414
E. J. Brändas
Without going into technical details, it is clear that rigorous analytic continuation [37]
implicates an inevitable loss of information due to the mandatory modifications (i–
vii). Take e.g. point (v). The complex scaling operation is unbounded and its domain
must be restricted to a smaller class of functions, or to a dense subspace of Hilbert
space [69], before the dilated non-normal operator again is extended back to the full
Hilbert space,
42 see [70]. This is the essential reason behind the previously mentioned
deficiency rule that governs the conversion of stationary QM to the dynamics of
evolution.
For the interested reader one should mention, as previously brought up, the issues
related to the positivity condition in the theory of irreversibility [71]. It was soon
realized that the advantage of rigorously applying complex scaling techniques to
atomic and molecular systems revealed a possible drawback in that the quantum
chemical Hamiltonian, containing the attractive Coulomb potential, does not generate
a contractive semigroup or in essence would violate the positivity condition (5.3). The
problem was discussed and solved at the time [20] although the topic unfortunately
appeared a bit confused for a while. The proof involves the knowing that the scaling
operator can be identified as a Lyapunov converter on the Nelson class [72], i.e.
mapping an isometric- to a contractive semigroup.
43 The theorem is of principal
importance, since the long-range Coulomb potential is inevitable for the emergence
of so-called coherent-dissipative structures in condensed amorphous phases [21, 73].
6 Life and Quantum Theory
In the discussions of the physical origin of the time asymmetry in the Universe,
44 a
general solution to the problem is often suggested as the appearance of a low entropy
initial condition of a system governed by time symmetric mathematical equations
[74]. A different view is proposed and discussed in Ref. [75] and references therein.
We will not discuss these matters in any detail here, however, as an illustration of
our intentions one should say that a piece of ice in a glass of water, an example of
Reichenbach’s notion of branch systems, would not qualify as an UD, since it would
not satisfy d S U D < 0.
With the cosmological aspect in mind, it is inevitable that the issue of the appropriate boundary condition turns up. Traditional QM is based on Schrödinger’s quantization as an eigenvalue problem, which, combined with the unique behaviour of the
42 This is a technical problem, yet of crucial importance. Suffice it to say that Nelson’s class of
analytic vectors, dense in Hilbert space, is a key ingredient in the proof of the Balslev-Combes
theorem [37], see also Ref. [70] for more details on its use.
43 The proof involves the notion of quasi-isometry. The problem is due to the fact that the numerical
range of a complex scaled attractive Coulomb potential cannot be proven to be restricted to the lower
complex halfplane, a necessary and a sufficient condition for the fulfillment of the Hille-Yosida
theorem [20].
44 The discussions occur in an interdisciplinary poll [74] that concerns the reconciliation of
profoundly conflicting facts regarding time and the fundamental processes going on in the world.
E. J. Brändas
Without going into technical details, it is clear that rigorous analytic continuation [37]
implicates an inevitable loss of information due to the mandatory modifications (i–
vii). Take e.g. point (v). The complex scaling operation is unbounded and its domain
must be restricted to a smaller class of functions, or to a dense subspace of Hilbert
space [69], before the dilated non-normal operator again is extended back to the full
Hilbert space,
42 see [70]. This is the essential reason behind the previously mentioned
deficiency rule that governs the conversion of stationary QM to the dynamics of
evolution.
For the interested reader one should mention, as previously brought up, the issues
related to the positivity condition in the theory of irreversibility [71]. It was soon
realized that the advantage of rigorously applying complex scaling techniques to
atomic and molecular systems revealed a possible drawback in that the quantum
chemical Hamiltonian, containing the attractive Coulomb potential, does not generate
a contractive semigroup or in essence would violate the positivity condition (5.3). The
problem was discussed and solved at the time [20] although the topic unfortunately
appeared a bit confused for a while. The proof involves the knowing that the scaling
operator can be identified as a Lyapunov converter on the Nelson class [72], i.e.
mapping an isometric- to a contractive semigroup.
43 The theorem is of principal
importance, since the long-range Coulomb potential is inevitable for the emergence
of so-called coherent-dissipative structures in condensed amorphous phases [21, 73].
6 Life and Quantum Theory
In the discussions of the physical origin of the time asymmetry in the Universe,
44 a
general solution to the problem is often suggested as the appearance of a low entropy
initial condition of a system governed by time symmetric mathematical equations
[74]. A different view is proposed and discussed in Ref. [75] and references therein.
We will not discuss these matters in any detail here, however, as an illustration of
our intentions one should say that a piece of ice in a glass of water, an example of
Reichenbach’s notion of branch systems, would not qualify as an UD, since it would
not satisfy d S U D < 0.
With the cosmological aspect in mind, it is inevitable that the issue of the appropriate boundary condition turns up. Traditional QM is based on Schrödinger’s quantization as an eigenvalue problem, which, combined with the unique behaviour of the
42 This is a technical problem, yet of crucial importance. Suffice it to say that Nelson’s class of
analytic vectors, dense in Hilbert space, is a key ingredient in the proof of the Balslev-Combes
theorem [37], see also Ref. [70] for more details on its use.
43 The proof involves the notion of quasi-isometry. The problem is due to the fact that the numerical
range of a complex scaled attractive Coulomb potential cannot be proven to be restricted to the lower
complex halfplane, a necessary and a sufficient condition for the fulfillment of the Hille-Yosida
theorem [20].
44 The discussions occur in an interdisciplinary poll [74] that concerns the reconciliation of
profoundly conflicting facts regarding time and the fundamental processes going on in the world.