Bohmian Trajectories for Hamiltonians with Interior–Boundary Conditions
Recently, there has been progress in developing interior-boundary conditions (IBCs) as a technique of avoiding the problem of ultraviolet divergence in non-relativistic quantum field theories while treating space as a continuum and electrons as point particles. An IBC can be expressed in the particle-position representation of a Fock vector as a condition on the values of on the set of collision configurations, and the corresponding Hamiltonian is defined on a domain of vectors satisfying this condition. We describe here how Bohmian mechanics can be extended to this type of Hamiltonian. In fact, part of the development of IBCs was inspired by the Bohmian picture. Particle creation and annihilation correspond to jumps in configuration space; the annihilation is deterministic and occurs when two particles (of the appropriate species) meet, whereas the creation is stochastic and occurs at a rate dictated by the demand for the equivariance of the distribution, time reversal symmetry, and the Markov property. The process is closely related to processes known as Bell-type quantum field theories.
Key words: regularization of quantum field theory; particle creation and annihilation; Bohmian mechanics; Bell-type quantum field theory; Schrödinger operator with boundary condition; Galilean transformation.
- 1 Introduction
- 2 Simple Example
- 3 Particle Creation in 3 Dimensions
- 4 Symmetries
- 5 Comparison with Known Bell-Type QFTs
- 6 Process on a Lattice
- 7 General Setup for Codimension-1 Boundaries
A new type of Hamiltonian has recently been proposed  for quantum field theories (QFTs), defined using interior–boundary conditions (IBCs). See [39, 43, 44, 45, 46, 61, 69, 28] for earlier work involving IBCs, but with rather different purposes than in . These Hamiltonians do not suffer from an ultraviolet (UV) divergence problem although they do not involve a UV cut-off such as would be provided by discretizing space or smearing out particles over a positive radius. On the contrary, in this new type of Hamiltonian, all particles are taken to have radius zero. The Hamiltonians involve particle creation and annihilation and have been shown [38, 37, 35], for various examples of non-relativistic QFTs, to be free of a UV (or any other) divergence problem; viz., they have been shown to be rigorously defined and self-adjoint.
An IBC is a type of boundary condition on the wave function that relates the value or derivative of the wave function on the boundary of configuration space to its value at a certain interior point. For particle creation, the relevant boundary consists of those configurations in which two particles meet at the same location.
We define here the Bohmian trajectories naturally associated with such Hamiltonians and describe what they look like; that is, we develop an extension of Bohmian mechanics [7, 29, 25, 63] to Hamiltonians with IBCs. This extension is no longer deterministic but has the form of a stochastic Markov process in the appropriate configuration space, a process that is -distributed at every time .
The theories that we develop here can be regarded as instances of Bell-type quantum field theories [4, 19, 20, 22], versions of QFTs that, like Bohmian mechanics, provide particle trajectories; the possible paths in configuration space are piecewise solutions of Bohm’s equation of motion, interrupted by jumps in configuration space, with the jump usually connected to the creation or annihilation of a particle. The QFTs considered in the references [4, 19, 20, 22] just cited involved a UV cut-off, usually implemented by smearing out the particles over a positive radius . For example, in a theory in which -particles can emit and absorb -particles, the emission of a -particle corresponds to the occurence of a further “Bohmian” particle at any point within the -particle, i.e., at any point within the 3-ball of radius around the center of the -particle. In contrast, for Hamiltonians defined by means of an IBC, the -particle has radius 0, and the -particle gets created at the location of the -particle, see Figure 1. Such a picture seems physically reasonable, in support of the IBC approach.
To study the combination of Bohm’s trajectory picture with IBC Hamiltonians is useful from both perspectives: For the Bohmian picture, it provides an extension of the Bell-type QFTs known so far to a further class of Hamiltonians, indeed perhaps more reasonable and plausible Hamiltonians than the ones based on UV cut-offs. For the IBC approach, it provides a welcome visualization and clarification of the physical meaning of the Hamiltonians based on IBCs.
