A new proof for the convergent iterative solution of the degenerate quantum double-well potential and its generalization

We present a new and simpler proof for the convergent iterative solution of the one-dimensional degenerate double-well potential. This new proof depends on a general theorem, called the hierarchy theorem, that shows the successive stages in the iteration to form a monotonically increasing sequence of approximations to the energy and to the wavefunction at any point x. This important property makes possible a much simpler proof of convergence than the one given before in the literature. The hierarchy theorem proven in this paper is applicable to a much wider class of potentials which includes the quartic potential.     

An Ergodic Theorem for the Quantum Relative Entropy

We prove the ergodic version of the quantum Steins lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equipartition Property (AEP) is given.     

The Poincaré-Lyapounov-Nekhoroshev Theorem

We give a detailed and mainly geometric proof of a theorem by N. N. Nekhoroshev for hamiltonian systems in n degrees of freedom with k constants of motion in involution, where 1≤k≤n. This state's persistence of k-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincaré-Lyapounov theorem (corresponding to k=1) and the Liouville-Arnold one (corresponding to k=n) and interpolates between them. The crucial tool for the proof is a generalization of the Poincaré map, also introduced by Nekhoroshev.     

Ising spin system on a Cayley tree: i-δ relations

The i-δ relations, used in recent theories of the transition region of Ising ferromagnets, are shown to be exact on a Cayley tree. The proof uses a theorem due to Falk, and the Callen-Suzuki identity.     

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.     

Luttinger theorem and imbalanced Fermi systems

The proof of the Luttinger theorem, which was originally given for a normal Fermi liquid with equal spin populations formally described by the exact many-body theory at zero temperature, is here extended to an approximate theory given in terms of a “conserving” approximation also with spin imbalanced populations. The need for this extended proof, whose underlying assumptions are here spelled out in detail, stems from the recent interest in superfluid trapped Fermi atoms with attractive inter-particle interaction, for which the difference between two spin populations can be made large enough that superfluidity is destroyed and the system remains normal even at zero temperature. In this context, we will demonstrate the validity of the Luttinger theorem separately for the two spin populations for any “Φ-derivable” approximation, and illustrate it in particular for the self-consistent t-matrix approximation.     

Bell’s Theorem Without Inequalities and Only Two Distant Observers

A proof of Bell's theorem without inequalities and involving only two observers is given by suitably extending a proof of the Bell-Kochen-Specker theorem due to Mermin. This proof is generalized to obtain an inequality-free proof of Bell's theorem for a set of n Bell states (with n odd) shared between two distant observers. A generalized CHSH inequality is formulated for n Bell states shared symmetrically between two observers and it is shown that quantum mechanics violates this inequality by an amount that grows exponentially with increasing n.     

A complex minkowski space approach to twistors

This paper is basically a review of known results in twistor theory. Its value is intended to lie in the connections presented between twistor concepts and structures in complex Minkowski space. The relationship of twistor theory to complex null infinity and a new proof of the Kerr theorem are presented; these results are to some extent original.     

The spectral property of time delay

This paper discusses the spectral property of time delay. If one considers the scattering by a potential v, then the spectral property is the statement that the trace of the time-delay operator is proportional to the change in state density produced by the interaction v. We give a new and elementary proof of this result. As a consequence it is shown that Levinson's theorem is implied by the spectral property.     

Impedanz und Lagrange-Funktion linearer dissipativer Systeme

A simple proof is given for the theorem, that the difference of electrical and magnetic energies is the same for equivalent 2n-terminal networks, to which the same electromotive forces are applied. For this difference, called the Lagrange function, an explicit expression in terms of the admittance matrix, is given. An analogous theorem is proved in the thermodynamic theory of relaxation phenomena and an extension is given to the case of general linear dissipative systems.     

Stability in Yang-Mills theories

At a solution of the Yang-Mills equations onS ⁴, or the Yang-Mills-Higgs equation on ?³, the hessian of the action functional defines a natural second order, elliptic operator. The number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge. The proof of this assertion requires a relative index theorem for Dirac-type operators on ? ⁿ ,n≧3.     

