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Arrival Times, Complex Potentials and Decoherent Histories

We address a number of aspects of the arrival time problem defined using a complex potential of step function form. We concentrate on the limit of a weak potential, in which the resulting arrival time distribution function is closely related to the quantum-mechanical current. We first consider the analagous classical arrival time problem involving an absorbing potential, and this sheds some light on certain aspects of the quantum case. In the quantum case, we review the path decomposition expansion (PDX), in which the propagator is factored across a surface of constant time, so is very useful for potentials of step function form. We use the PDX to derive the usual scattering wave functions and the arrival time distribution function. This method gives a direct and geometrically appealing account of known results (but also points the way to how they can be extended to more general complex potentials). We use these results to carry out a decoherent histories analysis of the arrival time problem, taking advantage of a recently demonstrated connection between pulsed measurements and complex potentials. We obtain very simple and plausible expressions for the class operators (describing the amplitudes for crossing the origin during intervals of time) and show that decoherence of histories is obtained for a wide class of initial states (such as simple wave packets and superpositions of wave packets). We find that the decoherent histories approach gives results with a sensible classical limit that are fully compatible with standard results on the arrival time problem. We also find some interesting connections between backflow and decoherence.


A. The Arrival Time Problem
The arrival time problem has attracted some considerable interest in recent years [1]. In the one-dimensional statement of this problem, one considers an initial wave function |ψ concentrated in the region x > 0 and consisting entirely of negative momenta. The question is then to find the probability Π(τ )dτ that the particle crosses x = 0 between time τ and where |ψ f (τ ) is the freely evolved state, which arises from elementary considerations of the Schrödinger equation. (For convenience we work in units in which = 1). This is sensible classically but can be negative in the quantum case for certain states consisting of superpositions of different momenta [2,3,4]. Second, a simple operator re-ordering of J(t) gives the "ideal" arrival time distribution of Kijowski [5] Π K (τ ) = 1 m ψ f (τ )||p| 1/2 δ(x)|p| 1/2 |ψ f (τ ) (1.2) which is clearly positive. Both of these distributions are measurable [6,7].

B. Complex Potentials
An interesting question is the extent to which such expressions emerge from more elaborate measurement or axiomatic schemes. There are many such schemes. Here, we will focus on expressions for the arrival time distribution arising from the inclusion of a complex potential V (x) = −iV 0 θ(−x) (1.3) in the Schrödinger equation. With such a potential, the state at time τ is |ψ(τ ) = exp (−iH 0 τ − V 0 θ(−x)τ ) |ψ (1.4) where H 0 is the free Hamiltonian. The idea here is that the part of the wave packet that reaches the origin during the time interval [0, τ ] should be absorbed, so that N(τ ) = ψ(τ )|ψ(τ ) (1.5) is the probability of not crossing x = 0 during the time interval [0, τ ]. The probability of crossing between τ and τ + dτ is then Complex potentials such as Eq.(1.3) were originally considered by Allcock in his seminal work on arrival time [8] and have subsequently appeared in detector models [6,9]. (See also Ref. [7,10] for further work with complex potentials). A recent interesting result of Echanobe et al. is that under certain conditions, evolution according to Eq.(1.4) is essentially the same as pulsed measurements, in which the wave function is projected onto x > 0 at discrete time intervals [11].
For large V 0 , the wave function defined by the evolution Eq.(1.4) undergoes significant reflection, with total reflection in the "Zeno limit", V 0 → ∞ [12]. Here, we are interested in the opposite case of small V 0 , where there is small reflection and Eq.(1.6) can give reasonable expressions for the arrival time distribution. A number of different authors [6,7,8] indicate that the resulting distribution is of the form where J(t) is the current Eq.(1.1) and It is therefore closely related to the current J(t) but also includes the influence of the complex potential via the "apparatus resolution function" R(V 0 , t).
The first aim of this paper is to look in some detail at the calculation and properties of the arrival time distribution Eq.(1.7) defined using a complex potential. In particular, we will use path integral methods, which in some ways give are more concise and transparent than previous derivations (and also suggest generalizations to complex potentials more general than Eq.(1.3)). We will also explore some of the properties of the result, Eq.(1.7).
The second aim of this paper is to carry out a decoherent histories analysis of the arrival time problem. This turns out to be closely related to the complex potentials definition of arrival time and was in fact the original motivation for exploring complex potentials. In brief, our aim is to see if standard results for Π(τ ), such as Eq.(1.1) or Eq.(1.7) have a natural place in the decoherent histories approach.

