The quantum measurement approach to particle oscillations

The quantum measurement approach to particle oscillations

Charis Anastopoulos1
Department of Physics, University of Patras, 26500 Greece


Ntina Savvidou2
Theoretical Physics Group, Imperial College, SW7 2BZ, London, UK

The LSND and MiniBoone seeming anomalies in neutrino oscillations are usually attributed to physics beyond the Standard model. It is, however, possible that they may be an artefact of the theoretical treatment of particle oscillations that ignores fine points of quantum measurement theory relevant to the experiments. In this paper, we construct a rigorous measurement-theoretic framework for the description of particle oscillations, employing no assumptions extrinsic to quantum theory. The formalism leads to a non-standard oscillation formula; at low energy it predicts an ‘anomalous’ oscillation wavelength, while at high energy it differs from the standard expression by a factor of 2. The key novelties in the formalism are the treatment of a particle’s time of arrival at the detector as a genuine quantum observable, the theoretical precision in the definition of quantum probabilities, and the detailed modeling of the measurement process. The article also contains an extensive critical review of existing theoretical treatments of particle oscillations, identifying key problems and showing that these are overcome by the proposed formalism.

1 Introduction

In this paper we construct a solid measurement-theoretic framework for the description of particle oscillations [1, 2], i.e., a framework that (i) contains no ad-hoc assumptions extrinsic to the general rules of quantum theory, (ii) allows for a precise correspondence between experimental operations and elements of the formalism, and (iii) is sufficiently general to incorporate fine details in the description of particle oscillations, when such are needed. The proposed formalism predicts an ‘anomalous’ oscillation wavelength at the low end of the neutrino energy spectrum, and, as such, it might provide an explanation for excess of events at low energy in the MiniBooNE experiment and of the LSND anomaly [3, 4, 5].

Existing treatments of particle oscillations, as a rule, introduce assumptions extrinsic to quantum theory, hence important problems arise. In particular, existing methods contain ambiguities in the relation of elements of the formalism to the experimental setup and in their treatment of time and quantum probabilities. These problems originate from the fact that particle oscillations touch upon fine points of quantum measurement theory (the quantum treatment of the time of arrival, the joint measurement of incompatible observables and the role of decoherence and coarse-graining in the measurement process), whose comprehensive treatment had been unavailable at the time the phenomenon was first predicted [1, 2, 6].

The original intent of this paper was to employ the techniques and ideas from quantum measurement theory, in order to resolve the problems above and to place the standard results on particle oscillations on a firm conceptual and mathematical footing. Impressively, we find that our method leads to a non-standard expression for the wavelength of particle oscillations, i.e., to an expression that disagrees with the result that has been obtained by the vast majority of existing approaches to the treatment of particle oscillations. We have verified our result to be robust: it is not an artefact of any calculational approximation, its derivation involves no ad hoc assumptions and it persists when more details are added in the models of the measurement process. The approach we propose is based on a novel treatment of the concept of time, as presented in the development of the History Projection Operator theory as a new approach to quantum gravity [7, 8]. As far as the theory presented in this paper, the distinction between the kinematics and dynamics on the definition of time resulted to a new treatment of the time of arrival as a genuine quantum observable [9].

Claims of non-standard oscillation formulas have appeared sporadically in the literature—for example, Refs. [10, 11]—and they have been repeatedly refuted in favor of the standard result [12, 13, 14, 15, 16]. We do agree with the critique of such non-standard formulas, in the sense that the methodologies through which they are produced can be found wanting. However, we claim that the standard derivations are also wanting on the same grounds. To show this, we have undertaken an extensive critical review of existing theoretical treatments of particle oscillations (Sec. 4), identifying their shortcomings and showing that these shortcomings are, indeed, overcome with the method we develop here (Sec. 5.3).

One argument in favor of the non-standard result obtained here is that its derivation is not plagued by any mathematical ambiguities or conceptual inconsistencies. More importantly, our expression for the oscillation wavelength can be experimentally distinguished from the standard one, at least in principle. For neutrinos, the standard expression for the oscillation wavelength at energy is


where is the difference of squared neutrino masses. The non-standard oscillation formula that has been obtained before differ from Eq. (1) by a factor of two, i.e.,


In absence of an independent precise measurement of neutrino masses Eqs. (1) and (2) cannot be experimentally distinguished. In contrast, here we find


where is the energy threshold for the process through which the neutrino is detected. While for , Eq. (3) coincides with Eq. (2), for near , Eq. (3) predicts that the ratio is not a constant but increases with energy.

Hence, the method we develop here leads to the following predictions.

  • At low energies, the oscillation wavelength depends strongly on the value of the threshold energy , hence, it may vary according to the process through which neutrinos are detected333To be precise, Eq. (3) is obtained from the study of reactions in which the neutrino is annihilated, i.e., charged current interactions—see, Sec. 5.2. Neutrino detection through scattering processes can also be studied in the present framework and it will be treated in a different publication.. In other words, data subsets for different reaction channels may exhibit different oscillation patterns.

  • For a given reaction channel, the expected number of events as a function of energy will differ strongly from what is predicted by the standard oscillation formula at low energies. This difference can be seen heuristically from the plots in Fig. 2 (Sec. 5.3).

The prediction of ‘anomalous’ neutrino oscillations at low energies might provide an explanation for the excess of neutrino detection events in the LSND [3] and MiniBooNE [4] experiments. In particular, our approach makes no recourse to physics beyond the Standard Model and introduces no additional undetermined parameters. Hence, it can be directly tested against the standard expression for particle oscillations in terms of better fit to existing data.

We will refer to the formalism developed here as the quantum measurement approach to particle oscillations. Its main points are the following.

  • Particle oscillation experiments do not correspond to measurements taking place at a predetermined moment of time. The time of arrival of the particle at the detector is a genuine physical observable that should be described in terms of quantum theory. In particular, the time of arrival is to be distinguished, both physically and mathematically, from the time parameter of Schrödinger’s equation [7].

  • For a quantum measurement the most general consistent assignment of probabilities is through the use of Positive-Operator-Valued-Measures (POVMs)—see Sec. 3.1.3 for definition. The probability densities for particle oscillation are represented as , in terms of a family of POVMs , where is the time of arrival, defined in terms of the detection of a particle produced through a channel of flavor , and is the distance from the production region. Hence, we describe particle oscillation experiments as joint measurements of the time of arrival and the position . This is a key difference of our approach from existing ones, and it leads to a non-standard expression for the oscillation wavelength.

  • The family is constructed in terms of the detection of product particles, i.e., in terms of directly observable quantities. Moreover, the method for constructing involves only the Hamiltonian of the system and the specification of the measured variables. In particular, it does not depend on properties of the initial state and contains no assumptions extrinsic to quantum theory.

The paper is structured as follows. Sec. 2 describes our motivation in undertaking this work, in order to correct certain conceptual and mathematical inadequacies in existing treatments of particle oscillations. We identify theoretical assumptions that enter into existing derivations that are not justified by the rules of quantum theory and, sometimes, they are erroneous. In that section we also present the main ideas of our approach. The presentation is largely non-technical, and it requires little background to the issue.

