The question of time in quantum mechanics has been a topic of intense discussion since its inception. Dating back to the works of the founders of the subject, the notion of time and how it fit in the theory was already a point with a lack of a formal explanation, especially in aspects such as the uncertainty principle and canonical commutation relations. A theorem by Pauli set in stone that any attempt at trying to understand time with the formalism in use at the time for quantum mechanics was impossible. Through studies on the logical basis of the subject and establishing different sets of axioms, more modern works were able to relax sufficiently the impositions on the concept of observables so that time could then be understood as a generalized observable. Not only that, but it was shown that the correct way to interpret eventual time measurements was through the formalism of Positive Operator Valued Measures. Latest works then were focused in the direction of constructing either direct measures and time operators, or, in the case that was followed throughout this work, to establish a procedure to construct measures. This work was then an in-depth study of the work done by Brunetti and Fredenhagen, with the intent to do the first steps in generalizing their ideas by establishing their method to be covariant with respect to Poincaré transformations.
The question of time in quantum mechanics has been a topic of intense discussion since its inception. Dating back to the works of the founders of the subject, the notion of time and how it fit in the theory was already a point with a lack of a formal explanation, especially in aspects such as the uncertainty principle and canonical commutation relations. A theorem by Pauli set in stone that any attempt at trying to understand time with the formalism in use at the time for quantum mechanics was impossible. Through studies on the logical basis of the subject and establishing different sets of axioms, more modern works were able to relax sufficiently the impositions on the concept of observables so that time could then be understood as a generalized observable. Not only that, but it was shown that the correct way to interpret eventual time measurements was through the formalism of Positive Operator Valued Measures. Latest works then were focused in the direction of constructing either direct measures and time operators, or, in the case that was followed throughout this work, to establish a procedure to construct measures. This work was then an in-depth study of the work done by Brunetti and Fredenhagen, with the intent to do the first steps in generalizing their ideas by establishing their method to be covariant with respect to Poincaré transformations.
The question of time in quantum mechanics has been a topic of intense discussion since its inception.Dating back to the works of the founders of the subject, the notion of time and how it fit in the theory was already a point with a lack of a formal explanation, especially in aspects such as the uncertainty principle and canonical commutation relations.A theorem by Pauli set in stone that any attempt at trying to understand time with the formalism in use at the time for quantum mechanics was impossible.Through studies on the logical basis of the subject and establishing different sets of axioms, more modern works were able to relax sufficiently the impositions on the concept of observables so that time could then be understood as a 'generalized' observable.Not only that, but it was shown that the correct way to interpret eventual time measurements was through the formalism of Positive Operator Valued Measures.Latest works then were focused in the direction of constructing either direct measures and time operators, or, in the case that was followed throughout this work, to establish a procedure to construct measures.This work was then an in-depth study of the work done by Brunetti and Fredenhagen, with the intent to do the first steps in generalizing their ideas by establishing their method to be covariant with respect to Poincaré transformations.
Um seculo apos a sua descoberta, o eletron continua a ser um estranho dentro do eletromagnetismo classico. Varios foram os modelos propostos para descreve-lo. Pode-se dizer que a teoria mais bem sucedida foi formulada por Dirac em 1938, muito embora apresentando problemas tais como as pre-aceleracoes e solucoes auto-acelerantes. Uma alternativa a esta teoria foi proposta na decada de 50 pelo fisico italiano P. Caldirola, tendo por hipotese basica a existencia de um intervalo fundamental de tempo, o cronon. Com base nesta hipotese Caldirola propos uma descricao baseada em equacoes a diferencas finitas equivalentes a de Dirac mas livre dos problemas por ela apresentados. A extensao da hipotese do cronon a mecânica quântica nao-relativistica, resultando em formulacoes discretizadas da equacao de Schrodinger, produziu resultados relevantes como a derivacao da massa do muon e do tau a partir da massa do eletron.
Neste trabalho, analisamos as equacoes discretas com base no formalismo de Feynman e estudamos algumas de suas solucoes. Partindo dos operadores de evolucao associados as equacoes discretas, estendemos o procedimento a representacao de Heisenberg e encontramos uma representacao que, como no caso continuo, e equivalente a de Schrodinger .
Fazemos uma analise de compatibilidade entre as varias representacoes: Schrodinger, Heisenberg e operadores de densidade. Sao estudados alguns exemplos tipicos como, por exemplo, o oscilador harmonico simples. O formalismo de operadores de densidade e utilizado para analisar o problema da medida na mecânica quântica.
