However, there is a correspondence limit in which thenew quantum logic becomes standard logic in the macroscopic world ofeveryday experience, and the laws of classical mechanics emerge as agood approximation to an underlying, and in principle more exact,quantum description.
Jacob Barandes - “A Deflationary Account of Quantum Theory & Implications for the Complex Numbers”
‘Shut up and calculate’ does a disservice to quantum mechanics | Aeon Essays
The cliché has it that the Copenhagen interpretation demands adherence without deep enquiry. That does physics a disservice
Jacob Barandes | Quantum Theory as a New Kind of Stochastic Process
Jacob Barandes | Quantum Theory as a New Kind of Stochastic Process
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in philosophy of mathematics. The theorems are interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
Terence Tao (@tao@mathstodon.xyz)
In traditional mathematics (or "Math 1.0"), a breakthrough proof of a long-standing open conjecture generates a large amount of subsequent activity and excitement in the field. The authors of the proof are invited to give many talks, and meet with other experts in the area. Workshops are set up to discuss the proof, as well as other recent developments. As a result, new connections and collaborations are formed, followup problems are shared, and other mathematicians (both junior and senior) are attracted to the field (including in some cases the authors themselves, if they initially came from a different subject). As a result of all this activity, the proofs naturally become digested, streamlined, placed in context with other results in the field, and ultimately become part of the textbooks and lecture notes for the next generation of mathematicians in the field. (1/4)
General Theory of Natural Equivalences
Samuel Eilenberg, Saunders MacLane, General Theory of Natural Equivalences, Transactions of the American Mathematical Society, Vol. 58, No. 2 (Sep., 1945), pp. 231-294
Theory Beyond Theorems and Proofs: A Guest Post
Scott’s foreword: I’m extremely grateful to my brilliant colleagues, Pravesh Kothari, Raghu Meka, and Prasad Raghavendra, for sharing the guest post below about how theoretical computer…
