Quantum jumps, superpositions, and the continuous evolution of quantum states

TL;DR

This paper explores how the development of photon quantization and the scattering matrix formalism reconcile the apparent dichotomy between quantum jumps and continuous evolution, impacting interpretations of quantum mechanics and quantum computing.

Contribution

It demonstrates that quantum jumps correspond to transitions in Fock space sectors and that continuous evolution is a sum over jump amplitudes, linking field quantization with quantum state dynamics.

Findings

Quantum jumps are transitions between Fock space sectors.

Continuous evolution is a sum over jump amplitudes.

Implications for quantum interpretations and qubit superpositions.

Abstract

The apparent dichotomy between quantum jumps on the one hand, and continuous time evolution according to wave equations on the other hand, provided a challenge to Bohr's proposal of quantum jumps in atoms. Furthermore, Schroedinger's time-dependent equation also seemed to require a modification of the explanation for the origin of line spectra due to the apparent possibility of superpositions of energy eigenstates for different energy levels. Indeed, Schroedinger himself proposed a quantum beat mechanism for the generation of discrete line spectra from superpositions of eigenstates with different energies. However, these issues between old quantum theory and Schroedinger's wave mechanics were correctly resolved only after the development and full implementation of photon quantization. The second quantized scattering matrix formalism reconciles quantum jumps with continuous time evolution through the identification of quantum jumps with transitions between different sectors of Fock space. The continuous evolution of quantum states is then recognized as a sum over continually evolving jump amplitudes between different sectors in Fock space. In today's terminology, this suggests that linear combinations of scattering matrix elements are epistemic sums over ontic states. Insights from the resolution of the dichotomy between quantum jumps and continuous time evolution therefore hold important lessons for modern research both on interpretations of quantum mechanics and on the foundations of quantum computing. They demonstrate that discussions of interpretations of quantum theory necessarily need to take into account field quantization. They also demonstrate the limitations of the role of wave equations in quantum theory, and caution us that superpositions of quantum states for the formation of qubits may be more limited than usually expected.

Equations

$$E_{\gamma }=\hslash ck=E_{n,\ell }-E_{n^{\prime },{\ell }^{\prime }}.$$

$$|\{C\}(t)⟩=\sum _{n,\ell ,m_{\ell }}{C}_{n,\ell ,m_{\ell }}|n,\ell ,m_{\ell }⟩\exp (-\mathrm{i}{E}_{n,\ell }t/\hslash ),$$

$$⟨E⟩(\{C_{n,\ell ,m_{\ell }}\})=\sum _{n,\ell ,m_{\ell }}|C_{n,\ell ,m_{\ell }}{|}^2{E}_{n,\ell },$$

$${\varrho }_e(\mathbf{x},t)=-e|⟨\mathbf{x}|\{C\}(t)⟩{|}^2$$

$${\mathbf{j}}_e(\mathbf{x},t)$$

$$H_I=-\int d^3\mathbf{x}\mathbf{A}(\mathbf{x},t)\cdot {\mathbf{j}}_e(\mathbf{x},t),$$

$$S_{fi}=\frac{\mathrm{i}}{\hslash }{\int }_{-\infty }^{\infty }dt\int d^3\mathbf{x}\mathbf{A}(\mathbf{x},t)\cdot {\mathbf{j}}_e(\mathbf{x},t),$$

$$\mathrm{i}\hslash \frac{\partial \psi (\mathbf{x},t)}{\partial t}+\frac{{\hslash }^2}{2m}{\left(\nabla +\mathrm{i}\frac{e}{\hslash }\mathbf{A}(\mathbf{x},t)\right)}^2\psi (\mathbf{x},t)$$

$$+\frac{{\alpha }_S\hslash c}{|\mathbf{x}|}\psi (\mathbf{x},t)=0,$$

$$\displaystyle\!\!\!\left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\Delta\right)\bm{A}(\bm{x},t)=-\,\mu_{0}\,\frac{e\hbar}{2\mathrm{i}m}\Big{(}\psi^{+}(\bm{x},t)\bm{\nabla}\psi(\bm{x},t)$$

$$\displaystyle-\,\bm{\nabla}\psi(\bm{x},t)^{+}\cdot\psi(\bm{x},t)+2\mathrm{i}\frac{e}{\hbar}\psi^{+}(\bm{x},t)\bm{A}(\bm{x},t)\psi(\bm{x},t)\Big{)}$$

$$-\frac{{\mu }_{0}e}{4\pi }\frac{\partial }{\partial t}\int d^3{\mathbf{x}}^{\prime }\frac{\mathbf{x}-{\mathbf{x}}^{\prime }}{|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^3}|\psi ({\mathbf{x}}^{\prime },t){|}^2.$$

$$H=H_0+H_I+H_{II},$$

$$H_0$$

$$H_I$$

$$H_{II}=\int d^3\mathbf{x}\frac{e^2}{2m}{\psi }^{+}(\mathbf{x}){\mathbf{A}}^2(\mathbf{x})\psi (\mathbf{x}).$$

$$\mathbf{A}(\mathbf{x},t)$$

$$|\{C\}⟩=\sum _{n,\ell ,m_{\ell }}{C}_{n,\ell ,m_{\ell }}\int d^3\mathbf{x}{\psi }^{+}(\mathbf{x})|0⟩{\Psi }_{n,\ell ,m_{\ell }}(\mathbf{x})$$

$$|\{C^{\prime }\};\mathbf{k},\alpha ⟩$$

$$|n(t\to -\infty )⟩=\underset{t\to -\infty }{\lim }\exp (-\mathrm{i}{H}_{0}t/\hslash )|n⟩,$$

$$S_{n^{\prime }n}$$

$$U_D(t,t^{\prime })$$

$$|\psi (t\to \infty )⟩=\sum _{n^{\prime }}|n^{\prime }(t\to \infty )⟩S_{n^{\prime }n},$$

$$|\psi (t)⟩=\sum _n|n⟩⟨n|\psi (t)⟩$$

$$S_{fi}$$

$$⟨n^{\prime },\ell +\Delta \ell ,m_{\ell }+\Delta m_{\ell }|{\mathbf{ϵ}}_{\alpha }(\mathbf{k})\cdot \mathbf{x}|n,\ell ,m_{\ell }⟩$$

$$=\int d^3\mathbf{x}{\Psi }_{n^{\prime },\ell +\Delta \ell ,m_{\ell }+\Delta m_{\ell }}^{+}(\mathbf{x}){\mathbf{ϵ}}_{\alpha }(\mathbf{k})\cdot \mathbf{x}{\Psi }_{n,\ell ,m_{\ell }}(\mathbf{x}),$$

$$\frac{d{\Gamma }^{(\alpha )}(\mathbf{k})}{d\Omega dk}$$

$${\frac{d{\tilde{\Gamma }}^{(\alpha )}(\mathbf{k})}{d\Omega dk}|}_{|\{C^{\prime }\};\mathbf{k},\alpha ⟩\to |\{C\}⟩}={\frac{d{\Gamma }^{(\alpha )}(\mathbf{k})}{d\Omega dk}|}_{|\{C\}⟩\to |\{C^{\prime }\};\mathbf{k},\alpha ⟩}.$$

$$\frac{d\mathbf{j}(\mathbf{k})}{k^{2}d\Omega dk}=\frac{\mathbf{E}\times \mathbf{B}}{{\mu }_0\hslash ck}=\frac{c}{(2\pi {)}^3}\hat{\mathbf{k}}.$$

Full text

However, these issues between old quantum theory and Schrödinger’s wave mechanics were correctly resolved only after the development and full implementation of photon quantization. The second quantized scattering matrix formalism reconciles quantum jumps with continuous time evolution through the identification of quantum jumps with transitions between different sectors of Fock space. The continuous evolution of quantum states is then recognized as a sum over continually evolving jump amplitudes between different sectors in Fock space.

