T. Weber

A quantum approach to static games of complete information

Abstract We extend the concept of a classical two-person static game to the quantum domain, by giving an Hilbert structure to the space of classical strategies and studying the Battle of the Sexes game. We show that the introduction of entangled strategies leads to a unique solution of this game.

On the quantum foundations of the exponential decay law

SummaryBy assuming that in every actual decay process the unstable system is subject to repeated processes of measurement which ascertain whether the system is decayed or not, we prove that the decay law is exponential, obtaining a consistent reconciliation of the quantum and the classical statistical descriptions of decay experiments. We also show that the measurement process is a localization of the decay products and that, consequently, the unstable system is reduced to a unique quantum state. We are able to give, within the framework of ordinary quantum mechanics, a natural definition of the wave function of the unstable system.RiassuntoNell’ipotesi che in ogni effettivo processo di decadimento il sistema instabile sia soggetto a ripetuti processi di misura che accertano se il sistema è decaduto o no, si dimostra che la legge di decadimento è esponenziale, giungendo ad una coerente riconciliazione delle descrizioni statistica e quantistica degli esperimenti di decadimento. Si mostra inoltre che il processo di misura è una localizzazione dei prodotti di decadimento e che, conseguentemente, il sistema instabile viene ridotto ad un unico stato quantistico. Nel quadro della meccanica quantistica ordinaria si dà una definizione naturale della funzione d’onda del sistema instabile.РеэюмеПредполагая, что в каждом реальном процессе распада нестабильная система подвергается повторным процессам иэмерения, которые устанавливают распалась ли система или нет, мы докаэываем, что эакон распада является зкспоненциальным, что приводит непротиворечивому примирению квантового и классического статистического описаний зкспериментов по распаду. Мы также покаэываем, что процесс иэмерения представляет локалиэацию продуктов распада и, следовательно, нестабильная система приводится к единственной квантовой системе. В рамках обычной квантовой механики мы можем дать естественное определение волновой функции нестабильной частицы.

The puzzling entanglement of Schrödinger's wave function

A brief review of the conceptual difficulties met by the quantum formalism is presented. The main attempts to overcome these difficulties are considered and their limitations are pointed out. A recent proposal based on the assumption of the occurrence of a specific type of wave function collapse is discussed and its consequences for the above-mentioned problems are analyzed.

WHICH KIND OF TWO-PARTICLE STATES CAN BE TELEPORTED THROUGH A THREE-PARTICLE QUANTUM CHANNEL?

The use of a three-particle quantum channel to teleport entangled states through a slight modification of Bennetts procedure is studied. It is shown that it is not possible to perform successful teleportation of an arbitrary and unknown two-particle entangled state, following our version of Bennetts procedure. On the contrary, it is shown which, and in how many different ways, particular classes of two-particle states can be teleported.