C. The Decoherent Histories Approach to the Arrival Time Problem
We now briefly review the decoherent histories approach to the arrival time problem [13,14,15,16,17,18]. In the decoherent histories approach [19,20,21,22,23,24,25], probabilities are assigned to histories via the formula, p(α 1 , α 2 , · · · ) = Tr C α ρC † α (1.9) where the class operator C α denotes a time-ordered string of projectors P α interspersed with unitary evolution, C α = P αn e −iH(tn−t n−1 ) P α n−1 e −iH(t 2 −t 1 ) · · · P α 1 (1. 10) and α denotes the string α 1 , α 2 , · · · α n . The class operator satisfies the condition α C α = e −iHτ (1.11) where τ is the total time interval, τ = t n − t 1 . Interference between pairs of histories is measured by the decoherence functional, D(α, α ′ ) = Tr C α ρC † Decoherence implies the weaker condition of consistency, which is that ReD(α, α ′ ) = 0 for α = α ′ , and this is equivalent to the requirement that the above probabilities satisfy the probability sum rules. In most situations of physical interest, both the real and imaginary parts of D(α, α ′ ) vanish for α = α ′ , a condition we shall simply call decoherence, and is related to the existence of records [20,25]. Decoherence is only approximate in general which raises the question of how to measure approximate decoherence. The decoherence functional satisfies the inequality [24] |D(α, This suggests that a sensible measure of approximate decoherence is The approach also permits other types of class operators which are not simply strings of projections, but sums of such strings. These are often called inhomogeneous histories [26] and are relevant to questions involving time in a non-trivial way. For example, for a given class operator C α , the object 1 − C α is also a class operator but it not equal to a simple string of projections. Unlike homogenous histories, inhomogeneous histories do not satisfy condition Eq.(1.15) in general, except when there is decoherence.
A quantity closely related to the probabilities is the quasi-probability Using Eq.(1.11), this satisfies (where α is fixed in the sum). This means that when there is decoherence, the probabilities for histories p(α) are equal to the simpler expression q(α). (The converse is generally not true except for in very simple circumstances). Note that q(α) is not positive (or even real) in general, but it is positive and real when there is decoherence. These facts turn out to be relevant to our analysis of the arrival time problem.
We ultimately seek a decoherent histories account of the arrival time probability Π(t)dt, the probability for the particle to arrive in an infinitesimal interval [t, t + dt]. However, for simplicity, we first consider the simpler problem of computing the probability for arriving during a finite (possibly large) interval [0, τ ]. We consider an incoming state entirely localized in x > 0, and we partition the system's histories into two classes: histories that either cross or do not cross x = 0 during the time interval [0, τ ]. We seek class operators C c , C nc corresponding to these two classes of histories.
Some earlier papers on the decoherent histories approach adopted the following definition of the class operators [13,14,15,16,17]. (This definition is problematic, as we shall see, but sets the stage for the corrected version we shall use here.) Let P = θ(x) denote the projection onto the positive x-axis. To define the class operator for histories which do not cross x = 0, we split the time interval into N parts of size ǫ, and the class operator is defined by where there are N + 1 projections and N unitary evolution operators, and N → ∞ in such a way that τ = Nǫ is constant. The limit actually yields the so-called restricted propagator, This object is also given by the path integral expression where the integral is over all paths from x(0) = x 0 to x(τ ) = x 1 that always remain in x(t) > 0. The class operator for crossing the surface is then defined by However, as indicated, there is a problem with this definition. The class operator Eq.(1.21) suffers from the quantum Zeno effect [12] -projecting continually in time onto the region x > 0 prevents the system from leaving it and the probability for not crossing x = 0 is unity, for any initial state. This is easily seen from the observation that the restricted propagator defined by the limit of Eq.(1.21) may actually be written, g r (τ, 0) = P exp (−iP HP τ ) (1.26) so is unitary in the Hilbert space of states with support only in x > 0 [27,28]. Differently put, an incoming wave packet evolving according to the restricted propagator undergoes total reflection, so never crosses x = 0. These results are clearly unphysical and have no sensible classical limit.
The problem here is that the system is monitored too closely to allow the wave packet to actually cross x = 0. The resolution is therefore to relax the monitoring in such a way that something interesting can happen. There are two obvious ways in which this may be achieved.
The first option is to simply decline to take the limit ǫ → 0 in Eq.(1.21), so we define the class operator for not crossing to be, where again τ = Nǫ, but N is finite and ǫ > 0. Clearly if ǫ is large enough the system will be monitored sufficiently infrequently to let the wave packet cross x = 0 without too much reflection. Studies of the quantum Zeno effect suggest that the appropriate lower limit on ǫ where H 0 is the free Hamiltonian [12].
The second option is to retain the limit in Eq.(1.21), but "soften" the projections to POVMs, that is, to replace P = θ(x) with a function which is approximately 1 for large positive x, approximately 0 for large negative x, and with a smooth transition in between.
What we will do in this paper is adopt a definition of the class operator which involves elements of both of these options. In particular, we exploit the useful observation of Echanobe et al [11], which is that the string of projections interspersed with unitary evolution, Eq. (1.27) is approximately equivalent to evolution in the presence of the complex potential Eq.(1.3), This connection follows from first noting that as long as We now note that the approximate equivalence will hold as long as Echanobe et al. put an upper bound on the left-hand side using the Schrödinger-Robertson inequality and the two inequalities together read which can be satisfied as long as will hold. This equivalence and demonstration of it is somewhat heuristic, but appears to be backed up by numerical evidence [11,29]. We will accept it in what follows, but it would be of interest to obtain a more rigorous proof of this connection.
This is a very useful result, since the problem of evolution in the presence of the complex potential Eq.(1.3) is straightforward to solve, but the evolution described by Eq.(1.27) could be difficult to solve analytically. In our decoherent histories analysis of the system, we will therefore define the class operator for not crossing to be with the crossing class operator defined by Eq.(1.24). These definitions will be extended to class operators C k c for crossing during any one of a set of small intervals [t k , t k+1 ] of size ∆. These are the class operators we need to give a decoherent histories account of the origin of Π(τ ).
Our main result is that for intervals of size ∆ ≫ 1/V 0 , these class operators are given approximately by Significantly, the dependence on the complex potential has dropped out entirely. We will show that there is decoherence for an interesting class of states, and, for such states the probabilities are then given by Eq. (1.19), which has the form This, we will show, coincides with the expected result Eq.(1.7), when Π(t) is integrated over a range of time much greater than 1/V 0 . Hence there is complete agreement with standard results on the arrival time distribution at sufficiently coarse-grained timescales.
Furthermore, our results shed some light on the problem of backflow -we find that the situations when Eq.(1.38) is negative are those in which there is no decoherence in which case probabilities cannot be assigned.
In another paper, we compute the crossing class operators in a simpler but more heuristic way, by exploring a semiclassical approximation to Eq.(1.27) keeping ǫ finite. [30]. The results are essentially the same.

