How long does a quantum particle take to traverse a classically forbidden energy barrier? In other words, what is the correct expression for quantum tunnelling time? This seemingly simple question has inspired widespread debate in the physics literature. I argue that we should not expect the orthodox interpretation of quantum mechanics to provide a unique correct expression for quantum tunnelling time, because to do so it would have to provide a unique correct answer to a question whose assumptions are in tension with its core interpretational commitments. I explain how this conclusion connects to time’s special status in quantum mechanics, the meaningfulness of classically inspired concepts in different interpretations of quantum mechanics, the prospect of constructing experimental tests to distinguish between different interpretations, and the status of weak measurement in resolving questions about the histories of subensembles.

In both classical and quantum physics, it is easy to calculate the time taken by a free particle to travel from point A to point B. Motivated by this simple picture, a whole community of quantum theorists have asked: how long does a quantum particle take to traverse a classically forbidden energy barrier? Or, in their terminology,

Various expressions for quantum tunnelling time have been proposed. Some track internal properties of the tunnelling system, while others rely on coupling between the tunnelling particle and an external physical system. The proposals are weighed against each other on mostly pragmatic grounds, and tunnelling time, as a result, is often described as a topic riddled with “confusion” and “controversy” (see e.g. Sokolovski and Connor

Many authors see this ambiguity in the definition of tunnelling time as part of the fallout from how quantum mechanics treats time in general: as a parameter, not an operator (see e.g. Challinor et al.,

Against this varied backdrop, recent high-profile experiments at the University of Toronto claim to have used weak measurement to extract a traversal time of 0.61ms for Bose-condensed ^{87}Rb atoms tunnelling through a 1.3-micrometre-thick optical barrier (Ramos et al.,

The state of the literature on quantum tunnelling time therefore motivates four questions of both physical and philosophical import. First, does the confusion about tunnelling time really stem from the more general “problem of time” in quantum mechanics—namely, the fact that time lacks an operator? Second, is tunnelling time really a meaningless concept in the orthodox interpretation of quantum mechanics? If so, why, and in what sense? Third, is it possible to use quantum tunnelling time to gain concrete evidence for the de Broglie-Bohm interpretation? And finally, are weak measurement techniques—and experimental realisations thereof—able to resolve the decades-long debate and provide clear, unique and experimentally verifiable values for tunnelling time?

This paper aims to answer each question in turn. Section

In Section

Answers to the four questions follow in Section

These questions and answers are not all new. Each has been raised in some form in the literature, but they have not yet been tied together, and where they do appear they are inserted as brief comments within much longer technically-focused expositions. Furthermore, they are all controversial: some existing works agree, either implicitly or explicitly, with the position I present here, and some stand in opposition.

To the extent that I am drawing together ideas that have already been expressed, I aim to offer a simple and focused explanation of why those ideas—and not their proposed alternatives—are correct. To the extent that I am presenting new ideas, I aim to show how quantum tunnelling time can shed new light on the relationship between de Broglie-Bohm theory and other, trajectory-free, interpretations of quantum mechanics.

Here I summarise the physical system on which discussions of tunnelling time are based. The system has two components: a free particle with momentum _{0} > 0 and initial position _{0} ≪ _{1}, and a classically forbidden barrier extending from _{1} to _{2}.

For _{1}, the wavefunction _{0}, but containing various momentum components:

When the wavefunction _{1} < _{2} by the transmitted part —by particles that are eventually transmitted through the barrier.

The stationary components of the wavepacket take the following form:
_{k} is the reflection coefficient for energy component _{k} is the transmission coefficient for energy component _{k}|^{2} and |_{k}|^{2} are the proportion of particles described by

Bringing time back into the picture, the tunnelling process can be broken into three steps (see Fig.

The three steps of the tunnelling process

A good expression for tunnelling time should provide the average time spent in the region _{1} < _{2} by particles detected at _{2}. But the question is: how can that kind of expression be extracted from the behaviour of

The duration of quantum tunnelling was first raised by Condon (

Soon after, in 1932, MacColl (_{2}] at about the time at which the incident packet reaches the point _{1}], so that there is no appreciable delay in the transmission of the packet through the barrier” (621).