Operations involving momentum variables in non-Hamiltonian evolution equations

SummaryNon-Hamiltonian evolution equations have been recently considered for the description of various physical processes. Among these types of equations the class which has been more extensively studied is the one usually referred to as quantum-dynamical semi-group equations (QDS). In particular an equation of the QDS type has been considered as the basis for a model, called quantum mechanics with spontaneous localization (QMSL), which has been shown to exhibit some very interesting features allowing us to overcome most of the conceptual difficulties of standard quantum theory. QMSL assumes a modification of the pure Schrödinger evolution by assuming the occurrence, at random times, of stochastic processes for the wave function corresponding formally to approximate position measurements. In this paper we investigate the consequences of modifying and/or enlarging the class of the considered stochastic processes, by considering the spontaneous occurrence of approximate momentum and of simultaneous position and momentum measurements. It is shown that the considered changes in the elementary processes have unacceptable consequences. In particular they either lead to drastic modifications in the dynamics of microsystems or are completely useless from the point of view of the conceptual advantages that one was trying to get from QMSL. The present work supports therefore the idea that QMSL, as originally formulated, can be taken as the basic scheme for the generalizations which are still necessary in order to make it appropriate for the description of systems of identical particles and to meet relativistic requirements.RiassuntoRecentemente sono state considerate equazioni di evoluzione di tipo non-Hamiltoniano per la descrizione di vari processi fisici. Una sottoclasse di queste è stata studiata piú in dettaglio: le cosiddette equazioni dei semigruppi dinamici quantistici. In particolare un’equazione di questo tipo è stata assunta per la formulazione di un modello, la meccanica quantistica con localizzazioni spontanee (QMSL), che si è rivelato estremamente interessante in quanto permette di superare le piú rilevanti difficoltà concettuali della teoria quantistica nella sua formulazione standard. La QMSL assume una modificazione dell’evoluzione alla Schrödinger ipotizzando che la funzione d’onda subisca a tempi a caso dei processi stocastici che corrispondono formalmente a localizzazioni spaziali approssimate. In questo lavoro si studiano le conseguenze di un ampliamento dei processi stocastici usati nella QMSL, considerando anche processi spontanei che coinvolgono le variabili impulso o simultaneamente le posizioni e gli impulsi. Si mostra che le modifiche considerate comportano conseguenze inaccettabili. In particolare esse o portano a drastiche modifiche nella dinamica dei sistemi microscopici, o risultano totalmente inefficaci nel conduire a un superamento delle difficoltà concettuali della teoria, un obiettivo che è il punto piú interessante del modello QMSL. Il presente lavoro rafforza quindi l’idea ehe la QMSL, come formulata originariamente, vada assunta come schema di base per tentarne le generalizzazioni che risultano necessarie per estenderla al caso di particelle identiche e per darne una versione relativistica.РезюмеНедавно были рассмот рены негамильтоновы уравнения эволюции д ля описания различных ф изических процессов. Среди этого типа уравнений имеет ся класс, который наибол ее широко исследуетс я и который обычно называют, как квантовые динамичес кие полугрупповые ур авнения. В частности, уравнение типа квантового динамиче ского полугрупповог о уравнения рассматривается как основа модели, называемой кв антовой механикой со спонтанной локализацией, котора я обнаруживает некото рые интересные особе ности, позволяющие преодол еть концептуальные труд ности стандартной кв антовой теории. Квантовая мех аника со спонтанной локализа цией предполагает мо дификацию Шредингеровской эво люции, допуская появление с тохастических проце ссов, в случайный момент вре мени, для волновой функции, что соответствует форма льно приближенным измере ниям положения. В этой стат ье мы исследуем следс твия модификации и/или рас ширения класса рассмотренны х стохастических про цессов, рассматривая спонта нное появление приближенного импул ьса и одно временные и змерения положения и импульса. Показывается, что рас смотренные изменени я в элементарных процес сах имеют неприемлемые следст вия. В частности, они пр иводят либо к существенным изменениям в динамик е микросистем или пол ностью бесполезны с точки зр ения концептуальных преи муществ квантовой ме ханики со спонтанной локализа цией. Данная работа подтве рждает идею, что перво начально сформулированная кв антовая механика со спонтанн ой локализацией може т рассматриваться как основная схема для обобщений, которые яв ляются необходимыми для соответствующего оп исания систем тождественны х частиц и для удовлет ворения релятивистским усло виям.

Spontaneous localization of a system of identical particles

SummaryQuantum mechanics with spontaneous localization is a recently proposed stochastic modification of theN-body Schrödinger equation which is consistent both with microphysics and macrophysics and leads to a natural solution of the problem of quantum measurement. The treatment is extended here to systems of identical particles in the framework of the first quantization formalism.RiassuntoLa meccanica quantistica con localizzazione spontanea è una modificazione stocastica dell’equazione di Schrödinger aN corpi recentemente proposta che è compatibile sia con la microfisica che con la macrofisica e conduce a una naturale soluzione del problema della misurazione quantistica. La trattazione è estesa qui a sistemi di particelle identiche nell’ambito del formalismo di prima quantizzazione.РезюмеКвантовая механика со спонтанной локализациеи представляет недавно предложенную стохастическую модификациюN-частичного уравнения Шредингера, которая согласуется с микрофизикой и макрофизикой и приводит к естественному решению проблемы квантового измерения. В этой работе производится обобщение на случай систем тождественных частиц в рамках формализма первичного квантования.

