Introduction and Some Problems Encountered in the Construction of a Relativistic Quantum Theory
One of the deepest and most difficult problems of theoretical physics in the past century has been the construction of a simple, well-defined one-particle theory which unites the ideas of quantum mechanics and relativity. Early attempts, such as the const
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Introduction and Some Problems Encountered in the Construction of a Relativistic Quantum Theory
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States in Relativistic Quantum and Classical Mechanics
One of the deepest and most difficult problems of theoretical physics in the past century has been the construction of a simple, well-defined one-particle theory which unites the ideas of quantum mechanics and relativity. Early attempts, such as the construction of the Klein-Gordon equation and the Dirac equation were inadequate to provide such a theory since, as shown by Newton and Wigner (1949), they are intrinsically non-local, in the sense that the solutions of these equations cannot provide a well-defined local probability distribution. This result will be discussed in detail below. Relativistic quantum field theories, such as quantum electrodynamics, provide a manifestly covariant framework for important questions such as the Lamb shift and other level shifts, the anomalous moment of the electron and scattering theory, but the discussion of quantum mechanical interference phenomena and associated local manifestations of the quantum theory are not within their scope; the one particle sector of such theories display the same problem pointed out by Newton and Wigner since they satisfy the same one-particle field equations. On the other hand, the nonrelativistic quantum theory carries a completely local interpretation of probability density; it can be used as a rigorous basis for the development of nonrelativistic quantum field theory, starting with the construction of tensor product spaces to build the Fock space, and on that space to define annihilation and creation operators (e.g., Baym 1969). The development of a manifestly covariant single particle quantum theory, with local probability interpretation, could be used in the same way to develop a rigorous basis for a relativistic quantum field theory which carries such a local interpretation. A central problem in formulating such a theory is posed by the requirement of constructing a description of the quantum state of an elementary system (e.g., a “particle”) as a manifestly covariant function on a manifold of observable coordinates which belongs to a Hilbert space. The essential properties of the quantum theory, such as the notions of probability, transition amplitudes, linear superposition, observables and their expectation values, are realized in terms of the structure of a Hilbert space. © Springer Science+Business Media Dordrecht 2015 L.P. Horwitz, Relativistic Quantum Mechanics, Fundamental Theories of Physics 180, DOI 10.1007/978-94-017-7261-7_1
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1 Introduction and Some Problems Encountered …
Nonrelativistic quantum mechanics, making explicit use of the Newtonian notion of a universal, absolute time, provides such a description in terms of a square integrable function over spatial variables at a given moment of this Newtonian time. This function is supposed to develop dynamically, from one moment of time to another, according to Schrödinger’s equation, with some model Hamiltonian operator
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