Abstract
The superposition principle forms the most fundamental kinematical concept of quantum theory. Its universality seems to have first been postulated by Dirac as part of the definition of his “ket-vectors” , which he proposed as a complete1 and general concept to characterize quantum states, regardless of any basis of representation. They were later recognized by von Neumann as forming an abstract Hilbert space. The inner product (also needed to define a Hilbert space, and formally indicated by the distinction between “bra” and “ket” vectors) is not part of the kinematics proper, but required for the probability interpretation, which may be regarded as dynamics (as will be discussed) . The third Hilbert space axiom (closure with respect to Cauchy series) is merely mathematically convenient, since one can never decide empirically whether the number of linearly independent physical states is infinite in reality, or just very large.
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This conceptual completeness does not, of course, imply that all degrees of freedom of a considered system are always known and taken into account. It only means that, within quantum theory (which, in its way, is able to describe all known experiments), no more complete description of the system is required or indicated. Quantum mechanics lets us even understand why we may neglect certain degrees of freedom, since gaps in the energy spectrum often “freeze them out”.
The empirically correct “pre-quantum” configurations for fermions are given by spinor fields on space, while the apparently observed particles are no more than the consequence of decoherence by means of local interactions with the environment (see Chap. 3) . Field amplitudes (such as Ψ(r) seem to form the general arguments of the wave function(al) Ψ, while space points r appear as their “indices” — not as dynamical position variables. Neither a “second quantization” nor a wave-particle dualism are required on a fundamental level. N-particle wave functions may be obtained as a non-relativistic approximation by applying the superposition principle (as a “quantization procedure”) to these apparent particles instead of the correct pre-quantum variables (fields), which are not directly observable for fermions. The concept of particle permutations then becomes a redundancy (see Sect. 9.4). Unified field theories are usually expected to provide a general (supersymmetric) pre-quantum field and its Hamiltonian.
Mere logic does not require, however, that the frequencies of events on the screen which follow the observed passage through slit 1 of a two-slit experiment, say, are the same as those without measurement, but with slit 2 closed. This distinction would be relevant in Bohm’s theory (Bohm 1952) if it allowed nondisturbing measurements of the (now assumed) passage through one definite slit (as it does not in order to remain indistinguishable from quantum theory) . The
fact that these two quite different situations (closing slit 2 or measuring the passage through slit 1) lead to exactly the same subsequent frequencies, which differ entirely from those that are defined by this theory when not measured or selected, emphasizes its extremely artificial nature (see also Englert et al. 1992, or Zeh 1999). The predictions of quantum theory are here simply reproduced by leaving the Schrödinger equation unaffected and universally valid, identical with Everett’s assumptions (Everett 1957) . In both these theories the wave function is (for good reasons) regarded as a real physical object (cf. Bell 1981).
It would be sufficient, for this purpose, to use an internal “environment” (unobserved degrees of freedom), but the assumption of a closed system is in general unrealistic.
The popular textbook argument that observables must be hermitean in order to have real expectation values is successful but wrong. The essential requirement for an observable is its diagonalizability, which allows even the choice of a complex scale an, if convenient.
Observables are axiomatically postulated in the Heisenberg picture and in the algebraic approach to quantum theory. They are also presumed (in order to define fundamental expectation values) in Chaps. 6 and 7. This may be pragmatically appropriate, but appears to be in conflict with attempts to describe measurements and quantum jumps dynamically — either by a collapse (Chap. 8) or by means of a universal Schrödinger equation (Chaps. 1–4) .
Some authors seem to have taken the phenomenological collapse in the microscopic system by itself too literally, and therefore disregarded the state of the measurement device in their measurement theory (see Machida and Namiki 1980, Srinivas 1984, and Sect. 9.1) . Their approach is based on the assumption that quantum states must always exist for all systems. This would be in conflict with quantum nonlocality, even though it may be in accordance with early interpretations of the quantum formalism.
Thus also Bohr (1928) in a subsection entitled “Quantum postulate and causality” about “the quantum theory” : “. . . its essence may be expressed in the socalled quantum postulate, which attributes to any atomic process an essential discontinuity, or rather individuality, completely foreign to classical theories and symbolized by Planck’s quantum of action” (my italics) . The later revision of these early interpretations of quantum theory (required by the important role of entangled quantum states for much larger systems) seems to have gone unnoticed by many physicists.
Bennett did not define physical entropy to include that of the microscopic ensemble a,b, since he regarded this variable as “controllable” — in contrast to the thermal (ergodic or irrelevant) property A’,B’.
Proposed decoherence mechanisms involving event horizons (Hawking 1987, Ellis, Mohanty and Nanopoulos 1989) would either have to postulate such a fundamental violation of unitarity, or merely represent a specific kind of environmental decoherence (entanglement beyond the horizon) — see Sect. 4.2.5. The most immediate consequence of quantum entanglement is that an exactly unitary evolution can only be consistently applied to the whole universe.
As Bell (1981) once pointed out, Bohm’s theory would instead require consciousness to be psycho-physically coupled to certain classical variables (which this theory postulates to exist). These variables are probabilistically related to the wave function by means of a conserved statistical initial condition. Thus one may argue that the “many minds interpretation” merely eliminates Bohm’s unobservable and therefore meaningless intermediary classical variables and their trajectories from this psycho-physical connection. This is possible because of the dynamical autonomy of a wave function that evolves in time according to a universal Schrödinger equation, and independently of Bohm’s classical variables. The latter cannot, by themselves, carry memories of their “surrealistic” histories. Memories are solely in the quasi-classical wave packets that effectively guide them, while the other myriads of “empty” Everett world components (criticized for being “extravagant” by Bell) exist as well in Bohm’s theory. Barbour (1999), in his theory of timelessness, proposed in effect a static Bohm theory. It eliminates the latter’s formal classical trajectories, while preserving a concept of memories without a history (“time capsules” — see also Chap. 6 of Zeh 2001).
Another aspect of this observer-relatedness of the observed world is the concept of a presence, which is not part of physical time. It reflects the empirical fact that the subjective observer is local in space and time.
In the Schmidt basis, interference terms are exactly absent by definition. Hepp (1972) used the formal limit N → ∞ to obtain this result in a given basis (though this may require infinite time) . However, the global state always remains one pure superposition. The Schmidt representation has therefore been used instead to specify the Everett branches, that is, to define the ultimate “pointer basis” (bserver) for each observer (cf. Zeh 1973, 1979, Albrecht 1992, 1993, Barvinsky and Kamenshchik 1995). It is also used in the “modal interpretation” of quantum mechanics (cf. Dieks 1995).
Such superluminal “phenomena” are reminiscent of the story of Der Hase und der Igel (the race between The Hedgehog and the Rabbit), narrated by the Grimm brothers. Here, the hedgehog, as a competitor in the race, does not run at all, while his wife is waiting at the end of the furrow, shouting in low German “Ick bin all hier!” (“I’m already here!”). Similar arguments hold for “quantum teleportation” , where an appropriate nonlocal state that contains the state to be ported as a component has to be well prepared (cf. also Vaidman 1998) . Experiments clearly support this view of a continuous evolution instead of quantum jumps (cf. Fearn, Cock, and Milonni 1995). Teleportation would be required if reality were local and physical properties entered existence “out of the blue” in fundamental quantum events. Therefore, this whole concept is another artifact of the Copenhagen interpretation.
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Zeh, H.D. (2003). Basic Concepts and Their Interpretation. In: Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05328-7_2
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DOI: https://doi.org/10.1007/978-3-662-05328-7_2
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