Thus, the first value ascribed to tunnelling time was

There have been three main developments in the literature on quantum tunnelling time since those early years. First, a time known as the

Second, multiple definitions of tunnelling time have been proposed by authors working within the ‘standard’ or ‘orthodox’ interpretation of quantum mechanics, on which particles exist and evolve as delocalized wavefunctions until they collapse indeterministically to some localized state on measurement (Section

Third, a minority of authors have assessed the tunnelling time problem from the perspective of de Broglie-Bohm theory, on which quantum particles follow entirely deterministic but epistemically inaccessible trajectories (Section

One well-established notion related to tunnelling time that does not depend on the difference between the orthodox interpretation and de Broglie-Bohm theory is the _{d}:

Introduced by Smith (

Although well-established and uncontroversial, _{d} it is not a contender for tunnelling time, because it does not distinguish between transmitted and reflected particles. Leavens has argued on this basis that it is precisely the attempt to distinguish between particles that will be eventually transmitted and particles that will be eventually reflected that has led to so much confusion about tunnelling time within the orthodox interpretation. He writes, “Since expression (5) [i.e. our Eq.

Later on, in Section

Many definitions of tunnelling time have been proposed by authors working in the orthodox interpretation. Here I will present the phase time, the Larmor clock time, the Salecker-Wigner-Peres clock time, Steinberg’s weak measurement time, and—very briefly—a few other proposals.

Hartman reevaluated the duration of quantum tunnelling in 1962 in light of technological developments that allowed for tunnelling across thin insulating layers. His interest stemmed from a desire to understand “The transmission times for metal-insulator-metal thin film sandwiches” (Hartman

Using the stationary phase approximation—and working, as with all the authors in this Subsection, within the orthodox interpretation—he came up with an expression for quantum tunnelling time now known as the

The phase time _{T}(

Given _{a} is some point to the left of the barrier, _{b} is some point to the right of the barrier, and

Although appealing in its apparent simplicity, this expression for tunnelling time suffers from significant problems. First of all, and most importantly, the phase time is defined in terms of the characteristics of _{T}(

However, we have assumed that we are dealing with a wavepacket which is narrow in _{a} ≪ _{1} and _{b} ≫ _{2} (924).

It might nevertheless be tempting to extrapolate back from _{1} and _{2} by subtracting from _{0} would take to traverse the distance (_{1} − _{a}) + (_{b} − _{2}). This would, however, be ill-advised, as Hauge & Støvneng (

In addition to this serious failure to answer the question at hand, the phase time comes along with a strange and problematic feature: it predicts that tunnelling time will approach a constant value as the width of the barrier increases towards infinity, implying faster-than-light tunnelling for particles traversing a wide barrier (Hartman,

The _{a}, _{b}) including the tunnelling barrier. The idea is that the particle will then precess at a constant rate in the plane perpendicular to the applied field as it moves between _{a} and _{b}.

For example, if the magnetic field is applied in the _{a} and _{b}.

In the limit where the characteristic frequency of the field _{L} goes to zero, and considering only one momentum component _{y}〉_{T}, defines a time (Hauge and Støvneng

Supporters of this Larmor clock time emphasize that it makes tunnelling time accessible to experiment, not just pen and paper analysis (Landauer and Martin

However, Falck and Hauge (

In particular, this means that the Larmor clock time, like the phase time, must be treated asymptotically (i.e. with _{a} ≪ _{1} and _{b} ≫ _{2} ). This is necessary in order “[t]o prevent interference between the processes of entering and leaving the field region on the one hand, and the tunneling process on the other” (3288). Entering and leaving the magnetic field will affect the behaviour of the wavepacket, and if the three steps of the tunnelling process are not all contained within the magnetic field region, resulting values for traversal time will be contaminated by those affects.

The only way to infer a

In a serious way, the Larmor clock time therefore fails to answer the question it set out to answer. This perhaps should not be surprising, given that _{a}, _{b}) (3287, 3292).

Furthermore, when the tunnelling particle interacts with the applied magnetic field, there are really two relevant time scales. As explained above, _{z,T} corresponding to the spin’s tendency to align with the applied field (Hauge and Støvneng

Büttiker (

The Salecker-Wigner-Peres (SWP) clock, like the Larmor clock, couples the particle of interest with an external observable. In fact, the SWP clock is equivalent to the Larmor clock in the limit of large spin (Sokolovski and Connor _{SWP} in the region of interest and tracks the resulting change in phase.