Preliminary considerations in the direction of this paper were reported in [28, 60]. For an introduction to IBCs, see ; for a study of IBCs in 1 dimension, see ; a brief overview of the results of this paper is given in Section 2 of ; other recent and upcoming works on IBCs include [26, 42, 50, 49, 65]. While the ideas of the IBC approach can also be applied to Dirac operators , we focus here on the non-relativistic case and give only a brief discussion of the analogous construction for the Dirac equation in Section 7.2. It is a future goal to develop a model analogous to the ones described here for full quantum electrodynamics, with particle trajectories for electrons, positrons, and photons. Other notable approaches to a version of QFTs with local beables (“hidden variables”) are based on either using configurations of infinitely many particles while avoiding the actual creation and annihilation of particles [9, 10, 11, 13, 14] or using, instead of an actual particle configuration, an actual field configuration, see  and references therein, or assuming that fermions have beables but bosons do not  (or vice versa ); we will not consider these approaches here.
Like Bohmian mechanics and Bell-type QFTs, the models we describe here entail, as we will show on a non-rigorous level, that the actual configuration at time is always distributed (and we then say that the process is equivariant). Like Bell-type QFTs and unlike Bohmian mechanics (for a conserved number of particles), these models involve a stochastic motion of . We regard it as a serious possibility that the fundamental dynamical laws of physics may be stochastic in nature (i.e., that the time evolution may be inherently random). The main advantage of Bohmian mechanics is not so much its determinism as the clear picture of reality, independent of observation, that it provides.
So, in the models developed in this paper, the path is random, and thus a stochastic process (in fact, a Markov process), which we call the Bell-type process with IBC because it naturally fits among the processes of Bell-type QFTs, or shorter the IBC process. The stochastic element in the process is connected to the jumps; between jumps, the trajectory follows Bohm’s deterministic equations of motion. For example, the emission of a -particle by an -particle (in the model of Section 3.4 below) occurs at a random time and in a random direction in space, with a probability distribution governed by the wave function according to one of the laws of the theory that we propose.
This situation is similar to the one in Bell-type QFTs with a UV cut-off, where a particle is created at a random time and a random location within radius of the center of the -particle. A difference is that the absorption event, which in Bell-type QFTs with UV cut-off is also stochastic (as it occurs at a random time), is deterministic in Bell-type processes with IBC: a -particle gets annihilated when it hits an -particle. While it may seem to break time reversal invariance that emission is stochastic and absorption deterministic, this is not so, as we will elucidate in Section 2.5. On the contrary, time reversal symmetry fixes uniquely the stochastic law governing the rate of particle creation. While Bell-type QFTs with UV cut-off can also obey time reversal symmetry, this symmetry does not dictate the rate for them, as several laws for the jump rate are compatible with it and with the quantum mechanical formula for the probability current, although one possibility for this law, the one chosen in Bell-type QFTs, is naturally selected by a minimality property. By the way, this choice of law now receives further support because it corresponds to the only possible law in IBC models. We also discuss how, in the limit of removing the UV cut-off (if the limit exists), the stochastic process of a Bell-type QFT approaches the process introduced here for IBC models.
We will use four models for our discussion:
Model 1. This is a non-relativistic QFT involving two species of particles, -particles and -particles, both spinless and moving in , such that the -particles can emit and absorb -particles. The Hilbert space is the tensor product of the fermionic Fock space of for the -particles and the bosonic Fock space of for the -particles.
Model 2. This is a simplified version of Model 1 in which the -particles are fixed at certain locations in space, as would arise in the limit in which the mass of the -particles tends to . We consider here only the case of a single -particle, and choose its location as the coordinate origin . So -particles can be created and annihilated at the origin, and move around in between.
Model 3. This model is a version of Model 2 that is further simplified by cutting off the sectors of the bosonic Fock space with 2 or more particles.
Model 4. This model is even simpler and does not have much to do with particle creation any more. Its configuration space is the disjoint union of and = the upper half-plane in (see Figure 2); the boundary of , to which the IBC refers, is the horizontal axis in . Away from the boundary, the Hamiltonian is just the free Schrödinger operator.
We give a full definition of these models below, including the IBC approach to them. Model 1 and Model 2 were discussed, under these names, in , and Models 3 and 4 were described in . Model 1 is adapated from [40, 53, 47] (where similar models were considered without IBCs and without Bohmian trajectories), and also models like Model 2 have long been considered [66, 16] (without IBCs and without Bohmian trajectories), sometimes under the name “Lee model.” We will consider Models 1–4 in reverse order, the order of increasing complexity. For some of these models, the Hamiltonians of their IBC versions are known [38, 35] to be bounded from below (as would be physically reasonable), although the IBC approach in general neither requires nor guarantees that Hamiltonians are bounded from below.
