4 On random variables

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 ℝ . This set ℝ may also be identified with the possible values of a random variable “position measurement” (whose name should strictly speaking differ from x but we shall neglect such differences here). The basic probability measure which assigns a probability to each event (subspace of ℝ ) is given by ρ (x,   t ) . According to standard probability theory the field p (x,   t) defined by (8) is itself a random variable. We may consider it as a function of the random variable X (denoting “position measurement”) or as a random variable defined independently on the fundamental event space ℝ ; it makes no difference. Its probability density is uniquely determined by ρ (x,   t) and the function p (x,  t ) . In order to avoid confusion of names it may be useful to denote the derivative of S  (x,   t) with respect to x by g (x,  t) instead of p(x,   t) . Thus p (x,   t) =   g (x,   t) and the notation p  =   g (x,  t ) indicates that a random variable p defined by the function g (x,   t) 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 p  =   g (x,   t) 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 x
  i , a probability measure P , and a invertible function g (x  ) , the probability that an event p   =   g (x   )
  i           i occurs is obviously given by W   (p  )  =   P  (g -  1(p   ))
       i                     i . This result is the starting point to obtain w  (p  ) , the probability density of a continuous random variable p  =   g (x  ) , which is defined by a non-invertible function g (x  ) . It is given by [43]
                                   |              |
            n∑ (p )                 |     -  1     |
                        -  1       | ∂-g-i---(p-)-|
w  (p ) =          ρ (g     (p  )) |              | ,
                        i          |     ∂ p      |

where g - 1 (p )
  i denotes the n (p ) solutions (the number of solutions depends on p ) of the equation p  -   g (x  ) =   0 . Using a well-known formula for Dirac‘s delta function δ , applied to the case where the argument of δ is an arbitrary function, Eq.(9) may be rewritten in the form
                ∫                      (                      )

w  (p,  t ) =        dx   ρ (x,  t )δ     p  -                   ,
                                                     ∂ x

where we came back to our original notation, writing down the t -dependencies of ρ and w and replacing g (x  ) by ∂ S  (x,  t )∕ ∂ x .

The representation (10) reveals very clearly a hybrid nature of random variables defined as (nontrivial) functions on the event space ℝ . They are partly defined by a probabilistic quantity [namely ρ (x ) ] and partly by a deterministic relation [namely g (x ) ]. 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 x of the classical phase space probability density ρ (x,  t) δ (p  -   ∂ S  (x,  t) ∕ ∂ x ) . Considered from an operational point of view, the hybrid nature of random variables may be described as follows. Deterministic predictions for random variables p  =   g (x ) are impossible, as are deterministic predictions for the original variables x . But once a number x has been observed in an experiment, then the value of p  =   g (x ) is with certainty given by the defining function g (x ) . If no such relation exists, this does not necessarily imply that x and p 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 F  (x  ) , has already been anticipated in Eq. (6). If in a position measurement of a particle a number x is observed, then the particle is - at the time of the measurement - with certainty under the influence of a force F  (x ) . Thus, an allowed class of deterministic relations might contain “given” functions, describing externally controlled influences like forces F  (x  ) or potentials V  (x  ) .

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 p2 ∕2m as a standard random variable insofar as it is a prescribed function of p (but it is, because it is a function of p , not a standard random variable with respect to the fundamental probability measure ρ ). 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 w  (p,  t) , and its relation to ρ (x,  t) , 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 ρ (x,  t ) ] may still determine (in a manner still to be clarified) the set of all momentum measurements [the probability w  (p,  t ) ]. 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 p , and will add the attributes ”‘standard”’ or “nonstandard” if required.