Sec. 3 provides background material necessary for the development of our approach. In particular, we focus on quantum measurement theory providing a review of main results that are relevant to our critique of existing approaches to particle oscillations.

Sec. 4 presents a specific and more technical critique of existing approaches to particle oscillations, in light of the material presented in Sec.3.

The development of a new framework for particle oscillations is contained in Sec. 5. In Sec. 5.1 we construct the necessary tools in the context of an abstract Hilbert space formalism. In particular, we develop a methodology for defining probabilities for general measurement schemes, where the time of detection is treated as a physical observable. Particle oscillation experiments constitute a special case of such schemes, and they are treated in Sec. 5.2, where we derive the formula for the oscillation wavelength. In Sec. 5.3 we undertake a critical analysis of our method, demonstrating that it resolves the problems we identified in existing approaches to the issue. In Sec. 5.4 we further develop the formalism, applying it to more general settings, and we demonstrate that our results are robust.

2 Motivation for a quantum measurement approach to particle oscillations

The problems in the existing treatments of particle oscillations arise from their identification of probabilities as the modulus squared of matrix elements of the evolution operator . While this is the most common probability assignment in quantum theory, it is not universal. In particular, it does not apply to particle oscillations: the resulting probabilities depend on the time and there is no rigorous way to relate them to what is actually being measured in an experiment. When particle oscillations were first predicted, the researchers did not have the benefit of a mature quantum theory of measurement that could provide a solution in terms of first principles. As a result, the predictions relied on heuristic arguments and assumptions that are not warranted by quantum theory. These created ambiguities in the relation of the formalism to experimental procedures.

Later work on the quantum theory of measurement provided a precise language for the treatment of the subtle issues that enter the theoretical description of particle oscillations. As it turns out, the more precise treatment yields information inaccessible from the heuristic ones, namely, the prediction of ‘anomalous’ oscillation wavelength at low energies that could be relevant for the explanation for the LSND and MiniBooNE experiments.

In section 2.1, we provide an extensive discussion of problems in existing treatments of particle oscillations, explaining why they arise and how they affect the existing treatments. In section 2.2, we describe the main ideas of our approach, and how they resolve these problems.

2.1 Main problems in particle-oscillations theory

We proceed to an analysis of the following issues: (i) the improper treatment of the time of arrival, (ii) the lack of a first principles derivation of the relation between amplitudes and probabilities, and (iii) the improper elimination of the time variable in the probability assignment. In our opinion, these issues constitute the most important problems in existing treatments of particle oscillations.

2.1.1 The time-of-arrival issue

Current methods for the treatment of particle oscillations do not establish a sharp correspondence between mathematical formalism and experimental operations. The reason is that these methods conflate two distinct notions of time as it appears in quantum theory: time as a parameter of evolution in Schrödinger equation and the time of arrival (or time of event, or time of transition) [7].

Time as a parameter of evolution, like in Schrödinger equation, is an essentially classical parameter [17], i.e., it does not correspond to a Hilbert space operator; hence, it is not an ‘observable’. On the other hand, the time of arrival is a variable defined in terms of coincidences of events: we observe an event (e.g., the creation of a lepton in a detector) and we correlate its occurrence with the reading of an external clock. The time of arrival is, therefore, a physical observable.

In existing treatments of particle oscillation the distinction above is ignored. This results into the definition of probabilities in terms of the Schrödinger time parameter which, unlike time of arrival, is not a physical observable. To obtain physical predictions from these probabilities it is necessary to remove this time dependence; this is achieved by procedures that are not justified from first principles—see Sec. 2.1.3.

The time of arrival is defined in terms of the physical event that a particle is detected by a measuring device. Since particle detection is a quantum process, time of arrival is a genuine quantum observable associated to physical processes on the measured system. (See, Sec. 3.3 for details and references.) As such, time of arrival may exhibit interferences. In a hypothetical experiment where a source prepares particles in a superposition of states with different mean momentum, the times of arrival recorded at a detector, at distance from the source, will be distributed according to an oscillatory pattern (i.e., interference) around a mean value of arrival time.

2.1.2 The physical interpretation of amplitudes

The vast majority of the theoretical studies of particle oscillations concentrate on the evaluation of amplitudes between certain vector states on a Hilbert space that describes the physical system. For example, these states may be ‘eigenstates’ for flavors and , leading to amplitudes of the form


In Eq. (4) the states refer to particles localized around the point and they correspond to detection of particles characterized by flavor . In other approaches that employ quantum field theory, analogous amplitudes are computed between states of incoming and product particles, rather than neutrinos or neutral bosons, as in Eq. (4). The dependence of the amplitude on the flavor indices arises through the flavor dependence of an interaction Hamiltonian. The critique in this section applies to those approaches, as well. The critique refers to a structural point of the quantum formalism (i.e., the relation of matrix elements of to the probabilities of measurement outcomes), and not to properties of specific physical systems or calculational techniques. It also applies to cases that the amplitude is defined in terms of elements of the -matrix rather than matrix elements of the finite-time propagator .

Having computed an amplitude of the form Eq. (4), the modulus square is then interpreted as a measure of probability, from which physical predictions about particle oscillation experiments are extracted. In Sec. 4, we will examine the various methods employed to extract physical predictions from such amplitudes.

A key question then arises: in what sense are amplitudes of the form Eq. (4) relevant to the description of particle oscillation experiments? In particular, Eq. (4) defines the matrix elements of the unitary evolution operator between states of the form . How does this relate to probabilities relevant to particle oscillation experiments?

The squared amplitude defines probabilities for the measurement of a particle’s position, at a specific, predetermined and sharply defined moment of time . Besides the idealization involved in talking about measurements taking place at a sharply defined moment of time, the main problem in the use of the squared amplitude is that in particle oscillation experiments the timing of the detection event is not predetermined by the experimentalist. What is controlled is the location of the detector. In other words, particle oscillation experiments do not fall in the domain of validity of the expression for probabilities of measurement outcomes. We shall elaborate on this point in Sec. 3.1.

In order to construct a proper probability assignment that fully reflects the setup of particle oscillation experiment, it is necessary to work within the broader context of quantum measurement theory [18, 19, 20]. The latter allows for probability assignments more general than squared amplitudes (for example, probabilities for joint measurements [18], continuous-time measurements [21], weak measurements [22], time-of-arrival measurements [23, 24, 9], and so on). In particular, it is necessary to derive the relevant probability from first principles, taking into account the physics of the detection scheme; we shall provide such a derivation in Sec. 5.1.

In most physical applications, a first-principles derivation of the probability assignment through quantum measurement theory makes little difference to the theory’s predictions, hence, it might be dismissed as too fine to be relevant. However, this is not the case for particle oscillation experiments, in which the measured quantities are directly correlated with tiny microscopic variations (oscillations) in the time of arrival of the particles. A proper treatment of the probabilities does affect the physical predictions of the theory.

2.1.3 Elimination of the time variable

The problems in the use of the squared amplitude for the description of particle oscillations have been noticed early on in the theory, as the presence of the time in addition to the position appears superfluous when relating the theoretical expression to experimental results. However, this issue has not led to a reconsideration of the starting point, that is, to the derivation of the probabilities for particle oscillations from first principles. Instead, the relevance of the squared amplitudes has been taken for granted, and various procedures have been invoked in order to eliminate time from the probabilities. Clearly, since the starting point is accepted without justification, any remedy to the consequent problems can be introduced only ad hoc, i.e., without justification on the basic principles of quantum theory.

