The algebra of physical magnitudes

We define a physical magnitude as an equivalence class of measurement procedures and formulate sufficient restrictions on the equivalence relation to guarantee meaningful algebraic operations between magnitudes. These restrictions are not sufficient to let the Kochen and Specker argument go through. They are, however, stronger than mere statistical equivalence of measurement procedures and thus are relevant to the problem of the completeness of quantum mechanics. In fact, they give rise to a strong argument for the incompleteness of quantum mechanics.     

Computational complementarity

Interactivity generates paradox in that the interactive control by one systemC of predicates about another system-under-studyS may falsify these predicates. We formulate an “interactive logic” to resolve this paradox of interactivity. Our construction generalizes one, the Galois connection, used by Von Neumann for the similar quantum paradox. We apply the construction to atransition system, a concept that includes general systems, automata, and quantum systems. In some (classical) automataS, the interactive predicates aboutS show quantumlike complementarity arising from interactivity: The interactive paradox generates the quantum paradox. Some classicalS's have noncommutative algebras of interactively observable coordinates similar to the Heisenberg algebra of a quantum system. SuchS's are “hidden variable” models of quantum theory not covered by the hidden variable studies of Von Neumann, Bohm, Bell, or Kochen and Specker. It is conceivable that some quantum effects in Nature arise from interactivity.     

Embedding Quantum Universes in Classical Ones

Do the partial order and ortholattice operations of a quantum logic correspond to the logical implication and connectives of classical logic? Rephrased, How far might a classical understanding of quantum mechanics be, in principle, possible? A celebrated result of Kochen and Specker answers the above question in the negative. However, this answer is just one among various possible ones, not all negative. It is our aim to discuss the above question in terms of mappings of quantum worlds into classical ones, more specifically, in terms of embeddings of quantum logics into classical logics; depending upon the type of restrictions imposed on embeddings, the question may get negative or positive answers.     

Conway–Kochen and the Finite Precision Loophole

Recently Cator and Landsman made a comparison between Bell’s Theorem and Conway and Kochen’s Strong Free Will Theorem. Their overall conclusion was that the latter is stronger in that it uses fewer assumptions, but also that it has two shortcomings. Firstly, no experimental test of the Conway–Kochen Theorem has been performed thus far, and, secondly, because the Conway–Kochen Theorem is strongly connected to the Kochen–Specker Theorem it may be susceptible to the finite precision loophole of Meyer, Kent and Clifton. In this paper I show that the finite precision loophole does not apply to the Conway–Kochen Theorem.     

Sum rule inequalities and subtractions in π π dispersion relations

Sum rule inequalities on the π π scattering amplitudes are derived from analyticity, s ? u crossing properties and positivity of these amplitudes. They connect the real and imaginary parts of the amplitude in the energy region where they may be calculated from phase-shift analysis, and do not require knowledge of these quantities at low energies or in the high energy region. The sum rules are used to confirm earlier results that two subtractions are needed in π π dispersion relations and this conclusion is now much more definite since in contrast to the earlier work, no knowledge of the magnitude of the absorptive part of the amplitude in the low energy region is needed.     

Kochen-Specker Theorem in Krein Space

The well known Kochen-Specker’s theorem (Kochen and Specker J. Math. Mech. 17:59–87, 1967) is devoted to the problem of hidden variables in quantum mechanics. In the paper we present a geometric proof for an indefinite analogy of Kochen-Specker’s theorem. On the real three-dimensional Krein space there exists unique two-valued probability measure.     

Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell

It is shown how the 300 rays associated with the antipodal pairs of vertices of a 120-cell (a four-dimensional regular polytope) can be used to give numerous “parity proofs” of the Kochen–Specker theorem ruling out the existence of noncontextual hidden variables theories. The symmetries of the 120-cell are exploited to give a simple construction of its Kochen–Specker diagram, which is exhibited in the form of a “basis table” showing all the orthogonalities between its rays. The basis table consists of 675 bases (a basis being a set of four mutually orthogonal rays), but all the bases can be written down from the few listed in this paper using some simple rules. The basis table is shown to contain a wide variety of parity proofs, ranging from 19 bases (or contexts) at the low end to 41 bases at the high end. Some explicit examples of these proofs are given, and their implications are discussed.     

Gluon condensate from superconvergent QCD sum rule

Sum rules for the nonperturbative piece of correlators (specifically, the vector current correlator) are discussed. The sum rule subtracting the perturbative part is of the superconvergent type. Thus it is dominated by the bound states and the low-energy production cross section. It leads to a determination of the gluon condensate _sG². We find _sG²0.048±0.030 GeV⁴.     

