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Precisely Predictable Dirac Observables

Part of the Fundamental Theories of Physics Book Series (volume 154)

DOI: 10.1007/1-4020-5169-7

ISBN: 978-1-4020-5168-5

Publisher: Springer Netherlands - Springer Journals

Jan 1, 2007

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Dirac Observables and ψdo-s

Cordes, Heinz Otto

Pages: 1 –35

[The Dirac equation is a first order 4×4-system of partial differential equations of the form1]
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Why Should Observables be Pseudodifferential?

Cordes, Heinz Otto

Pages: 37 –53

[The present chapter will be irrelevant1 for the mathematical deployment in succeeding chapters. We offer this material only to provide a motivation for our claim that a Dirac observable should be a self-adjoint pseudodifferential operator.]
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Decoupling with ψdo-s

Cordes, Heinz Otto

Pages: 55 –81

[Spectral theory of the Dirac Hamiltonian H of (1.0.2) has been vigorously pursued since the early 1930-s.]
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Smooth Pseudodifferential Heisenberg Representation

Cordes, Heinz Otto

Pages: 83 –116

[In this and the following chapter we will investigate time-dependence of ψdoobservables when physical states are kept constant in time. In particular we look for “smooth” dependence on t in uniform operators norms (of our weighted Sobolev spaces). Clearly the exponential operator e-iHt - for...
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The Algebra of Precisely Predictable Observables

Cordes, Heinz Otto

Pages: 117 –148

[In this chapter we will start by discussing a precise theorem giving a necessary and sufficient condition for smoothness of the (inverse) Heisenberg transform, with some “framing conditions” added. Note, the symbol classes ψcm carry a “topology” (in fact, a Frechet topology), defined by the sup...
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Lorentz Covariance of Precise Predictability

Cordes, Heinz Otto

Pages: 149 –191

[It is known and very essential to Dirac’s theory that it is compatible with a transformation of coordinates under the laws of special relativity. In other words, if we change the (space-time) coordinate system by a Lorentz transform then Dirac’s equation remains intact - except for the...
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Spectral Theory of Precisely Predictable Approximations

Cordes, Heinz Otto

Pages: 193 –235

[In this chapter we will analyze the spectral theory of a few of our “precisely predictable approximations” of dynamical observables, which are not precisely predictable. Let us emphasize again: We are not attempting to redefine these observables. The approximations only are good for calculating...
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Dirac and Schrödinger Equations; a Comparison

Cordes, Heinz Otto

Pages: 237 –257

[In this chapter we shall venture beyond the Dirac equation - so far our only object of study - and try reflecting on other wave equations in Quantum Mechanics. Perhaps we have fortified our opinion that - for the hydrogen atom - and, more generally, any “one-particle problem” considering a...

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