5 Statistical theories

We study now the implications of the second statistical condition (3). Using the variables ρ,   S it takes the form
    ∫   ∞                        ∫   ∞
 d                 ∂  S                         ∂  V
----         dx  ρ -----  =   -           dx  ρ ------,
dt                  ∂ x                          ∂ x
      -  ∞                         -  ∞

if --
p is replaced by the integral on the l.h.s. of (13). Making again use of (7), we replace in (22) the derivative of ρ with respect to t by a derivative with respect to x . Then, after an integration by parts, the left hand side of (22) takes the form
-d--                ∂-S--
             dx   ρ       =
dt     - ∞          ∂  x
∫              [               (        )                    ]
    ∞                                      2
                    --1---∂-ρ--    ∂-S--           -∂---∂-S--
         dx      -                           +   ρ              .
   - ∞              2m    ∂ x      ∂ x             ∂ x  ∂  t

Performing two more integrations by parts [a second one in (23) substituting the term with the time-derivative of S , and a third one on the right hand side of (22)], condition (3) takes the final form
                    [                                      ]
∫                           (        )2
    ∞         ∂ ρ       1      ∂  S           ∂ S
         dx   -----   ------   -----      +   ----- +   V     =   0.

  -  ∞        ∂ x     2m        ∂ x            ∂ t

Equation (24) can be considered as an integral equation for the real function L (x,   t) defined by
                                (               )
                ∂-S--    --1---    ∂-S--(x,--t)-
L (x,  t ) =         +                               +   V  (x,  t ).
                ∂ t      2m            ∂ x

Obviously, (24) admits an infinite number of solutions for L  (x,  t ) , which are given by
∂ ρ (x,  t )                 ∂ Q
-------------L (x,  t ) =    -----,
    ∂ x                      ∂ x

The function Q  (x,  t) in (26) has to vanish at x   →    ±  ∞ but is otherwise completely arbitrary.

Equation (26), with fixed Q and L as defined by  (25), is the second differential equation for our variables S and ρ we were looking for, and defines - together with the continuity equation (7) - a statistical theory. The dynamic behavior is completely determined by these differential equations for S and ρ . On the other hand, the dynamic equation - in the sense of an equation describing the time-dependence of observable quantities - is given by (2) and (3).

From the subset of functions Q which do not depend explicitely on x and t we list the following three possibilities for Q and the corresponding L . The simplest solution is
Q   =   0,            L   =   0.

The second Q depends only on ρ ,
          n                               n -  1
Q   ~   ρ   ,  n   ≥   1,     L   ~   n ρ       .

The third Q depends also on the derivative of ρ ,
           (            )                             √  --
         1      ∂  √  --   2                 1    ∂  2   ρ
         --   -----                        -----------------
Q   ~                 ρ       ,    L  ~      √  --      2   .
         2    ∂  x                         2    ρ   ∂ x

We discuss first (27). The statistical theory defined by (27) consists of the continuity equation (7) and [see (25)] the Hamilton-Jacobi equation,
                (                )2
∂ S         1      ∂  S (x,   t)
----- +   ------   -------------      +   V  (x,  t ) =   0.
 ∂ t      2m            ∂ x

The fact that one of these equations agrees with the Hamilton-Jacobi equation does not imply that this theory is a type 1 theory (making predictions about individual events). This is not the case; many misleading statements concerning this limit may be found in the literature. It is a statistical theory whose observables are statistical averages. However, Eq. (30) becomes a type 1 theory if it is considered separately - and embedded in the theory of canonical transformations. The crucial point is that (30) does not contain ρ ; otherwise it could not be considered separately. This separability - or equivalently the absence of ρ in (30) - implies that this theory is a classical (type 2) statistical theory [37]. The function S may be interpreted as describing the individual behavior of particles in the given environment (potential V ). Loosely speaking, the function S may be identified with the considered particle; recall that S is the function generating the canonical transformation to a trivial Hamiltonian. The identity of the particles described by S is not influenced by statistical correlations because there is no coupling to ρ in (30). The classical theory defined by (7) and (30) may also be formulated in terms of the variables ψ and ψ  ⋆ [ but not as a single equation containing only ψ ; see the remark at the end of section (5)]. In this form it has been discussed in several works [383734].

All theories with nontrivial Q , depending on ρ or its derivatives, should be classified as “non-classical” (or type 3) according to the above analysis. In non-classical theories any treatment of single events (calculation of trajectories) is impossible due to the coupling between S and ρ . The problem is that single events are nevertheless real and observable. There must be a kind of dependence (correlation of non-classical type) between these single events. But this dependence cannot be described by concepts of deterministic theories like “interaction”.

The impossibility to identify objects in type 3 theories - independently from the statistical context - is obviously related to the breakdown of the concept of standard random variables discussed in the last section. There, we anticipated that a standard random variable (which is defined as a unique function of another random variable) contains an element of determinism that should be absent in type 3 theories. In fact, it does not make sense to define a unique relation between measuring data - e.g. of spatial position and momentum - if the quantities to be measured cannot themselves be defined independently from statistical aspects.

The theory defined by Eq. (28) is a type 3 theory. We will not discuss it in detail because it may be shown (see the next section) to be unphysical. It has been listed here in order to have a concrete example from the large set of insignificant type 3 theories.

The theory defined by Eq. (29) is also a type 3 theory. Here, the second statistical condition takes the form
                (        )                               √  --
                            2                2       ∂  2   ρ
∂--S-     --1---    ∂-S--                 -ℏ-----1-------------
      +                       +   V   -         √  --      2    =   0,
 ∂ t      2m        ∂ x                   2m       ρ   ∂ x

if the free proportionality constant in (29) is fixed according to   2
ℏ  ∕m . The two equations (7) and (31) may be rewritten in a more familiar form if the transformation (16) (with s  =   ℏ ) to variables ψ,   ψ  ⋆ is performed. Then, both equations are contained (as real and imaginary parts) in the single equation
                      2    2
   ℏ  ∂ ψ           ℏ    ∂   ψ
-  -------- =   -  -------------+   V  ψ,
    i ∂  t         2m    ∂ x2

which is the one-dimensional version of Schrödinger’s equation [39]. Thus, quantum mechanics belongs to the class of theories defined by the above conditions. We see that the statistical conditions (2), (3) comprise both quantum mechanical and classical statistical theories; these relations express a “deep-rooted unity” [40] of the classical and quantum mechanical domain of physics.

We found an infinite number of statistical theories which are all compatible with our basic conditions and are all on equal footing so far. However, only one of them, quantum mechanics, is realized by nature. This situation leads us to ask which further conditions are required to single out quantum mechanics from this set. Knowing such condition(s) would allow us to have premises which imply quantum mechanics.

The above analysis shows that Schrödinger’s equation (32) can be derived from the condition that the dynamic law for the probabilities takes the form of a single equation for ψ (instead of two equations for ψ and ψ  ⋆ as is the case for all other theories). Our previous use of the variables ψ and ψ ⋆ instead of S and ρ was entirely a matter of mathematical convenience. In contrast, this last condition presents a real constraint for the physics since a different number field has been chosen [23]. Recently, Schrödingers equation including the gauge coupling term has been derived [26] from this condition (which had to be supplemented by two further conditions, namely the existence of a continuity equation and the assumption of a linear time evolution law for ψ ). Of course, this is a mathematical condition whose physical meaning is not at all clear. This formal criterion will be replaced in section 9 by a different condition which leads to the same conclusion but may be formulated in more physical terms.