D. This Paper
The initial motive and overall aim of this paper are to give a decoherent histories account of the arrival time question which has a sensible classical limit. As stated, we seek a decoherent histories analysis account of standard results, such as Eq.(1.7). However, along the way we have found some new derivations of known results and will explore these areas too.
In Section 2, we consider the classical analogue of the arrival time defined by a complex potential. This sheds light on the form of the result Eq.(1.7) and in particular, the origin of the resolution function Eq.(1.8). In Section 3, we review the path decomposition expansion (PDX), which is a useful path integral technique for factoring the propagator across the surface x = 0, so is very useful for systems with a step function potential. In Section 4, we use the PDX to derive the scattering wave functions corresponding to evolution with a complex step potential. These are of course known, but the PDX gives a useful and concise derivation of them. In Section 5, we use the PDX to calculation the arrival time distribution function Eq.(1.6), and very readily obtain the expected result of the form Eq.(1.7). In Section 6 and 7, we use the results of the previous sections to carry out the decoherent histories analysis, as outlined in the previous subsection. We summarize and conclude in Section 8.

II. THE CLASSICAL CASE
Before looking at the quantum arrival time problem, it is enlightening to look at the corresponding classical arrival time problem defined using an absorbing potential. This gives some understanding of the expected form of the result in the quantum case, Eq.(1.7).
We consider a classical phase space distribution w t (p, q), with initial value w 0 (p, q) concentrated entirely in q > 0 with only negative momenta. The appropriate evolution equation corresponding to the quantum case Eq.(1.4) is Following the corresponding steps in the quantum case, the survival probability is and the arrival time distribution is where we have made use of Eq.(2.1). This expression is conveniently rewritten by noting that, again using Eq.(2.1), Π(τ ) obeys the equation This may be solved to yield From Eq.(2.2), we see that but since the momenta are all negative the exponential factor makes no contribution. We thus obtain The current J(t) is the usual classical arrival time distribution that would have been obtained in the absence of the absorbing potential. . This is given by Rearranging the order of integration and integrating over τ yields, We will see in what follows that 1/V 0 plays a role as a fundamental short timescale in the problem. So now suppose we assume that τ 1 , τ 2 and (τ 2 − τ 1 ) are all much greater than 1/V 0 . It follows that all the exponential terms may be dropped in Eq.(2.12) and we obtain the very simple result, That is, all dependence on the resolution function R(t) and the complex potential parameter V 0 completely drops out when we look at probabilities defined on timescales much greater than 1/V 0 . This result is very relevant to the decoherent histories analysis considered later where it is natural to look at the arrival time during a finite time interval.