Here we describe the two most common such procedures, for eliminating time from squared amplitudes.

Use of the classical arrival time.

The oldest approach consists in using as a probability density for the detection, by substituting with a classical time of arrival ; is the distance from the detector to the source and the mean velocity of the wave-packet.

The obvious problem with this method is that it is mathematically inconsistent: the time , that a measurement takes place, is substituted by an ‘average’ time-of-arrival, which depends on the system’s initial state, through . The resulting probabilities are then not linear functionals of the initial state of the system , as required by quantum theory. Hence, this solution can only be a heuristic device, perhaps valid for a specific class of initial states.

The method above could be viewed as a kind of semi-classical approximation. As such, it can only be justified by demonstrating that it is a good approximation, i.e., by comparing with a full quantum mechanical treatment of the problem. Heuristic arguments based on estimations of uncertainties have pitfalls, and they are no substitute for a full calculation.

Integration of probabilities over time.

A second commonly employed procedure for the elimination of time from the squared amplitude is integration. The related argument is that, since time is not a directly measured quantity in the experiment, we integrate over the single-time probabilities . We then obtain a probability density for position alone, i.e.,


where is an integration time for the experiment.

The argument above is invalid for the following reasons. First, the squared amplitudes are probability densities with respect to position and not with respect to time, and thus time-integration does not lead to meaningful probabilities. This can be seen even on dimensional grounds: a density with respect to time as well as space ought to have dimensions , while the squared modulus has dimensions . Moreover, it does not transform as a density under a time-rescaling .

This obvious problem is sidestepped by a procedure invoked for the solution of another problem, namely, the fact that is not normalized to unity. One is then forced to normalize it by dividing by a constant , for example, . However, depends on the initial state , hence, the probability assignment is not a linear functional of , thus violating a fundamental property of quantum mechanics. For an elaboration of these points, see, Sec. 4.2.3.

The mathematical and conceptual consistency of this procedure apart, there are also physical problems with the integration over time in Eq. (5). Suppose we were able to properly define amplitudes , different from Eq. (4), in which the parameter stands for the time of arrival which is a genuine observable (we shall see how this is done in Sec. 5.1). Would we be justified then in defining a probability density for position


if we assume that the time of arrival is not directly observable? In classical probability, the answer would be affirmative. However, the defining feature of quantum probability, clearly manifested in the two-slit experiment, is that alternatives are summed over at the level of amplitudes. Therefore interference terms appear. Only if a macroscopic distinction of alternatives is possible are the interference terms suppressed. Hence, Eq. (6) for the probability would be justified only if time of arrival is measured very finely. In particular, Eq. (6) is justified only if the resolution of the time measurements captures the full dependence of on time. Unless there is no time measurement—-or if the time measurements are too coarse to distinguish oscillating terms in the distribution of the time of arrival—then one should expect that the relevant probabilities are of the form


where interference terms will be in general present.

In other words, even if all the mathematical problems associated to integration over time at the level of squared amplitudes were resolved, Eq. (6) does not follow from the basic principles of quantum theory, unless one postulates a decoherence mechanism that destroys all interferences for the time of arrival in the amplitudes .

2.2 The main ideas of the quantum measurement approach to particle oscillations

In Sec. 2.1, we argued that the existing methods for particle oscillations suffer from problems of consistency with the basic principles of quantum theory. These problems can be concisely summarized in the following statements.

  1. A key issue in the theoretical study of particle oscillations is the definition of the time of arrival as a genuine quantum variable. In existing treatments, time of arrival is either conflated with the time-parameter of Schrödinger’s equation, or it is treated classically. The former case contains a conceptual error that leads to a discrepancy between the formalism and the experiment. The latter case involves an approximation, whose validity can only be ascertained by comparison with the results of a full quantum description.

  2. There is no first-principles derivation of the probabilities that pertain to particle oscillation experiments. The modulus square of the amplitude Eq. (4) is introduced without justification as a probability measure relevant to particle oscillation experiments. Ad-hoc procedures are employed in order to extract physical predictions.

These issues are resolved in the present paper: the quantum measurement approach to particle oscillations is based on two key ideas which can be concisely summarized as follows.

  1. The distinction between the time of arrival and the time parameter of Schrödinger’s equation is the groundwork of our approach. The importance of this distinction originates from previous work on the concept of time in quantum theory [7, 8], as we stated in the Introduction. The formalism presented here strongly separates between those two different notions of time, by representing them by different mathematical objects. In particular, the formalism treats the time of arrival as a physical observable, directly relevant to particle oscillation experiments and, it provides a probabilistically consistent treatment of its quantum fluctuations.

  2. Treating the time of arrival as a quantum observable, implies that the probabilities relevant to particle oscillations correspond to the joint measurement of two observables: the time of arrival and the location of a detection event . In quantum theory, probabilities for joint measurements cannot, in general, be obtained by squared amplitudes. However, they are described, with full consistency and rigor, in terms of Positive Operator Valued Measures (POVMs) [18, 20]. Hence, we obtain the probabilities relevant to particle oscillation experiments through the construction of a POVM that describes the probabilities for the detection of -flavor events at location and time . The corresponding probabilities are then of the form


    where is the initial state of the system. Since in Eq. (8) the time of arrival is an observable, it can be consistently integrated out, leading to a probability density that is a function of alone. The oscillation wavelength Eq. (3) is then obtained.

The implementation of the ideas above involves the construction of amplitudes that depend on the time of arrival rather than the time parameter of Schrödinger’s equation. These amplitudes are then employed for the construction of the POVM using a method that was developed in Ref. [9]. In fact, we do not construct a single POVM for the task at hand but we provide a general methodology for their construction. The results, therefore, do not depend on simple idealizations; there exists a systematic procedure that allows us to incorporate as many details in their derivation as needed.

The measurement process is described in terms of the detection of particles produced by reactions of the oscillating particles at the detector, hence the use of quantum field theory is necessary. We also take into account the fact that the detector is a macroscopic system with a large number of degrees of freedom. In spite of the complexity involved in a realistic description of the measurement process, our main result Eq. (3) for the oscillation wavelength is robust, and it does not depend on variations in the modeling of the measurement. Moreover, it does not require any fine-tuning in the properties of the initial state.

3 Background

This section consists of two parts that address issues important for the critique of existing approaches to particle oscillation. It also serves as a theoretical background for the formalism we develop in Sec. 5. In particular, in Sec. 3.1 we review the main ideas of quantum measurement theory, placing special emphasis on measurements of von Neumann type, explaining why they are not suitable for the description of particle oscillations and discussing the properties and wide applicability of POVMs. In Sec. 3.2 we briefly review the quantum treatment of time of arrival in the language of POVMs, showing that, like any other quantum observable, the time of arrival may be characterized by interferences.