The remainder of this paper is organized as follows. In Section 2, we introduce and discuss the Bell-type process with IBC for Model 4. In Section 3, we apply IBCs to particle creation and annihilation for Models 3, 2, and 1; we define the appropriate processes, show (non-rigorously) that they are equivariant, and compare the IBC approach to renormalization on the level of the Hamiltonians, the wave functions, and the Bell-type process. After Section 3, we turn to more technical aspects of IBC processes. In Section 4, we discuss the symmetries of the processes (particularly for Model 1), with particular attention to Galilean boosts. In Section 5, we compare the IBC process to the known processes (“Bell-type QFTs” [4, 57, 67, 19, 20, 22, 68]) for QFTs with UV cut-off. In Section 6, we formulate a lattice version of the IBC process and argue that in the continuum limit, the continuum version of the IBC process described in Section 2 is recovered. In Section 7, we consider general IBC processes for codimension-1 boundaries (which are simpler than the physically realistic codimension-3 boundaries) and characterize such processes in general and abstract terms for Schrödinger and Dirac operators.
2 Simple Example
We begin with the simplest of our four models, Model 4.
2.1 Model 4 Comes First
As mentioned, , , , wave functions are complex-valued functions on , and volume in is understood as the measure defined by
for , where means the -dimensional volume (Lebesgue measure). For the restriction of a wave function to we write ; so, for a point , we can either write or (depending on whether we want to emphasize the number of the sector). The Hilbert space of the model is , whose inner product is
The IBC reads :
for every . Here, is a mass parameter and a coupling constant.666The dimension of is (energy)(length) if we take to have the dimension of the square root of a probability density, i.e., (length) for and (length) for . The corresponding Hamiltonian is:
(The reasons for setting up the equations this way will become clearer once we have described the Bell-type process and its probability current.) One can show that is self-adjoint on a suitable dense domain in consisting of functions satisfying the IBC (3), so that is a unitary operator on , and is the solution of the Schrödinger equation
We will assume in the following that (and thus also ) lies in the domain of and in particular satisfies the IBC.
2.2 Process for Model 4
The Bell-type process for this model, for any solution to (5), is defined as follows. The initial configuration is chosen with the distribution. At any time , if lies in the interior of , then it moves according to Bohm’s equation of motion [7, 29, 25],
in terms of the quantities
that are usually called the probability current and probability density in quantum mechanics (and that will turn out to indeed be the probability current and density for our process).
As soon as the configuration hits the boundary , say at , it jumps to , and continues moving there according to Bohm’s equation of motion (6), now understood on . The motion in will be interrupted at a random time whose distribution is specified below. At time the configuration jumps from to the boundary of ; if is the position immediately before the jump,777The notation means the limit with ; means with . , then the point it jumps to is . After the jump, the configuration moves again according to Bohm’s equation of motion (6), etc. The distribution of can be expressed by specifying the jump rate , i.e., the probability per time of a jump to occur: the probability of a jump between and , given that , is
It is one of the laws of the theory that the jump rate is
with the notation
for the positive part of . (We write for to emphasize that we are evaluating in sector .)
For the law (10) to be meaningful and sufficient, we need that whenever the process jumps to , a unique trajectory begins there and leads away from the boundary. This is in fact the case: Since the jump rate (10) can be written as
where means the -component of the current in , the process can only jump to if the current has positive -component, and thus points away from the boundary. But then also the Bohmian velocity (7) points away from the boundary (i.e., has positive -component), and there is a unique solution of the ODE (7) beginning at ; see Figure 3. In contrast, if , then a solution moving towards the boundary ends there, so it would not be possible to jump to and then move along the trajectory passing through at that time. The case is more complicated but irrelevant here because the jump rate (12) vanishes in this case. What if ? Then the jump rate (10) is ill defined, but this is not a problem because the process should be expected to have probability zero to ever reach such an [5, 27, 28, 58].
This completes the definition of the process (or just for short). It is clear that is a Markov process, i.e., a stochastic process for which the probability distribution of the future path depends on the past only through the present configuration. (Bohmian mechanics, the known Bell-type QFTs , and the analogous process on a graph  are also Markov processes.)