In today’s terminology, this suggests that linear combinations of scattering matrix elements are epistemic sums over ontic states. Insights from the resolution of the dichotomy between quantum jumps and continuous time evolution therefore hold important lessons for modern research both on interpretations of quantum mechanics and on the foundations of quantum computing. They demonstrate that discussions of interpretations of quantum theory necessarily need to take into account field quantization. They also demonstrate the limitations of the role of wave equations in quantum theory, and caution us that superpositions of quantum states for the formation of qubits may be more limited than usually expected.

††journal: Studies in History and Philosophy of Modern Physics, October 1, 2016

1 Introduction

The question for the interpretation of quantum states remains an unsolved and still controversial issue ever since Schrödinger introduced his evolution equation for quantum states as a function of time, which also opened the door for deterministic interpretations [1]. We know how to combine causal evolution of quantum states with Born’s prescription to relate quantum states to observables, and the most precise physical constants have been calculated using this formalism. However, the final step of projecting out one particular part of a continually evolved quantum state, purportedly in response to a question that we ask through an experimental probe, prima facie seems to introduce a somewhat mysterious discontinuous dynamical aspect into quantum mechanics. This puzzling aspect of quantum mechanics has been investigated from many angles and in many experimental settings, most notably in particle diffraction and Stern-Gerlach type experiments. The atomic emission problem does not enjoy the same level of prominence, although it has played an important role in Schrödinger’s concerns about the foundations of quantum mechanics, and there are aspects in the explanation of the emergence of spectral lines in spontaneous emission which are still challenging from the point of view of the Copenhagen interpretation111The notion of a Copenhagen interpretation of quantum mechanics can mean different things for different experts in the field. Here “Copenhagen interpretation” refers to Born’s interpretation of projections of ontic quantum states onto eigenstates of hermitian operators as probability amplitudes (or amplitudes for probability densities), combined with the assertion that the resulting probabilities measure the likelihood that individual quantum systems (e.g. single particles or few particle systems) collapse into the eigenstate if stimulated through a corresponding observation..

Many of the more problematic aspects of the Copenhagen interpretation can be avoided or resolved in epistemic interpretations of quantum states as records of collective or individual knowledge of observers performing an experiment [2, 3, 4, 5]. There is no need for apparent spontaneous or even retroactive collapse of ontic quantum states in epistemic interpretations, and Schrödinger cat states do not refer to ontological properties of a system. Epistemic interpretations therefore challenge the realist or ontic interpretation of quantum states as records of complete, objective information of ontological properties of physical systems. However, it is likely a fair assessment that “quantum practitioners”, while mostly remaining agnostic with respect of the difficult philosophical aspects of quantum mechanics, tacitly use a realist interpretation of quantum states. Furthermore, epistemic interpretations have recently been confronted with numerous counterarguments on the basis of quantitative and observational analyses of entangled states [7, 6, 8, 9, 10, 11, 12, 13], which in turn have been challenged through full locality in the QBism framework [14].

Another interesting way to avoid puzzling aspects of the traditional Copenhagen interpretation is the statistical ensemble interpretation of quantum mechanics [15, 16]. This emphasizes that the Born rule relates predictions from single particle wavefunctions to experiments with many particles, but that we cannot infer predictions for single particles from the wavefunction. In particular, there is no need for quantum state collapse if single particle wavefunctions cannot be directly associated with the behavior of individual particles. This proposal in itself does not necessarily favor either an ontic or an epistemic interpretation of quantum states. The flipside of the statistical ensemble interpretation (in my understanding) is that we would have to resign ourselves to the fact that we do not have access to any single particle dynamics, or otherwise assume that quantum mechanics is only a statistical approximation to a yet to be discovered single particle dynamics.

The amount of intellectual effort and capacity spent on the interpretation of quantum mechanics, and the fact that the discussion has intensified rather than subsided over the past 90 years, provide testimony to the difficulty of the underlying interpretational problems. The present paper tries to shed light on these questions from the perspective of the theory of photon emission and absorption by atoms. The underlying calculations are well-known standard applications of quantum optics. However, I would like to argue that paying attention not only to the overall results of the calculations, but to the derivation and to particular details of the results, helps to put the quest for a universally acceptable interpretation of quantum states into even stronger focus. I hope to convince the reader that revisiting the atomic emission problem as a case study for what everyday practical calculations reveal about quantum systems, yields interesting insights into the nature of quantum states, including the possibility of epistemic combinations of ontic states in Fock space.

For example, we will see that the scattering matrix elements for optical emission and absorption both for single photons and for coherent states show that the formalism of calculating quantum mechanical observables assigns special significance to energy eigenstates even if the observables do not include the energy of any component of a system. More specifically, quantum jumps between energy eigenstates seem to be an integral part of the response of every quantum system no matter how we probe the system. How can that be if all unitarily equivalent bases of quantum states are physically equivalent? What makes energy eigenstates a preferred part of the response of a system even if we are not probing for energies? On the other hand, how is it possible to reconcile quantum jumps and continuous evolution of quantum states?

Besides illuminating the problem of the interpretation of quantum states and the measurement process from a different angle, reviewing quantum optical calculations with a critical view on what they really tell us is also of renewed practical interest. Demonstrations of single photon emissions from quantum dots [17, 18, 19] and the prospects to use these devices for quantum cryptography and quantum information processing force us to sharpen our understanding of the basic quantum optical processes of photon emission, absorption, and observation. Moreover, progress in attosecond spectroscopy [20, 21] now affords observations of electrons with atomic time scale resolutions [22, 23, 24, 25, 26]. The non-perturbative nature of the strong electromagnetic fields involved in these experiments led to theoretical analyses of the observations through continuous evolution of atomic wavefunctions between different energy levels within the confines of semi-classical approximations. This is born out of practical calculational necessity and should be justified on the basis of high photon numbers in the laser pulses. However, semi-classical calculations for optical responses of small numbers of atoms, described through continuously evolving wavefunctions, are also at the heart of de Broglie-Bohm or Schrödinger type interpretations of quantum dynamics222I will denote theories or interpretations which assume that formulations of quantum dynamics should be possible in terms of continually evolving wavefunctions without any need for quantum jumps as “de Broglie-Schrödinger type”. This is motivated by de Broglie’s early proposal of de Broglie-Bohm type pilot wave theories for continuous particle trajectories and by Schrödinger’s early interpretations of redundancy of wave-particle duality in a theory of continually evolving wavefunctions. Indeed, Schrödinger’s early work on quantum mechanical wave equations was influenced by de Broglie’s ideas, but the vagueness of his early statements on the interpretation of the wavefunction seem to indicate a readiness to move beyond a matter wave picture and could even be interpreted as early hints at a statistical interpretation, see in particular the paragraph on pp. 134-135 in Ref. [1]. However, later in life Schrödinger clearly embraced de Broglie’s ideas on matter waves [27, 28], although he did not consider them as pilot waves for particles. The early developments of de Broglie-Bohm theories and of Schrödinger’s interpretation are beautifully reviewed in [29] and [30], respectively., which assume that continually evolving wavefunctions can provide a complete description of quantum evolution without a need for quantum jumps [27, 28]. This is in contrast to the well-established second quantized scattering matrix formalism, which yields continually evolving probability amplitudes for quantum jumps.