III. THE PATH DECOMPOSITION EXPANSION
In this section we describe some useful path integral techniques. We wish to evaluate the for arbitrary x 1 and x 0 > 0. This may be calculated using a sum over paths, and the sum is over all paths To deal with the step function form of the potential we need to split off the sections of the paths lying entirely in x > 0 or x < 0. The way to do this is to use the path decomposition expansion or PDX [31,32,33]. Consider first paths from x 0 > 0 to x 1 < 0. Each path from x 0 > 0 to x 1 < 0 will typically cross x = 0 many times, but all paths have a first crossing, at time t 1 , say. As a consequence of this, it is possible to derive the formula, Here, g r (x, t|x 0 , 0) is the restricted propagator given by a sum over paths of the form (3.2) but with all paths restricted to x(t) > 0. It vanishes when either end point is the origin but its derivative at x = 0 is non-zero (and in fact the derivative of g r corresponds to a sum over all paths in x > 0 which end on x = 0 [32]).
It is also useful to record a PDX formula involving the last crossing time t 2 , for x 0 > 0 and x 1 < 0, These two formulae may be combined to give a first and last crossing version of the PDX, This is clearly very useful for a step potential since the propagator is decomposed in terms of propagation in x < 0 and in x > 0, essentially reducing the problem to that of computing the propagator along x = 0, g(0, t 2 |0, t 1 ). (See Figure 2.) For paths with x 0 > 0 and x 1 > 0, Eq.(3.4) is modified by the addition of a term g r (x 1 , t|x 0 , 0), corresponding to a sum over paths which never cross x = 0, so we have Again a further decomposition involving the last crossing, as in Eq.(3.6) can also be included.
The various elements of these expressions are easily calculated for a potential of simple . The restricted propagator in x > 0 is given by the method of images expression where g f denotes the free particle propagator The restricted propagator in x < 0 is given by Eq.(3.8), multiplied by exp(−V 0 τ ). The only complicated propagator to calculate is the propagation from the origin to itself along the edge of the potential, and in the case V (x) = V 0 θ(−x) this is given by [34], Using these results we may write down the full solution to the evolution with a complex potential, described by Eq.(3.1), for an initial state ψ(x) with support only in x > 0 and with negative momenta. It has the form, Here, ψ tr is the transmitted wave function and is given by the propagation of the initial state ψ(x) using the PDX formulae Eq. a certain heuristic path integral approximation to be tested. The results will also be useful for the decoherent histories analysis in Section 6.
The transmitted and reflected wave functions are defined above in Eq. (3.12). For large τ , a freely evolving packet moves entirely into x < 0 so that the free part, ψ f (x, τ ) is zero, leaving just the transmitted and reflected parts. Following the above definition, the transmitted wave function is given by where |0 denotes the position eigenstate |x at x = 0. Also, we have introduced s = τ − t 1 is the total Hamiltonian. The scattering wave functions concern the regime of large τ , so we let the upper limit of the integration ranges extend to ∞.
Writing the initial state as a sum of momentum states |p , and introducing E = p 2 /2m, To evaluate the s integral, we use the formula [35], from which it follows by differentiation with respect to x and setting The v integral may be evaluated using the explicit expression for the propagator along the edge of the potential, Eq.(3.11), together with the formula, m 2πi We thus obtain the result, where ψ tr (p) = 2 Note that in this final result, it is possible to identify the specific effects of the different sections of propagation: the propagation along the edge of the potential corresponds to the coefficient in the transmission amplitude Eq.(4.7) (which is equal to 1 when V 0 = 0), and the propagation from final crossing to the final point produces the V 0 dependence of the exponent. These observations will be useful below.
The reflected wave function ψ ref is defined above using the PDX Eq.(3.7) (rewritten using Eq.(3.6). The first term in Eq.(3.7), the crossing part, is the same as the transmitted case, Eq.(4.2), except that V 0 = 0 in the last segment of propagation, from x = 0 to the final point, and also the sign of x is reversed. We must also add the effects of the reflection part of the restricted propagator, and this simply subtracts the reflection of the incoming wave packet. The reflected wave function is therefore given by We thus see that the PDX very readily gives the standard stationary wave functions [8], without having to use the usual (somewhat cumbersome) technique of matching eigenfunctions at x = 0. In fact, this procedure is in some sense already encoded in the PDX.
We now use these exact results to check the validity of a useful approximation in the path integral representation of the propagator. In the PDX, Eq. It is not entirely clear that this is the case, however. On the one hand, the usual semiclassical approximation indicates that paths close to the straight line paths dominate, but on the other hand, paths in x < 0 are suppressed, so maybe the wiggly paths that spend less time in x < 0 make a significant contribution. Since this approximation is potentially a useful one, it is useful to compare with the exact result for the transmitted wave packet calculated above.
We therefore evaluate the following approximate expression for the transmitted wave function, This is the PDX, Eq.(3.4), in operator form with the semiclassical approximation described above and we have set s = τ − t 1 . We now take τ → ∞ in the integration and evaluate.
The key integral is, given by to denote the total non-Hermitian Hamiltonian). We are interested in calculating this expression for the case in which V 0 is much smaller than the energy scale of the initial state. (The very different limit, of V 0 → ∞, the Zeno limit, has been explored elsewhere [36]).
One way to evaluate Eq.(5.1) is to use the transmitted wave functions, Eq.(4.2). However, we give here instead a different and more direct derivation using the PDX. We use the first crossing PDX, Eq.(3.4), which is conveniently rewritten as the operator expression, We are aiming to show that this coincides with Eq. Consider first the x integral. Since we are assuming small V 0 , we may use the semiclassical approximation Eq.(4.10), which reads The x integral may now be carried out, with the result, where |ψ τ denotes the free evolution of the initial state.
We now carry out one of the time integrals. Note that, In the first integral, we set u = s ′ , v = s − s ′ , and in the second integral we set u = s, v = s ′ − s. We thus obtain The factors of 1/(±i) 1 2 in front of each term come from a careful consideration of the square root in the free propagator prefactor (and must have this form because Π(τ ) is real).
We will assume that V 0 τ ≫ 1, which means that the integrals are concentrated around u = v = 0. This means that we may take the upper limit of the v integral to be ∞, and it may be carried out, to yield, where the operator Σ(p) is given by For V 0 much less than the energy scale, so Σ(p) is simply the sign function of the momentum, which is −1 in this case, since the initial state consists entirely of negative momenta. Finally, writing u = τ − t, we obtain We therefore have precise confirmation of the classical result Eq. It should also be added that it would be misleading to explore the first order corrections in V 0 in going from Eq.(5.10) to Eq.(5.13), since comparable correction terms have already been dropped in using the semiclassical approximation Eq.(4.10).