Finalmente, fazemos uma abordagem da equivalencia entre o formalismo discreto e a introducao de operadores nao-hermitianos na equacao de Schrodinger continua
Abstract
Nós interpretamos a extensão space-time-symmetrical (STS) da mecânica quântica (MQ) proposta em (DIAS; PARISIO, 2017) e exploramos as previsões de seus estados “espaço-condicio nais” (EC) para potenciais arbitrários. Seguindo uma quantização alternativa, onde o tempo se torna um operador auto-adjunto e a posição um parâmetro, a extensão STS postula a existência de um novo estado quântico (intrínseco à partícula), |φ(x)⟩, definido a cada ponto no espaço. |φ(x)⟩ obedece à uma equação de Schrödinger EC que, na base do tempo, prevê o tempo ideal de chegada da partícula em x. Neste trabalho investigamos o comportamento para um potencial arbitrário da equação de autovalor do momento, que é análoga à equação de autovalor da energia na MQ usual. Verificamos que para potenciais dependentes do espaço, estados com momento bem definido dependem da posição, assim como estados com energia bem definida na MQ usual dependem do tempo para potenciais dependentes do tempo. Posteriormente, interpretamos a equação de Schrödinger EC de forma análoga à equação de Schrödinger: Dada uma função de onda EC “inicial”, φ(t|x0), a solução φ(t|x) é a amplitude de probabilidade da partícula chegar no instante t, dado que o detector é movido para uma nova posição x. Neste contexto, comparando |ψ(t)⟩ e |φ(x)⟩, os quais descrevem dados estatísticos coletados em t e x, respectivamente, concluímos que eles fornecem informações complementares. Finalmente, resolvemos a equação Schrödinger EC para um potencial arbitrário dependente do espaço e aplicamos esta solução à uma barreira de potencial. Comparando esse resultado com uma generalização da distribuição de Kijowski, concluímos que a equação de Schrödinger EC talvez deva ser reformulada para acoplar as componentes de φ(t|x), levando em consideração a interferência entre momentos positivos e negativos.
Na teoria quântica é usual se referir a probabilidade de encontrar uma partícula entre as posições x e x + dx no instante t, embora a teoria não forneça quando ocorre a detecção. Nos primórdios da teoria, Pauli apresentou um argumento refutando a ideia de introduzir um operador que descreva a probabilidade do instante em que ocorre a detecção. Mais tarde, Page e Wootters, motivados pelo fato de não se observar autoestados de superposição de energia, apresentaram um formalismo quântico não-relativista num espaço estendido em que espaço e tempo desempenham papéis equivalentes, sendo consistente com as indagações feitas por Pauli. Isso leva a uma descrição dada pela probabilidade condicional de encontrar uma partícula entre x e x + dx em relação ao tempo [t, t + dt] dado por outra partícula, chamada de partícula relógio. Essa dissertação consiste de uma revisão da proposta de Page e Wootters para abordar o problema do tempo na mecânica quântica bem como de uma análise detalhada desta utilizando de ferramentas recentes introduzidas na área de informação quântica para o tratamento de simetrias em física.
This thesis is concerned with the notions of complementarity and uncertainty encountered in quantum mechanics. Its starting point is an assessment of how these concepts have been represented and illustrated by various writers dating back to their inception. Following the survey a coherent account of the connections and contrasts between complementarity and uncertainty is developed in the context of Mach-Zehnder interferometry. The effect on the interference pattern contrast of path detection via entanglement with a probe system, is explored and a joint unsharp measurement scheme of the complementary pairs, path and interference, described. The Mach-Zehnder set-up proves sufficiently versatile to show that quantum erasure and quantitative quantum erasure constitute instances of joint unsharp measurement of complementary observables. The analysis uses the representation of observables as positive operator valued measures.Path detection and interference observation require different experimental set-ups but can be reconciled in the simultaneous unsharp measurement and preparation. This reconciliation is expressed as an uncertainty relation however the mutually exclusive feature of complementarity is not discarded. It is possible to recover strict complementarity as a limit case of the appropriate uncertainty relation.One motivation for this study is the effort some authors have made in trying to express the founding features of quantum mechanics in the form of a hierarchy of significance. Here it is shown that complementarity and uncertainty have separate identities but are not completely independent of each other. Consequently, establishing a hierarchy of these features within the present formalism of quantum mechanics is not possible.