Therefore, a particular topic that we will reconsider for its relevance to single photon generation and detection, and for its potential to afford us a better understanding of the foundations of quantum dynamics and the meaning of quantum states, is the emergence of line spectra from atomic states. We will see that the issues of quantum jumps, photon quantization, scattering matrices for photon emission or absorption, and interpretation of quantum states in atom-photon systems are all intertwined. The second quantized scattering matrix will play a central role in our analysis, but we need to develop a brief historic perspective on Schrödinger’s wave ontology in relation to line spectra to fully appreciate what the scattering matrix has to tell us. This will be done in Section 2. We will see in particular that combinations of the established wave equation for non-relativistic particles, viz. Schrödinger’s equation, with Maxwell’s equations for radiation cannot support Schrödinger’s proposal of continuous (i.e. without quantum jumps) generation of radiation from atoms. This observation reconfirms the central role of the second quantized scattering matrix formalism for the description of emission from atoms.

The emission rates and absorption cross sections are therefore revisited for monochromatic photon states in Section 3, and for coherent photon states in Section 4. Section 5 discusses our conclusions.

Details of the standard derivations and results for photon emission and absorption in quantum theory are particularly relevant for the present investigation. They will tell us that quantum jumps are unavoidable for the formation of line spectra. The scattering matrices for coherent photon states will also demonstrate that collapse into energy eigenstates is an inevitable phenomenon in quantum optics even in cases where we do not observe the energy of any component of the system at any time. This is clearly worrisome for traditional proposals of observer or apparatus induced state collapse. On the other hand, there is also no space here for any critical participation of the environment in the formation of line spectra. The results rather indicate a mixed ontic and epistemic interpretation of quantum state evolution as encoded in scattering matrices: Scattering matrices generically evolve ontic $N$-particle states into epistemic sums spanning several sectors of Fock space, i.e. scattering matrices describe epistemic sums over ontic states.

The phrases “quantum jump” and “quantum leap” are usually interchangeable synonyms in physics. Here the word “jump” will be used to designate sudden, discontinuous transitions between ontic quantum states with the underlying assumption that the transition happens between the states without any necessary involvement of an observer. On the other hand, the word “leap” will be associated with a discontinuous reassignment of a quantum state to a system by an observer, as a consequence of a change of knowledge about the system. This is not a clear call in many instances, and in cases where neither an ontic nor an epistemic interpretation appears warranted, the word “jump” will be used as the default option.

2 Schrödinger’s explanation for line spectra

It is prudent to revisit Schrödinger’s proposal for formation of line spectra from continually evolving wavefunctions before taking the second quantized scattering matrix formalism and quantum jumps for granted.

If the time-independent Schrödinger equation did not have a time-dependent counterpart, the existence of discrete energy eigenstates in atoms would be sufficient to explain the observation of line spectra because we could only form superpositions of eigenstates within one energy level. The time-dependent Schrödinger equation invalidates this simple reasoning, and therefore the postulate that a single quantum system (like an atom) always responds with an eigenvalue of the observed quantity [31] is usually invoked to argue that observation of emitted radiation yields observed values in terms of eigenvalues of atomic Hamiltonians,

[TABLE]

The underlying problem is the superposition principle for solutions of the time-dependent Schrödinger equation. If hydrogen atoms can exist in a unitarily evolving superposition of bound energy eigenstates333We neglect a possible admixture of Coulomb waves for simplicity. The purpose is to study the conceptual problem of line spectra, for which linear combinations of bound states are sufficient.

[TABLE]

with $\sum_{n,\ell,m_{\ell}}|C_{n,\ell,m_{\ell}}|^{2}=1$ and energy expectation value

[TABLE]

why does a transition to a new state with expansion coefficients $C^{\prime}_{n,\ell,m_{\ell}}$ and lower energy expectation value not lead to observation of photons of energy $E_{\gamma}=\langle E\rangle(\{C_{n,\ell,m_{\ell}}\})-\langle E\rangle(\{C^{\prime}_{n,\ell,m_{\ell}}\})$? Stated differently: Why does emission or absorption of photons by atoms only involve transitions between energy eigenstates in agreement with Bohr’s explanation of atomic transitions in old quantum theory? Is it possible to demonstrate that unperturbed atoms should only exist in energy eigenstates in spite of the fact that (2) is a perfectly viable solution of the time-dependent Schrödinger equation? Or otherwise, is it possible to explain observation of line spectra without inferring that the atomic transitions involve pure energy eigenstates as initial and final states of the emitting or absorbing atoms?

These questions about the role of unitarily evolving superpositions of energy eigenstates arose as soon as Schrödinger published the time-dependent version of his wave equation [1], when superposition of atomic eigenstates for different energies instead of stationary Bohr orbits became an apparent possibility for the state of an atom. However, Schrödinger did not consider this as a potentially problematic challenge to the direct transition between Bohr orbits in old quantum theory, but rather as an opportunity to propose an alternative explanation for line spectra through the beats in the electric fields of atomic electric charge and current densities from the time-dependent interference terms in

[TABLE]

and

[TABLE]

see pp. 121 and 129-130 in Ref. [1].

Indeed, the photon-matter interaction term in leading order in minimal coupling is

[TABLE]

and this yields the leading order scattering matrix element for photon absorption or emission,

[TABLE]

where $\bm{A}(\bm{x},t)$ is the vector potential corresponding to the absorbed or emitted photon. The time integration in (5) forces the Fourier components of $\bm{A}(\bm{x},t)$ to match the atomic transition frequencies in (3), whereas all the other modes suffer destructive interference. This yields the line spectra even for transitions between superpositions of atomic eigenstates and even if the photon states are not monochromatic. From a de Broglie-Schrödinger perspective this looks like the absorption or emission of radiation arises as a consequence of transient beats in the intra-atomic electric current. In stark contrast to the traditional interpretation of the very same scattering matrix element (5), it would not arise as a consequence of a quantum jump from one energy eigenstate to another energy eigenstate, but rather as a consequence of a temporary superposition of both energy eigenstates which may arise as a consequence of a perfectly smooth time evolution of the atomic state.

Schrödinger was mostly concerned with applications of his time-dependent wave equation to light scattering in Ref. [1], i.e. Schrödinger offered a qualitative explanation for emission and absorption of light through beats in the wavefunction, but he did not explicitly write down the coupling term (4) nor did he calculate corresponding emission and absorption probabilities. Emission and absorption probabilities were calculated by Dirac in the semi-classical dipole approximation two months later [32]. However, Dirac abstained from any speculation as to the origin of the transition, i.e. he did not address the question whether he was calculating probabilities for quantum jumps in agreement with Bohr’s proposal from old quantum theory, or whether he was dealing with consequences of quantum beats as Schrödinger had mused two months earlier. On p. 264 of his follow-up paper [33], which went beyond the semi-classical approximation by coupling a quantized electric field into the Schrödinger equation, Dirac does explicitly refer to quantum jumps. Dirac does not offer an explanation for his choice of interpretation, but there may be two reasons for his preference for quantum jumps: The actual calculation of emission and absorption coefficients through integration of the time-dependent Schrödinger equation does not require a superposition of energy eigenstates for the initial state, but naturally evolves even a pure energy eigenstate into a superposition of eigenstates. The intuitive (although not logically inevitable) conclusion from this is that quantum beats are not necessary for emission and absorption of radiation444The conclusion is not logically inevitable, because coupling to the electromagnetic field immediately evolves a pure energy eigenstate into a superposition of atomic eigenstates, which could be read to imply quantum beats in the sense of Schrödinger within the confines of semi-classical approximations.. Furthermore, Schrödinger himself cautioned on p. 130 of [1] that a non-linear feedback mechanism would have to be added to his equation to terminate emission from quantum beats and explain how the system relaxes into the groundstate, and this may have stalled further pursuit of his idea. Schrödinger reiterated the idea of radiation from quantum beats in atoms much later in [27, 28], but again only in a qualitative sense.

