The Pauli Exclusion Principle. Can It Be Proved?

The modern state of the Pauli exclusion principle studies is discussed. The Pauli exclusion principle can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is a so-called spin-statistics connection. The reasons why the spin-statistics connection exists are still unknown, see discussion in text. On the other hand, according to the Pauli exclusion principle, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, all other types of permutation symmetry are forbidden; although the solutions of the Schrödinger equation may belong to any representation of the permutation group, including the multi-dimensional ones. It is demonstrated that the proofs of the Pauli exclusion principle in some textbooks on quantum mechanics are incorrect and, in general, the indistinguishability principle is insensitive to the permutation symmetry of the wave function and cannot be used as a criterion for the verification of the Pauli exclusion principle. Heuristic arguments are given in favor that the existence in nature only the one-dimensional permutation representations (symmetric and antisymmetric) are not accidental. As follows from the analysis of possible scenarios, the permission of multi-dimensional representations of the permutation group leads to contradictions with the concept of particle identity and their independence. Thus, the prohibition of the degenerate permutation states by the Pauli exclusion principle follows from the general physical assumptions underlying quantum theory.     

Does the PBR Theorem Rule out a Statistical Understanding of QM?

The PBR theorem gives insight into how quantum mechanics describes a physical system. This paper explores PBRs’ general result and shows that it does not disallow the ensemble interpretation of quantum mechanics and maintains, as it must, the fundamentally statistical character of quantum mechanics. This is illustrated by drawing an analogy with an ideal gas. An ensemble interpretation of the Schrödinger cat experiment that does not violate the PBR conclusion is also given. The ramifications, limits, and weaknesses of the PBR assumptions, especially in light of lessons learned from Bell’s theorem, are elucidated. It is shown that, if valid, PBRs’ conclusion specifies what type of ensemble interpretations are possible. The PBR conclusion would require a more direct correspondence between the quantum state (e.g., \( \left| {\psi \rangle } \right. \)) and the reality it describes than might otherwise be expected. A simple terminology is introduced to clarify this greater correspondence.     

The Schrödinger-HJW Theorem

A concise presentation of Schrödinger's ancilla theorem (Proc. Camb. Phil. Soc. 32, 446 (1936)) and its several recent rediscoveries.     

Über ein Theorem von Bloch

It is pointed out that the theorem ofBloch saying that the state of lowest free energy corresponds to zero current is a direct consequence of the symmetry of a problem against inversion of motion (?Bewegungsumkehr“). By this it becomes possible to enlarge and delimit its sphere of validity. As a conclusion to theory of superconductivity follows thatBlock's theorem does not contradict the hypothesis of spontaneous currents.     

The Steep Nekhoroshev’s Theorem

Revising Nekhoroshev’s geometry of resonances, we provide a fully constructive and quantitative proof of Nekhoroshev’s theorem for steep Hamiltonian systems proving, in particular, that the exponential stability exponent can be taken to be \({1/(2n\alpha_1\cdots\alpha_{n-2}}\)) (\({\alpha_i}\)’s being Nekhoroshev’s steepness indices and \({n \ge 3}\) the number of degrees of freedom). On the base of a heuristic argument, we conjecture that the new stability exponent is optimal.     

Who is whistling? Localizing and identifying phonating dolphins in captivity

Acoustic communication through whistles is well developed in dolphins. However, little is known on how dolphins are using whistles because localizing the sound source is not an easy task. In the present study, the hyperbola method was used to localize the sound source using a two-hydrophone array. A combined visual and acoustic method was used to determine the identity of the whistling dolphin. In an aquarium in Mexico City where two adult bottlenose dolphins were housed we recorded 946 whistles during 22 days. We found that a dolphin was located along the calculated hyperbola for 72.9% of the whistles, but only for 60.3% of the whistles could we determine the identity of the whistling dolphin. However, sometimes it was possible to use other cues to identify the whistling dolphin. It could be the animal that performed a behavior named “observation” at the time whistling occurred or, when a whistle was only recorded on one channel, the whistling dolphin could be the animal located closest to the hydrophone that captured the whistle. Using these cues, 15.4% of the whistles were further ascribed to either dolphin to obtain an overall identification efficiency of 75.7%. Our results show that a very simple and inexpensive acoustic setup can lead to a reasonable number of identifications of the captive whistling dolphin: this is the first study to report such a high rate of whistles identified to the free swimming, captive dolphin that produced them. Therefore, we have a data set with which we can investigate how dolphins are using whistles. This method can be applied in other aquaria where a small number of dolphins is housed; though, the actual efficiency of this method will depend on how often dolphins spend time next to each other and on the reverberation conditions of the pool.     