First introduced by Salecker and Wigner (

This

Steinberg, in 1995, introduced a new definition of tunnelling time based on the theory of weak measurement developed by Aharonov et al. (

Weak measurement was developed precisely to deal with questions about subensembles: questions about the average state of some operator for particles prepared in a state |

For tunnelling time, we want to ask: on average, how long did a particle spend in the barrier region,

The operator of interest, in this case, is the projection operator which measures whether a particle is in the barrier region (Steinberg

The resulting value for tunnelling time

Indeed one of the main benefits of this weak measurement approach, from Steinberg’s perspective, is that it can be applied to the Larmor clock and thereby render tunnelling time accessible to experiment.

Steinberg describes his approach as “a sensible way to define conditional probabilities in quantum mechanics, assuming only Bayes’s theorem and standard quantum theory” (Steinberg

Sokolovski and Akhmatskaya (

In their words,

“[i]f you consider all possible experiments of this type, some of them will give seemingly reasonable outcomes, whereas other ‘times’ would be negative, too short, too long, etc. This is necessary, and is possible because such ‘times’ can be expressed as the combinations of probability amplitudes which, unlike probabilities, have few restrictions on their signs and magnitudes. Though your result of 0.61

Sokolovski’s critique of the weak measurement approach to tunnelling time fits with his stance on weak measurement in general; Sokolovski (

The expressions described above are just four of the main contenders for tunnelling time on the orthodox interpretation—there are many more. A time based on Feynman path integrals is supported by Sokolovski and Baskin (

Alongside the development of this varied landscape in the orthodox literature, a handful of authors have assessed the quantum tunnelling time problem from the perspective of de Broglie-Bohm theory.

By reinterpreting the wavefunction and postulating the existence of underlying point particles, de Broglie-Bohm theory is able to maintain the mathematical formalism of the standard interpretation but simultaneously adopt a completely deterministic underlying dynamics.

The wavefunction is reinterpreted as a field that guides the time-evolution of underlying localised quantum particles, but simultaneously conceals information about which trajectory any particle is actually following. Only a measurement can reveal that information; and the collapse that we see during measurement is thereby demoted from a genuine indeterministic change in the state of the system to something that only seems indeterministic because it reveals information about the system that we previously could not access.

The tunnelling time problem, when viewed through this lens, becomes much more tractable. For a given wavefunction

Following Leavens (_{0} will later spend a time in the region (_{a}, _{b}) given by:
_{0}, _{0}.

This can be used to construct an expression for tunnelling time, simply by setting _{a} = _{1} and _{b} = _{2} and taking the average value <>_{T} of _{0}, _{1}, _{2}) only over the subensemble of initial positions _{0} that lead to transmission through the barrier:

This expression is unique and uncontroversial within de Broglie-Bohm theory; it has even been called “trivial” (Muga et al.,

The de Broglie-Bohm interpretation, in its traditional form, therefore provides a clear and unique answer to a question that the standard interpretation has not been able to answer clearly or uniquely. Many existing reviews of the history of quantum tunnelling time do not even consider this perspective—including, for example, Winful (

Four themes, which I label (1)

Various authors have blamed the apparent ambiguity of quantum tunnelling time on the fact that time enters quantum mechanics as a parameter, not a self-adjoint operator. Abolhasani and Golshani (

The proliferation of mutually exclusive definitions highlighted in Section

The existence of a unique and clear expression for quantum tunnelling time in de Broglie-Bohm theory, contrasted with the total lack of consensus on the definition of tunnelling time in the orthodox interpretation, has led to speculation about whether tunnelling time could provide evidence for de Broglie-Bohm theory. Cushing (

Recent experimental realisations of Steinberg’s proposal for probing tunnelling time have raised the idea that weak measurement might be able to resolve the long-standing controversy and provide empirical measurements of quantum tunnelling time that come along with a clear interpretation. In a 2020 Nature paper, Ramos et al. (^{87}Rb atoms tunnelling through a 1.3-micrometre-thick optical barrier”. They apply a weak magnetic field within the barrier region and postselect for eventually transmitted particles, using the Larmor clock to read off the time spent in the barrier region by particles that are eventually detected on its right hand side. The authors write, “This experiment lays the groundwork for addressing fundamental questions about history in quantum mechanics: for instance, what we can learn about where a particle was at earlier times by observing where it is now” (529). However, the physical meaning of these experimental results has been directly questioned by Sokolovski and Akhmatskaya (

In the next Section I will attempt to provide the groundwork for answering these questions by looking more closely at how tunnelling time features in the orthodox interpretation (Section

The difference between the orthodox intepretation and de Broglie-Bohm theory that will weigh most heavily on the issues just raised is revealed by the question:

These answers clearly do not permit the same analysis of a tunnelling particle’s behaviour before detection. The orthodox interpretation must claim that immediately before a particle corresponding to Step 3 of the tunnelling system is detected either to the right of the barrier or to the left of the barrier (see Fig.

