**2. The Initial Phase of Measurement as A Scattering Process**

Here we study the interaction between the small system *μ* and a larger system *A* with a large number of degrees of freedom. The larger system is assumed to be characterized by an ensemble of possible *initial* microstates of *A*. We consider this interaction to be the first part of a measurement process.

Since we are dealing with a two-level system *μ*, the Pauli matrices provide a suitable formalism with *σ*<sup>3</sup> = 1 0 0 −1 representing the observable to be measured, with eigenstates <sup>|</sup>+*<sup>μ</sup>* <sup>=</sup> <sup>1</sup> 0 and |−*<sup>μ</sup>* <sup>=</sup> <sup>0</sup> 1 .

Let us investigate the characteristics of the interaction between *μ* and *A* in scattering theory for the case with *A* in a state with (unknown) microscopic details that are summarized in a variable *α*. We then denote the normalized initial state of *A* by |0, *α<sup>A</sup>* (with 0 indicating a state of preparedness). This means that we assume *α* to represent one microstate in an ensemble of possible initial states.

A basic requirement is that if *μ* is initially in the state |*j<sup>μ</sup>* (*j* = + or −), after the interaction with *A*, its state remains the same. In this process *A* changes from the initial state |0, *α<sup>A</sup>* to a final state |*j*, *<sup>β</sup>j*(*α*)*A*, also normalized. The first *<sup>j</sup>* here indicates that *<sup>A</sup>* has been marked by the state |*j<sup>μ</sup>* of *<sup>μ</sup>*. All other characteristics of the final state of *A* are collected in *βj*(*α*). The interaction thus transforms the system *A* from an initial state of readiness, characterized by *α*, to a final state, marked by |*j<sup>μ</sup>* and characterized by *βj*(*α*).

For a general normalized state of *μ*, |*ψ<sup>μ</sup>* = *ψ*<sup>+</sup> |+*<sup>μ</sup>* + *ψ*<sup>−</sup> |−*<sup>μ</sup>* (with |*ψ*+| <sup>2</sup> <sup>+</sup> <sup>|</sup>*ψ*−| <sup>2</sup> = 1), the combined initial state of *μ* ∪ *A* is

$$\left| \left| \psi \right\rangle\_{\mu} \otimes \left| 0, a \right\rangle\_{A} = \left( \psi\_{+} \left| + \right\rangle\_{\mu} + \psi\_{-} \left| - \right\rangle\_{\mu} \right) \otimes \left| 0, a \right\rangle\_{A} \,. \tag{1}$$

A measurement of *σ*<sup>3</sup> on *μ* leads to a certain result. Since two different results are possible, the *μA*-interaction should in general result in a transition to one of the following states,

$$|+\rangle\_{\mu}\otimes|+\rangle\_{\prime}\beta\_{+}(a)\rangle\_{A} \qquad \text{or} \qquad |-\rangle\_{\mu}\otimes|-\rangle\_{\prime}\beta\_{-}(a)\rangle\_{A} \,. \tag{2}$$

The conclusion is then that the outcome must depend on the initial state of *A*, i.e., on *α*.

In scattering theory, the interaction between *μ* and *A* is characterized by a transition operator *M*, and this leads to the (non-normalized) final state (see Figure 1),

$$M|\psi\rangle\_{\mu}\otimes|0,a\rangle\_{A} = b\_{+}(a)\psi\_{+}| + \rangle\_{\mu}\otimes| + \rangle\_{}\beta\_{+}(a)\rangle\_{A} + b\_{-}(a)\psi\_{-}| - \rangle\_{\mu}\otimes|-\rangle\_{-}\beta\_{-}(a)\rangle\_{A} \tag{3}$$

In general, the amplitudes, *b*+(*α*) and *b*−(*α*), are not equal and therefore the proportions between + and − can change in a way that depends on the initial state |0, *α<sup>A</sup>* of *A*. (Please note that *M* must not to be confused with the unitary scattering operator *S*, see Appendix A).

**Figure 1.** Schematic Feynman diagram for a transition from the initial state |*ψ<sup>μ</sup>* ⊗ |0, *α<sup>A</sup>* to the final state |*j<sup>μ</sup>* ⊗ |*j*, *<sup>β</sup>j*(*α*)*A*, *<sup>j</sup>* = ±. The transition amplitude *<sup>ψ</sup>jbj*(*α*) depends on the microscopic details of the initial state |0, *α<sup>A</sup>* of the larger system *A* and on the initial state |*ψ<sup>μ</sup>* of *μ*.

The requirement of a statistically unbiased measurement means that |*b*+| <sup>2</sup> <sup>=</sup> |*b*−| <sup>2</sup>, where denotes mean value over the ensemble of initial states |0, *α<sup>A</sup>* of *A*.

Equation (3) describes a mechanism of the measurement process in which von Neumann's dilemma is not present. Relativistic quantum mechanics, in the form of scattering theory of quantum field theory, is a more correct theory than the non-relativistic Schrödinger equation, as used in the 1930s, and we choose to use Equation (3) as our starting point.

In the Feynman-diagram language of quantum field theory, transitions between the two channels, + and −, are possible via returns to the initial state. This is a way to understand how the proportions of the channels can change as described by Equation (3). A formulation based on perturbation theory to all orders, leads to an explicitly unitary description of the whole process. (This is shown in Appendix B)

In scattering theory, transition rate (transition probability per unit time) is a central concept as we have reviewed in Appendix A. The transition rate from the initial state (1) to the final state (3) is (2*π*)−1*w*(*α*), where *w*(*α*) is the squared modulus of (3),

$$w(\mathfrak{a}) = |\psi\_+|^2 |b\_+(\mathfrak{a})|^2 + |\psi\_-|^2 |b\_-(\mathfrak{a})|^2 \,. \tag{4}$$

Each term in (4) represents the partial transition rate for the corresponding channel. Because of our assumption of common mean values for |*b*±(*α*)| 2, the mean value of (4) is the same, *w* <sup>=</sup> |*b*±| <sup>2</sup>.

For equation (3) to properly represent a measurement process, i.e., a bifurcation that leads to a final state with *μ* in either of the eigenstates of *σ*3, it is necessary that the squared moduli of the amplitudes satisfy either |*b*+(*α*)| <sup>2</sup> >> <sup>|</sup>*b*−(*α*)<sup>|</sup> <sup>2</sup> or <sup>|</sup>*b*−(*α*)<sup>|</sup> <sup>2</sup> >> <sup>|</sup>*b*+(*α*)<sup>|</sup> 2. If this holds for (almost) all microstates *α* in the resulting ensemble of final states, then it can function as a mechanism for the bifurcation of the measurement process.

Part of the basis for the von Neumann dilemma was the assumption that *A* is in a given initial state |0, *αA*. Before the *μA*-interaction, *A* can be in any of the states of the available initial ensemble. These states are ready to influence the recording process in different ways. To reach a final state, given by Equation (3), they compete with their transition rates, (2*π*)−1*w*(*α*), which can differ widely between different values of *α*. The competition can lead to a selection and to a statistical distribution over *α* of the final states that is very different from the distribution in the initial ensemble.

In the following section, we will construct a mathematical model of how *A* influences the *μA*-interaction, including a description of the ensemble of possible initial states *α*. We then make a statistical analysis to show how both the bifurcation and the proper weights for the different measurement results can be understood in this simple setting.
