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.     

HOW REYE'S CONFIGURATION HELPS IN PROVING THE BELL–KOCHEN–SPECKER THEOREM: A CURIOUS GEOMETRICAL TALE

It is shown that the 24 quantum states or rays used by Peres (J. Phys. A 24, 174-8 (1991)) to give a proof of the Bell–Kochen–Specker (BKS) theorem have a close connection with Reye's configuration, a system of twelve points and sixteen lines known to projective geometers for over a century. The interest of this observation stems from the fact that it provides a ready explanation for many of the regularities exhibited by the Peres rays and also permits a systematic construction of all possible non-coloring proofs of the BKS theorem based on these rays. An elementary exposition of the connection between the Peres rays and Reye's configuration is given, following which its applications to the BKS theorem are discussed.     

Parity Proofs of the Kochen-Specker Theorem Based on the 24 Rays of Peres

A diagrammatic representation is given of the 24 rays of Peres that makes it easy to pick out all the 512 parity proofs of the Kochen-Specker theorem contained in them. The origin of this representation in the four-dimensional geometry of the rays is pointed out.     

Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present two proofs of this theorem which are both elementary and economical. Central to our proofs is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain two-dimensional subspaces of the Hilbert space under the action of a given Wigner symmetry, but the relevance of this behaviour to the larger picture of the whole Hilbert space is made transparent at every stage.     

Probabilistic generation of quantum contextual sets

We give a method for exhaustive generation of a huge number of Kochen-Specker contextual sets, based on the 600-cell, for possible experiments and quantum gates. The method is complementary to our previous parity proof generation of these sets, and it gives all sets while the parity proof method gives only sets with an odd number of edges in their hypergraph representation. Thus we obtain 35 new kinds of critical KS sets with an even number of edges. We also give a statistical estimate of the number of sets that might be obtained in an eventual exhaustive enumeration.     

Brillouin's theorem and the Hellmann-Feynman theorem for Hartree-Fock wave functions

It is shown that there is an intimate relation between the Hellmann-Feynman theorem and Brillouin's theorem. A more general form of Brillouin's theorem is provided, which applies to excited states of arbitrary symmetry and multiplicity. This new form leads to a simple proof of the Hellmann-Feynman theorem. This theorem is valid when all the orbitals that occur in the wave function are determined by a complete, and not a partial, variational procedure. Arguing in the opposite direction it is shown that the complete satisfaction of the generalized Brillouin's theorem provides an alternative scheme for obtaining the Hartree-Fock orbitals.     

Isomorphism between the Peres and Penrose Proofs of the BKS Theorem in Three Dimensions

It is shown that the 33 complex rays in three dimensions used by Penrose to prove the Bell-Kochen-Specker theorem have the same orthogonality relations as the 33 real rays of Peres, and therefore provide an isomorphic proof of the theorem. It is further shown that the Peres and Penrose rays are just two members of a continuous three-parameter family of unitarily inequivalent rays that prove the theorem.     

静电场环路定理的非抵消法证明

在普通物理学中,"静电场环路定理"是一个重要的基本定理.与一般教科书中的证明方法--"抵消法"不同,提出了另外两种巧妙而更为简单的非抵消法的证法.     

Lie algebra bases for the pseudo-orthogonal groups

The isomorphic mappings between the canonical bases appearing in the general structure theory of semi-simple real Lie algebras and the bases obtained directly from the pseudoorthogonal groups are investigated in detail, and it is shown that these mappings can be cast in a remarkable simple form which is valid for all cases.     

GHZ paradoxes based on an even number of qubits

GHZ paradoxes are presented for all even numbers of qubits from four up. They are obtained from proofs of the Kochen–Specker (KS) theorem by showing how the assumption of noncontextuality can be justified on the basis of locality. The nature of the entangled states involved in our paradoxes is discussed. Some multiqubit proofs of the KS theorem are also presented in the form of diagrams from which they are visually obvious. The implications of our results are discussed.     

What price the spin-statistics theorem?

We examine a number of recent proofs of the spin-statistics theorem. All, of course, get the target result of Bose-Einstein statistics for identical integral spin particles and Fermi-Dirac statistics for identical half-integral spin particles. It is pointed out that these proofs, distinguished by their purported simple and intuitive kinematic character, require assumptions that are outside the realm of standard quantum mechanics. We construct a counterexample to these non-dynamical kinematic ‘proofs’ to emphasize the necessity of a dynamical proof as distinct from a kinematic proof. Sudarshan’s simple non-relativistic dynamical proof is briefly described. Finally, we make clear the price paid for any kinematic ‘proof’.     