3.1 Quantum measurement theory

3.1.1 von Neumann measurements

First, we study the simple model about the measurement of a variable corresponding to the self-adjoint operator , with generalized (continuous) eigenvalues . This class of models originates with von Neumann [25, 26]—see also [20]—the variation we present here is from Ref. [27].

To this end, let be the Hilbert space of the microscopic system and the Hilbert space corresponding to the degrees of freedom of a macroscopic apparatus. The values of the observable of the quantum system are correlated to the values of an operator on . is usually called the ‘pointer variable’, its values are accessible to macroscopic observation. Hence, its spectrum is highly degenerate. We denote the generalized eigenstates of as ; labels orthonormal bases in the degeneracy subspaces of .

is the Hamiltonian operator describing the self-dynamics of the quantum system. The interaction between the measured system and the apparatus is described by a Hamiltonian of the form . is the ‘conjugate momentum’ of , i.e., the generator of translations for the pointer variable , where . The function in describes the ‘switching-on’ of the interaction between system and measurement device.

For simplicity, we assume that the self-dynamics of the apparatus during the interaction are negligible, so that the total Hamiltonian is . The corresponding evolution operator is , where denotes the time-ordered exponential.

The microscopic system is prepared in a state and the initial state of the apparatus is . The initial state of the combined system is then , and it evolves into , under the action of the unitary operator .

With the above assumptions, the probability distribution over the value of the pointer variable at time is


where is a family of positive operators on , and where


A common approximation is to assume that the function is narrowly concentrated around a value , i.e., . In that case,




and stands for the positive operator . The function determines the correlation between the pointer variable and the values of the operator .

Eq. (11) corresponds to the probability for a measurement of the observable , at a single moment of time [28]. While the degrees of freedom of the apparatus have been included in the description, the end result for the probabilities is expressed solely in terms of operators on the Hilbert space of the microscopic system.

3.1.2 Coarse-graining and measurement uncertainties

Next we elaborate on the physical origin of uncertainties and the role of coarse-graining in quantum measurements. These quantum measurement theory issues are crucial for the theoretical treatment of particle oscillations.

The accuracy by which the pointer variable determines the value of is given by the weight function in Eq. (12); depends on the properties of the measurement device. The resolution in the measurement of is at least of order , where is the mean deviation of for the state . In general, is of macroscopic magnitude, since the apparatus has a large number of degrees of freedom. For example, cannot be smaller than the thermal fluctuations of .

A measurement is a macroscopic amplification of a microscopic event. The inability to distinguish between values of at a scale smaller than is an objective fact of the system. There is no way in standard quantum theory—without introducing hidden variables—to infer properties of a microscopic system at a scale smaller than the resolution [29, 30, 31, 32]. In quantum theory, microscopic system and macroscopic apparatus become entangled during measurement and the association of definite properties requires a sufficient degree of coarse-graining.

Eq. (11) reduces to the usual probability assignment, in terms of squared amplitudes only when sufficiently coarse alternatives are considered, i.e., at scales much larger than . In that case, the weight , in Eq. (11), can be substituted by a delta function and Eq. (11) corresponds to a squared amplitude .

Measurements of von Neumann type are not universally applicable because they involve special assumptions and idealizations. In particular, the description of time in von Neumann measurements is artificial: we assumed an interaction between microscopic system and apparatus determined by a specific profile function . The timing of the interaction between microscopic system and apparatus is not controlled by the experimentalist and it is not independent of the initial state. For example, the time of arrival of particle at a detector, located at distance from the particle source is not predetermined by the experimentalist and it varies according to the energy distribution of the incoming particles. In this setup, the parameter controlled by the experimentalist is the distance . Hence, an accurate modeling of such measurements should involve a profile function for position rather than time.

The idealization in the treatment of time in von Neumann measurements is a good approximation when the precise timing of the event is not significant to the measurement outcomes. However, for measurements where the time of detection is a physical variable (particle oscillations), a description in terms of von Neumann measurements is insufficient.

A realistic measurement scheme has a finite duration that corresponds to the time it takes for the pointer variable to settle to a definite value, correlated to a property of the microscopic system. This time-scale is determined by the physics of the apparatus, it typically lies in the macroscopic domain, and it sets a lower limit to the temporal resolution of the apparatus.

In order to provide an estimation for the duration of the measurement we generalize the model presented in Sec. 3.1.1, by including a Hamiltonian term for the apparatus’s degrees of freedom, so that the total Hamiltonian is . A good pointer variable must be stable with respect to the self-dynamics of the apparatus, so that evolution under does not significantly affect the recorded values after the measurement’s completion. This means that , or more precisely


and a similar condition must hold for the operator that generates translations in . The condition above can be satisfied if and are coarse-grained quasi-classical variables, i.e., if they have large degeneracy eigenspaces that remain approximately invariant under time evolution. In general, the evolution of quasi-classical variables involves noise [32, 33, 34, 35, 36, 37], so in Eq. (13) can be modeled by a classical stochastic process. When the effects of the noise can be ignored in the evolution of the system, the previous analysis passes unchanged and Eq. (11) is obtained. However, this involves a specific scale of coarse-graining both for the uncertainties and for the duration of the measurement.

The effect of noise, such as above, in the evolution on the quasi-classical variables is often studied within the quantum theory of open systems [18, 38]. The ignored degrees of freedom of the apparatus are then treated as an environment and they are modeled by a thermal reservoir—see, Refs. [29, 39, 40, 41, 42]. It is natural to expect that the characteristic timescale of such measurement corresponds to the time that it takes a variable to settle to an asymptotic value, characterized by purely thermal fluctuations. Therefore, is of the order of the relaxation time of the reservoir.

The limit above is essentially classical, in the sense that it arises from the statistical fluctuations of a system with a large number of degrees of freedom. There is a more stringent, unavoidable lower limit for the resolution time scale arising from the quantum nature of the pointer variable. This limit is the localization time , i.e., the time it takes the pointer variable to establish classical behavior due to its interaction with the remaining degrees of freedom [36, 43, 44]. Modeling the pointer variable by an oscillator undergoing quantum Brownian motion in a thermal reservoir of temperature [38, 45, 46], one finds a localization time of order , where is the relaxation time [43, 44, 47]. Such systems are characterized by a time scale separation . For , this implies that . We note that the temporal resolution in a solid state detector is much coarser than these lower bounds: it is at best of the order of .

Models such as the above are simplistic in comparison to the physics of actual particle detectors. However, they serve to identify the irreducibly quantum origin of the temporal resolution in a measurement scheme, namely, that the pointer variables can be assigned definite classical properties only if a substantial degree of coarse-graining is involved.

3.1.3 Positive-Operator-Valued Measures

In Sec. 3.1.1, we constructed the positive operators for a measurement device by considering a single pointer variable , whose values are correlated with one quantum variable . In general, a measurement may involve many pointer variables each correlated with a different property of the quantum system. Again, the probabilities for the measurement outcomes can be expressed in terms of a POVM. Here, we briefly describe the basic properties and uses of POVMs in quantum measurement theory.

Let be the Hilbert space of a quantum system and let be the values of observables correlated to pointer variables of a measurement device. The possible outcomes of the measurements span a space . A POVM is a map that assigns a positive operator on to each (measurable) subset of , such that (i) , for , and (ii) . Usually, the POVM can be expressed as an integral over in terms of operators , such that .

