**Appendix C. Factorization of the Transition Amplitudes**

We can think of the interaction between the small system *μ* and the measurement apparatus as an electromagnetic interaction with small energy and momentum transfer. In Quantum Electrodynamics (QED), it is described in terms of an exchange of soft photons. Emission and exchange of soft photons is an old and well-known example of factorizable processes in QED.

To show the factorization of soft photon exchange, we consider an outgoing electron (charge −*e*, mass *m*) with final momentum *p*, described by a spinor *u*(*p*) (we follow rather closely the conventions of Ref. [17]),

$$\begin{aligned} \, \, \, p^2 + m^2 = 0 \,, \, \, \overline{u}(p)(ip \cdot \gamma - m) = 0 \,, \end{aligned} \tag{A12}$$

after emitting two soft photons with momenta *k*1, *k*2, and polarizations *τ*1, *τ*2,

$$k\_1^2 = k\_2^2 = 0 \; ; \quad k\_1 \cdot \tau\_1 = k\_2 \cdot \tau\_2 = 0 \; ; \tag{A13}$$
  $|\mathbf{k\_1}| \; |\; |\mathbf{k\_2}| < \zeta \; m \; .$ 

In the evaluation of the Feynman diagram of Figure A3, the spinor *u*(*p*) for the outgoing electron of an original diagram (without soft photons) is replaced by an expression proportional to

$$\begin{split} &c^2 \overline{\boldsymbol{\pi}}(p) \left[ \boldsymbol{\tau}\_1 \cdot \gamma \frac{\boldsymbol{i}(p+k\_1) \cdot \gamma + m}{(p+k\_1)^2 + m^2} \boldsymbol{\tau}\_2 \cdot \gamma + (1 \leftrightarrow 2) \right] \frac{\boldsymbol{i}(p+k\_1+k\_2) \cdot \gamma + m}{(p+k\_1+k\_2)^2 + m^2} \approx \\ &c^2 \frac{1}{2(p \cdot k\_1 + p \cdot k\_2)} \overline{\boldsymbol{\pi}}(p) \left[ \frac{\boldsymbol{\tau}\_1 \cdot \gamma(\boldsymbol{i}p \cdot \gamma + m) \boldsymbol{\tau}\_2 \cdot \gamma(\boldsymbol{i}p \cdot \gamma + m)}{2p \cdot k\_1} + (1 \leftrightarrow 2) \right] = \\ &c^2 \frac{-p \cdot \boldsymbol{\tau}\_1 p \cdot \gamma \boldsymbol{2}}{p \cdot k\_1 + p \cdot k\_2} \left( \frac{1}{p \cdot k\_1} + \frac{1}{p \cdot k\_2} \right) \overline{\boldsymbol{\pi}}(p) = (\boldsymbol{s}(k\_1) \cdot \boldsymbol{\tau}\_1)(\boldsymbol{s}(k\_2) \cdot \boldsymbol{\tau}\_2)\overline{\boldsymbol{\pi}}(p), \end{split} \tag{A14}$$

where we have used the basic relation for the Dirac gamma matrices, *γμν* + *γνμ* = 2*gμν*, where *gμν* is the metric tensor. In (A14), *sμ*(*k*) is the Fourier transform of the current density of a classical point charge −*e* moving from **x** = 0 at time zero with the velocity **p**/*p*0,

$$s\_{\mu}(k) = -c \frac{ip\_{\mu}}{p \cdot k} = -c \int\_{0}^{\infty} dt \int d^{3} \mathbf{x} \, e^{i(\mathbf{k} \cdot \mathbf{x} - |\mathbf{k}|t)} \, \delta\left(\mathbf{x} - \frac{\mathbf{p}}{p\_{0}}t\right) \frac{p\_{\mu}}{p\_{0}} \,. \tag{A15}$$

The contribution of the diagram that *μ* comes from is unchanged in the limit of small *k*1, *k*2. Equation (A14) states that the emission of the two photons is described by one scalar emission factor for each photon. This can be extended to also include absorption.

**Figure A3.** Feynman diagram for the emission of two soft photons.

The case of only one photon is, of course, similar and even simpler with just one emission factor. We took the example of two photons to show how the algebra gives the two factors also in this case. For the *n*-photon case one uses the algebraic identity

$$\sum\_{i\_1(i\_1i\_2\dots i\_n)} \frac{1}{a\_{i\_1}(a\_{i\_1} + a\_{i\_2})\dots(a\_{i\_1} + a\_{i\_2} + \dots + a\_{i\_n})} = \frac{1}{a\_1a\_2\dots a\_na} \,. \tag{A16}$$

Thus, we can think of the classical current density

$$j\_{\mu}(k) = -e \,\delta\left(\mathbf{x} - \frac{\mathbf{p}}{p\_0}t\right) \frac{p\_{\mu}}{p\_0} \tag{A17}$$

as representing the electron, i.e., the system *μ* as felt by other systems. *μ* can come as a very small wave-packet travelling along the *x*1-axis (**p** = (*p*1, 0, 0)) through a substance that can be ionized. Then we take *A* to be a thin cylinder of this substance around the path of the *μ* wave-packet.

It is convenient to consider *A* as consisting of a chain of *N* shorter cylinders *A*1, *A*2, ..., *AN* along the *x*1-axis with *Ak* defined by

$$\begin{aligned} x\_1^{(0)} + \frac{k-1}{2} \Delta l \le x\_1 \le x\_1^{(0)} + \frac{k+1}{2} \Delta l \end{aligned} \tag{A18}$$
  $x\_2^2 + x\_3^2 \le \Delta r^2$  ,

where Δ*l* and Δ*r* are small.

For our model, we can neglect the interaction between the small cylinders and we can consider *μ* to interact with *Ak* only during its passage, i.e., during the time interval

$$\frac{p\_0}{p\_1} \left( \mathbf{x}\_1^{(0)} + \frac{k-1}{2} \Delta l \right) < t \le \frac{p\_0}{p\_1} \left( \mathbf{x}\_1^{(0)} + \frac{k+1}{2} \Delta l \right) \,. \tag{A19}$$

Because of their independence, each of the cylinders acted on by *μ* via the current density (A17), contributes an independent factor to the total transition rate. In the mean, the factor from the *μAk*-interaction should not change anything but in the single case, it can contribute an unknown factor close to one, as assumed in Equation (5). Thus, we have given here a rationale for this assumption.
