2.11 Quantum theory, historical
53
How clearly Bohr saw from the very outset the principal problem can be
attested by a quote from his 1913 report to the Danish Academy of Sciences:
Before closing, I only wish to say that I hope I have expressed myself
su
4.3 Space and time
95
elements of the symmetry group of the ur. Consequently position space ought
to be a homogeneous space of SU (2).
The most natural homogeneous space of SU (2) is SU (2) itself. It is an
S3 , thus isomorphic to the spatial part of the
62
3 Probability and abstract quantum theory
using a preconceived concept of probability. These two concepts, experience
and probability, are not in a relationship of hierarchical subordination.
In practice every application of the theory of errors implie
4.2 Reconstruction of quantum theory via variable alternatives
87
subobject. Then the state space of a free particle is the direct sum of infinitely
many finite-dimensional state spaces of subobjects. We will interpret the open
finitism so that the state
4.3 Space and time
97
wish to describe this in the tensor space of the urs. As the simplest model
(2)
we consider T , the space of symmetric tensors over the binary vector space
V (2) . We consider the largest Lie group for which unitary representations i
5.1 Open questions
E. alternatives.
one
expansion postulate
107
108
5 Models of particles and interaction
tensor space of the urs
p
p
p>
p
p
p
interaction
p>
p
uniquely
program
homogeneous space
T
model of quantum electrodynamics
1
elementary particles
a
4.3 Space and time
99
Here the bar denotes complex conjugation and the are the Pauli matrices
with 0 = 1. The k satisfy the relation
k k = (k 0 )2 (k)2 = 0
(4.29)
with k = (k 1 , k 2 , k 3 ). SU(2) rotates the vector k, two elements of opposite
sign produ
4.3 Space and time
103
for the special Lorentz transformations Ni4 , and in (4.38) for the special conformal transformations Ki . In Minkowski space it also holds for the space and
time translations Pi (1 = 1, 2, 3, 4), in the anti-de Sitter space for the
4.1 Concrete quantum theory
83
also evolutionary as the reason for our form of perception.3 But precisely these
empirical facts about the spatiality of empirically known objects, as well as
about the form of our perception, are what we do not want to pres
4.2 Reconstruction of quantum theory via variable alternatives
89
4.2.2 Ur alternatives
1. Theorem of the logical decomposition of alternatives. An n-fold
alternative can be mapped into the Cartesian product of k binary alternatives
with 2k n.
Comment. Th
4
Quantum theory and spacetime
4.1 Concrete quantum theory
In the previous chapter we indicated a path to the reconstruction of abstract
quantum theory, i.e., the quantum theory of arbitrary alternatives and objects,
and arbitrary forces. Now our goal is
4.3 Space and time
93
4.3 Space and time
Du siehst, mein Sohn, zum Raum wird hier die Zeit.
Wagner, Parsifal, 1. Akt5
4.3.1 Realistic hypothesis
If in the context of abstract quantum theory, especially from the ur hypothesis, there should arise a universa
On Weizs
ackers philosophy of physics
xxxiii
Books on Weizs
ackers philosophy of physics
P. Ackermann, W. Eisenberg, H. Herwig, and K. Kannegieer (eds.)
(1989). Erfahrung des DenkensWahrnehmung des Ganzen: Carl
Friedrich von Weizs
acker als Physiker und
6
1 Introduction
equilibrium of a continuum, explains the stability and identity of the atoms
of an element, and oers a universal framework for physics.
The physics of the past century also began to fuse the other two foundations of classical mechanics, s
2.8 The relativity problem
37
same time of a Euclidean space (per physical semantics). The physical theory
created this way would therefore be semantically inconsistent.
Any physicist knows how to resolve this apparent problem. The new theory
is supposed
2.2 Classical point mechanics
!
21
fkij ,
(2.3)
xjk xik mi mj
,
2
rij
rij
(2.4)
fik =
j=i
fkij = G
with
2
rij
=
!"
k
#2
xik xjk ,
(2.5)
and a universal constant G. Their positions at a certain time t0 can be determined empirically.6 These positions are th
2.10 General theory of relativity
49
Viewed in this way, the theory is then a stronger version of Einsteins intention,
insofar as it no longer needs to postulate the equivalence principle, deriving
it instead from topology and local Lorentz invariance. Ei
4.2 Reconstruction of quantum theory via variable alternatives
91
and which allow a distinction between types of urs. Only with this reservation is the ur hypothesis an epistemologically justified consequence of abstract
quantum theory.
About the terminol
2.12 Quantum theory, plan of reconstruction
55
the latest closed theory oers the best hope of being reconstructed from completely general principles. We will propose postulates which do not utilize at
all any of the special concepts of mechanics, field th
76
3 Probability and abstract quantum theory
that an event e
simultaneously).
j from each alternative occurs (not necessarily !
This is an element of the combined alternative which has n = n elements.
Now N objects also define a total object of which they
78
3 Probability and abstract quantum theory
We can denote this postulate equivalently by the more abstract term postulate of expansion. The connection between the two names is as follows. Every
alternative of final propositions is expanded through this p
5
Models of particles and interaction
5.1 Open questions
5.1.1 Recapitulation
We begin with a review of the system of theories. What in addition have we
learned by reconstructing quantum theory and the special theory of relativity?
How does the unity of p
60
3 Probability and abstract quantum theory
interpretation of probability stems from the idea that experience can be
treated as a given concept and probability as a concept to be applied to experience. This is what I call a mistaken epistemological hiera
4.2 Reconstruction of quantum theory via variable alternatives
85
4.1.2 Particles
Here we only remark that the existence of particles which can be described
as point masses follows according to Wigner (1939) from relativistic quantum
theory: their state s
2.12 Quantum theory, plan of reconstruction
57
fundamental classical physics. With this topic we begin Chap. 7. Second, directly to the quantum theoretical amplification of the theory of probability,
to the theory of evolution, which then leads directly t
2.11 Quantum theory, historical
51
because it is only a small part of the whole. Perhaps our present concepts are
utterly inadequate to describe the entirety of the universe; perhaps the very
concept of the entirety of the universe is inconsistent.
This w
70
3 Probability and abstract quantum theory
(p1 + p2 )N =
!
n1
we obtain
N!
pn1 pn2 = 1
n1 ! n2 ! 1 2
|cn1 n2 |2 =
N!
.
n1 ! n2 !
(3.12)
(3.13)
The numbers n1 and n2 can be interpreted as eigenvalues of the operators n1
and n2 , whose action on consists
72
3 Probability and abstract quantum theory
evolutionary knowledge is, as human knowledge, subject to the conditions of
such knowledge, as studied in epistemology. Also, the back of the mirror we
only see in the mirror. But the mirror in which we see the
64
3 Probability and abstract quantum theory
For the mathematical theory we can adopt Kolmogorovs text literally,
changing only some notation:
Let M be a set of elements , , , . . . which we call elementary events, and
F a set of subsets of M ; the elemen
4.3 Space and time
101
The choice of this group can be justified as follows. It is the largest subgroup of Sp(4,R ) that leaves invariant the operator
s=
1
2
(n1 + n2 n3 n4 ).
(4.34)
(is itself belongs as generator to SP(4). Thus, strictly speaking, SO(4,