An Attempt at a Unified Description of Microscopic and Macroscopic Systems

When quantum mechanics is applied to a macroscopic particle, most features of the behaviour of such an object are accounted for: mean values of position and momentum evolve according to the classical laws and quantum effects such as the spread of wave packets and the tunnel effect are negligible. However, in the behaviour of a macroscopic particle, we do not find any trace of a stable existence of superpositions of macroscopically distinguishable (e.g. localized in far-away spatial regions) states. One can say that the superposition principle breaks down or, at least, that it suffers serious limitations. The discrepancy in this respect between quantum mechanics and the actual behaviour of a macroscopic particle would be dramatic if we would admit (as it is usual for microobjects) that any selfadjoint operator corresponds to an observable. On the other hand, if we are willing to accept serious limitations to the observability of macroobjects the discrepancy can be cured. We do not want to discuss here this attitude. We only note that it implies accepting a partition of all objects into two classes, those for which standard quantum mechanics is fully valid and those for which a limitation of the superposition principle is introduced which hardly can be considered as consistent with the conceptual framework of the theory.

On proton decay

Abstract The problem of proton decay is considered taking into account that in actual experiments there is an interaction of the proton with its environment which could imply an increase of its theoretical lifetime. It is seen that, by application of the time-energy uncertainly relation, no prolongement of the lifetime is obtained in this case.

Quantum limitations for spin measurements on systems of arbitrary spin

SummaryThe existence of additive conserved quantities implies, as is well known, an unavoidable error in the measurement of those observables which do not commute with the conserved quantities. We introduce a formal procedure to derive a lower bound for the error in the case of a measurement of a spin component for a quantum system of arbitrary spin.RiassuntoL’esistenza di quantità additive e conservate implica, come ben noto, un errore inevitabile nella misura di quelle osservabili che non commutano con le quantità conservate. Si introduce un procedimento formale per ottenere un limite inferiore per l’errore nel caso della misura di una componente dello spin per un sistema quantistico di spin qualsiasi.РезюмеСуществование аддитивных сохраняющихся величин подразумевает, как мы знаем, неизбежную ошибку при измерении тех наблюдаемых величин, которые не коммутируют с сохраняющимися величинами. Мы вводим формальную процеуру для определения нижней границы для ошибки в случае измерения спиновой компоненты в квантовой системе с произвольным спином.

Quantumvs. classical laws for sequential-decay processes

SummaryThe problem of the deviations of the quantum from the classical laws for the occupation numbers of the various levels in a sequential-decay process is discussed in general. A factorization formula for the matrix elements of the complete Greens function entering in the expression of the occupation numbers of the levels is obtained. Through this formula and using specific forms of the quantum nondecay probability for the single levels, explicit expressions for the occupation numbers of the levels are obtained and compared with the classical ones.RiassuntoSi studia del tutto in generale il problema delle deviazioni dalle previsioni classiche della legge quantistica di decadimento sequenziale. Si ricava una formula di fattorizzazione per gli elementi di matrice della funzione di Green che entra nella espressione dei numeri di occupazione dei livelli. Per mezzo di questa formula, ed usando forme già ricavate in un precedente lavoro per le probabilità quantistiche di non decadimento dei singoli livelli, si calcolano i numeri di occupazione e si confrontano con quelli ottenuti classicamente.РезюмеОбсуждается проблема отклонения классических предсказаний от квантовых законов для чисел заполнения различных уровней в просессе последовательного распада. Получается формула факторизации для матричных элементов полной функции Грина, входящей в выражение для чисел заполнения уровней. С помощью этой формулы и, используя выражения, полученные в предыдущей работе для квантовой вероятности нераспада для синтлетных уровней, вычисляются числа заполнения уровней и проводится сравнение с классическими выражениями.