We now turn to the derivation of the equivariance of the distribution (i.e., of the process defined in Section 2.2).
Consider an arbitrary probability density (instead of ) for the initial configuration , and let this density be denoted by . Then we can formulate the transport equations for , or probability balance, as follows. In the interior of , gets transported according to the continuity equation
where is the Bohmian velocity vector field,
Some amount of gets lost due to trajectories that hit the boundary and jump to . In , evolves according to
The first term on the right-hand side represents the change in due to transport of along , the second term represents the gain due to jumps coming from , and the third the loss due to jumps from to . Note that is the amount of probability arriving at due to motion in between and .
Now the following equations for the time evolution of follow from the Schrödinger equation with the Hamiltonian (4). In the interior of ,
with as in (8a), and in ,
The IBC (3) allows us to replace on the right-hand side by , so we obtain that
Thus, it follows from (10) and (14) that whenever , the right-hand side of (13) agrees with that of (16), and the right-hand side of (15) agrees with that of (18). Thus, is a solution to (13) and (15), establishing the equivariance of the distribution.
This calculation also conveys how the conservation of works for this Hamiltonian: The second term on the right-hand side of (4a) ensures, together with the IBC (3), that the continuity equation (18) for contains an additional term (the second term on the right-hand side) that compensates exactly the loss of due to flux into the boundary while yielding the gain of due to the jumps described by (10).
At the time of jump. Two types of jumps occur: (deterministic) or (stochastic). Let denote the time of either jump. We have not specified whether or . For the sake of a complete mathematical definition of the process, various choices for can be adopted. For example, we could define that always (in the lower sector), or that always (in the upper sector). Both choices define Markov processes, and the differences between them seem physically irrelevant.
Choice concerning the boundary. Another mathematical fine point that is physically irrelevant is whether boundary points should be regarded as elements of or not. For clarity, let us write for the interior of (the set of non-boundary points) and for the completion of (the set of boundary and non-boundary points).888The definition of should not be conflated with the definition of the interior of a subset in a topological space, as the interior of the whole space is always the whole space. Above, we took , but we could equally well have defined (with a sign instead of ), which would have led to . Since is a -null set, we have that . Since is then not defined on , we would have to write instead of in (3) and (10) and instead of in (4a) and (10). Moreover, since we want that for all , we would need to demand that lies in the lower sector (see Remark 1). Except possibly for the choice of , the IBC process is the same as for the previous choice , as every trajectory in that hits the boundary has a unique limiting arrival point , and conversely, there is no more than one trajectory whose limit backwards in time at a given time , , is a given boundary point. Thus, the choice vs. actually does not matter.
Moreover, if we wish, we can even take the wave function to be defined on and the process to move in . This choice will be convenient in Section 3.
Another topology on configuration space. One can take a somewhat different view of the same process by introducing a different topology on , which we call the radical topology. It is obtained by identifying with , viz., with . This means, for example, that an open neighborhood of contains not only nearby points in but also points near in .999So in a sense, the topology is not very radical at all: In the present example (Model 4), it is the standard topology of a closed half plane. The name “radical topology” should not be over-interpreted. In this topology, is a connected space, and the process has continuous paths. Since can now be pictured as the line in , a typical path starting with may reach sooner or later, stay on the line for a random duration, then leave that line into the upper half plane, etc.; see Figure 4.
The radical topology may seem natural in view of (3), particularly in units in which , as then the IBC is satisfied as soon as the wave function is continuous. Note, however, that the appropriate measure of volume in is still given by (1), so the line (identified with ) is not a null set, and on is given by (4a), not (4b). It may also seem confusing that, for on the boundary, there are two conflicting Bohmian equations of motion, one using and the other ; the way the process uses them is that for a random duration, the first equation governs the motion, and then, spontaneously at a random time , the second equation takes over.
2.5 Time Reversal Symmetry
Time reversal symmetry plays a bigger role for the IBC process than for ordinary Bohmian mechanics for two reasons: first, it may be counter-intuitive that the IBC process is time reversal symmetric at all, and second, the IBC process can be characterized as the unique time reversal symmetric process in a suitable class of processes, as elucidated below.
