The Physical Basis of The Direction of Time (The Frontiers ...

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172 6 The Time Arrow in Quantum Cosmology tation with respect to external observers is obviously ruled out, since the very concept of cosmology presumes an objective (though in principle hypothetical) reality. Quantum field theory has instead traditionally been used and confirmed as a method for calculating S-matrix elements, which describe probabilities for scattering events. This amounts to applying a collapse of the wave function after each elementary scattering process, and it would be insufficient for consistently describing objects which make up the Universe, such as condensed matter, complex systems (including measurement devices and observers), macroscopic fields, and global spacetime structure. The general quantum framework is usually applied in the form of a ‘quantization’ of a classical theory (see Sect. 4.1.1) – in particular of the mechanics of particles, which are kinematically described as space points. By quantization I mean here 1 the application of the superposition principle to the elements of a classical configuration space (thus defining a wave function on it), and the construction of the corresponding quantum Hamiltonian by replacing variables and their canonical momenta by operators acting on wave functions. The second part is ambiguous because of the factor ordering problem. We can now re-interpret this quantization procedure as the conceptual reversal of a physical decoherence process that led to the classical appearance of the system under consideration. This explains why this quantization cannot be expected to define a unique result, but requires further empirical input. The quantization of many-particle mechanics leads non-relativistically ‘back’ to a consistent and successful quantum theory: quantum mechanics. Some other ‘particle’ properties (such as spin or isotopic spin) have no similar classical correspondence. The quantization of classical fields in this canonical way leads to wave functionals on the configuration space for field amplitudes. It does not in general directly define a consistent quantum theory, although it can often be rendered consistent by a mere renormalization of its fundamental parameters. This is evidence that a fundamental quantum theory may be quite independent of any classical theory that could be quantized in this way. For example, relativistic quantum mechanics led to the discovery that field amplitudes of not classically observed fermion fields rather than particle positions define the correct arena for the wave function(al) – an approach that is somewhat misleadingly called a ‘second quantization’, since the fermion fields were first discovered as effective ‘single-particle wave functions’ (see Zeh 2003). The underlying fields (on space) define a local basis (the ‘stage’ for quantum 1 This interpretation is quite different from the original and literal meaning of the term ‘quantization’ as a discretization of certain quantities. For example, ‘light quanta’ can be understood as a consequence of the eigenvalue problem in terms of wave functions for the amplitudes of free field modes, dynamically described as harmonic oscillators. These fundamental aspects of quantum theory are often hidden behind a collection of recipes to perform calculations (such as perturbation theory in terms of Feynman graphs). In particular, a ‘quantization of time’ (Sect. 6.2) does not require a quantum of time – just as the quantization of particle motion does not require a quantum of length (or a spatial lattice).
6 The Time Arrow in Quantum Cosmology 173 dynamics) that spans the required Hilbert space. This structure permits the formulation of local dynamics by means of a Hamiltonian density in spite of generically nonlocal states. It may therefore be useful – though also dangerous and certainly insufficient – to investigate mathematical models for a unified quantum field theory solely by investigating certain classical fields on three- or higher-dimensional spaces, rather than consistently taking into account their quantum nature from the beginning (for instance in terms of wave functionals of these fields as representing the true quantum reality). Extrapolating unitary dynamics to the whole Universe requires an Everett type interpretation (see Sect. 4.6). Hugh Everett (1957) seems to have first seriously considered a wave function of the Universe, 2 that must then include internal observers. Although he may have had in mind the quantization of general relativity with its cosmological aspects, Everett applied his ideas, which were based on a time-dependent Schrödinger equation, to nonrelativistic quantum theory. His main interpretational obstacle was the entanglement arising from measurements described by means of von Neumann’s unitary interaction (4.32). This led him to his ‘extravagant’ interpretation (in Bell’s words) in terms of many quasi-classical ‘branches’ of the world, which are separately experienced, but are all assumed to exist in one superposition that defines the true and dynamically consistent quantum world. Beyond measurements proper and occasional interactions he does not seem to have regarded entanglement as particularly important (see Tegmark 1998). The quantitative considerations reviewed in Sect. 4.3 demonstrate that uncontrollable ‘measurement-like’ interactions with the environment are essential and unavoidable for almost all systems under all realistic circumstances. Strong entanglement is, therefore, a generic aspect of quantum theory. The more macroscopic a system, the stronger its entanglement with its environment. The concept of a (pure) quantum state can be consistently applied only to the Universe as a whole (Zeh 1970, Gell-Mann and Hartle 1990). This seems to be a far more powerful argument for the need of quantum cosmology than an attempt to construct a unified quantum field theory. The second pillar of physical cosmology is general relativity. It is empirically confirmed only as a classical theory, but this fact can be well understood by decoherence again (see Sects. 4.3.5 and 6.2.2). Exactly classical gravity would lead to inconsistencies with the uncertainty principle. Applying the quantization rules to the Hamiltonian formalism of general relativity (described in Sect. 5.4) leads to a non-renormalizable ‘effective’ quantum gravity that cannot be exact, but may be expected to be appropriate as a low energy limit. This readily allows us to discuss a number of important novel conceptual problems that must come up, in particular the need for a ‘quantization of time’ (Sect. 6.2). 2 Thibault Damour (2006) has recently presented evidence that Everett was originally stimulated by remarks Albert Einstein made about quantum theory during his last seminar, given at Princeton in 1954.
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the frontiers collection
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H. Dieter Zeh THE PHYSICAL BASIS OF
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Preface Four previous editions of t
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Contents Introduction .............
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Introduction The asymmetry of Natur
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Introduction 3 tion of time (thus a
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Introduction 5 1. Radiation. In mos
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Introduction 7 language is loaded w
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Introduction 9 between temporal and
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12 1 The Physical Concept of Time i
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18 2 The Time Arrow of Radiation Th
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References 223 Smoluchowski, M. Rit
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References 225 Zeh, H.D. (1971): On
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Index absorber, 18, 24-28, 36-38, 5
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Index 229 forward determinism, 66,
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Index 231 stochastic process, see d
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