By the ‘orthodox interpretation’, I mean the interpretation—broadly construed—on which quantum particles do not have well-defined trajectories and collapse indeterministically to seemingly determinate values only when they are measured.

In particular, I refer to any interpretation in which a normalised quantum state
_{0} to

As reviewed in Section

If we are working within the orthodox interpretation, however, then we have no choice but to take the state of a quantum particle to be identical to the amplitude of its wavefunction; it evolves deterministically when not being measured but collapses indeterministically on measurement to any of the observable states for which the previous amplitude was nonzero. For the tunnelling time system, this means that the state of an incident particle evolves as outlined in Fig.

The orthodox account of the state of a particle described by the tunnelling time problem, in three stages: during interaction with the barrier, after interaction with the barrier but before detection, and after detection

If the point of collapse is instead to the right of the barrier, we appear to have detected a particle that tunnelled through the barrier. But immediately before detection the particle had not reflected or tunnelled—rather it existed as a complex probability amplitude including a reflected peak and a transmitted peak. It is therefore in contradiction with the basic tenets of the orthodox interpretation to attempt to extract an expression for the average duration-behaviour-in-the-barrier of eventually transmitted particles, because according to those basic tenets, any property corresponding to a particle detected on the right hand side of the barrier was attained not by a particle destined to be detected as such, but by the probability distribution in the ‘During interaction with barrier’ step of Fig.

Any proposal for extracting a tunnelling time from the orthodox interpretation will encounter this problem.

Within the orthodox interpretation, the question “

A loose analogy with the familiar double slit experiment illustrates the force of this point. For the double slit experiment, we want to ask: given many individual runs that produce an interference-pattern-shaped probability distribution on the detection screen, which of the two slits did each individual particle go through? For the tunnelling time problem, we want to ask: given that a particle is detected on the right hand side of the barrier, on average how much time did it spend in the barrier region? Both are questions about how to infer information about a quantum particle’s previous behaviour given information about the state to which it collapsed on measurement.

But the orthodox interpretation explicitly prohibits this kind of inference.

There are two sides to the underlying dynamics of de Broglie-Bohm theory, both of which can be derived by mathematical manipulation of the orthodox formalism: the wave dynamics, and the particle dynamics. In three spatial dimensions, they are given by:

^{2}

For

Each underlying particle

The interpretation as a whole can be criticized on various grounds: among other issues, it is strongly nonlocal, and relies on the controversial assumption that the probability distribution of underlying particles matches the density of the wavefunction. But, as introduced in Section

Within this interpretation of quantum mechanics, each tunnelling particle travelled through the barrier in a state that was destined to end up being transmitted, or in a state that was destined to end up being reflected. See Fig.

Space-time diagram showing a representative sample of possible particle trajectories for the case of a plane-wave packet incident from the left on a rectangular potential barrier. Image and caption reproduced with permission from Norsen (

It is the uncertainty principle that blocks us from being able to see, by looking at a particle’s behaviour in the barrier region, which of these trajectories it is following: we only have access to the behaviour of the wavefunction, which permits both eventually transmitted and eventually reflected trajectories.

Thus, whereas in the orthodox interpretation, a tunnelling particle interacts with the barrier in a superposition state that cannot be broken into eventually transmitted and eventually reflected components, in de Broglie-Bohm theory it makes sense to speak of a particle interacting with the barrier in a state that is destined to be eventually transmitted.

This makes the question at the core of the tunnelling time debate —“

The same double slit analogy can be introduced here. On the Bohmian view, each particle incident on a double slit goes through either the left slit or the right slit on a localized, deterministic trajectory. It is the uncertainty principle that blocks us from being able to see, by tracking a particle’s behaviour at the double slit, which slit it is going through: we only have access to the behaviour of the wavefunction, which permits both left-slit and right-slit trajectories.

