## Abstract

Quantum theory determines the evolution of quantum states between quantum jumps. Quantum theory also allows us to calculate rates of quantum jumps and, on a probabilistic level, the outcomes of those quantum jumps. Both quantum jumps and the continuous evolution of quantum states are important in the time evolution of quantum systems, and the scattering matrix ties those seemingly disparate concepts together. Indeed, quantum jumps are so essential in quantum dynamics that we should refocus discussion of a quantum ontology on the power and principal limitations of our knowledge about quantum jumps as encoded in the scattering matrix. On the one hand, one might argue that the lack of a dynamical resolution of quantum jumps indicates an inherent incompleteness of the theory. However, we would rather submit that quantum theory is complete and that the observations indicate a principal limitation to the description of the universe as a smoothly evolving dynamical system. The modern understanding of quantum jumps therefore calls for updates to the Copenhagen interpretation instead of modifications of quantum theory.

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## Notes

On the other hand, spreadings of electron wave functions from millimetre-aperture electron guns or of neutron wave functions from centimetre-aperture neutron guides are completely negligible on the time scales of lab experiments, even without any environmental stabilization.

We will define the notion of a “traditional Copenhagen interpretation”in Sect. 2.

We denote a signal in a particle detector as “pointlike” if the resolution of the spatial location of a single-particle signal is only determined by the spatial resolution of the detector, but not by the wave function of the incident particle that is observed. High-energy physics allows to calculate limits on elementary particle radii from the agreement of observed properties of charged particles with radiative corrections which are calculated under the assumption of point particles. This leads e.g. to a limit \(R<4\times 10^{-19}\) m for the electron radius from the comparison of high-precision measurements with high-precision calculations for the fine structure constant (Morel et al. 2020). A limit of \(R<2\times 10^{-20}\) m has been inferred from the absence of non-renormalizable contact interactions of the electron in measurements at the LEP collider (Bourilkov 2001). However, these theoretical limits on a possible electron radius are model dependent with respect to how substructure of an electron should relate to its anomalous magnetic moment and to non-renormalizable couplings, respectively, and they do not constitute position measurements. Nevertheless, it is noteworthy that all observations to date are compatible with a point particle interpretation of the electron. However, the

*point*about*pointlike signals*concerns the observation that single-particle position signals in diffraction experiments are determined by detector resolution and are much smaller than the widths of the wave functions.Fock space is

*a priori*a representation space for the Schrödinger picture quantum fields and the freely evolving quantum fields of the Dirac picture. The transformation from the Dirac picture fields into the Heisenberg picture fields is defined through a similarity transformation which involves the interaction terms of the theory expressed through the Dirac picture fields. As such, the transformation into the Heisenberg picture is singular since it generates the same self-energy effects that affect higher orders in the scattering matrix. The expression of time-evolved quantum states through Heisenberg picture quantum fields therefore always comes with the cautionary remark that renormalization is required if we apply higher powers of interaction terms on quantum states. We will not go deeper into the renormalization program, since wave-particle duality, quantum jumps, and the ontic-epistemic boundary in Fock space all show up already in first order.We recall that a pointlike signal is a signal for which the spatial resolution is only determined by the resolution of the detector.

The non-local relation of the relativistic particle creation operators in position space to the full quantum fields had caused a lot of confusion about the concept of relativistic wave functions in position space. However, the position space operators allow for the formulation of local energy-momentum densities, \({\mathcal {P}}_\mu ({\varvec{x}},t)=-\,\mathrm {i}\hbar \sum _s\varPsi ^+_s({\varvec{x}},t)\partial _\mu \varPsi _s({\varvec{x}},t)\), and local charge densities, \(\varrho ({\varvec{x}},t)=q\sum _s\varPsi ^+_s({\varvec{x}},t)\varPsi _s({\varvec{x}},t)\), which after integration yield the same results as the full interaction picture quantum fields after neglecting the anti-particle contributions. It is clear from the observation of relativistic electron and meson tracks, and from the creation and observation of single-photon states (Kuhn et al. 2002; Eisaman et al. 2011; Rueckner and Peidle 2013; Aspden et al. 2016; Aharonovich et al. 2016), that position space wave functions and their corresponding operators must exist. The reason why we cannot observe photon tracks is that photons are annihilated in interactions and therefore cannot participate in consecutive interactions. Furthermore, the differential cross section for elastic photon scattering off electrons, \(d\sigma /d\varOmega =(\alpha _S\hbar /m_ec)^2(1+\cos ^2\theta )/2\), implies that emitted photons from scattering have the same emission probability in forward and backward direction relative to the incident photon. Sequences of photons from scattering of a single incident photon cannot even yield the illusion of a photon track. Light rays in a transparent medium arise through interference from coherent scattering of many incident photons off many scattering centers in the medium, i.e. we cannot interprete them as manifestations of photon tracks.

This can obviously be a transient notion. Protons were considered elementary particles in the 1940s.

Equation (17) provides the \({\mathcal {O}}(\alpha _S)\) probability amplitude for the electron-electron scattering between initial and final normalized wave packets in \({\varvec{k}}\)-space.

The term on the right hand side of Eq. (33) satisfies the consistency condition \({\varvec{\nabla }}\cdot {\varvec{J}}=0\) as a consequence of the local charge conservation law that follows from Eq. (31). We also note that the driving terms on the right hand side of Eq. (33) show that there is no continuous wave-like photon emission from energy eigenstates in the first-quantized theory: substition of wave functions of energy eigenstates of Eq. (31) with \({\varvec{A}}=0\), \(\varphi _{a,s,n}({\varvec{x}},t)=\varphi _{a,s,n}({\varvec{x}})\exp (-\,\mathrm {i}E_n t/\hbar )\), \(\varphi _{a,s,n}({\varvec{x}})\in {\mathbb {R}}\), for the Schrödinger operators \(\varPsi _{a,s}({\varvec{x}},t)\) yields \({\varvec{A}}=0\) as a solution Eq. (33).

The scattering matrix element for photon emission is proportional to \(1/\sqrt{\alpha _S}\) although the photon interaction term is proportional to \(\sqrt{\alpha _S}\). This is because the matrix element \(\langle 1,0,0|z|2,1,0\rangle\) of the hydrogen states is proportional to \(a=\hbar /\alpha _S\mu c\).

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Dick, R. The Role of Quantum Jumps in Quantum Ontology.
*J Gen Philos Sci* (2023). https://doi.org/10.1007/s10838-022-09635-0

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DOI: https://doi.org/10.1007/s10838-022-09635-0

### Keywords

- Quantum ontology
- Copenhagen interpretation
- Ontic states
- Epistemic states