The most general probability assignment for measurement outcomes in ,compatible with the rules of quantum mechanics is


in terms of a density matrix that incorporates the initial state and preparation of the system and some POVM . If a procedure provides probabilities for a measurement outcome that cannot be brought into the form Eq. (14), then these probabilities are incompatible with quantum theory. They may approximate a valid quantum mechanical expression (for a special class of initial states), but they are not themselves fundamental and cannot be taken as a standard for the theoretical description.

3.2 The quantum description of the time of arrival

This section contains a brief review on the quantum treatment of the time of arrival, with an emphasis on its description in terms of POVMs. The key point relevant to particle oscillations is to demonstrate that the probabilities associated to the time of arrival, like those of any other quantum observable, may exhibit quantum interferences.

3.2.1 Key issues in the quantum description of the time of arrival

An idealized description of a time-of-arrival measurement is the following. At time a source emits particles prepared at an initial state . At a macroscopic distance from the source a particle detector is located. An external clock determines the time coincident with each detection event. Given a large number of events, a probability distribution for the time of arrival is constructed. The issue is to determine this probability distribution from first principles in quantum theory.

The difficulty in describing the time of arrival in a quantum system stems from the fact that given a solution to Schrödinger’s equation , its modulus square is a probability density with respect to at each time , and not a probability density with respect to . Moreover, there is no operator representing time in the system’s Hilbert space. In that case one would use the Born rule to determine a probability density for the time of arrival. The existence of a time operator , conjugate to the Hamiltonian (so that the Hamiltonian generates time-translations: ), is ruled out by the requirement that the Hamiltonian is bounded from below [48, 49]. For some systems, one may still define a quantum time variable, by choosing some degrees of freedom of the system as defining an internal ‘clock’. However, clock times fail to be conjugate to the Hamiltonian of the system, with the result that they do not forward under Hamiltonian evolution. In other words, quantum fluctuations invariably force clocks to move occasionally ‘backwards in time’.

There are various approaches for the description of the time of arrival [24, 50, 52], and related problems, e.g., transversal time in tunneling [53]. Some of the approaches go beyond the standard quantum mechanical formalism by employing properties of ‘paths’ identified either through path integrals, or the Wigner function, or Bohmian mechanics. In this paper we are interested in well-defined probabilities for the time of arrival in concrete measurement situations, which means the identification of a POVM for the time of arrival [9, 23, 51, 57, 58]. We also note that the method we adopt in this paper for the construction of a POVM for the time of arrival shares many common features with the studies based on the decoherent histories approach [54, 55, 56].

3.2.2 Time-of-arrival interferences

Next, we present Kijowski’s POVM [23] for the time of arrival of a non-relativistic particle of mass in one spatial dimension. Given an initial state at time , Kijowski’s POVM assigns the following probabilities for the time of arrival of the particle at a detector located at


The POVM Eq. (15) is uniquely constructed by the requirements of Galilean invariance, of invariance under parity and time-reversal transformations, and of having the correct classical limit. It is defined on wave functions with support on either positive or negative values of momentum, and the values of extend to the whole real axis. It can be extended to all states if we also include the alternative of no detection [9], as .

In order to demonstrate that the POVM Eq. (15) may lead to interferences in the time of arrival, we evaluate Eq. (15) for an initial superposition of two Gaussian wave packets with the same mean position , but different mean momentum and


We assume that so that the two wave packets do not significantly overlap.

For , , and ignoring dispersion, the leading contribution to the probability distribution obtained by the POVM Eq. (15) is


i.e., there are oscillations of frequency extending for a time interval with width of order around the mean value of arrival time . If the momentum difference is much smaller than the mean momentum , then and . Clearly , but this has nothing to do with the measurability of the interferences in the time of arrival. The interference are observable if , where is the temporal resolution of the detector. For measurements of the time of arrival, see Ref. [24] and references therein.

The discussion above demonstrates that there exist regimes in which the semiclassical description of the time of arrival is inadequate. A common theme in the study of particle oscillations is to identify the relevant probabilities by substituting into the square amplitude Eq. (4) the classical value of time of arrival at the detector. The meaning of such an expression apart, this procedure may gravely misrepresent the probabilities of the system. Indeed, for the initial superposition state Eq. (16) substituting, for example, into the square amplitude , leads invariably to an expression that does not exhibit an interference pattern.

4 Critique of existing theoretical treatments of particle oscillations

In this section, we undertake a more detailed critique of existing approaches to particle oscillations. First, we present the existing approaches for the treatment of particle oscillations, separated into three broad categories as in the reviews [59, 60, 61, 62], namely, plane wave methods, wave packet methods and quantum field theoretic methods. We then proceed to a critique of their assumptions and formalism, elaborating on the issues sketched in Sec. 2.

4.1 A brief presentation of existing formalisms for the description of particle oscillations

4.1.1 The plane wave description of particle oscillations

We present the elementary treatment of particle oscillations based on plane waves, which is the most well-known and widely used due to its simplicity. This approach is known to be fundamentally inconsistent [6, 60], as it relies on infinitely extended wave packets with zero momentum spread in each mass eigenspace. The wave packet methods, presented in Sec. 4.1.2, and the quantum field theory methods, presented in Sec. 4.1.3, were developed partly in order to resolve outstanding issues in the plane wave treatment and to place the derivation of the oscillation formulas on a firmer mathematical ground.

By definition, an oscillating particle is described in terms of a Hilbert space , where the subspaces carry irreducible unitary representation of the Poincaré group with the same spin but different masses . Ignoring for simplicity the spin degrees of freedom and particle decay, each mass eigenspace is spanned by the generalized eigenstates of the Hamiltonian for a free particle of mass . One defines ‘flavor eigenstates’


where is the mixing matrix appearing in the interaction Hamiltonian for the oscillating particles. The transition amplitude between a generalized eigenstate of flavor ,at time and location , and a generalized eigenstate of flavor , at time and location , is


where . The term is really infinite, but in the plane wave method we are interested primarily in the relative phases between the mass eigenstates, so this issue can be ignored.

The squared amplitude


depends explicitly only on . It would also carry a dependence on if we had assumed different values of in the definition of flavor eigenstates, Eq. (18), but this is not necessary for the present demonstration. In order to relate the square amplitude Eq. (20) with a measurable quantity, it is necessary to convert its time dependence into spatial dependence. There is no rule of quantum mechanics that allows one to do so, so one has to rely on ad hoc prescriptions. To this end, we note that each subspace is characterized by a velocity , where . At the classical level, this corresponds to the time of arrival , for a particle starting at to a detector at , where .

The standard oscillation phase is obtained by substituting the time parameter with a mean time of arrival at the detector, e.g., for the time of arrival calculated for a mean energy ,—this procedure is usually called an ‘equal times prescription’. We then obtain a phase difference


which renders the square amplitude Eq. (20) an oscillating function of , from which the standard oscillation wave-number follows. In the ultra-relativistic regime, relevant to neutrinos, the oscillation wave-number is


On the other hand, if we substitute the different values for arrival time in each mass eigenspace (unequal-times prescription), one obtains , corresponding to an oscillation wavenumber


which is twice the standard expression Eq. (22) in the ultra-relativistic limit.