VI. DECOHERENT HISTORIES ANALYSIS FOR A SINGLE LARGE TIME IN-TERVAL
We now consider the decoherent histories analysis of this system. We consider an incoming wave packet approaching the origin from x > 0 and ask for the probability of crossing during a given time interval. We do this in two parts: first in this section, using a large time interval [0, τ ] and second in the next section, using a set of intervals of arbitrary size interval.

A. Class Operators and Probabilities
We consider first the following simple question. What is the probability of crossing or not crossing during the time interval [0, τ ]? The class operators for not crossing and crossing are C nc = exp (−iH 0 τ − V (x)τ ) (6.1) and they satisfy We are interested in the probabilities for not crossing and crossing,  When there is decoherence, Eq.(6.9) holds and we may deduce that Hence the decoherent histories analysis is compatible with the standard result, but only when there is decoherence.

B. Calculation of the Decoherence Functional
We now give two methods for checking for the decoherence of histories. The first involves expressing the probabilities and decoherence functional in terms of the transmitted and reflected waves defined in Eq.(3.12), which implies that 14) The probabilities and decoherence functional are therefore given by There is a second method of demonstrating decoherence which gives a different picture and will be useful later. Following the general pattern described in Eqs. (1.19), (1.20), consider the quantity, q nc (τ ) = Tr C nc ρe iH 0 τ (6.18) From Eq.(6.6), we see that the decoherence functional may be written, D c,nc = q nc (τ ) − p nc (τ ) (6.19) This means that q nc = p nc when there is decoherence. Or to put it the other way round, decoherence of histories may be checked by comparing q nc with p nc and this is what we now do. Recall that p nc is given by Eqs.(6.10), (6.11) (which hold in the absence of decoherence).
We may write q nc in a similar form: The integrand is similar to Π(t) defined in Eq.(5.1), so we definẽ We now have that It then follows that the decoherence functional is To compute the decoherence functional we need to calculateΠ(t), which is given bỹ This is almost the same as Π(t) except that the exponential on the left involves only H 0 and not the complex potential (and also an overall factor of 2). We therefore follow the calculation of Π(t) in Section 5 with small modifications. With a little care, one may see that the final result is the same as that for Π(t), Eq.(5.13), except that 2V 0 is replaced with as long as V 0 τ ≫ 1. Since this result is independent of V 0 ,Π(t) will satisfy the same relation.
We thus deduce that D c,nc ≈ 0 (6.27) hence there is decoherence, under the above conditions.

VII. DECOHERENT HISTORIES ANALYSIS FOR AN ARBITRARY SET OF TIME INTERVALS
We now turn to the more complicated question of much more refined histories, that may cross the origin at any one of a large number of times, during the time interval [0, τ ]. This corresponds more directly to the standard crossing probability, Π(t)dt, the probability that the particle crosses during an infinitesimal time interval [t, t + dt].

A. Class Operators
We have defined class operators Eqs.(6.1), (6.2) describing crossing or not crossing during a time interval [0, τ ]. We now split this time interval into n equal parts of size ǫ, so τ = nǫ and we seek class operators describing crossing or not crossing during any one of the n intervals. We first note that where C nc (ǫ) and C c (ǫ) are defined as in Eqs.(6.1), (6.2) except that here they are for a time interval [0, ǫ]. We now use this to decompose e −iH 0 τ into the desired class operators. We Repeating the same steps on the first term, this yields, Repeating more times eventually yields, From this expression, we see that the class operator for crossing x = 0 for the first time during the time interval [kǫ, (k + 1)ǫ] is given by the summand of the second term, We will not in fact work with the class operator Eq.(7.5), since a more useful similar but alternative expression can also be found. Taking the continuum limit of Eq.(7.4) (and inserting the explicit expression for C nc ), we obtain, ). This indicates that the class operator for first crossing during We do not, however, expect histories characterized by such precise crossing time to be decoherent, so it is natural to consider coarser-grained class operators, which represents crossing during one of the N time intervals [t k , t k+1 ] of size ∆, where t k = k∆ with k = 0, 1 · · · N − 1 and τ = N∆. The complete set of class operators C α for crossing and not crossing is the set of N + 1 operators and Eq.(7.6) implies that they satisfy To check for decoherence of histories we need to calculate two types of decoherence functional and this will be carried out below.