Any attempt to try to measure which slit a given particle is going through destroys the interference pattern on the screen, because the measurement apparatus interacts with the underlying wave field

So we cannot know whether a given particle went through the left or right slit just by subscribing to de Broglie-Bohm theory; but at least it makes sense to ask the question.

Tunnelling time, I have argued, is therefore an inherently unclear and ambiguous concept on the orthodox interpretation, because it stems from a question about particle histories that the orthodox interpretation is not set up to answer. It is an inherently clear and unambiguous concept in de Broglie-Bohm theory, because de Broglie-Bohm theory has the trajectory infrastructure to be able to answer a wide range of questions about particle histories.

These conclusions shed light on the four issues raised in Section

As reviewed in Section

Even before considering the conclusions of the previous section, this view is called into question by the fact that quantum clock observables can be defined: they just cannot be represented by PVMs (Pashby

Furthermore, quantum theory has no problem providing a clear and unique expression for other duration-based concepts, including, for example, the time of flight of a free particle. In most cases we can simply exploit functional relationships between time and other quantities, like position and momentum, that can be represented by self-adjoint operators.

The conclusions established in Section

Tellingly, the time spent within the barrier region is not ambiguous even for tunnelling, provided no attempt is made to distinguish between eventually reflected and eventually transmitted particles. As shown in Section

The confusion surrounding quantum tunnelling time

As discussed in Section

Meaningless is perhaps too strong a term. What we can assert with confidence is that tunnelling time requests an answer to a question that the orthodox interpretation is unable to provide, having committed itself to the state of a particle

Within the de Broglie-Bohm interpretation, tunnelling time is not problematic in this sense, simply because the de Broglie-Bohm interpretation does not commit itself to the wavefunction being a full and complete description of a particle’s state. Rather, it posits an underlying dynamics that includes deterministic and localised trajectories.

I suggest that we should, nevertheless, hesitate to call tunnelling time “meaningless” in the orthodox interpretation. Various definitions have been proposed, and some are even experimentally measurable. Even if we admit that no unique ‘correct’ definition will ever be established, we can weigh the existing expressions on pragmatic grounds, as authors have indeed done for decades.

Authors have assessed proposed orthodox expressions based on their generality, experimental accessibility, convergence on classically expected results, on whether they involve stationary or time-dependent treatment, asymptotic or local treatment, the absence or presence of the Hartman effect, the satisfaction or violation of a weighted average identity that reduces to the dwell time, and on whether the values they yield are real or complex. Each of these features might be considered desirable or undesirable by authors with different aims.

In a sense, the proliferation of different orthodox definitions for tunnelling time is similar to the proliferation of interpretations of quantum mechanics as a whole. They have provoked the same kind of debate: a debate that is largely considered to be unresolvable, but which is not thereby rendered meaningless. We can identify desirable and undesirable criteria for orthodox definitions of tunnelling time, just like we can identify desirable and undesirable criteria for different interpretations of quantum mechanics.

Various authors have suggested that tunnelling time could be used to provide experimental evidence for de Broglie-Bohm theory (Section

Two versions of this idea have been raised: first, the idea that tunnelling time could be used to set up a crucial test of the Bohmian program, capable of confirming or disconfirming its predictive accuracy, and second, the somewhat weaker idea that desirable features of tunnelling time could or should be used to provide evidence in support of de Broglie-Bohm theory.

These suggestions, especially of the first kind, have been controversial—even Cushing, the first to provide a detailed proposal for a crucial experimental test based on tunnelling time, eventually gave up the idea. But some speculation remains—and besides, where authors have dismissed the possibility they have not always based their arguments on the same reasoning.

Bedard (

Belousek (

“Regarding the question of whether the ‘transit’ or ‘tunneling’ times in Bohmian mechanics constitute excess empirical content over ‘orthodox’ quantum mechanics (cf. Ref. 31), I am of the view that, while the ontology of particles following definite trajectories does constitute surplus physical content, this does not generate any excess empirical content in the sense of novel predictions. Instead of novel prediction, Bohmian mechanics allows a more detailed interpretation, and perhaps a more satisfactory explanation, of the measurement outcomes of certain experiments in terms of the dynamical quantities definable within its own theoretical framework. What one has here is a case not of excess empirical content but rather of the well-known ‘theory-ladenness of observation’.”