As a first step in our study of w  (p,  t ) , we will now investigate the integral equation (2) and will derive a relation for w  (p,  t ) which will be used again in section 6. In the course of the following calculations the behavior of ρ and S at infinity will frequently be required. We know that ρ (x,  t ) is normalizable and vanishes at infinity. More specifically, we shall assume that ρ (x,  t) and S  (x,  t ) obey the following conditions:
                ∂--ρ-              -1 ∂-ρ--∂-ρ--
ρ  A   →    0,        A   →    0,               →    0,     for x  →    ∞,
                ∂  x               ρ  ∂ x  ∂ t

where A is anyone of the following factors
                              (        )
           ∂-S--      ∂-S--       ∂-S--
1,   V  ,       ,   x      ,                .
            ∂ t       ∂ x         ∂ x

Roughly speaking, condition (11) means that ρ vanishes faster than 1 ∕x and S 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 ρ,   S which are compatible with (2)-(7). According to the above discussion we are not allowed to identify (8) with the random variable p . Using (7) we replace the derivative with respect to t in (2) by a derivative with respect to x and perform an integration by parts. Then, (2) takes the form
∫   ∞
                       ∂  S (x,   t)      --
         dx  ρ (x,  t )-------------  =   p.
                            ∂ x
  -  ∞

Eq. (13) shows that the averaged value of the random variable p is the expectation value of the field p (x,   t ) . In the next section we shall insert this expression for --
p in the second statistical condition (3). More specific results for the probability density w  (p,  t ) will be obtained later (in section 10). As an intermediate step, we now use (13) and (5) to derive a relation for w  (p,  t) , introducing thereby an important change of variables.

We replace the variables ρ,  S by new variables ψ    , ψ
   1     2 defined by
             --                         --
          √           S--            √           S--
ψ1    =      ρ  cos        ψ2    =      ρ  sin     .
                      s                          s

We may as well introduce the imaginary unit and define the complex field ψ   =   ψ1   +   iψ2 . Then, the last transformation and its inverse may be written as

                      √  --  iS-
            ψ    =       ρe   s                              (15)

            ⋆                 s       ψ
ρ  =   ψ  ψ   ,       S  =   ----ln  -----.                  (16)
                             2i      ψ  ⋆
We note that so far no new condition or constraint has been introduced; choosing one of the sets of real variables { ρ,  S  }, { ψ1,   ψ2  }, or the set { ψ,  ψ  ⋆ } of complex fields is just a matter of mathematical convenience. Using          ⋆
{ψ,   ψ   } the integrand on the left hand side of (13) takes the form
   ∂  S           s  ∂           s   ∂
ρ  ----- =   ψ  ⋆ -------ψ   -   --------|ψ  |2.

   ∂  x           i ∂  x         2i ∂ x

The derivative of      2
|ψ  | may be omitted under the integral sign and (13) takes the form
∫                               ∫
    ∞                               ∞
                ⋆ s--∂---
         dx  ψ           ψ  =            dpw    (p,  t)p.
  -  ∞            i ∂ x           -  ∞

We introduce the Fourier transform of ψ , defined by

                  ---1------                         -ipx
ψ (x,   t)   =    √  -----           d p φ  (p,  t)e s                (19)
                     2 πa     -  ∞
                          ∫   ∞
                    1                                  i-
φ  (p,  t)   =    √-------         dx  ψ  (x,  t )e -  spx.           (20)

                     2 π    -  ∞
The constant s , introduced in Eq. (14), has the dimension of an action, which means that p has the dimension of a momentum. Performing the Fourier transform one finds that the momentum probability density may be written as
w (p,  t ) =    --|φ (p,  t) |2 +   h (p,  t ),

where the integral over ph  (p,  t ) has to vanish. Using Parseval’s formula and the fact that both ρ (x,   t) and w  (p,  t) are normalized to unity we find that the integral of h (p,  t) 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 h (p,  t ) =   0 to characteristic relations of the quantum mechanical formalism. However, the function h (p,  t) , as well as the probability density w  (p,  t) 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 w  (p,  t ) . We shall come back to the problem of the determination of w  (p,  t) in section 7.