B. An Important Simplification of the Class Operator
There turns out to be a very useful simplification in the class operator Eq.(7.7). Consider the amplitude for any x. The right-hand side is very similar to Eq.(5.1), except that there is no complex potential in one of the exponential terms and also the "final" state is |x not |ψ . (And there is also an overall factor of 2 different). Despite these differences, we may once again make use of the details of the calculation of Section 5, and we deduce from the analogous result Eq.(5.13), that Like the derivation of Eq.(5.13), this is valid under the conditions that all energy scales are much greater than V 0 and all time scales much greater than 1/V 0 . Now we integrate this over time to obtain the coarse-grained crossing time operator, Eq.(7.8), and again use approximations of the form Eqs.(2.10)-(2.13) (again using the assumption of timescales much greater than 1/V 0 ), to yield, the remarkably simple and appealing form, This may also be written even more simply e iH 0 τ C k c = θ(x(t k )) − θ(x(t k+1 )) (7.16)

C. Probabilities for Crossing
The above expressions for the crossing time class operator are the most important results of the paper and provide an immediate connection to the standard expression for the arrival time distribution. Supposing for the moment that there is decoherence of histories, we may assign probabilities to the histories. The probability for crossing during the time interval However, as noted in Eqs. (1.19), (1.20) when there is decoherence of histories, this expression for the probabilities for histories is equal to the simpler expression which is precisely the standard result! The expression for the probability q(t k , t k+1 ) is not positive in general (although is real in this case, as it happens), but when there is decoherence, it is equal to p(t k , t k+1 ), which is positive. Hence the decoherent histories result coincides with the standard result under the somewhat special conditions of decoherence of histories.

D. Decoherence of Histories and the Backflow Problem
There is an interesting connection between decoherence of histories and backflow. To see this, consider the following simple case. We consider histories which either cross or do not cross the origin during the time interval [t 1 , t 2 ]. So the crossing and not crossing class operators are C and 1 − C, where where we have adopted the notationx k =x(t k ) (and for convenience we have dropped the exponential factor which is just a matter of definition and drops out of all expression of interest). The decoherence functional is This may also be written a form we will use below to check decoherence. When there is decoherence, D = 0 and the probability for crossing is As noted above, C is the standard result, Eq.(7.18), for the probability of crossing.
There is an interesting connection here between backflow and decoherence. If there is decoherence, D is zero so C must cancel C 2 in Eq. (7.20), which means that C ≥ 0, so there is no backflow. Or we may make a logically equivalent statement: if there is backflow, C < 0, then there cannot be decoherence, since |D| is then greater than the probability C 2 . Hence, states with backflow do not permit decoherence of histories. (Note that absence of backflow, C ≥ 0, is not itself enough to guarantee decoherence -the stronger condition D = 0 must be satisfied).
This is an important result. The quantity C is regarded as the "standard" result for crossing time probability and its possible negativity is disturbing. Here, the decoherent histories approach sheds new light on this issue. In the decoherence histories apporach, the true probability for crossing is the manifestly positive quantity C 2 and this is equal to C only when there is decoherence. In particular, when there is significant backflow, there cannot be decoherence, so probabilities cannot be assigned and C 2 is not equal to C .

E. The Decoherence Conditions
The crossing probabilities described above are only valid when all components of the decoherence functional, Eqs.(7.11), (7.12), are zero. We therefore address the issue of finding those states for which there is negligible decoherence.
We consider first the simpler case, of the decoherence functional Eq.(7.12). The noncrossing class operator C nc is given in general by Eq.(6.1). However, it simplifies considerably in the approximations used to derive Eq.(7.16), which we adopt here. In particular, Eq.(7.10) with Eq.(7.16) imply that Since we are interested only in initial states with support entirely in x > 0, we have θ(x)|ψ = |ψ , which means that effectively, The decoherence functional of interest is then This is conveniently rewritten, We will take τ to be very large and it is pretty clear that this object will be approximately zero, since we expect all the initial state to end up in x < 0 at large times. However, we will see this below in more detail.
The more important decoherence condition is that D kk ′ defined in Eq.(7.11) vanishes, so we now focus on that. We write the class operator Eq.(7.16) for crossing during the time For an initial state |ψ , the quantity C k c |ψ is a quantum state representing the property of crossing of the origin in the time interval [t k , t k+1 ]. The decoherence condition D kk ′ = 0 is simply the condition that the "crossing states" C k c |ψ for different time intervals have negligible interference. The states C k c |ψ consist of an initial state which has been localized to a range of time at x = 0. This is closely related to the interesting question of diffraction in time [37] and this connection will be explored in more detail elsewhere [38].
The decoherence functional is given by where without loss of generality we take t j+1 < t k . It is a sum of terms each of the form, where t k < t j . Note that The key thing is that d 2 m has the form of a probability -it is the probability to find the particle in x < 0 at t k and then in x > 0 at t j . Semiclassical expectations suggest that this is small in general for the states considered here, which are left-moving wave packets, and indeed the classical limit of this probability is zero. So this is a useful object to calculate in terms of checking decoherence. (Although it may not be small for states with backflow).
Note that it also implies that the other parts of the decoherence functional, Eq.(7.26) will also be small. In detailed calculations, the upper bound Eq.(7.30) must be compared with the probabilities, as in Eq.(1.18).