Cushing, in a response published as a postcript to Bedard’s paper, agrees that his original proposal is unviable, but for a slightly different reason. The problem, in his view, lies not only with how to interpret a successful observation of de Broglie-Bohm tunnelling time, but with whether de Broglie-Bohm tunnelling time is observable at all (Bedard

I suggest, based on the conclusions of Section

The double slit, once again, provides a useful analogy. De Broglie-Bohm theory tells us whether a particle went through the upper or lower slit based on where it appears on the screen far beyond the slits. But it is still not possible to put a detection device at the slits, and measure which slit each individual particle is going through, without changing the particle trajectories and destroying the interference pattern on the screen. Even though the theory distinguishes left-slit from right-slit trajectories, we cannot experimentally isolate either set of trajectories without running into exactly the same problems that we would run into in the standard interpretation.

The situation is similar for tunnelling time. De Broglie-Bohm theory tells us, based on whether a particle ends up being detected as eventually transmitted or eventually reflected, how long on average it spent in the barrier region. But it is still not possible to use a measurement device to pick out the eventually transmitted particles before they have been transmitted, and track their duration behaviour, without changing the particle trajectories themselves. Even though the theory distinguishes between eventually transmitted and eventually reflected trajectories, we cannot experimentally isolate either set of trajectories and keep the underlying dynamics intact.

This aligns closely with Cushing’s own last word on the topic. He writes: “for Bohm all measurements are ultimately position measurements and the distribution of these results must be given by |^{2}, both in Bohm and in Copenhagen. Hence, this does not appear to be a promising avenue for the resolution of our underdetermination” (Cushing and Bowman

On the weaker question of whether we should take the reasonableness of tunnelling time in de Broglie-Bohm theory as evidence for the Bohmian program, I suggest that we should not even go that far, because we should not expect the orthodox interpretation to provide a reasonable answer to a question it is not well-posed to answer. We should not be surprised that de Broglie-Bohm theory offers a clear and unique definition for tunnelling time while the orthodox interpretation does not: its ability to provide such a definition stems from its ability to provide particle trajectories. Its ability to provide particle trajectories, in itself, can be seen as a desirable feature of de Broglie-Bohm theory. But the clear tunnelling time that stems from those trajectories should not be interpreted as additional evidence for the Bohmian program.

Experiments like the one reported in Ramos et al. (

I suggest, however, that they do not, as long as their results are still interpreted from an orthodox perspective. A weak measurement approach to tunnelling time does make progress, in some sense, precisely because it uses postselection. At least it leaves the quantum tunnelling time question well-posed: we can reasonably ask an experiment based on weak measurement to provide a time spent in the barrier

The problem lies in the interpretation of those results, because it is not clear what measurements like those in Ramos et al. (

This is a general problem for weak measurement, and it is the reason why weak measurement is not taken to “resolve” the double slit which-way question, despite having revealed average trajectories for single photons in a double slit interferometer.

The point stands quite aside from concerns about whether the Larmor clock is a reliable clock for measuring tunnelling, and about whether the implementation of the Larmor clock reported in Ramos et al. (

Thus weak measurement offers a new perspective on the tunnelling time problem, not a resolution. It would, nevertheless, be interesting to see a weak measurement experiment extract average trajectories for particles incident on a tunnelling barrier—and to see whether they coincide with the Bohmian trajectories, as was shown to be the case for the double slit experiment (see Kocsis et al.,

The orthodox interpretation does not deny us the information we rely on to calculate expectation values for position, or momentum. In general, it does not even deny us the information we need to track duration. But it does deny us exactly the information we need to distinguish between eventually transmitted and eventually reflected behaviour of a tunnelling particle within the barrier region, just like it denies us the information we need to distinguish between went-through-left-slit and went-through-right-slit behaviour in the double slit experiment.

On this basis, I have argued for answers to four controversial questions about quantum tunnelling time. Motivated by the links that have been drawn in the literature between the tunnelling time problem and the more general “problem of time” in quantum mechanics, I asked whether the confusion and ambiguity surrounding tunnelling time on the orthodox interpretation can really be traced back to time’s status as a parameter rather than an operator. I argued ‘No’: the lack of clarity stems from the status of superposition in the orthodox interpretation, not the fact that time cannot be represented by a self-adjoint operator. In fact it does not have much to do with time at all.