We first describe the extension of the IBC process to negative times: The laws governing are such that they define a unique process not only for all positive , but also for all negative ; put differently, they define, for any solution (with ) of (5), a probability distribution over paths . To see this, choose a random -distributed configuration at an “initial time” and let the process evolve for all . Since for any , will be -distributed, and since it is a Markov process, the restriction of the process to a time interval with has the same distribution as the one obtained by starting at time . Thus, the family with parameter of processes is consistent, and by the Kolmogorov extension theorem, each such process is the restriction to the time interval of some process .
Let us now turn to time reversal. Notwithstanding the fact that “downward” jumps (from to ) are deterministic (they occur when hits the boundary) while “upward” jumps are stochastic, the process is invariant under time reversal.101010Strictly speaking, to ensure reversibility, we need a reversible rule for the choice of (see Remark 1). Two such rules would be: (i) always lies in the lower sector (i.e., ); or (ii) always lies in the higher sector (i.e., ). This means the following: if evolves according to the Hamiltonian , i.e., , and if is the associated process, then defined by also evolves according to , and is the process associated with . To see this, note first that (i) if satisfies the IBC then so does ; and (ii) ; (i) and (ii) together imply that evolves according to . (iii) As is well known and obvious from (6), the Bohmian velocity field changes sign when is replaced by . Now, we need to consider the time reversal of the jumps.
The downward jumps erase certain information. That is, if denotes a solution of Bohm’s equation of motion (6) in , then a history could arrive at in various ways, including specifically a downward jump at time from to , followed by motion along without an upward jump. It is this many-to-one evolution that becomes, when time-reversed, stochastic (i.e., one-to-many), see Figure 5. The distribution induces a distribution over those histories ending up at at time , and this determines the distribution of the time of the upward jump in the time-reversed histories.
The mathematical criterion for the reversibility of the jumps is that the amount of probability transported by jumps from the interval in to during the time interval (or, so to speak, the number of histories with an upward jump from during ), given the wave function , is equal to the amount of probability transported from the interval between and in to during , given the wave function . The former quantity is
the latter is
Conversely, this reasoning determines the jump rate formula. That is, the law (10) is uniquely selected by the conjunction of the following requirements: time reversal symmetry, the Markov property, Bohm’s equation of motion, deterministic jumps from to , and equivariance. Indeed, consider any Markov process in such that (i) is distributed for every ; (ii) in , obeys Bohm’s equation of motion until it hits the boundary, at which time it jumps from to ; (iii) in , obeys Bohm’s equation, except that at any time it may jump anywhere on . Then, by the Markov property, the jumps occur spontaneously with some rate ; reversibility requires that the only transitions that occur are the reverse of possible jumps from to , and thus with ; reversibility requires further that (19) is equal to (20), so, using ,
which implies (10).
Alternatively, the jump rate formula (10) also follows without assuming time reversal symmetry if we assume instead (in addition to equivariance, the Markov property, Bohm’s equation of motion, and deterministic jumps from to ) that the upward jumps can only be of the form . Indeed, if a given Bohmian trajectory in begins at at time , then the process can reach it only by jumping to at time , and it can only jump there from . For equivariance, the process’s -current out of at time must agree with , which implies (10) whenever ; further jumps cannot occur since no trajectories begin at at time if .
2.6 Neumann-Type and Robin-Type Boundary Conditions
As pointed out already in  for Model 4 and in [59, 38, 65] for other models, other IBCs are possible that involve derivatives of normal to the boundary. While the IBC (3) is of Dirichlet type in that it involves, like a Dirichlet boundary condition, the value but not the normal derivative of on the boundary, an IBC of Neumann type involves the normal derivative but not the value of , and one of Robin type involves both.111111However, some care is required with this terminology. For example, the “Neumann-type” IBC for Model 1–3, which replaces, e.g., (27) by , has the property that for those that do not diverge at , the left-hand side just yields , the expression that would appear in a Dirichlet condition. A general scheme is [60, 65]
with constants such that
Given constants satisfying (23), a configuration process can be defined in the same way as before, using Bohm’s equation of motion again and literally the same formula (10) for the jump rate, and postulating again that the configuration, upon reaching the boundary at , jumps to . Then equivariance of holds again, and so does time reversal symmetry. In general, complex phases such as in the coefficients of an IBC lead to violations of time reversal symmetry when the phases at different boundaries are neither equal nor opposite ; this does not happen for Model 4 because it has only one boundary. However, the action of time reversal on is now not merely complex conjugation but involves in addition a different phase factor on each sector, viz., and .