F. Checking the Decoherence Condition for Wave Packets
We now consider the particular case of an initial state consisting of a wave packet where q 0 > 0 and p 0 < 0. We first consider a heuristic analysis of decoherence. In the simplest case, the wave packet crosses the origin almost entirely during the time interval [t k , t k+1 ] (of size ∆) for some k, without any substantial overlap with any other time intervals.
(See Figure 1.) This means that and it follows that D kk ′ ≈ 0. The key time scales here are the classical arrival time for the centre of the packet, t a = mq 0 |p 0 | (7.33) and the Zeno time (which is also approximately equal to 1/(∆H 0 )). The Zeno time is the time taken for the wave packet to move a distance equal to it spatial width σ, or equivalently, it is the size of the packet's "temporal imprint" at the origin. Therefore, the above approximations work if, firstly, t z ≪ ∆ (7.35) and secondly, if the classical arrival time t a lies inside the interval [t k , t k+1 ] and is at least one or two Zeno times away from the boundaries.
It is easy to see that similar conclusions hold for superpositions of initial states of the form Eq.(7.31) as long as they are approximately orthogonal. Loosely, this is because under the above conditions, the class operators do not disturb the states and the only non-zero components of the off-diagonal terms of the decoherence functional will be proportional to the overlap of pairs of initial wave packets, so will be approximately zero. (See Figure 3.) More general, non-orthogonal superpositions may, however, produce backflow, so there may be no decoherence.

G. Checking Decoherence for a Detailed Model
Decoherence starts to become lost as the the size ∆ of the time intervals [t k , t k+1 ] is reduced to close to the Zeno time. This is because the wave packet will split into parts that cross during different time intervals and the effect of diffraction in time mentioned above [37] will cause these different parts to be non-orthogonal. These effects will be explored in more detail elsewhere [38]. Here, we give a more detailed calculation to check for decoherence.
For simplicity we work with the simple case considered in Subsection 7D above. We take the initial state to be the wave packet Eq.(7.31) and we note that the decoherence functional Eq.(7.21) satisfies with d 2 m given by Eq.(7.30) with k = 1, j = 2. We need some probabilities to compare this with. We have that The interesting case is that in which the crossing probability C 2 is somewhat less than 1, less than about 1/2, say, in which case the non-crossing probability (1 − C) 2 will be of order 1. It is therefore sufficient to compare d 2 m with C 2 . Now note that This means that C 2 is in fact equal to the probability up to terms which vanish when D = 0. This is useful since it is now identical in form to the expression for d 2 m and our goal is to show that It is useful to work in the Wigner representation [39], defined, for a state ρ(x, y) by The probabilities p 12 and d 2 m are then given by p 12 = 2π dpdq W 12 (p, q) W 0 (p, q, t 1 ) (7.42) d 2 m = 2π dpdq W D (p, q) W 0 (p, q, t 1 ) (7.43) Here, W 0 (p, q, t 1 ) is the Wigner function of the initial state, evolved in time to t 1 , The objects W P and W D are the Wigner transforms of the θ-function combinations appearing in Eqs.(7.30), (7.39) and are given by dy sin y y (7.45) where u(p, q) = 2q p + mq (t 2 − t 1 ) (7.47) We see that the only difference between the expressions for p 12 and d 2 m is in the sign in the θ-functions. We now compare the size of p 12 and d 2 m .We assume that the wave packet is spatially broad, so σ is large and the Wigner function Eq.(7.44) is therefore concentrated strongly about p = p 0 < 0. We therefore integrate out p and set p = p 0 throughout. The most important case to check is that in which the wave packet is reasonably evenly divided between x > 0 and x < 0 at time t 1 so that both p 12 and d 2 m have a chance of being reasonably large. This means that q 0 + p 0 t 1 /m should be close to zero (to within a few widths σ), so for simplicity we take it to be exactly zero.
With these simplifications, we have Here, since q ≤ 0 and p 0 < 0, we have u(p 0 , q) ≥ 0. We can evaluate this expression by examining the comparative effects of f (u) and the exponential term. From the plot of f (u) (see Figure 4), we see that it drops to zero at u = u 0 (which is of order 1) and oscillates rapidly around zero for u > u 0 , so we expect the integral to be dominated by values of q for which 0 ≤ u ≤ u 0 . The value u = u 0 corresponds to q = q 0 where where E 0 = p 2 0 /2m. In the complex potential calculations, we have assumed that E 0 ≫ V 0 and we also assumed that all timescales are much greater than 1/V 0 , and these together imply that E 0 ∆ ≫ 1. We may therefore expand the square root to leading order and obtain We have assumed that the wave packet is sufficiently broad that σp 0 ≫ 1, and this means that |q 0 | ≪ σ (7.53) Since the right-hand side is already ≪ 1, this is easily satisfied even for time intervals whose size is ∆ is smaller than the Zeno time. In fact this condition is equivalent to the condition E 0 ∆ ≫ 1, which is satisfied by the assumptions of the complex potential model, as stated above.
In brief, we therefore get decoherence of histories for a single wave packet under a wide variety of circumstances.