Carathéodory's principle and the existence of global integrating factors

A proof is given of a theorem on the integrability of Pfaffian forms which is used in Carathéodory's approach to thermodynamics. It is pointed out that Carathéodory's original proof of the existence of entropy and of absolute temperature is incomplete, since it fails to take into account the local nature of this theorem. By combining the theorem with the results ofBuchdahl andGreve on the existence of continuous empirical entropy functions, it is shown that the First and Second Laws of Thermodynamics imply the existence of a globally defined differentiable empirical entropy function for every simple thermodynamic system. This result supplies the missing step in Carathéodory's argument and makes a separate proof of the principle of increase of entropy unnecessary.     

Proof of factorization using background field method of QCD

Factorization theorem plays the central role at high energy colliders to study standard model and beyond standard model physics. The proof of factorization theorem is given by Collins, Soper and Sterman to all orders in perturbation theory by using diagrammatic approach. One might wonder if one can obtain the proof of factorization theorem through symmetry considerations at the lagrangian level. In this paper we provide such a proof.     

Symmetric semistable measures on Rn and symmetric semistable distribution operators in quantum mechanics

The paper contains two theorems. The first one deals with the classic probability, provides the form of the characteristic function of a symmetric, semistable probability distribution in Rⁿ. By a semistable probability distribution we mean a certain class of limit distributions of the sequence of random variables, containing the classic stable probability distribution and included in a set of infinitely divisible distributions.The result will be applied in the proof of the second theorem on non-commutative probability, in which we deal with the generalization of the central limit theorem in quantum mechanics.In the paper we define the symmetric semistable probability distributions for pairs of canonical observables in a state ?₀. In the second theorem we indicate that the symmetric semistable distribution operator is a ground state. By a ground state we mean a state for which Heisenberg's inequality becomes an equality.     

Realizations of pseudo bosonic theories with non-diagonal automorphisms

Pseudo conformal field theories are theories with the same fusion rules, but different modular matrix as some conventional field theory. One of the authors defined these and conjectured that, for bosonic systems, they can all be realized by some actual RCFT, which is that of free bosons. We complete the proof here by treating the non-diagonal automorphism case. It is shown that for characteristic p≠2, they are equivalent to a diagonal case, fully classified in our previous publication. For p=2ⁿ we realize the non-diagonal cases, establishing this theorem.     

Non-integrability of the 3-D hydrogen atom under motional Stark effect or circularly polarized microwave combined with magnetic fields

Limits of the motional Stark effect on Rydberg atoms under the crossed magnetic and electric fields in the case of identical masses, and the hydrogen atom with combined circularly polarized microwave field and a magnetic field that is perpendicular to the plane of polarization have been shown recently to be integrable in the manifold z=Z=0. In this letter we prove the non integrability, in the Liouville–Arnold sense, of the Hamiltonian system corresponding to those limits in 3-dimensions. The proof makes use of a theorem of Morales and Ramis about non integrability based on differential Galois theory.     

Confinement,the Wilson loop and the gluon propagator

We show that if the gluon propagator has a highly singular infrared behavior (such as $\text{1}\text{q}{}^4$) in any one gauge then the Wilson loop must fall exponentially to zero with an area law, indicative of confinement. The proof of this claim follows from an inequality derived via a mean value theorem.     

Two proofs of Fine's theorem

Fine's theorem concerns the question of determining the conditions under which a certain set of probabilities for pairs of four bivalent quantities may be taken to be the marginals of an underlying probability distribution. The eight CHSH inequalities are well-known to be necessary conditions, but Fine's theorem is the striking result that they are also sufficient conditions. Here two transparent and self-contained proofs of Fine's theorem are presented. The first is a physically motivated proof using an explicit local hidden variables model. The second is an algebraic proof which uses a representation of the probabilities in terms of correlation functions.