Motivated by claims in the literature about the meaninglessness of tunnelling time on the orthodox interpretation, I asked whether tunnelling time is in fact meaningless on the orthodox view, and if so, in what sense. I argued that the question at the basis of the tunnelling time problem is one which the orthodox interpretation is ill-posed to answer. But I suggested that this does not make tunnelling time meaningless in an absolute sense; the various competing orthodox definitions for tunnelling time might all be useful in different contexts.

Several authors have suggested that tunnelling time could be used to extract evidence for the Bohmian program, and I asked whether this is something we can or should aim to do. I answered that we cannot, and should not, in two senses. First, any experimental test of tunnelling time will be predicted to provide the same results on both de Broglie-Bohm theory and the standard interpretation, and therefore we cannot hope to be able to construct a crucial test of the Bohmian program using tunnelling time as the deciding factor. Second, pragmatically desirable features of tunnelling time in the de Broglie-Bohm interpretation (including, for example, its simplicity and clarity) should not be interpreted as outright evidence for that interpretation, because we should not expect the orthodox interpretation to provide a reasonable value for a concept that it is ill-posed to define.

Finally, I asked whether recent experiments based on weak measurement offer a solution to the orthodox interpretation’s inability to track duration behaviour specific to eventually transmitted particles. I argued that they do not, because the meaning of weak measurement results —within a framework of quantum mechanics that does not endorse the veridicality of particle trajectories—is bound to remain unclear.

These answers will, I hope, bring some clarity on a topic that remains genuinely controversial amongst physicists, and largely unknown within philosophy of physics.

I am grateful to Mike Miller and Jeremy Butterfield for their feedback and guidance on many previous versions of the paper. I would also like to thank John Sipe, Daniel James, Bryan Roberts and Hasok Chang for helpful discussions in the early stages of this project. Audiences at Caltech and ETH Zürich provided helpful comments and constructive criticism on draft versions, as did two anonymous referees.

This work has been supported by a University of Toronto National Scholarship, the University of Cambridge Harding Distinguished Postgraduate Scholars Programme, and funding from the Social Sciences and Humanities Research Council of Canada. None of these funding bodies played any role in the writing of the article or in the decision to submit the article for publication.

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For more on the history of quantum tunnelling and the early development of quantum tunnelling time as a concept, see Perovic (

I will explain what I mean by ‘orthodox interpretation’ in more detail in Section

I will introduce the formalism at the basis of de Broglie-Bohm theory in more detail in Section

_{0} is the magnitude of the applied magnetic field.

The recent University of Toronto weak measurement experiments, reported in Ramos et al. (

Note that this implies equivalence between the SWP clock time and the phase time for large spin _{a}, _{b}) much wider than (_{1}, _{2}), based on the equivalence between the Larmor clock time and the phase time for (_{a}, _{b}) much wider than (_{1}, _{2}).

This is exactly what Ramos et al. (

The reference to “0.61

For more on the interpretation of weak measurement, see e.g. Svensson (

Throughout I restrict myself to the traditional version of de Broglie-Bohm theory that I describe here—other proposals for the ontology underlying the pilot wave program, although fascinating in their own right, do not bear on the conceptual points I aim to make in this paper.

Since this expectation value is taken only over the eventually transmitted components of the original wavefunction, it needs to be renormalized by a factor inversely proportional to the probability of transmission |^{2}. This renormalization factor is absorbed into my notation for the restricted expectation value, <>_{T}.

See e.g. Challinor et al. (

Cushing entertains the same possibility, although in less detail, in Chapter 4 of his 1994 book on the historical dominance of standard quantum mechanics over de Broglie-Bohm theory (see pages 54–55 and 72–75 of Cushing (

For authors who argue that we should not expect to find a unique expression within the standard interpretation, see e.g. Hauge and Støvneng (

Definitions based on weak measurement might at first sight seem to present an exception; this possibility will be discussed in Section

The correspondence between the tunnelling and double slit scenarios is not exact, but it is not intended to be; the analogy is enough to demonstrate this key point.

It is disputed whether this assumption needs a dynamical justification or should just be taken as a matter of postulate. For a recent review, see Valentini (

See e.g. Bohm (

For more details on the double slit experiment in de Broglie-Bohm theory, see e.g. Bricmont(

For a consideration of generality, see e.g. Sokolovski and Baskin (

For the paper which reports these average trajectories, see Kocsis et al. (

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