We note that the non-standard result Eq. (23) includes the contribution from interferences in the time of arrival corresponding to the different values in the different mass eigenstates. The derivation of the standard result Eq. (22) involves an assumption that such interferences are somehow suppressed.

4.1.2 Wave-packet methods

The natural generalization of the plane wave method is the consideration of amplitudes between properly square-integrable wave-packets [6, 63, 64, 65, 59]. That is, one defines Hilbert space vectors


in terms of profile functions and centered around the production ( ) and detection () region respectively; stand for the generalized eigenstates of position in the -th mass eigenspace. One then computes the amplitude


The simplest case that allows for analytic calculations corresponds to isotropic Gaussian wave-packets


where one assumes that the production region is centered around with spread equal to and the detection region is centered around with spread equal to .

The amplitude Eq. (25) can be computed in the no-dispersion regime, defined by the approximation . Restricting for simplicity to one spatial dimension, one finds


where .

In order to express the probabilities in terms of observable quantities alone, one needs to remove the time-dependence from the amplitude in Eq. (27). This can be done in two ways.

  1. The interference phases in the modulus square of the amplitude Eq. (27)


    are the same with the ones obtained from the plane wave methods. They can, therefore, be evaluated with the same methods, i.e., by making a specific choice of time corresponding to a ‘mean’ time of arrival. The standard expression for the oscillation wavelength necessitates the postulate of an additional prescription for the terms in Eq. (28). The most usual form of such prescription is the assumption of equal-times , i.e. an assumption that interference takes place only for equal propagation times [12]. Alternatively, the standard oscillation formula may be obtained by an equal energy assumption [13, 66], by an equal velocity assumption [67], or by a specific choice of ‘mean velocity’ for the wave packet [15, 16, 84].

  2. The introduction of ad hoc prescriptions is avoided in the wave-packet method by integrating the squared amplitude over time [63], i.e., by defining a probability for particle detection


    in terms of some long integration time . The constant is introduced for purposes of normalization of the probability to unity.

4.1.3 Quantum field theoretic methods

The oscillating particles are not directly observable, therefore, the natural framework for the treatment of particle oscillations is quantum field theory [69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 60, 62]. In the latter, oscillating particles are described to internal lines of Feynman diagrams, whose external lines correspond to incoming particles in the production region and outcoming particles at the detector.

In this approach, one computes amplitudes of the form


where describes the state of the incoming particles in the production region, describes the state of the particles produced in the detection process, denotes time-ordering and is the interaction Hamiltonian for the total process. Typically, correspond to wave-packets localized at the production region at a specific time and corresponds to wave-packets localized at the detector region at a time . The calculation of the amplitude Eq. (30) to leading order in perturbation theory introduces the characteristic dependence of the amplitude on the mixing matrix in Eq. (27). Usually, one constructs the relevant probabilities from an integration of the amplitudes Eq. (30) squared over time .

Ref. [70] avoids the use of integration over time in squared amplitudes by following a different procedure. One computes the -matrix elements for the weak process , for the production of the antineutrino localized around and the subsequent scattering process localized around for the antineutrino detection without reference to an initial and final time. The -matrix formalism in this case implies that the initial and final states are ‘in’ and ‘out’ respectively, i.e., they are defined at and . The authors then construct the scattering cross-section at the limit recovering the standard result. Ref. [75] avoids the -matrix formalism, relying instead on time-dependent perturbation theory.

The quantum field theoretic method provides a substantial improvement in the theoretical treatment of particle oscillation, because one needs not introduce ill-defined flavor eigenstates and also because it allows for a more elaborate treatment of the production and detection of oscillating particles—for a comparison of the methods, see Ref. [62]. A quantum field theoretic treatment is essential to a first-principles treatment of the problem and we shall employ it in Sec. 5.2.

However, there is little difference at the abstract level of formalism between the quantum field theoretic and the wave packet methods. Both make the same use of vector states and evolution operators in the definition of probabilities, essentially presupposing a description in terms of von Neumann measurements. This elementary point may be obscured by the matrix language commonly employed in quantum field theory, so we elaborate here on the meaning of the amplitude Eq. (30). In quantum theory, the dependence of a vector state on time arises from unitary time evolution. The vector states are then states of the form for some vector states and a Hamiltonian . The choices for in the literature imply an identification of with the Hamiltonian of free particles. Then corresponds to single-particle states for outcoming and incoming particles, centered in the detection and production region respectively. The amplitude Eq. (30) is then equivalent to


where .

Eq. (31) is a meaningful quantum expression only if the integration in the time-ordered exponential is for the time interval , in which case, Eq. (31) equals . The squared amplitude then provides the usual expression for probabilities in von Neumann measurements. Hence, as far as the description of measurements is concerned, both wave packet and quantum field theoretic methods employ the same assumptions.

4.2 Critical analysis of the existing formalisms for particle oscillations

4.2.1 Justification in terms of first principles.

The first point of critique towards the particle-oscillations methods described above is that the construction of probabilities does not follow from the first principles of quantum theory. The vast majority of these methods identifies the relevant probabilities with the modulus square of amplitudes, such as Eq. (25). However, this expression is relevant only for measurements that take place at a pre-specified moment of time (or time-interval)—see, Sec. 3.1. In particle oscillations, the experimentally fixed quantity is the location of the detector, not the time that a particle is detected. The time of detection is a variable rather than an external parameter. This inconsistency in the definition of probabilities is the reason why, in order to obtain a connection with measurable quantities, one needs to remove the time dependence from the square amplitudes—either by introducing additional prescriptions or by time integration.

The construction of probabilities using squared amplitudes is the most common practice in applications of quantum theory, but it is neither necessary nor unique. The most general probability assignments in quantum theory correspond to POVMs (see, Sec. 3.1.3), and these can be applied to cases where the simple definition in terms of squared amplitudes fails. We maintain that the correct probability assignment for each particular case should be constructed from first principles, in order to reflect precisely the physics of the experiment. The use of a probability assignment without prior justification may lead to inconsistencies in the correspondence of elements of the formalism to experimentally determined variables. Hence, the modeling of particle oscillation experiments by von Neumann measurements is problematic, because the setup of the former violates a main assumption in the latter, namely, that measurements take place at a pre-determined instant of time.

Even should one attempt to model particle oscillations by von Neumann measurements, amplitudes such as Eq. (25) or Eq. (30) are insufficient: they ignore the effect of the detector, which is a macroscopic system with a large number of degrees of freedom. To see this, we note that the squared amplitude can be brought in the standard form by identifying the state and the projector , that describes the measurement scheme. The projector is one-dimensional (), i.e., it corresponds to an extremely fine measurement that determines a phase space volume for the particle of size [18, 29]. One-dimensional projectors may be relevant for sharp measurements (observables with discrete spectrum), but they are unsuitable for measurements of continuous variables. As explained in Sec. 3.1, the interaction with the measuring apparatus necessitates the use of much coarser projectors () for continuous variables. Hence, the use of the squared amplitude for the probabilities misrepresents the coarse-graining inherent in the detection process.