On a Theorem of Høegh-Krohn

A theorem proved by R. Høegh-Krohn in Comm. Math. Phys. 38(1974), 195–224, which yields a possibility to define states of systems of quantum particles by their values on the products , where \mathfraka _t , t are time automorphisms and F _j are multiplication operators, is generalized and extended. In particular, it is shown that the algebras generated by such products with F _j taken from the families of multiplication operators satisfying certain conditions are dense in the algebras of observables in the -weak topology, in which normal states are continuous. This result was obtained for the systems with two types of kinetic energy: the usual one expressed by means of the Laplacian; the relativistic kinetic energy defined by a pseudo-differential operator.     

Elementary Proof of Petz–Hasegawa Theorem

We give an elementary proof of the following important result first stated by Petz?CHasegawa: $$\begin{array}{ll}f_{p}(t)= p (1-p) \frac{(t-1)^2}{(t^{p}-1) (t^{1-p}-1)}\; {\rm is\; an\; operator\; monotone\; function\; for }\; -1 \le p \le 2\end{array}$$ .     

On Gleason’s Theorem without Gleason

The original proof of Gleason’s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason’s Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason’s Theorem, e.g. that on the quantum logic ℒ(H) of all closed subspaces of a Hilbert space H, dim H≥3, there is no finitely additive state whose range is countably infinite. In particular, if dim H=n, then on ℒ(H) there is a unique discrete state, namely m(A)=dim A/dim H, A∈ℒ(H). Dedicated to Pekka J. Lahti on the occasion of his 60th birthday. The paper has been supported by the Center of Excellence SAS–Physics of Information–I/2/2005, the grant VEGA No. 2/6088/26 SAV, by Science and Technology Assistance Agency under the contract APVV-0071-06, Bratislava, Slovakia.     

Globalization of Tamarkin’s Formality Theorem

The constructions appearing in the formality theorem by Kontsevich [9] and Tamarkin [13] are first made locally. In these references, sufficient conditions are given to globalize the formality maps. Kontsevich formality maps satisfy these conditions. In this Letter, we show that Tamarkins maps can also be constructed so as to satify these conditions, thus can be globalized.     

Bell’s Theorem and Entangled Solitons

Entangled solitons construction being introduced in the nonlinear spinor field model, the Einstein—Podolsky—Rosen (EPR) spin correlation is calculated and shown to coincide with the quantum mechanical one for the 1/2–spin particles.     

Extensions of Lieb’s Concavity Theorem

The operator function (A,B)→ Trf(A,B)(K ^*)K, defined in pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. We obtain, as a special case, a new proof of Lieb’s concavity theorem for the function (A,B)→ TrA ^p K ^* B ^q K, where p and q are non-negative numbers with sum p+q ≤ 1. In addition, we prove concavity of the operator function

A Strong Szegő Theorem for Jacobi Matrices

We use a classical result of Golinskii and Ibragimov to prove an analog of the strong Szegő theorem for Jacobi matrices on . In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that and lie in , the linearly-weighted l ² space. An erratum to this article can be found at     

Xanthopoulos Theorem in the Kaluza–Klein Theory

It is shown that if _AB is an exact solution of the Einstein vacuum field equations in 4 + 1 dimensions, R^ _AB = 0, and l _A is a null vector field, then _AB + l _A l _B is also an exact solution of the Einstein equations R^ _AB = 0 if and only if the perturbation l _A l _B satisfies the Einstein equations linearized about _AB. Then, making use of the Kaluza–Klein approach, it is shown that this result allows us to obtain exact solutions of the Einstein–Maxwell equations (possibly coupled to a scalar field) by solving a system of linear equations.