Experimental verification of the GDH sum rule

Sum rules involving the spin structure of the nucleon like those due to Bjorken, Ellis and Jaffe and the one due to Gerasimov, Drell and Hearn offer the opportunity to study the structure of strong interactions. At long distance scales in the confinement regime the Gerasimov-Drell-Hearn (GDH) Sum Rule connects static properties of the nucleon like the anomalous magnetic moment κ and the nucleon mass m, with the spin dependent absorption of real photons with total cross sections 0gs_3/2 and σ_1/2:

Here 3/2 and 1/2 identify relative spin orientation of the photon and the nucleon parallel or anti-parallel respectively in the nucleon rest frame; denotes the fine-structure constant and ν the energy of the photon. Hence the full spin-dependent excitation spectrum of the nucleon is related to its static properties. The sum rule has not been investigated experimentally until recently. For the first time this fundamental sum rule is verified by the GDH-Collaboration with circularly polarized real photons and longitudinally polarized nucleons at the two accelerators and . The investigation of the response of the proton as well as of the neutron allows to perform an isospin decomposition. Data from the resonance region up to the onset of the Regge regime are shown. The “sum” on the left hand side of the GDH Sum Rule can be generalized to the case of virtual photons. This allows to establish a Q² dependency and to study the transition to the perturbative regime of QCD. This is the subject of several experiments e.g. at for the resonance region and of the experiment at for higher Q². Moreover, this paper covers the status of theory concerning the GDH Sum Rule, the different experimental approaches and the results for the absorption of real and virtual photons will be reviewed.     

Cost Optimization Technique for Quantum Circuits

In this paper, an attempt is made to present a method of quantum cost minimization or optimization technique for quantum reversible circuits using proposed merger rules in Exclusive Sum of Product (ESOP) method. These modified ESOP methods are used to minimize the quantum circuits. We found that the quantum cost is drastically decreased than the previous ESOP method. It will be easy to find the quantum cost and quantum gate optimized quantum circuits implementation. It will also reduce quantum error while the quantum circuit is executed in real quantum processor.

     

Kochen–Specker ϵ-obstruction for position and momentum using a single degree of freedom

The Bell–Kochen–Specker theorem shows that, in any Hilbert space of dimension of at least 3, it is impossible to assign noncontextual definite values to all observables in such a way that the quantum-mechanical predictions are reproduced. This leaves open the issue of what subsets of observables may be assigned definite values. Clifton has shown that, for a system of at least two continuous degrees of freedom, it is not possible to assign simultaneous noncontextual values to two coordinates and their conjugate momenta. In this Letter, it is shown that, for a system of a single continuous degree of freedom, it is not possible to assign noncontextual values to the coordinate and its conjugate momenta that satisfy a continuity assumption herein called the ‘ϵ-Product Rule’.     

State-independent proof of Kochen-Specker theorem with 13 rays

Quantum contextuality, as proved by Kochen and Specker, and also by Bell, should manifest itself in any state in any system with more than two distinguishable states and recently has been experimentally verified. However, for the simplest system capable of exhibiting contextuality, a qutrit, the quantum contextuality is verified only state dependently in experiment because too many (at least 31) observables are involved in all the known state-independent tests. Here we report an experimentally testable inequality involving only 13 observables that is satisfied by all noncontextual realistic models while being violated by all qutrit states. Thus our inequality facilitates a state-independent test of the quantum contextuality for an indivisible quantum system. We also provide a record-breaking state-independent proof of the Kochen-Specker theorem with 13 directions determined by 26 points on the surface of a magic cube.     

Effects and Propositions

The quantum logical and quantum information-theoretic traditions have exerted an especially powerful influence on Bub’s thinking about the conceptual foundations of quantum mechanics. This paper discusses both the quantum logical and information-theoretic traditions from the point of view of their representational frameworks. I argue that it is at this level—at the level of its framework—that the quantum logical tradition has retained its centrality to Bub’s thought. It is further argued that there is implicit in the quantum information-theoretic tradition a set of ideas that mark a genuinely new alternative to the framework of quantum logic. These ideas are of considerable interest for the philosophy of quantum mechanics, a claim which I defend with an extended discussion of their application to our understanding of the philosophical significance of the no hidden variable theorem of Kochen and Specker.     

What is a hidden variable theory of quantum phenomena?

The purpose of this paper is to clarify the relationship between existing so-called hidden variable theories of quantum phenomena and some well-known proofs, such as those of von Neumann, Jauch and Piron, and Kochen and Specker, which purport to establish that no such theory is possible. The proof of Kochen and Specker, which is a stronger version of von Neumann's result, demonstrates the impossibility of embedding the algebraic structure of physical parameters of the quantum theory, represented by the self-adjoint Hubert space operators, into the commutative algebra of real-valued functions on a phase space of hidden states. This is a necessary condition for a hidden variable extension of the quantum theory in the usual sense of a statistical mechanical derivation of the statistical theorems of the quantum theory in the classical manner. No existing so-called hidden variable theory is a counter-example to von Neumann's proof. The early 1951 hidden variable theory of Bohm and the recent theory of Bohm and Bub are not in fact hidden variable theories in the usual sense of the term. Since the term hidden variable theory is justifiably used to denote the kind of theory rejected by von Neumann, Jauch and Piron, and Kochen and Specker, it is suggested that the term should not be used as a label for the theories considered by Bohm and other workers in this field. Such theories could be regarded as fundamentally compatible with the original Copenhagen interpretation of the quantum theory, as expressed by Bohr.Supported by the National Science Foundation.     