For example, a projector describing an unsharp measurement of position is


where is a positive function centered around with a spread of order [81]. The corresponding probabilities are


Such modifications in the derivation of probabilities would not affect the phases relevant to the oscillations (as long as ), but it would make a difference in the expression for the so called ‘coherence length’ [6, 63, 68, 79, 82] i.e., a scale such that, if no oscillations are observable.

4.2.2 Times of arrival and the different prescriptions

In Sec. 4.2.1, we argued that the definition of probabilities from the squared amplitude is not suitable for particle oscillation experiments. Its use presents an immediate problem of reconciling the theoretical description with the experimental procedure. In particular, it is necessary to remove the time dependence from the squared amplitudes . In this section, we examine the most common procedure employed for this purpose, namely, the substitution of the time parameter in the squared amplitude by the ‘mean’ or classical time of arrival for a particle propagating from the source to the detector. The second common procedure of eliminating the time variable, by time-integration, is considered in Sec. 4.2.3.

The substitution of a mean time of arrival in the squared amplitudes has no fundamental justification. In fact, this procedure is inconsistent with quantum theory. The time of arrival depends on the initial state (for example, its mean energy), and its substitution into the squared amplitudes leads to a probability assignment that is not a linear functional of the system’s density matrix, thus contradicting a fundamental requirement for any probability assignment in quantum theory. In particular, as we explain in more detail in Sec. 4.2.3, a probability assignment that does not respect linearity does not have a consistent statistical interpretation.

The lack of a solid mathematical justification for the aforementioned procedure implies that there is no independent theoretical criterion to settle any ambiguities that may arise in its application. This is a problem, because, as we saw in Sec. 4.1.1, a variation of this procedure that is equally plausible a priori—the so-called unequal-times prescription—leads to a prediction for the particle oscillation wavelength that differs from the standard one by a factor of two. Hence, a significant part of the related bibliography is devoted in showing that the unequal-times prescription cannot be valid, aiming to strengthen the standard result by the exclusion of its alternative.

In what follows, we will show that (i) that the physical arguments developed against the non-standard oscillation formulas do not apply to our result which is obtained by a different method, (ii) the standard derivations ignore the probabilistic and quantum aspects of the time of arrival, and (iii) the timescales involved in particle oscillation experiments suggest that interferences in the time of arrival contribute to the oscillation wavelength thus leading to a non-standard particle oscillation formula.

A strong argument against the unequal-times prescription is that it leads to an unphysical prediction of oscillations for recoil particles; for instance, the muon in the reaction or the in the reaction . This is highly counterintuitive, and it is generally considered as a severe deficiency of the method. The formalism we develop here leads to a non-standard oscillation formula, but it does not predict oscillations for recoil particles—see Sec. 5.3. A careful description of the measurement process, in terms of particles that are actually being detected, shows that oscillations are only possible for particles with different mass eigenstates. Therefore, we maintain that the oscillations for recoil particles is the artefact of an improper mathematical framework for the evaluation of quantum probabilities. It is not a consequence of the general statement that different values of the time of arrival contribute coherently into the particle oscillation probability.

Common arguments against the unequal prescription involve statements that quantum interference only occurs between states defined at the same spacetime point [12, 13, 83], or that measurements take place in space and not in time [66]. However, the restrictions implied by such statements do not follow from the basic principles of quantum theory. Any observable is recorded by pointer variables on a measuring apparatus, and interferences are observed through the distribution of values of these variables in a large number of runs. Nothing in quantum theory forbids a pointer variable from becoming correlated with properties of the measured system that correspond to different instants of time. As shown in Sec. 3.2.2, the time of arrival may exhibit interferences, just as any other quantum observable. In general, we maintain that any statements about what cannot be measured in quantum theory can be made only on the basis of a precise modeling of the measurement process.

In Sec. 2.1, we stated that a key problem in existing approaches to particle oscillations is that they conflate the time-parameter of Schrödinger’s equation with the time of arrival, which is a physical observable. In the present context, this problem is manifested by the ad hoc substitution of the time parameter —arising from the solution of Schrödinger’s equation—in the squared amplitudes by a ‘mean’ time of arrival. This procedure ignores the fact that the time of arrival is a genuine quantum observable. Indeed, the time of arrival is determined by the coincidence of a microscopic quantum event (for example, a particle reaction in the detector) with the reading of an external clock. While the value of the time of arrival is a classical variable, it is correlated to a microscopic process that is governed by the rules of quantum theory. Consequently, the values of the time of arrival exhibit fluctuations, and these must be described by quantum probabilities. The substitution of a mean value for the time of arrival in the squared amplitudes ignores all effects pertaining to quantum fluctuations; in particular, it excludes a priori the possibility of quantum interference in the values of the time of arrival.

Acknowledging the time of arrival as a quantum observable, brings new light in the distinction between the standard oscillation formula and the non-standard one following from the unequal-times prescription. The former essentially assumes that quantum interferences in the time of arrival are either suppressed or negligible; the latter implies that such interferences persist and they contribute significantly to the particle oscillation probability. As with any other quantum variable, the issue whether such interferences persist or not can be resolved only through a careful study of the physical situation at hand, namely, of the quantum system and the measuring apparatus. To this end, a fully quantum description of the time of arrival is necessary. This, we undertake in Sec. 5.

An additional problem in the use of a ‘mean’ time of arrival in the squared amplitudes is that the very notion of a ‘mean’ time of arrival is ambiguous in absence of a full quantum treatment. For example, in Refs. [15, 16, 84] the standard expression for particle oscillation wavelength is obtained by substituting a mean time of arrival in Eq. (20), where is the source-detector distance, and the mean velocity of the wave-packet is defined as


where the indices and refer to two mass eigenspaces. Eq. (34) has no fundamental justification in quantum theory, where the only notion of a mean velocity is the expectation value of the operator . There is no reason why the mean velocity should not be defined by any other expression, such as the the arithmetic or the geometric mean of the velocities . A justification for Eq. (34) is that it guarantees that no oscillations for recoil particles are predicted, without making an equal-times assumption. If, however, oscillations of recoil particles is an artefact of a mathematically ambiguous formalism, as our results in Sec. 5 indicate, this argument loses much of its power.

In our opinion, one of the most important arguments in support of prescriptions for the standard oscillation formula (it is also quoted as definitive in the Review of Particle Physics [5]) is essentially the following [16]. Consider Eq. (28) for the oscillation phases for two mass eigenspaces, . Substitute for the expression


where is given by Eq. (34) and is “a correction to describe any discrepancies, including quantum fluctuations”. Then,


The first term in the right-hand-side of Eq. (36) is the standard oscillation phase, while the second term is a correction much smaller than the first term. The correction is deemed to be also much larger than the time-scale , that corresponds to the different centers of the wave-packets in the different mass eigenspaces. The term is responsible for the different prediction of the unequal-times prescription. One then concludes that the effect of in the oscillation phase is negligible and thus the standard oscillation formula holds.

