10 Derivation of L

For a general L0 , the Euler-Lagrange equation belonging to the functional C  [ρ ] [see (58)] depends on derivatives higher than second order since the integrand in (58) depends on   ′′
ρ . This is a problem, since according to the universal rule mentioned above all differential equations of physics are formulated using derivatives not higher than second order. If we are to conform with this general rule (and we would like to do so) then we should use a Lagrangian containing only first order derivatives. But this would then again produce a conflict with Eq. (53) because the variational procedure increases the order of the highest derivative by one. We postpone the resolution of this conflict and proceed by calculating the Euler-Lagrange equations according to (58), which are given by
                      L  (x,  t)  -   L   (ρ,  ρ ′, ρ ′′)
   -d----   ∂-L0---      ′∂-L0---       -d--∂--L0--       ∂-L0---
-        ρ      ′′ +   ρ       ′  +  ρ           ′  -   ρ         =   0.
   dx2      ∂ ρ            ∂ ρ          dx   ∂ ρ           ∂  ρ

Using the second basic condition (53) we see that the first line of Eq. (59) vanishes and we obtain, introducing the abbreviation β  =   ρL
            0 , the following partial differential equation for the determination of the functional form of L
  0 with respect to the variables ρ,  ρ ′, ρ ′′ ,
     d     ∂ β        d   ∂ β       ∂  β      β
-   -------------+   ------------   ----- +   --- =   0.
    dx2   ∂ ρ ′′     dx   ∂ ρ ′     ∂  ρ      ρ

Expressing the derivatives of L
  0 in terms of the derivatives of ρ, ρ ′ and ρ ′′ leads to a lengthy relation which will not be written down here. Since L0 does not contain higher derivatives than   ′′
ρ , the sums of the coefficients of both the third and fourth derivatives of ρ have to vanish. This implies that β may be written in the form
β (ρ,  ρ ′, ρ ′′) =   C  ( ρ, ρ ′) ρ ′′ +  D   (ρ,  ρ ′),

where           ′
C  (ρ,  ρ  ) and           ′
D  ( ρ, ρ  ) are solutions of

                   2                 2
     ∂ C         ∂   C     ′       ∂  D         C
-  2 ------ -   ---------ρ   +   ---------- +   ----=     0             (62)
      ∂ ρ       ∂ ρ ∂ ρ ′        ∂  (ρ ′)2       ρ

      2                    2
    ∂---C--   ′  2     -∂---D----  ′     ∂-D---     D---
 -         (ρ  )   +             ρ   -          +        =    0.        (63)
    ∂  ρ2              ∂  ρ ∂ ρ ′         ∂ ρ       ρ
Thus, two functions of       ′
ρ,  ρ have to be found, instead of a single function of       ′    ′′
ρ,  ρ  , ρ . Fortunately, the solution we look for presents a term in a differential equation. This allows us to restrict our search to relatively simple solutions of (62), (63). If the differential equation is intended to be comparable in complexity to other fundamental laws of physics, then a polynomial form,
                       ∞                                                        ∞
                     ∑                                                         ∑
          ′                              n     ′ m                 ′                               n     ′ m
C  (ρ,  ρ  ) =                  cn,m   ρ   ( ρ  )   ,    D  ( ρ, ρ  )  =                 dn,m    ρ   ( ρ  )   ,

                 n,m=    -  ∞                                              n,m=    - ∞

preferably with a finite number of terms, will be a sufficiently general Ansatz.

Eqs. (62) and (63) must of course hold for arbitrary       ′
ρ,  ρ . Inserting the Ansatz (64), renaming indices and comparing coefficients of equal powers of ρ and ρ ′ one obtains the relations

(nm     +   2n   +  m    +  1 )cn+1,m        =   (m    +   1 )(m    +  2 )dn,m+2   (65)

    (n   +   1 )(n   +  2 )cn+2,m      - 2   =   (nm     -   n   +  m   )dn+1,m    (66)
to determine cn,m and dn,m . These relations may be used to calculate those values of n,  m which allow for non-vanishing coefficients and to calculate the proportionality constants between these coefficients; e.g. Eq. (65) may be used to express d
  n,m+2 in terms of c
  n+1,m provided m    ⁄=   -  1 and m    ⁄=   -  2 . One obtains the result that the general solution β ( ρ,  ρ ′, ρ ′′) of (60) of polynomial form is given by (61), with
C  ( ρ, ρ ′)   =          C    ρn  (ρ ′) - n                                 (67)

                   n ∈I
                             ∑      n  -   1
D  ( ρ, ρ ′)   =   A  ρ  -          --------- C    ρn  -  1 (ρ ′)-  n+2,     (68)
                                    n  -   2
                             n ∈I
where A and Cn are arbitrary constants and the index set I is given by
I  =   {n   |n  ∈   Z  , n  ≤   0,  n  ≥   3 }  .

While the derivation of (67), (68) is straightforward but lengthy, the fact that (61), (67), (68) fulfills (60) may be verified easily.

At this point we are looking for further constraints in order to reduce the number of unknown constants. The simplest (nontrivial) special case of (61), (67), (68) is Cn   =   0,   ∀n   ∈   I. The corresponding solution for L    =   β ∕ ρ
  0 is given by L    =   A
   0 . However, a solution given by a nonzero constant A may be eliminated by adding a corresponding constant to the potential in L (x,   t) . Thus, this solution need not be taken into account and we may set A   =   0 .