Critique of Home and Sengupta's derivation of a Bell inequality

Home and Sengupta claim to derive a Bell-type inequality without assuming locality. I show that their derivation implicitly assumes determinism and a weak locality condition. Their derivation also relies upon further, highly implausible conditions that severely limit the physical implications of their argument.1. If by dispersion-free H&S mean that measurement results depend deterministically on the system's state, then they agree with my argument in this section.2. The first term in inequality (1) implicitly assumes that measurement results of two incompatible (noncommuting) observables, L_z and L_a, are simultaneously well-defined. This assumption is not problematic given an underlying determinism, because L_z ^t can be defined as the result wewould obtain upon measuring L_z. Similarly for L_a ^t. Since we can define those two values counterfactually, it is irrelevant for H&S's derivation that we cannot measure them simultaneously.3. Formally, in going from (1) to (2), H&S make the followingperfect correlations assumption: If QM ascribes zero probability to a joint measurement result, then that joint measurement result does not occur. Specifically, H&S assume that, since QM assigns (for state ) zero probability to the joint measurement result S_z ^tL_z ^t, it therefore is not the case that S_z ^tL_z ^t. Standard Bell derivations (cf. Redhead [2]) do not assume that the hidden-variable theory incorporates QM's perfect correlations. But an alternate, Stapp-Eberhard style derivation of an H&S inequality, using four spatial directions instead of three, also avoids this perfect correlations assumption.For deterministic theories in which all observables possess values and measurement always faithfully reveals those values (independent of which other measurements occur simultaneously), Kochen and Specker's functionality condition (FUNC) follows from the perfect correlations assumption. Since an H&S derivation need not assume perfect correlations, Home and Sengupta rule out a slightly broader class of possessed-values theories than Kochen and Specker do.4. This criticism of local noncontextuality also applies under certain conditions to Kochen and Specker's FUNC condition. FUNC, applied to the class of theories discussed in the second paragraph of footnote 3, implies LNC. For those theories, therefore, my critique of LNC also constitutes an attack on the physical plausibility of FUNC.     

Model independent constraints from vacuum and in-medium QCD Sum Rules

We discuss QCD sum rule constraints based on moments of vector meson spectral distributions in the vacuum and in a nuclear medium. Sum rules for the two lowest moments of these spectral distributions do not suffer from uncertainties related to QCD condensates of dimension higher than four. We exemplify these relations for the case of the ω meson and discuss the issue of in-medium mass shifts from this viewpoint. Received: 22 December 1998 / Revised version: 13 January 1999     

Algebraic constraints on hidden variables

In the contemporary discussion of hidden variable interpretations of quantum mechanics, much attention has been paid to the no hidden variable proof contained in an important paper of Kochen and Specker. It is a little noticed fact that Bell published a proof of the same result the preceding year, in his well-known 1966 article, where it is modestly described as a corollary to Gleason's theorem. We want to bring out the great simplicity of Bell's formulation of this result and to show how it can be extended in certain respects.Work on this paper was partially supported by National Science Foundation Grants SOC 76-82113 and SOC 76-10659.     

1H/31P distance determination by solid state NMR in multiple-spin systems

The results of two techniques of dipolar recoupling, REDOR and CPMAS, are compared in the case of a coupled multiple-spin system. A fundamentally different behavior is observed for these two techniques. In REDOR, the terms associated with each interaction S-I(k) commute with each other and no truncation takes place so that each addition of spin I(k) causes a splitting with its dipolar frequency. In CPMAS, the flip-flop terms of the dipolar Hamiltonian do not commute with the dominant term from the strongly coupled spin pair so that the weak coupling terms from the neighboring spin I(k) are effectively truncated by the dominant pair interaction. Spin dynamics calculations are in agreement with the experimental data in a cubane shaped cluster.     

Spin eigen-states of Dirac equation for quasi-two-dimensional electrons

Dirac equation for electrons in a potential created by quantum well is solved and the three sets of the eigen-functions are obtained. In each set the wavefunction is at the same time the eigen-function of one of the three spin operators, which do not commute with each other, but do commute with the Dirac Hamiltonian. This means that the eigen-functions of Dirac equation describe three independent spin eigen-states. The energy spectrum of electrons confined by the rectangular quantum well is calculated for each of these spin states at the values of energies relevant for solid state physics. It is shown that the standard Rashba spin splitting takes place in one of such states only. In another one, 2D electron subbands remain spin degenerate, and for the third one the spin splitting is anisotropic for different directions of 2D wave vector.     

Adjoint-based optimization of PDEs in moving domains

In this investigation we address the problem of adjoint-based optimization of PDE systems in moving domains. As an example we consider the one-dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping the problem to a fixed domain, whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus. We show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that, while the gradient information contained in both systems is the same, the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically. This approach is then used to solve a moving boundary optimization problem for our model system.