We note that the introduction of the correction in the phase difference is essential for the argument to work. However, if corresponds to the fluctuations of the time of arrival, its introduction into the phase difference is not justified. For the argument’s sake, let us assume that the parameter in the amplitude can indeed be substituted by the time of arrival. Let be the probability distribution describing any fluctuations of the time of arrival— corresponds to the mean deviation of this distribution. According to standard probability theory, the incorporation of these fluctuations involves weighting the squared amplitudes with , i.e., one should consider the integral


Then, the integration Eq. (37) does not lead to the introduction of the mean deviation into the phase difference . For example, if is a Gaussian , the integration in Eq. (37) would only lead to a suppression of the oscillation’s amplitude. Hence, the fact that or that has nothing to do with the relative size of terms in the oscillation phase, and, hence, with the oscillation wavelength.

Moreover, the specification of a single parameter for the description of the fluctuations in the time of arrival is insufficient if these fluctuations are quantum. A single parameter does not allow for distinction between incoherent and coherent fluctuations because these have very different physical behavior444For example, a Gaussian wave function has approximately the same spread with the superposition , if , but their physical properties are very different.. The former tend to suppress oscillations while the latter would contribute additional terms to the oscillation phase. Ref. [16] emphasizes that the understanding of time fluctuations is fundamental for the proper theoretical description of particle oscillations. We strongly agree on this point. Furthermore, we maintain that such a treatment should follow precise probabilistic reasoning and it should take into account the quantum nature of the time of arrival.

The treatment of the time of arrival as a physical observable has important implications for the description of particle oscillations. A particle-detection event takes place at a specific locus and a specific moment of time. In the context of particle oscillations the detection time corresponds to the time of arrival. Since the latter is a quantum observable, the detection process corresponds to a joint measurement of time of arrival and position, which can be described in terms of a suitable POVM—see, Sec. 3.1.3. For a joint measurement, the separation of the timescales and , noted in Ref. [16], leads to the opposite conclusion from the one reached in that reference. If the sampling of one of the two variables in a joint measurement is much coarser than the relevant interference scales in the quantum state, then the measurement of this variable does not affect the probability distribution for the other variable. In other words, the fact that the timescale and , defined earlier, satisfy , implies that the measurement of the time of arrival does not disturb the contribution of the time-of-arrival interferences in the oscillation probability.

To see this, one may consider the two-slit experiment. Let be the distance between the two slits. Consider now a special case of a joint measurement: first, a recording of the position of a particle as it passes through the slits and its later position on a screen. If the accuracy of the first measurement satisfies , the slit through which the particle crossed is undetermined, so the interference pattern in the second measurement will persist. Only for a finer-grained first measurement are the interferences suppressed555This property is seen most clearly in the so-called “which-way” experiments [85], where a microwave cavity is used in order to perform a measurement of position as the particle crosses through the slits: the state of the electromagnetic field in the cavity determines the accuracy of this measurement.. The key point in this version of the two-slit experiment, is that we also have a joint measurement of two observables. A coarse measurement of one observable does not affect the coherence in the measurement of the second observable.

4.2.3 Integration of probabilities over time

In Sec. 4.2.2, we examined the procedure of removing the time-dependence from the squared amplitude by substitution with a ‘mean’ value for the time of arrival. In this section we consider the alternative procedure of integrating the squared amplitude over time, as in Eq. (29). This procedure is more common in the recent works on particle oscillations, and it is often applied in a field theory setting. The arguments presented here do not depend on the way such amplitudes are calculated, but follow from basic properties of the formalism of quantum theory.

From a mathematical point of view, is not a density with respect to , so integration with respect to is ill-defined. This problem arises because in is an external parameter and not a random variable (i.e, an observable). The integration over parameters (rather than over random variables) does not define proper probability densities, neither in classical nor in quantum probability theory. The reason for this is that the probabilities for different alternatives can be added only if the alternatives are disjoint. If the alternatives are not disjoint, the joint probability is not obtained by the sum of the individual probabilities666For two overlapping alternatives and and probability distribution , the joint probability ..

An elementary example from ordinary probability theory serves to illustrate that events distinguished only by the value of the time parameter fail to be independent. Let us consider a variable taking values in some set and consider probabilities defined on paths of such variables labeled by the time parameter . The event (alternative) “ at time ” is not disjoint to the event that “ at time ”. There are paths that satisfy both propositions above, for example a path that satisfies at all times . One cannot therefore obtain the joint probability for the event “ at or at time ” by summing the individual probabilities.

The same also holds in quantum theory [86]. As a result, the integral in Eq. (29) does not lead to well-defined probabilities. In particular, it does not preserve normalization: to normalize one has to divide by a constant . In general, the normalization constant depends on the initial state of the system, so the probability assignment is a non-linear functional of the initial density matrix , a property that is incompatible with the rules of standard quantum theory777One might respond to our argument by stating that the normalized probabilities Eq. (29) are in fact conditional probabilities at time , provided that detection has taken place. But in this case one should prove that the normalization constant is proportional to the total detection probability, as predicted by quantum theory. This is problematic because has no analogue in standard expressions for detection probabilities. Moreover, the recourse to conditional probabilities does not remove the problem of defining unconditioned probabilities. In particular, a proper normalization of unconditioned probabilities is necessary for the description of experiments at which neutrino detection takes place at two different locations—see, for example, the proposal in Ref. [87]..

Linearity with respect to the initial state is an essential property for any probability assignment. It guarantees that one can combine sub-ensembles through statistical mixing. This means the following: let be the density matrix corresponding to a sub-ensemble of events distinguished by the macroscopically determined quantities , and let be the probability distribution corresponding to an observable —see the Appendix B for details. The ensemble constructed by weighting each sub-ensemble with a factor is described by the convex combination . Linearity guarantees that the probability for the variable in the total ensemble will also be a convex combination . The lack of linearity in the probability assignment Eq. (29) implies the lack of a consistency in the combination of different subsets of data [88].

The requirement of linearity and positivity in the probability assignment is very restrictive. It implies that any probability assignment must correspond to a POVM. In the present context, the POVM should describe a joint measurement of position and arrival time . Given such a POVM we can integrate consistently over time of arrival in the probabilities obtaining a probability density that depends only on position


It is only in the context of equations of the form Eq. (38) that integration of probabilities over time can be justified in accordance with the rules of quantum theory.

The problems above would be resolved if one constructed amplitudes that depend on the time of arrival rather than the parameter of Schrödinger equation. However, neither in this case would Eq. (29) be justified. In quantum theory a summation of different alternatives at the level of probabilities is justified only when the alternatives are macroscopically distinct (for example, if they are distinguished by a measurement scheme). In absence of such distinction, the summation ought to take place at the level of amplitudes. This property is clearly manifested in the two-slit experiment. If we can determine macroscopically through which slit the particle passed, we sum over the corresponding probabilities. If we cannot, then we sum over amplitudes. In the latter case, we observe an interference pattern, in the former we do not. Hence, the assumption of an integral over time, at the level of probabilities, presupposes that we can distinguish temporal alternatives at a scale smaller than any interference terms in the system’s state. This is not the case in particle oscillation experiments. As also discussed in Sec. 4.2.2, the temporal resolution of any particle detection corresponds to a time-scale which is at best of the order of for a solid state detector. On the other hand, the characteristic timescale of interferences is of the order of