The ’next simplest’ solution, given by A   =   0 , Cn   =   0 for all n  ∈   I except n   =   0 , takes the form
                         (                      )
                               ′′           ′ 2                        2 √  --
           (0 )              ρ        1 ( ρ  )                   1   ∂      ρ
L0   =   L      =   C0       ---- -   ----------    =   2  C0  √--------------.
           0                  ρ       2   ρ2                      ρ   ∂  x2

Let us also write down here, for later use, the solution given by A   =   0 , C     =   0
   n for all n   ∈  I except n  =   -  1 . It takes the form
                               (                          )
                                     ′               ′  3
            ( - 1 )                ρ---  ′′     2-(ρ--)---
L0    =   L 0       =   C  - 1         ρ   -                 ,
                                   ρ2           3   ρ3

Comparison of the r.h.s. of Eq. (70) with Eq. (29) shows that the solution (70) leads to Schrödinger’s equation (32). At this point the question arises why this particular solution has been realized by nature - and not any other from the huge set of possible solutions. Eq. (70) consists of two parts. Let us consider the two corresponding terms in ρL
     0 , which represent two contributions to the Lagrangian in Eq. (56). The second of these terms agrees with the integrand of the Fisher functional (48). The first is proportional to   ′′
ρ . This first term may be omitted in the Lagrangian (under the integral sign) because it represents a boundary (or surface) term and gives no contribution to the Euler-Lagrange equations [it must not be omitted in the final differential equation (53), where exactly the same term reappears as a consequence of the differentiation of   ′
ρ ]. Thus, integrating the contribution ρL
     0 of the solution (70) to the Lagrangian yields exactly the Fisher functional. No other solution with this property exists. Therefore, the reason why nature has chosen this particular solution is basically the same as in classical statistics, namely the principle of maximal disorder - but realized in a different (local) context and expressed in terms of a principle of minimal Fisher information.

We see that the conflict mentioned at the beginning of this section does not exist for the quantum mechanical solution (70). The reason is again that the term in L0 containing the second derivative   ′′
ρ is of the form of a total derivative and can, consequently, be neglected as far as its occurrence in the term ρL0 of the Lagrangian is concerned. Generalizing this fact, we may formulate the following criterion for the absence of any conflict: The terms in L
  0 containing ρ ′′ must not yield contributions to the variation, i. e. they must in the present context take the form of total derivatives (for more general variational problems such terms are called ”‘null Lagrangians”’ [13]).

So far, in order to reduce the number of our integration constants, we used the criterion that the corresponding term in the Lagrangian should agree with the form of Fishers functional. This ‘direct’ implementation of the principle of maximal disorder led to quantum mechanics. The absence of the above mentioned conflict means that the theory may be formulated using a Lagrangian containing no derivatives higher than first order. As is well known, this is a criterion universally realized in nature; a list of fundamental physical laws obeying this criterion may e.g. be found in a paper by Frieden and Soffer [12]. Thus, it is convincing although of a ’formal’ character. Let us apply this ’formal’ criterion as an alternative physical argument to reduce the number of unknown coefficients the above solution. This criterion implies, that the derivatives of L0 with respect to   ′′
ρ do not play any role, i.e. the solutions of (60) must also obey
 d   ∂ β       ∂ β       β
---------- -   ----- +   ---=   0.
dx  ∂  ρ ′     ∂ ρ       ρ

This implies that only those solutions of (60) are acceptable, which obey
     d2    ∂ β
-   ------------- =  0.
        2 ∂ ρ ′′

Using (61) and (67) it is easy to see that the solution (70) belonging to n   =   0 is the only solution compatible with the requirement (73) [as one would suspect it is also possible to derive (70) directly from (72)]. Thus the ’formal’ principle, that the Lagrangian contains no terms of order higher than one, leads to the same result as the ’direct’ application of the principle of maximal disorder. The deep connection between statistical criteria and the form of the kinetic energy terms in the fundamental laws of physics has been mentioned before in the literature [11]. The present derivation sheds new light, from a different perspective, on this connection.

Summarizing, the shape of our unknown function L0 has been found. The result for L0 leads to Schrödinger’s equation, as pointed out already in section 5. This means that quantum mechanics may be selected from an infinite set of possible theories by means of a logical principle of simplicity, the statistical principle of maximal disorder. Considered from this point of view quantum mechanics is ’more reasonable’ than its classical limit (which is a statistical theory like quantum mechanics). It also means (see section 6) that the choice h  =   0 is justified as far as the calculation of expectation values of   n
p   , n  ≤   2 is concerned.

In closing this section we note that the particular form of the function L  (x,  t ) has never been used. Thus, while the calculation of L0 reported in this section completes our derivation of quantum mechanics, the result obtained is by no means specific for quantum mechanics. Consider the steps leading from the differential equation (53) and the variational principle (58) to the general solution (61), (67), (68). If we now supplement our previous assumptions with the composition law (41), we are able to single out the Fisher I among all solutions [compare e.g. (71) and (70)]. Thus, the above calculations may also be considered as a new derivation of the Fisher functional, based on assumptions different from those used previously in the literature.