A Kochen-Specker inequality from a SIC

Yu and Oh (eprint) [1] have given a state-independent proof of the Kochen-Specker theorem in three dimensions using only 13 rays. The proof consists of showing that a non-contextual hidden variable theory necessarily leads to an inequality that is violated by quantum mechanics. We give a similar proof making use of 21 rays that constitute a SIC (symmetric informationally-complete positive operator-valued measure) and a complete set of MUB (mutually unbiased bases). A theory-independent inequality is also presented using the same 21 rays, as required for experimental tests of contextuality.     

Levinson theorem for Dirac particles in one dimension

The scattering of Dirac particles by symmetric potentials in one dimension is studied. A Levinson theorem is established. By this theorem, the number of bound states with even(odd)-parity, n₊ (n_-), is related to the phase shifts [] of scattering states with the same parity at zero momentum as follows: The theorem is verified by several simple examples. Received 26 August 1998 and Received in final form 18 January 1999     

Equivalence of the (Generalised) Hadamard and Microlocal Spectrum Condition for (Generalised) Free Fields in Curved Spacetime

We prove that the singularity structure of all n-point distributions of a state of a generalised real free scalar field in curved spacetime can be estimated if the two-point distribution is of Hadamard form. In particular this applies to the free field and the result has applications in perturbative quantum field theory, showing that the class of all Hadamard states is the state space of interest. In our proof we assume that the field is a generalised free field, i.e. that it satisfies scalar (c-number) commutation relations, but it need not satisfy an equation of motion. The same arguments also work for anti-commutation relations and for vector-valued fields. To indicate the strengths and limitations of our assumption we also prove the analogues of a theorem by Borchers and Zimmermann on the self-adjointness of field operators and of a weak form of the Jost-Schroer theorem. The original proofs of these results make use of analytic continuation arguments. In our case no analyticity is assumed, but to some extent the scalar commutation relations can take its place.     

Eynard–Mehta Theorem, Schur Process, and their Pfaffian Analogs

We give simple linear algebraic proofs of the Eynard–Mehta theorem, the Okounkov-Reshetikhin formula for the correlation kernel of the Schur process, and Pfaffian analogs of these results. We also discuss certain general properties of the spaces of all determinantal and Pfaffian processes on a given finite set     

Directed random walks in random environments

We use holding time methods to study the asymptotic behavior of pure birth processes with random transition rates. Both the normal and slow approaches to infinity are studied. Fluctuations are shown to obey the central limit theorem for almost all sample-transition rates. Our results are stronger, and our proofs simpler, then those of recently published studies.     

Five-qubit contextuality,noise-like distribution of distances between maximal bases and finite geometry

Employing five commuting sets of five-qubit observables, we propose specific 160–661 and 160–21 state proofs of the Bell–Kochen–Specker theorem that are also proofs of Bell?s theorem. A histogram of the ‘Hilbert–Schmidt’ distances between the corresponding maximal bases shows in both cases a noise-like behavior. The five commuting sets are also ascribed a finite-geometrical meaning in terms of the structure of symplectic polar space W(9,2)$W(9,2)$.     

Casimir effect for the sphere revisited

In a recent work Brevik et al. have offered formal proofs of two results which figure prominently in calculations of the Casimir pressure on a sphere. It is shown by means of simple counterexamples that each of those proofs is necessarily incorrect.     

Birkhoff's theorem for ghost-free,tachyon-freeR + R 2 +Q 2 theories with torsion

We investigate the conditions under which the class of ghost-free, tachyon-freeR + R ² +Q ² theories with torsion satisfy Birkhoff's theorem. We prove a weakened Birkhoff theorem requiring an additional assumption of parity invariance for two Lagrangians one of which contains torsion squared terms in addition to curvature squared terms. For another Lagrangian, also containing torsion squared terms, a weakened Birkhoff theorem requiring the additional assumptions of parity invariance and constant scalar curvature is proven. A special case of this Lagrangian is shown to satisfy a weakened Birkhoff theorem requiring only the additional assumption of constant scalar curvature. In addition the explicit dependence of torsion on parity noninvariant quantities is displayed.     

Bases on multipunctured Riemann surfaces and interacting strings amplitudes

The Krichever-Novikov bases are studied on Riemann surfaces with more-than-two punctures. The bases are presented and the completness theorem is proven for the case of integer (up to a common constant) momenta. Then the interacting strings are considered, the amplitudes and partition functions are obtained, comparable with that of path-integral approach. For the amplitudes the simple geometric implication is proposed.