VIII. SUMMARY AND CONCLUSIONS
This paper was initially motivated by a desire to analyse the arrival time problem using the decoherent histories approach to quantum theory. But along the way we have reconsidered and derived a number of other useful related results.
We considered the arrival time problem using a complex potential to kill paths entering x < 0. In Section 2 we gave a classical analysis of the problem. We derived a result of the expected form exposing the resolution function as an essentially classical effect summarizing the role of the complex potential. We also showed that coarse graining over time scales much greater than 1/V 0 produces a formula for the arrival time of expect form and which is independent of the complex potential. This is an important result for the rest of the paper.
After reviewing the path decomposition expansion in Section 3, we used it in Section 4 to derive the standard results for stationary scattering states. This is simpler and more direct than the usual method, involving matching eigenfunctions. (In a sense this cumbersome machinery is already encoded in the PDX). In Section 5, we used the PDX to rederive the standard form of the arrival time distribution with a complex potential, in the limit of weak potential. The form of this calculation turned out to be useful for the subsequent work on the decoherent histories approach.
In Section 6, we considered the decoherent histories analysis for the simple case of a particle crossing or not crossing x = 0 during a large time interval [0, τ ]. We found the simple and expected result that the histories are decoherent as long as reflection by the complex potential is negligible. The resultant probabilities are consistent with the standard result for the arrival time.
The main part of the decoherent histories analysis was given in Section 7, where we first derived the class operators described crossing x = 0 for an arbitrary set of small time intervals. Here we obtained our most important result: the crossing class operator Eq. (7.15) for timescales much greater than 1/V 0 . This form of the class operator gives an immediate connection with the standard result for probabilities when there is decoherence. Indeed, one may have guessed the form of the class operator from the standard form of the probabilities, and this is pursued in another paper [30]. However, it is also gratifying that it can be derived in some detail using the complex potential approach used here.
To assign probabilities, the decoherence functional must be diagonal and we considered this condition. We found a variety of states for which there is decoherence, under certain more detailed conditions, which we discussed.
We also noted an interesting and important relationship between decoherence and backflow: If there is decoherence, the probabilities for crossing must be positive so there cannot be any backflow. If there is no decoherence, the integrated current may still be positive, but one can say that if there is backflow there will definitely be no decoherence. This means that the decoherent histories approach brings something genuinely new to the arrival time problem: it establishes the conditions under which probabilities can be assigned and in particular forbids the assignment of probabilities in cases where there is backflow.

IX. ACKNOWLEDGEMENTS
We are grateful to Gonzalo Muga and Larry Schulman for useful discussions. We are also very grateful to Chris Dewdney for supplying a computer programme representing the evolution of wave packets, which was very useful at an early stage in this work. JJH Here we explore the possibility that averaging it over time might improve the situation. On the one hand, the results of Bracken and Melloy [3] show that there is always some state for which p(0, T ) defined above is negative, for any T . On the other hand, for a given state, one might hope that p(0, T ) will be positive for sufficiently large T .
Here we give a brief argument for this, which also makes contact with the negativity of the Wigner function.
The current can be written in terms of the Wigner function W (p, x) as J(t) = − dp p m W (p, 0, t).
The Wigner function evolves freely according to W (p, x, t) = W (p, x − pt/m, 0). We assume it has support only on negative momentum states, with average momentum p 0 < 0 and momentum width σ p .
Consider a time interval 0 < t < T over which backflow occurs. It is clear that in order for this to occur the Wigner function must be negative for at least some of this interval. We So p(0, T ) is given by the average of the Wigner function over a region of phase space. We now recall a standard property of the Wigner function which is that, broadly speaking, it will tend to be positive when averaged over a region of phase space of size greater than order 1 (in the units used here where = 1). This region is of size of order |p 0 |T /m in the x-direction but infinite in the p-direction. However, the Wigner function has momentum spread σ p , so the effective size averaged over is σ p p 0 T /m which is approximately the same as ∆H T . This means that we expect that p(0, T ) will be positive as long as Hence, as expected, the integrated current will be positive for T sufficiently large and the key timescale is the Zeno time. This heuristic argument will be revisited in more detail    Figure 4. The function f (u). It oscillates around zero for u ≫ 0 and oscillates around π for u ≪ 0. As a function of q, f (u(p 0 , q)) differs from 0 or π only in a region of size 1/|p 0 | around q = 0.