3 Particle Creation in 3 Dimensions
We now turn to Model 3 and, later in this section, Models 2 and 1. In Model 3, a -particle can move in 3-dimensional space and be absorbed and emitted by an -particle fixed at the origin. We refer to the -configuration simply as “the configuration.” Thus, the configuration space consists of two sectors, and , corresponding to the number of -particles. That is, contains only a single configuration, namely the empty configuration , and , whose boundary contains only the origin. This model lies outside the framework discussed so far because the boundary now has codimension 3.
We will use spherical coordinates in with and (the unit sphere in ), so that the boundary corresponds to and looks like a surface in coordinate space. In fact, it will be convenient to revise the definition of in the previous paragraph a little bit and set with the Riemannian metric (with the Riemannian metric on , so becomes degenerate on the boundary , but the IBC approach works nevertheless). With this choice of , we have that is a sphere, not a point, and that in the sense of Remark 2. In the following, we will assume that is defined on , whereas moves in .
3.1 Model 3
The configuration space is equipped with the measure defined by
The Hilbert space of this model is with and . In spherical coordinates, the inner product in reads
where is the surface area element on , and the Laplace operator becomes
with the Laplace operator on the sphere.
The IBC demands that in Cartesian coordinates,
. Equivalently in spherical coordinates, for any sequence of positive numbers and any sequence ,
This condition implies that, whenever is nonzero, diverges as like . The Hamiltonian is
We now define the Bell-type process in . If then it moves according to Bohm’s equation of motion (6) until it hits the boundary , at which time it jumps to , where it remains for a random waiting time. The rate of jumping from to the surface element around on the boundary is
The factor , not present in the previous jump rate formula (10), can be thought of as arising in this way: Since the probability current density is
whose radial component is , the outward probability flux per time through a solid angle element of a sphere around the origin of radius is
Now the outward flux from the origin in directions in the solid angle element is the limit thereof as .
Let us continue the definition of . After jumping to , the process moves along the solution of Bohm’s equation of motion starting at , see Figure 6. In fact, it turns out (see Remark 10 below) that the velocity vector field in spherical coordinates possesses a continuous extension to , so that at most one trajectory starts at . As in Section 2, the positive sign of at guarantees that there actually is a solution of Bohm’s equation beginning at . This completes the definition of the process .
If we want to use instead of as the value space of , then, instead of “jumping to at time ,” we need to say “jumping to the trajectory whose limit backwards in time at is ” and stipulate that . If we want to use Cartesian coordinates, we need to say “jumping to the trajectory whose limit backwards in time at is the origin () and whose limiting direction is ().” We will often abbreviate this phrase and simply say “jumping to .”
We now derive the equivariance of , following the lines of the argument in Section 2.3 for Model 4. For to be distributed means that the distribution of has density relative to , i.e.,
for a volume element around . The probability transport equations for the process are
and the equations for implied by the Schrödinger equation with Hamiltonian (29) are
Location of creation. We see that the behavior depicted in Figure 1 occurs in Model 3: The -particle gets emitted at the location of the -particle (the origin). Likewise, it gets absorbed at the location of the -particle.
Dirichlet vs. Neumann vs. Robin conditions. We have used a Dirichlet-type IBC for Model 3, but Neumann-type or Robin-type conditions are equally possible [59, 38], also with respect to the Bohmian dynamics. The version of the theory with the Dirichlet-type condition seems to be the physically most natural and relevant .
Expansion by powers of . Let us assume for simplicity that can be expanded in powers of according to
does not depend on either. If it did, then and thus would not be square-integrable. To see this, note that contains contributions, arising from the third term on the right-hand side of (26), of the form
As seen from (25), a function of the form can be square-integrable near the origin only if
that is, if . Thus, in (41) the terms with and , if nonzero, will ruin the square-integrability (and one easily checks that this cannot be avoided by cancellations between summands of (41)); so for and , which is possible on the sphere only if (as the eigenfunctions of are the spherical harmonics, and every non-constant one of them has negative eigenvalue).
Unnecessary -integration. As a curious consequence of the previous remark, we can actually drop the -integration in the definition (29a) of