It is noteworthy that there is no need to invoke observers or probes in the quantum beat interpretation, nor is there any need to spontaneously collapse states to energy eigenstates at any time. In short, Schrödinger’s proposal amounted to a trade off between actual electronic oscillations in atoms on the one hand versus some of the more problematic aspects of the emerging Copenhagen interpretation on the other hand. In this regard it is amusing to note that on p. 133 of his famous textbook on atomic physics [34], Born also proposed a quantum beat interpretation of the dipole matrix element $e\langle n^{\prime},\ell^{\prime},m^{\prime}_{\ell}|\mathrm{\bf x}|n,\ell,m_{\ell}\rangle$ which appears in scattering elements for photon emission.

However, insurmountable weaknesses of Schrödinger’s proposal of radiation from quantum beats are the reliance on a semi-classical description of electromagnetic fields, and that he never developed it into a mathematical theory for emission line strengths. Schrödinger mentions the need for nonlinear coupling terms for wavefunctions in [28], but he never writes down a coupled Schrödinger-Maxwell system to see how far his ideas might carry. Indeed, he would have found that modifications of his wave equation would have been necessary. This point is easily illustrated by studying the Schrödinger-Maxwell system for the hydrogen atom,

[TABLE]

[TABLE]

This describes the coupling of the electromagnetic potential in Coulomb gauge, $\bm{\nabla}\cdot\bm{A}(\bm{x},t)=0$, to the wavefunction $\psi(\bm{x},t)$ for relative motion in the atom. It is a good approximation if the dominant wavelengths of the electromagnetic field are large compared to the Bohr radius. The last term in Ampère’s law (7) in Coulomb gauge comes from the longitudinal component of the electric field.

A severe difficulty with Schrödinger’s proposal is that the evolution equations (6,7) for the atom-photon system have solutions which correspond to excited atomic energy eigenstates without radiating a photon. Absence of radiation from excited states on the level of the wave equations (6,7) can be seen e.g. by observing that substitution of the atomic eigenstates $\psi_{n,\ell,0}(\bm{x},t)$ into the equations solves these equations for $\bm{A}(\bm{x},t)=0$. For $m_{\ell}\neq 0$, we note that the excited energy eigenstates $\psi_{n,\ell,m_{\ell}}(\bm{x},t)$ yield static vector potentials $\bm{A}(\bm{x})$, but no radiation fields. Indeed, equations (6,7) cannot describe emission of classical radiation under any circumstances, since a Poynting vector for radiation must drop with distance $r$ like $r^{-2}$, whereas any Poynting vector formed from solutions of (7) with atomic orbitals with principal quantum numbers $n$ and $n^{\prime}$ on the right hand side would be exponentially suppressed at large distance with a factor $\exp[-2(n+n^{\prime})r/(nn^{\prime}a_{0})]$. This implies that the coupling terms of the atom-photon system cannot explain spontaneous emission from excited atomic eigenfunctions as a consequence of continuous time evolution according to the wave equations (6,7). The continually evolving atomic energy eigenfunctions $\psi_{n,\ell,m_{\ell}}(\bm{x},t)$ would not spontaneously decay through emission of electromagnetic radiation at the level of wave equations. Even with the coupling terms included, the Schrödinger equation and Ampère’s law predict that excited atomic eigenstates would be stable within the semi-classical formalism, and no classical radiation could be emitted under any circumstances.

Instead, we will see in the following sections that the minimally coupled Schrödinger-Maxwell equations require quantized photon operators to describe spontaneous emission of radiation. We will also see in Sections 3 and 4 that quantum jumps appear as a generic property of any electromagnetic interaction of atoms, no matter whether any energies are resolved. To elucidate these points, the explicit or implicit appearance of line spectra will be addressed from the perspective of two different optical probes, viz. monochromatic photon states and coherent photon states, to see what the standard calculations of photon absorption and emission in quantum mechanics tell us about the dichotomy between superpositions of energy eigenstates on the one hand and line spectra on the other hand. This will lay the groundwork for explaining how second quantization reconciles the deterministic evolution of quantum states with quantum jumps in Section 5. It will also provide yet another piece of evidence for the importance of field quantization for interpretations of quantum mechanics, thus supplementing the recent observations of Myrvold on the relevance of quantum fields for the emergence of wavefunctions [35].

3 Absorption and emission of monochromatic photon states

We are interested in photon emission or absorption in the visible and near-ultraviolet wavelength range, where dipole approximation is excellent and the coupling of electromagnetic fields can effectively be described by minimal coupling to the atomic wavefunction for relative motion of the electron and proton, rather than coupling to the individual constituents of the atom. The relevant second quantized Hamiltonian for the study of electromagnetic absorption and emission from a gas of hydrogen atoms is therefore

[TABLE]

where

[TABLE]

is the Hamiltonian of decoupled hydrogen atoms and external photons in Coulomb gauge. The constant $\alpha_{S}=e^{2}/4\pi\epsilon_{0}\hbar c$ is the fine structure constant. The interaction terms are the minimal coupling terms in Schrödinger field theory,

[TABLE]

[TABLE]

The Hamiltonian (8) is written in terms of the Schrödinger picture field operators $\psi(\bm{x})$, $\bm{A}(\bm{x})$, and the electromagnetic field is quantized in Coulomb gauge, i.e. the photon field in the interaction picture is

[TABLE]

with $\bm{k}\cdot\bm{\epsilon}_{\alpha}(\bm{k})=0$, $\bm{\epsilon}_{\alpha}(\bm{k})\cdot\bm{\epsilon}_{\beta}(\bm{k})=\delta_{\alpha\beta}$, $[a_{\alpha}(\bm{k}),a_{\beta}(\bm{k}^{\prime})]=0$, $[a_{\alpha}(\bm{k}),a_{\beta}^{+}(\bm{k}^{\prime})]=\delta_{\alpha\beta}\delta(\bm{k}-\bm{k}^{\prime})$, and $\dot{\bm{A}}(\bm{x})\equiv\dot{\bm{A}}(\bm{x},0)$ in $H_{0}$.

Spin labels will be suppressed since we did not include the subleading Pauli term and the terms (9,10) do not induce spin transitions. We will first discuss emission rates before deriving absorption cross sections. In agreement with the superposition principle in quantum mechanics, no presumption will be made that initial or final atomic states should be energy eigenstates.

