### 9 Transition to N particles as final step to non-relativistic quantum theory

In this section the present derivation of non-relativistic QT is completed by deriving Schrödinger’s equation for an arbitrary number of particles or, more precisely, for statistical ensembles of identically prepared experimental arrangements involving particles.

In order to generalize the results of sections 2 and 5, a convenient set of coordinates and masses is defined by
 (105)

The index is used to distinguish particles, while indices are used here to distinguish the coordinates . No new symbol has been introduced in (105) to distinguish the masses and since there is no danger of confusion in anyone of the formulas below. However, the indices of masses will be frequently written in the form in order to avoid ambiguities with regard to the summation convention. The symbol in arguments denotes dependence on all . In order to generalize the results of section 8 a notation , , and (with and ) for coordinates, positions, and masses will be more convenient.

The basic relations of section 2, generalized in an obvious way to particles, take the form

Here, is a single-valued variable; the multi-valuedness will be added later, following the method of section 8.

The following calculations may be performed in complete analogy to the corresponding steps of section 2. For the present dimensional problem, the vanishing of the surface integrals, ocurring in the course of various partial integrations, requires that vanishes exponentially in arbitrary directions of the configuration space. The final conclusion to be drawn from Eqs. (106)- (109) takes the form
 (110)

The remaining problem is the determination of the unknown function , whose form is constrained by the condition defined in Eq. (110).

can be determined using again the principle of minimal Fisher information, see I for details. Its implementation in the present framework takes the form

where are shorthand notations for the two basic equations (110) and (106). As before, Eqs. (111), (112) represent a method to construct a Lagrangian. After determination of the three relations listed in (111), (112) become redundant and (112) become the fundamental equations of the particle theory.

The following calculation can be performed in complete analogy to the case reported in section 5. All relations remain valid if the upper summation limit is replaced by . This is also true for the final result, which takes the form
 (113)

If a complex-valued variable , defined as in (52), is introduced, the two basic relations may be condensed into the single differential equation,
 (114)

which is referred to as particle Schrödinger equation, rewritten here in the more familiar form using particle indices. As is well-known, only approximate solutions of this partial differential equation of order exist for realistic systems. The inaccessible complexity of quantum-mechanical solutions for large is not reflected in the abstract Hilbert space structure (which is sometimes believed to characterize the whole of QT) but plays probably a decisive role for a proper description of the mysterious relation between QT and the macroscopic world.

Let us now generalize the Arunsalam-Gould method, discussed in section 8, to an arbitrary number of particles. We assume, that the considered particle statististical ensemble responds in ways to the external electromagnetic field. This means we restrict ourselves again, like in section 67 to spin one-half. Then, the state function may be written as where . In the first of the steps listed at the beginning of section 8, a differential equation, which is equivalent to Eq. (114) for single-valued but may give non-vanishing contributions for multi-valued , has to be constructed. The proper generalization of Eq. (103) to arbitrary takes the form
 (115)

where the Pauli matrices operate by definition only on the two-dimensional subspace spanned by the variable . In the second step we perform the replacement
 (116)

using a multi-valued phase factor, which is an obvious generalization of Eq. (104). The remaining steps, in the listing of section 8, lead in a straightforward way to the final result
 (117)

where and . The mechanical potential describes a general many-body force but contains, of course, the usual sum of two-body potentials as a special case. Eq. (117) is the body version of Pauli’s equation and completes - in the sense discussed at the very beginning of this paper - the present derivation of non-relativistic QT.