Introducing standard notions of probability theory, the fundamental sample space of the present theory is given
by all possible results of position measurements, i.e. it may be identified with the set of real numbers . This
set
may also be identified with the possible values of a random variable “position measurement”
(whose name should strictly speaking differ from
but we shall neglect such differences here). The
basic probability measure which assigns a probability to each event (subspace of
) is given by
. According to standard probability theory the field
defined by (8) is itself a
random variable. We may consider it as a function of the random variable
(denoting “position
measurement”) or as a random variable defined independently on the fundamental event space
; it makes no difference. Its probability density is uniquely determined by
and
the function
. In order to avoid confusion of names it may be useful to denote the
derivative of
with respect to
by
instead of
. Thus
and the notation
indicates that a random variable
defined by the function
exists (the time variable will sometimes be omitted for
brevity).
In order to study this important point further, we rewrite the standard result for the probability density of
in a form more appropriate for physical considerations (a form apparently not easily
found in textbooks on probability). For the simplest possible situation, a denumerable sample space with
elements
, a probability measure
, and a invertible function
, the probability that an event
occurs is obviously given by
. This
result is the starting point to obtain
, the probability density of a continuous random
variable
, which is defined by a non-invertible function
. It is given
by [43]
![]() | (9) |
where denotes the
solutions (the number of solutions depends on
) of the equation
. Using a well-known formula for Dirac‘s delta function
, applied
to the case where the argument of
is an arbitrary function, Eq.(9) may be rewritten in the
form
![]() | (10) |
where we came back to our original notation, writing down the dependencies of
and
and
replacing
by
.
The representation (10) reveals very clearly a hybrid nature of random variables defined as (nontrivial)
functions on the event space . They are partly defined by a probabilistic quantity [namely
] and
partly by a deterministic relation [namely
]. The deterministic nature of the latter is expressed by the
singular (delta-function) shape of the associated probability. Such densities occur in classical statistics, i.e. in
type 2 theories; Eq. (10) may obviously be obtained by performing an integration over
of the classical phase
space probability density
. Considered from an operational
point of view, the hybrid nature of random variables may be described as follows. Deterministic
predictions for random variables
are impossible, as are deterministic predictions
for the original variables
. But once a number
has been observed in an experiment, then
the value of
is with certainty given by the defining function
. If no
such relation exists, this does not necessarily imply that
and
are completely independent.
Many other more complicated (’nonlocal’ or ’probabilistic’) relations between such variables are
conceivable.
We formulated general conditions comprising both type 2 and type 3 theories. Thus, as far as this general framework is concerned we can certainly not dispense with the standard notion of random variables, which are basic ingredients of type 2 theories; such variables will certainly occur as special (type 2) cases in our formalism. But, of course, we are essentially interested in the characterization of type 3 theories and the form of Eq. (10) shows that the standard notion of random variable is not necessarily meaningful in a type 3 theory. Thus we will allow for the possibility of random variables which are not defined by deterministic relations of the standard type, as functions on the sample space.
This situation leads to a number of questions. We may, e.g. ask: Can we completely dispense with the the
standard concept of random variables if we are dealing exclusively with a type 3 theory ? The answer is certainly
no; it seems impossible to formulate a physical theory without any deterministic relations. In fact, a
deterministic relation, corresponding to a standard random variable , has already been anticipated in
Eq. (6). If in a position measurement of a particle a number
is observed, then the particle is - at the time of
the measurement - with certainty under the influence of a force
. Thus, an allowed class of
deterministic relations might contain “given” functions, describing externally controlled influences like forces
or potentials
.
There may be other standard random variables. To decide on purely logical grounds which relations of a type
3 theory are deterministic and which are not is not an obvious matter. However, one would suspect that the
deterministic relations should be of an universal nature; e.g. they should hold both in type 2 and type 3 theories.
Further, we may expect that all relations which are a logical consequence of the structure of space-time should
belong to this class. Such a quantity is the kinetic energy. In fact, for the currently considered nonrelativistic
range of physics, the functional form of the kinetic energy can be derived from the structure of the Galilei group
both in the mathematical framework of classical mechanics [20] and quantum mechanics [19]. We refer to the
kinetic energy as a standard random variable insofar as it is a prescribed function of
(but it is, because it is a function of
, not a standard random variable with respect to the
fundamental probability measure
). Combining the standard random variables “kinetic energy” and
“potential” we obtain a standard random variable “energy”, which will be studied in more detail in
section 6.
Thus, in the present framework, particle momentum will, in general, not be considered as a
standard random variable. This means that an element of determinism has been eliminated from the
theoretical description. It seems that this elimination is one of the basic steps in the transition from
type 2 to type 3 theories. The functional form of the probability density , and its
relation to
, are one of the main objectives of the present study. According to the
above discussion a measurement of position does no longer determine momentum at the time of the
measurement. However the set of all position measurements [represented formally by the probability
density
] may still determine (in a manner still to be clarified) the set of all momentum
measurements [the probability
]. Interestingly, Torre [7], using a completely different
approach, arrived at a similar conclusion, namely that the quantum mechanical ’variables’ position and
momentum cannot be random variables in the conventional sense. For simplicity we will continue to use
the term random variable for
, and will add the attributes ”‘standard”’ or “nonstandard” if
required.
As a first step in our study of , we will now investigate the integral equation (2) and will
derive a relation for
which will be used again in section 6. In the course of the following
calculations the behavior of
and
at infinity will frequently be required. We know that
is
normalizable and vanishes at infinity. More specifically, we shall assume that
and
obey the following conditions:
![]() | (11) |
where is anyone of the following factors
![]() | (12) |
Roughly speaking, condition (11) means that vanishes faster than
and
is nonsingular at
infinity. Whenever in the following an integration by parts will be performed, one of the conditions (11) will be
used to eliminate the resulting boundary term. For brevity we shall not refer to (11) any more; it will be
sufficiently clear in the context of the calculation which one of the factors in (12) will be referred
to.
We look for differential equations for our fields which are compatible with (2)-(7). According to
the above discussion we are not allowed to identify (8) with the random variable
. Using (7) we replace the
derivative with respect to
in (2) by a derivative with respect to
and perform an integration by parts.
Then, (2) takes the form
![]() | (13) |
Eq. (13) shows that the averaged value of the random variable is the expectation value of the field
. In the next section we shall insert this expression for
in the second statistical condition (3).
More specific results for the probability density
will be obtained later (in section 10). As an
intermediate step, we now use (13) and (5) to derive a relation for
, introducing thereby an
important change of variables.
We replace the variables by new variables
defined by
![]() | (14) |
We may as well introduce the imaginary unit and define the complex field . Then,
the last transformation and its inverse may be written as
![]() | (17) |
The derivative of may be omitted under the integral sign and (13) takes the form
![]() | (18) |
We introduce the Fourier transform of , defined by
![]() | (21) |
where the integral over has to vanish. Using Parseval’s formula and the fact that both
and
are normalized to unity we find that the integral of
has to vanish
too.
Using the continuity equation (7) and the first statistical condition (2) we found two results
[namely (18) and (21)] which reduce for to characteristic relations of the
quantum mechanical formalism. However, the function
, as well as the probability
density
we are finally interested in, is still unknown, because the validity of the
deterministic relation (8) is not guaranteed in the present general formalism allowing for type 3
theories. In the next section the implications of the second statistical condition will be studied without
using
. We shall come back to the problem of the determination of
in
section 7.