To explore the tension between the superposition principle and the emergence of line spectra, we have to study the scattering matrix element for the transition from the initial state (now written in the language of second quantization and at $t=0$)

[TABLE]

to the final state with an emitted photon of momentum $\hbar\bm{k}$ and polarization $\bm{\epsilon}_{\alpha}(\bm{k})$,

[TABLE]

Here boldface Dirac notation $\bm{|}\ldots\bm{\rangle}$ is introduced for the Fock space states of the second quantized theory, to tell them apart from the states of the first quantized theory like (2).

Photon absorption and emission amplitudes for transitions in the form of scattering matrix elements between energy eigenstates are reported in many textbooks, see e.g. [36, 37, 38, 16, 39, 40], and just considering those scattering matrix elements in isolation can make us easily forget that they actually correspond to a superposition of energy eigenstates at least in the final state: Even if we presume that the initial state is an energy eigenstate of the unperturbed system,

[TABLE]

the scattering matrix

[TABLE]

with the interaction picture (or Dirac picture) time evolution operator

[TABLE]

always produces the final state as a superposition of eigenstates,

[TABLE]

i.e. quantum dynamics does not seem to describe photon emission or absorption as quantum jumps between Bohr levels. However, please note that the superposition (16) necessarily includes states from different sectors in Fock space, which implies that the state upon observation still includes a reduction e.g. to a single atom state or to a state containing both an atom and a photon. In that sense, $\bm{|}\psi(t\to\infty)\bm{\rangle}$ in (16) should be considered as an epistemic state, which reduces to an ontic component upon observation.

It is noteworthy that the first two papers on time-dependent perturbations, viz. Schrödinger’s early study of light scattering in Ref. [1] and Dirac’s first path breaking paper [32] on the development of the general theory of time-dependent perturbations, explicitly use linear superpositions of energy eigenstates both in the initial and in the final state (however, without addressing the potential contradiction with quantum jumps between energy levels in [32]). Dirac’s second paper [33], which was partly written in Copenhagen, assumes energy eigenstates as initial states, but still needs to treat the final states as superpositions of energy eigenstates in agreement with (16). Therefore the Schrödinger equation was a breakthrough in quantum theory by providing a dynamical rationale to Heisenberg’s transition matrix elements as scattering matrix elements, but it also created a problem by apparently taking the jumps out of the theory, which on the other hand seem inevitable to understand observations.

Viewing (16) only as a formal sum over transition probability amplitudes would suggest that the Schrödinger equation and its relativistic generalizations are just evolution equations for mnemonic devices

[TABLE]

which only remind us that the probability to encounter the eigenstate $\bm{|}n\bm{\rangle}$ at time $t$ is $|\bm{\langle}n\bm{|}\psi(t)\bm{\rangle}|^{2}$. Thinking of $\bm{|}\psi(t)\bm{\rangle}$ only as a mnemonic device for evolution of probability amplitudes makes the whole conundrum of the “dynamics of collapse of a state” redundant. This is the information explanation (or epistemic view) of quantum states [2, 3, 4, 5, 14], and it holds the promise to avoid any need for hidden dynamics or hidden variables without invoking problematic assumptions about the observer/system interaction or the quantum/classical divide555It is difficult to assess from Bohr’s writings on quantum states and observers how much importance he would have assigned to purely epistemic interpretations. This question is beautifully revisited in a recent investigation by Zinkernagel [41].. Rather than being a dynamical variable attributed to the system in the same sense as e.g. location $\langle\bm{x}\rangle(t)$, the state $\bm{|}\psi(t)\bm{\rangle}$ summarizes our knowledge about the system, without any connotation that there is any deficiency in the sense that there should be more to know. As already emphasized before, this approach has been challenged with numerous counterarguments [7, 6, 8, 9, 10, 11, 12, 13], but it offers particularly appealing perspectives on the problem of line spectra, and we will continue to use the epistemic interpretation for illustrative purposes. Indeed, both ontic and epistemic views of quantum states have so many respective advantages that the question may well be: If these opposed interpretations can be reconciled in some way, how much of each interpretation can and must be maintained?

If we suppose that the state $\bm{|}\psi(t)\bm{\rangle}$ just represents our best possible knowledge about a quantum system, can we do without quantum jumps to explain line spectra? The explicit forms of the scattering matrix elements between superpositions of atomic energy eigenstates with different kinds of photon states in the initial or final state will help us to shed some light on this question.

The scattering matrix between eigenstates of $H_{0}$ to any order (see e.g. Section 13.7 in Ref. [40], here $V=H_{I}+H_{II}$),

[TABLE]

implies that superposition of unperturbed states will always yield sums over energy preserving $\delta$ functions between unperturbed energy eigenstates (see the appendix for an explanation of the emergence of energy preserving $\delta$ functions in scattering matrix elements). This implies that the scattering matrix for bound atomic states will produce line spectra for single photons at every order of perturbation theory. We will exlicitly demonstrate the emergence of line spectra from superpositions for two different sets of observations in first order perturbation theory.

The differential photon emission amplitude for the states (12,13) is in first order

[TABLE]

In agreement with equations (14,16) and generally accepted conventions of applied quantum mechanics, this is a probability amplitude for the transition from the initial atomic state $\bm{|}\{C\}\bm{\rangle}$ (12) to the final state $\bm{|}\{C^{\prime}\};\bm{k},\alpha\bm{\rangle}$ (13) containing an atomic state with expansion coefficients $C^{\prime}_{n,\ell,m_{\ell}}$ and a photon state with momentum $\hbar\bm{k}$ and polarization $\bm{\epsilon}_{\alpha}(\bm{k})$.

Evaluation of (18) in dipole approximation yields the superposition of the familiar scattering matrix elements for transitions between atomic energy eigenstates,

[TABLE]

where $\omega_{n^{\prime},\ell+\Delta\ell;n,\ell}\equiv\omega_{n^{\prime},\ell+\Delta\ell}-\omega_{n,\ell}$ and dipole selection rules resulting from the matrix element are already taken into account. Evaluation of all the quantized field operators has transformed the second quantized matrix element in (18) into a remnant combination of first quantized matrix elements between atomic wavefunctions,

[TABLE]

[TABLE]

where we use upright notation $\mathrm{\bf x}$ for the operator for particle location in quantum mechanics.

The scattering matrix element (19) yields the differential emission rate666See e.g. [37, 40] for an explanation of the relation between scattering matrix elements and emission rates. The time $\Delta t$ is the observation time window, which is finally taken as $\Delta t\to\infty$, see the appendix. Equation (20) gives the differential emission rate in the sense that the number of photons with polarization $\bm{\epsilon}_{\alpha}(\bm{k})$, emitted by $n(\{C\})$ atoms in the state $\bm{|}\{C\}\bm{\rangle}$ into a solid angle $\Omega$ and with wavenumbers $k_{1}\leq k\leq k_{2}$, due to the atomic transition into the final state $\bm{|}\{C^{\prime}\}\bm{\rangle}$, is $n(\{C\})\int_{\Omega}d\Omega\int_{k_{1}}^{k_{2}}dkd\Gamma^{(\alpha)}(\bm{k})/d\Omega dk$.

$d\Gamma^{(\alpha)}(\bm{k})/d\Omega dk=k^{2}|S_{\!fi}|^{2}/\Delta t$ in the form

[TABLE]

where it is taken into account that the energy preserving $\delta$ function usually prevents additional interference besides the interference terms which arise from energy degeneracy with respect to $m_{\ell}$. The emergence of the energy preserving $\delta$ functions in (17,19,20) is briefly outlined in the appendix.

