144
5 Models of particles and interaction
1. Finding the field equations,
2. determining the constant .
1) As in other theories we expect the form of the equation to be already
determined to a large extent by invariance requirements. Here we need merely
r
6.3 Relativistic particles
lvac =
! ! (1)p[1]+p[2] "
#
"
#
f 1, 3, p[1] f 2, 4, p[2] |
p[1]!p[2]!
175
(6.57)
p[1] p[2]
where the sums over p[1] and p[2] extend in each case from zero to infinity.
Clearly the statement that there are no particles in Minkow
166
6 Cosmology and particle physics (by Th. G
ornitz)
Fig. 6.10. The probability distribution for H0 t0 given the SN Ia observations is
tightly constrained to 0.960.04, and an approximating Gaussian curve (from Tonry
et al., 2003, Fig. 15).
At present th
200
7 Irreversibility and entropy
on the system could make the inference from documents more dubious, the
knowledge of special causal connections can make them more precise. But in
the end, the validity of our argument is not aected, on the average over a
234
8 Information and evolution
More recent investigations of the authors show how large the domain of
biological and societal phenomena is that can be described by these qualitative
concepts.
8.7 Biological preliminaries to logic
8.7.1 Methodological
Thi
7.2 A model of irreversible processes
w
=
N2
n2
.
2n(N n)
191
(7.21)
The derived probabilities of the four indicated cases are products, where we
denote the denominator by N :
w = w
w
=
w = w =
n4
,
N 2
w
w+
w = w+
w+
=
n2 (N n)2
=
.
N 2
(N n)4
,
N 2
(7.
6.2 Ur-theoretic vacuum and particle states
169
state (6.18) of quantum bits for the interior. For according to Bekenstein and
Hawking the amount of information in the interior of a black hole can be
estimated which, as indicated, in its concrete form is
9.1 About the history of the interpretation
251
miracleand appears to me as a miracle even today. This is the highest form
of musicality in the sphere of thought (See Sect. 2.11a).
It was given to Heisenberg and not Bohr to find the solid ground on
which
9.3 Paradoxes and alternatives
281
have arrived on Earth, either the value of x1 or the value of p1 . In an admissible Lorentzian reference frame, e.g., the rest frame of the center of mass of
both objects, an observer measures on Sirius, at an instant th
8.5 Evolution as growth of potential information
219
In biology or in sociology, the idea of evolution is, on the contrary, closely
associated with an increase in organization giving rise to the creation of more
and more complex structures. (p. 287)
Are t
188
7 Irreversibility and entropy
the k-th move; the number of black balls in B is also nk , and the number of
black balls in A and white ones in B is N nk .
Our model in fact assumes the elementary laws b) to be statistical (equal
probability for each ba
10.2 The stream of information in quantum theory
303
excess information in the first answer is simply the information contained in
the concept of a living being (p. 353).6
Both answers thus explain meaningful but dierent concepts of information needed in
216
8 Information and evolution
information Ik measures the newsworthiness of the event xk if a decision
produces just this value xk . An event that has occurred is the less newsworthy,
the more likely it was before; if it occurred with certainty, one wil
6.1 Quantum theory of abstract binary alternatives and cosmology
157
"N
!
Decomposing the representation 2D1/2
of SU (2) into its irreducible
components, we obtain a ClebschGordan series of the following form
%
|N/2|
# N !(N + 1 2j) $
"N
!2
2|N/2|2j+1
D1/
228
8 Information and evolution
system is to be interpreted. From the original Kant-Laplace nebula where
hydrodynamic and chemical processes occur, there ultimately emerge separate
planets, acting upon one another solely via gravitaiton; the stability the
194
7 Irreversibility and entropy
theory were justified, then at that time it was legitimate to predict an average
increase in entropy for events which were then in the future. If afterwards
this increase in entropy had not happened on the average, then o
9.2 The semantic consistency of quantum theory
263
the dual rule for the change of .13 No contradiction arises if we follow these
simple arguments. But we can now see that the name state for is misleading. is a catalog of knowledge that follows from one o
160
6 Cosmology and particle physics (by Th. G
ornitz)
where we have included the constants in (6.23) and (6.24) (G is the gravitational constant). The constant K has the value K = 1 for a closed, K = 0 for
a flat, and K = 1 for a hyperbolic position spac
248
9 The problem of the interpretation of quantum theory
particle conforms to the laws of probability, but the probability itself is propagated in accordance with the law of causality (Jammer, p. 40). Orthodox
quantum theory followed him there.
Neverthel
6.1 Quantum theory of abstract binary alternatives and cosmology
163
In astronomical observations one uses the luminosity distance DL , which is
obtained from the absolute luminosity L and the apparent luminosity :
(6.45)
2
.
L = 4DL
DL is a function of t
206
7 Irreversibility and entropy
from the fact, as repeatedly emphasized, that thereby the distinction of the
respective present is not explained) here lies the circle that the initial state of
the universe guarantees the Second Law only for those physic
9.1 About the history of the interpretation
257
of phenomenon. Time should not be introduced primarily mathematically as a
continuum of numbers but in the grouping of present, past, and future which
forms the basis of our experience. Irreversibility is ex
7.1 Irreversibility as problem
185
attain a precise meaning only after an accurate specification of what must be
understood under object, environment, state, quantity, change,
A sucient mathematical condition for reversibility is that there are, as
state
138
5 Models of particles and interaction
with large p. The sharpness of the rest mases ought then to manifest itself as
the distinction of certain values of p. Only if such values are distinguished can
one hope to actually find field equations for distin
254
9 The problem of the interpretation of quantum theory
The question then was from which classical theory one ought to start. For
matter one started with classical point mechanics and obtained through quantization Schr
odingers wave mechanics in configu
9.1 About the history of the interpretation
245
is a complete alternative, if we only interpret both concepts broadly enough.
Particles or bodies are localized objects; fields, especially waves, are states
which in principle extend over all space. Accordi
5.5 Elementary particles
135
In Sect. 5.4e we found arguments for the conjecture that, at least in electrodynamics, the value of the coupling constant depends on the value of the
rest mass. This question, however, exceeds the scope of the present stage of
266
9 The problem of the interpretation of quantum theory
which one may disregard, in a certain approximation, the noncommutativity.
But we do not follow Neumanns leveling assumption (Sect. 9.1h) that every
bounded self-adjoint operator represents a permi