The differential emission rate can be readily integrated with respect to photon energy, but the spectrally resolved emission rate is more useful for our discussion. The important message in (19,20) is the appearance of the energy preserving $\delta$ function and the appearance of interference terms in (20) from $m_{\ell}$ degeneracy.

Although our initial and final atomic states are superpositions of energy eigenstates, standard quantum dynamics without any additional assumptions implies that the energy resolved spectrum (20) from transition between those initial and final states yields the same spectral lines that we get from transitions between energy eigenstates. Can we then conclude that observation of the first Balmer line in a spectrum implies that all those photons contributing to that line arise from actual quantum jumps from an $n=3$ state to an $n^{\prime}=2$ state? The derivation of (20) does not help to justify such a conclusion, since it allows for creation of a contribution to the Balmer line along with contributions to several other lines. We have to invoke energy conservation for the single event that created the observed photon to conclude that an actual transition from an $n=3$ state to an $n^{\prime}=2$ state has taken place to create each of those photons which contribute to the first Balmer line. From an epistemic point of view, the ensuing reduction of the quantum state is only puzzling if we do not take into account the information aspects of the quantum state. Observation of one particular component, viz. the emitted photon, has reduced the wavefunction of the atomic component of the atom-photon system to an $n=3$ state before emission and an $n^{\prime}=2$ state after emission, but this is only a reflection of our change of knowledge about the system due to observation of the photon. It does imply, however, collapse of the atoms into energy eigenstates during the interaction, since atomic collision will usually perturb the eigenstates before and after the emission, and therefore we cannot postulate (contrary to Bohr’s orbits) that atoms can only exist in energy eigenstates. Interpretation of the scattering matrix element (19) in terms of a collapse of atomic states therefore appears inevitable. This creates a problem if we associate quantum state collapse with observation, as in the traditional Copenhagen interpretation, since it would apparently introduce an acausal, retroactive aspect into the collapse. The matrix element (19) indicates that collapse is not a consequence of observation, but rather a hallmark of occurence of an elementary interaction, viz. spontaneous emission of a photon.

With respect to the photons, we have decomposed the emitted radiation into monochromatic photon states through decomposition in that basis of states, because Born’s rule implies that this is appropriate for the calculation of signals in an experiment where we would place a spectral polarimeter in direction $\hat{\bm{k}}$ from an emitting gas of atoms or molecules. The particular component that emerges from a prism or grating spectrograph in a certain direction consists of monochromatic photon states of momentum $\bm{k}$. However, that is not because the spectrograph collapsed photons into these states in a dynamical sense. Instead it scattered the incoming photons coherently into beams of outgoing photons, and the basic quantum electrodynamical processes involved imply that incoming photons are virtually absorbed by electrons in the spectrograph and re-emitted as outgoing photons. The corresponding reduction of the state only signifies what we know about the photons emerging in a particular direction from the spectrograph. There is a difference here compared with the atomic states, since we have to infer a collapse for the atomic states in ontic representations to make sense of the energy condition implied with the scattering matrix element (19), whereas the photon is spontaneously emitted in agreement with the energy condition and no collapse of any pre-existing photon state is needed.

In agreement with unitarity, the first order scattering matrix element for the absorption process $\bm{|}\{C^{\prime}\};\bm{k},\alpha\bm{\rangle}\to\bm{|}\{C\}\bm{\rangle}$ is the negative complex conjugate of the scattering matrix element for the inverse emission process (18). The absorption rate is therefore given by the same expression (20) as the corresponding emission rate,

[TABLE]

The differential absorption rate can be converted into an absorption cross section by dividing with the differential current density per $\bm{k}$ space volume for photons of momentum $\hbar\bm{k}$,

[TABLE]

This yields

[TABLE]

The sharp absorption lines in (22) are often approximated through Lorentzian profiles in practical applications of absorption cross sections. However, the important, although at this stage certainly not unexpected message from (22) is that quantum dynamics again synthesizes the response of the system through superposition of photon absorption between pairs of atomic energy eigenstates without any presumption that absorption actually proceeds through energy eigenstates. As before, we have to infer quantum state collapse and the direct involvement of the atomic energy eigenstates from energy conservation for single photons.

What we will see in spectral decompositions of emission or absorption spectra will be emission lines (20) or absorption lines (22) with photon energies corresponding exactly to the transition energies between atomic energy eigenstates, even when we suppose that the transitions happen between superpositions of energy eigenstates. The emission and absorption line spectra are in perfect agreement with observations, of course, but the emphasis is on the observation that direct quantum evolution on the basis of wave equations does not automatically imply direct emission or absorption through atomic energy eigenstates to predict the observed line spectra. Deterministic evolution of quantum states remains mute on that particular aspect. Within the epistemic view, we could nevertheless infer that emission or absorption of photons proceeds through the corresponding atomic energy eigenstates. That only seems to violate causality through retroactive reduction of the state of the atom, because changing the state only signifies our change in knowledge. We also do not have to postulate that the atom reacts to our observation, or that the photon acts as an observer. However, the photon certainly acts as a messenger of information, and there is nothing controversial about that. Therefore a purely epistemic view of quantum states seems to work just fine, but there are caveats: On the one hand it is difficult to conceive that a system must not have some unique and ultimately observable state at a given time and we would have to assign special significance to energy eigenstates.

We will re-examine the emergence of line spectra using other probes to gain further insights into the interpretational questions from the point of view of applied quantum mechanics.

4 Absorption and emission of coherent photon states

As a complementary approach to the interpretation of observations of absorption or emission of radiation by atoms or molecules, we will now assume that the experiment prepares or tests the electric or magnetic fields of the radiation.

The electric and magnetic field operators

[TABLE]

yield expectation values corresponding to a classical electromagnetic wave,

[TABLE]

[TABLE]

[TABLE]

if we use the coherent photon state [42]

[TABLE]

with expectation values for photon number, energy and momentum

[TABLE]

[TABLE]

Here the definitions $\zeta(\bm{k})\cdot a^{+}(\bm{k})=\sum_{\alpha=1}^{2}\zeta_{\alpha}(\bm{k})a^{+}_{\alpha}(\bm{k})$, $\left|\zeta(\bm{k})\right|^{2}=\sum_{\alpha=1}^{2}\zeta^{+}_{\alpha}(\bm{k})\zeta_{\alpha}(\bm{k})$ were used.

Coherent states are often associated with large photon numbers, when their relative quantum uncertainties $\Delta n/\langle n\rangle=\langle n\rangle^{-1/2}$ and

[TABLE]

are small. However, no principle of quantum mechanics prevents formation of coherent states for small photon numbers or interaction of coherent photon states with single atoms or molecules.

The first order scattering matrix element for emission of a coherent photon state is again the negative complex conjugate of the scattering matrix element for absorption,

[TABLE]

and evaluation in dipole approximation yields

[TABLE]

where the frequency in the condition on the bracket is the transition frequency, $k=\omega_{n,\ell;n^{\prime},\ell+\Delta\ell}/c$.

However, the calculation of the spectral photon emission rate (20) or the absorption cross section (22) is based on the completeness relation

[TABLE]

in the photon sector of Fock space. Therefore the calculation of probabilities for emission or absorption of coherent states requires the use of a cavity approximation since the classical decomposition of the identity through coherent states of harmonic oscillators can be formulated for coherent photon states like $\bm{|}\bm{\zeta}\bm{\rangle}$ only in the form

[TABLE]

The resulting discretization of momentum space amounts to the substitution

[TABLE]

in the mode expansion of the vector potential, and the amplitudes $\zeta_{\alpha}(\bm{k})$ become dimensionless. The matrix element

[TABLE]

therefore yields the $\mathcal{O}(\alpha_{S})$ differential transition probabilities

[TABLE]

between the in and out states.

The scattering matrix elements (27) contain already a summation over photon momentum, and therefore $|S_{\!fi}|^{2}$ does not contain any $\delta$ function any more which generates a line spectrum. Indeed, any signal that tests for the electric or magnetic field of emitted photons from a low density atomic or molecular gas would observe a response that will not yield a line spectrum but interference of photon energies from many different atomic or molecular transitions. Observation of emitted radiation through induced electric or magnetic polarization or electromagnetic forces on probes would apparently constitute such measurements. This part of the result is in agreement with the Copenhagen interpretation. It is significant, however, that the scattering matrix elements (27) still synthesize the signals (28) from interference of transitions between equidistant pairs of stationary states, $\hbar ck=E_{n,\ell}-E_{n^{\prime},\ell+\Delta\ell}$. They only combine those pairs differently through summation over the photon polarizations and summation over the momentum directions $\hat{\bm{k}}$. Why do quantum jumps between energy eigenstates play such an unseemingly prominent role in dynamical responses of quantum systems if we do not even care about energy of any component of the observed system at any time?

The apparent ubiquity of quantum jumps between Bohr levels even when no component of the system is prepared or observed in an energy eigenstate seems very surprising from the perspective of the traditional Copenhagen interpretation. It also presents a puzzle for purely epistemic interpretations of quantum states. Why should the dynamical response of atoms to coherent electromagnetic probes involve quantum jumps between energy eigenstates if these are not ontological aspects of atoms?

5 Conclusions

We can now return to our original question for the emergence of line spectra when we spectroscopically resolve the emitted or absorbed radiation from a thin gas of atoms or molecules. What we observe when using a spectrograph in direction $\hat{\bm{k}}$ from the sample is a projection onto photon momentum eigenstates $\bm{\langle}\bm{k},\alpha\bm{|}$, and is therefore described by the spectral emission rate $d\Gamma(\bm{k})/d\Omega dk=\sum_{\alpha}d\Gamma^{(\alpha)}(\bm{k})/d\Omega dk$, see equations (18-20). On the other hand, observing electric or magnetic polarization effects from the emitted radiation amounts to projection onto coherent final state components $\bm{\langle}\bm{\zeta}\bm{|}$ (24). However, both signals are synthesized from interference of transitions between equidistant pairs of stationary states of the unperturbed system, $\hbar ck=E_{n,\ell}-E_{n^{\prime},\ell+\Delta\ell}$. Technically this comes about because evaluation of the matrix elements of the operator (15) automatically projects out the $\omega_{fi}=0$ component in the Fourier decomposition of signal amplitudes at every level of perturbation theory (17), where $\omega_{fi}=(E_{f}-E_{i})/\hbar$ is the transition frequency between unperturbed eigenstates of the atom-photon system. This applies even if the observation does not imply any energy measurement of any component of the system in the initial or final state. This is certainly demonstrated by equations (24-28). These observations cast doubts on any notion of observer induced ontological collapse of wavefunctions of atoms or photons. On the other hand, the ubiquity of quantum jumps between energy eigenstates in optical experiments, even when we do not observe any energy of any component of the system, casts doubts on purely epistemic interpretations. Ontological quantum jumps between atomic energy eigenstates, accompanied by observer-independent collapse of atomic states, appears as the most credible interpretation of equations (24-28).

The sharp jump condition (1) follows only for very long time evolution from the initial unperturbed state to the final state, and one might suspect that jumps are an artifact of asymptotic behavior for large times. However, the scattering matrix for finite times777The scattering matrix for finite time was already implicit in the early papers on time-dependent perturbation theory by Dirac [32, 33] and by Born and Heisenberg (see [43, 44]), although the elegant formulation in terms of time evolution operators was given only much later by Dyson [45]. maps expansion coefficients of the quantum state at any initial time $t^{\prime}$ to expansion coefficients at any final time $t$,

[TABLE]

and this will still yield quantum jumps, albeit with a finite energy uncertainty of order $\hbar/(t-t^{\prime})$. Note that the derivation of the scattering matrix element (19) only involves the matrix formulation of standard quantum mechanical time evolution without any proposition about relative distances between components. Neither the completeness relations for eigenstates of $H_{0}$ nor the identification of the time evolution operator $U(t,t^{\prime})$ nor the evolution equation (29) require any assumption about asymptotics. Whether or not the freely evolving states $\bm{|}n(t)\bm{\rangle}$ are a good description of the system at time $t$ does not matter for the fact that the complete set of scattering matrix elements $S_{n^{\prime}n}(t,t^{\prime})$ provides a complete description of evolution from time $t^{\prime}$ to time $t$ as stated in equation (29). Furthermore, it appears most likely that experimentalists will soon devise single photon probing [17, 18, 19, 46, 47] of atoms over nanoscale distances and with sub-femtosecond time resolution [20, 21], and the results will still be described by the scattering matrix element (19) which automatically yields the jump equation (1), albeit with a finite energy uncertainty.

The results from Section 2 reaffirm that the established framework of photon quantization appears as the only viable option to describe photon emission and absorption by atoms. The consequences from the results of Sections 3 and 4 are then best stated in the reformulation of (29) in terms of interaction picture states, $\bm{|}\psi_{D}(t)\bm{\rangle}=\exp(\mathrm{i}H_{0}t/\hbar)\bm{|}\psi(t)\bm{\rangle}$,

[TABLE]

[TABLE]

Every power of the coupling constant $e$ in the expansion of the time evolution operator (15),

[TABLE]

corresponds to a change of the number of photons in $\bm{|}\psi_{D}(t)\bm{\rangle}$ by one unit. This produces $\bm{|}\psi_{D}(t)\bm{\rangle}$ as a sum over all sectors of the Fock space of photons which comply with energy conservation between the initial and final states, even if $\bm{|}\psi_{D}(t^{\prime})\bm{\rangle}$ was an atomic state. The creation or annihilation of each photon constitutes a quantum jump, but the probability amplitude for each of these jumps evolves continuously with time. The resolution of the dichotomy between continuous evolution of quantum states on the one hand and quantum jumps on the other hand then calls for a mixed interpretation: Quantum states evolving only within one sector of Fock space reflect ontological properties of the quantum system, but sums involving different sectors of Fock space are epistemic. This explains why scattering matrices of the second quantized theory evolve continuously with time, and yet they are transition amplitudes for quantum jumps. It also avoids the pitfall of Schrödinger’s cat: The sum over contributions from two sectors of Fock space

[TABLE]

with $|A(t)|^{2}+|B(t)|^{2}=1$ is an epistemic sum over ontic components, and therefore it does not describe a situation where there is and is not a photon in the system. In the absence of a decisive observation, the state (32) only reflects our best possible guess regarding the question whether a photon has been emitted (if we start with the 2p state) or absorbed (if we start with the 1s state). Note also that the difference in interpretation of quantum states within a single sector of Fock space versus states spanning sums over several sectors of Fock space does not contradict the recent no-go theorems for epistemic interpretations, which were based on measurements on entangled states. We cannot observe the state (32) per se because we cannot define a probe which tests for the simultaneous presence and absence of a photon, and therefore we also cannot observe the state as a component in entangled measurements.

Within the standard computational framework of applied quantum mechanics, the following picture emerges: The singular stochastic event of spontaneous emission (creation) or absorption (annihilation) of a photon by an electron forms the ontological basis for quantum jumps. The probability amplitudes for these singular stochastic events evolve continuously according to the minimally coupled equations of quantum optics, although the event itself is not continuous. The continuous evolution of the epistemic states (29) with the time evolution operator $U(t,t^{\prime})$ is unitary due to $U^{+}(t,t^{\prime})=U(t^{\prime},t)=U^{-1}(t,t^{\prime})$, but the final step of reduction upon observation will not be unitary in the mathematical sense since the reduction of the state is not invertible: An atom in a 1s state accompanied by a Lyman $\alpha$ photon can arise from spontaneous decay from an infinite multitude of 2p states,

[TABLE]

[TABLE]

Both the 2p state (33) and the resulting atom-plus-photon state after spontaneous decay,

[TABLE]

[TABLE]

are ontic states within sectors with defined particle numbers in Fock space. The observation in Sections 3 and 4 that atoms respond to any kind of optical measurement through quantum jumps between energy eigenstates, even if we do not observe the energy of any component of the system during any time of an experiment, certainly indicates that energy and energy eigenstates are ontological properties of atoms. This gives credibility to the ontic interpretation of energy eigenstates.

The states (29), on the other hand, are generically linear combinations of states from sectors in Fock space with different particle content. Since, in 90 years of applied quantum mechanics, we have not developed or encountered any probes or observations which amount to collapse into a state spanning different sectors in Fock space, it seems natural to conclude that the states (29) are epistemic sums over ontic components. The reduction of the state to a component in a single sector in Fock space upon observation, is also epistemic in the sense that it constitutes a leap in our knowledge regarding occurence of emission or absorption of photons. However, besides the epistemic leap in knowledge upon observation, there is also the underlying ontological quantum jump due to spontaneous emission or absorption of a photon. Reduction of the quantum state corresponds to the epistemic leap. The ontological jump corresponds to a spontaneous transition e.g. from a single atom in a 2p state to an atom in a 1s state and a Lyman $\alpha$ photon, without any connotation that the atom-photon system at any time existed in a superposition of the initial state and the final state. Please note that both kinds of discontinuous transitions do not need to appear simultaneously.

From a different angle, the proposal that states spanning several sectors in Fock space are epistemic can also be considered as an ontological superselection rule888Traditional superselection rules forbid particular linear combinations of single-particle states, e.g. with different masses or spins, as these states were not considered meaningful in quantum theory. See [48, 49] for discussions of the history and philosophical aspects of these superselection rules. Linear combination of single-particle states with different quantum numbers would be interpreted as spontaneous transition of one particle into another particle in the scattering matrix framework. The modern situation regarding the formulation of traditional superselection rules has become complex due to the possibility of inequivalence of mass and flavour eigenstates in spontaneously broken gauge theories (flavour is the particle physics quantum number underlying nuclear isospin). This leads to oscillations between different flavour eigenstates. The effect has initially been observed in the form of oscillations of hadrons as bound quark states, and more recently through neutrino oscillations [50, 51]. Violations of the most basic Wigner type superselection rule in terms of oscillations between states of different spin requires spontaneous breaking of Lorentz invariance and has not been observed. However, quantum field theories with broken Lorentz symmetry are an active research field in theoretical particle physics, see e.g. [52, 53] and references there. in the sense that we suppose well-defined particle content of ontological states at any time, whereas the quantum state that we can know and calculate via the scattering matrix only provides us with probabilities for any particular particle content at any time.

As for the necessary inclusion of Fock space, one might worry that Haag’s theorem [54, 55, 56, 49] on the non-existence of creation operators following the time evolution under full interactions might be a cause for concern. However, note that the beauty of quantum field theory in the interaction picture is that it only uses free field operators for all calculations, and therefore also only employs a Fock space created by field operators which are free999In popular terms, Haag’s theorem is sometimes phrased as a statement of non-existence of the interaction picture. This is not correct. The only actual requirement for the interaction picture is the existence of quantized free field operators, combined with a prescription on how to move between free Fock space states through Dirac’s version of Dyson’s time evolution operator. The actual implication of Haag’s theorem is that the mapping from the freely evolving interaction picture operators into the Heisenberg picture operators does not imply that the Heisenberg picture field operators can be used to generate a corresponding Fock space, or as Fraser [56] and Ruetsche [49] have correctly observed, the Heisenberg picture operators cannot be directly associated with particles or quanta. in the sense of following linear wave equations (this also includes wave equations including classical potentials). The second quantized interaction terms enter as polynomials in free field operators into the scattering matrix. Indeed, the impossibility to map free Fock space unitarily into a Fock space of states which individually follow the full time evolution of an interacting field theory lends further credibility to the proposal that the sums over states from different sectors of Fock space, as created by the scattering matrix, are epistemic, while states within single sectors of Fock space can be ontic.

In summary, both epistemic and ontic properties appear as inherent aspects of quantum states and their dynamics. The observations of the present paper indicate that states within sectors of Fock space with well defined particle numbers are likely ontic, whereas states like (29), which evolve under interactions involving spontaneous annihilation or creation of particles, generically span several sectors of Fock space and should be considered epistemic. However, there are certainly many more physical and philosophical aspects to this conjecture that need to be studied. Identifying and resolving the boundaries (and epistontic overlaps?) of epistemic and ontic domains in Fock space and in quantum dynamics promises to be an interesting frontier in quantum foundations and philosophy.

Appendix: Energy conservation in

decay rates and cross sections

The emergence of the energy preserving $\delta$ function in scattering matrix elements $S_{\!fi}$ like (17,19) is a consequence of Dirichlet’s equation [57, 58]

[TABLE]

which holds if the function $f(\omega)$ is continuous. Dirichlet’s result motivates the notion of Dirac’s $\delta$ function (better addressed as a distribution in the mathematical literature) in the form

[TABLE]

such that equation (35) can be written as

[TABLE]

The energy preserving $\delta$ function appears in scattering matrix elements through the representation (37).

Proofs of Dirichlet’s equation can be found e.g. in Sec. 4.13c in [58] or Sec. 2.1 in [40]. Indeed, there are many generalizations of Dirichlet’s result through convolution integrals with other normalized kernel functions [40], and one of particular relevance for quantum mechanics is

[TABLE]

This can be used to justify Fermi’s trick to deal with the squares of $\delta$ functions, wich appear in expressions like $|S_{fi}|^{2}/\tau$ in the calculations of particle reaction rates. Particle reaction rates enter into the calculations of decay rates like (20) and cross sections like (22), and equation (39) can be used to show that after division by $\tau$, the square of $\delta$ functions expressed in the form (36) yields again one remaining energy preserving $\delta$ function through

[TABLE]

The standard textbook version $\delta^{2}(\omega)/\Delta t\to\delta(\omega)/2\pi$ of Fermi’s trick follows from (40) if we take into account that the observation time in (37) is $\Delta t=2\tau$.

Acknowledgements

This work was supported in part by NSERC Canada. I would also like to thank the anonymous referees for comments and questions which helped to improve the paper.