#### 3.1. Flow Lines Constructed from Weak Values

In quantum mechanics, the uncertainty principle does not allow us to give meaning to the ‘trajectory’ of a single particle so we are left with the question: “How does a particle get from A to B?”. Rather than taking two points, consider two small volumes, $\Delta {V}^{\prime}\left({x}^{\prime}\right)$ surrounding the point $A={x}^{\prime}$ and $\Delta V\left(x\right)$ surrounding $B=x$. We assume these volumes are initially large enough to avoid problems with the uncertainty principle.

Now imagine a sequence of particles emanating from $\Delta {V}^{\prime}\left({x}^{\prime}\right)$, each with a different momentum. Over time we will have a spray of possible momenta emerging from the volume $\Delta {V}^{\prime}\left({x}^{\prime}\right)$, the nature of this spray depending on the size of $\Delta {V}^{\prime}\left({x}^{\prime}\right)$. Similarly there will be a spray of momenta over time arriving at the small volume $\Delta V\left(x\right)$ surrounding the point x.

Better still let us consider a small volume surrounding the midpoint

X. At this point there is a spray arriving and a spray leaving a volume

$\Delta V\left(X\right)$ as shown in

Figure 1. To see how the local momenta behave at the midpoint

X, we will use the real part of

${S}_{\u03f5}^{\prime}(x,{x}^{\prime})$ defined by

Let us define a quantity

then using the action (Equation (

14)), we find

Not surprisingly, this is exactly what Feynman [

9] is averaging over at the point

X, agreeing with the term between the brace

$[\dots ]$.

What is more important is the relation of Equation (

16) to Equation (

10) which is the real part of the weak value of the momentum operator. Thus, the mean momentum of a set of Feynman paths at

X is clearly the real part of this weak value. However, this weak value is just the Bohm momentum. Thus the Bohm ‘trajectories’ are simply an ensemble of the average of the ensemble of individual Feynman paths.

To see how this unexpected result also emerges from a different perspective, let us consider the process in

Figure 1 which we regard as an image of an ensemble of actual individual quantum processes. We are interested in finding the average behaviour of the momentum,

${P}_{X}$, at the point

X. However, we have two contributions to consider, one coming from the point

${x}^{\prime}$ and one leaving for the point

x. We must determine the distribution of momenta in each spray to produce a result that is consistent with the wave function

$\psi \left(X\right)$ at

X. Feynman suggests [

9] that we can think of

$\psi \left(X\right)$ as ‘information coming from the past’ and

${\psi}^{\ast}\left(X\right)$ as ‘potential information appearing in the future’. This suggests that we can write

The $\varphi \left({p}^{\prime}\right)$ contains information regarding the probability distribution of the incoming momentum spray, while ${\varphi}^{\ast}\left(p\right)$ contains information about the probability distribution in the outgoing momentum spray. These wave functions must be such that in the limit $\u03f5\to 0$ they are consistent with the wave function $\psi \left(X\right)$.

Thus, we can define the mean momentum,

$\overline{\overline{P}}\left(X\right)$, at the point

X as

where

$\rho \left(X\right)$ is the probability density at

X. We have added the restriction

$\delta (P-({p}^{\prime}+p)/2)$ since momentum is conserved at

X. We can rewrite Equation (

17) and form

or equivalently taking Fourier transforms

which means that

$\overline{\overline{P}}\left(X\right)$ is the conditional expectation value of the momentum weighted by the Wigner function. Equation (

17) can be put in the form

an equation that appears in the Moyal approach [

30], which is based on a different non-commutative algebra. If we evaluate this expression for the wave function written in polar form

$\psi \left(x\right)=R\left(x\right)\mathrm{exp}\left[iS\right(x\left)\right]$, we find

$\overline{\overline{P}}\left(X\right)=\nabla S\left(X\right)$ which is identical to the expression for the local (Bohm) momentum used in the Bohm interpretation.

This then confirms the conclusion we reached above, namely, that the set of Bohm ‘trajectories’ is an ensemble of the average ensemble of individual paths. Notice, once again, that this gives a very different picture of the Bohm momentum from the usual one used in Bohmian mechanics [

31]. It is not the momentum of a single ‘particle’ passing the point

X, but the mean

momentum flow at the point in question.

This conclusion is supported by the experiments of Kocsis et al. [

5]. They construct the flow lines from an average made over many individual input photons. Thus, the so-called ‘photon’ flow-lines are constructed

statistically from an ensemble of individual events. As was shown in Flack and Hiley [

26], these flow lines are an average of the momentum flow as described by the

weak value of the Poynting vector. This agrees with what one would expect from standard quantum electrodynamics, where the notion of a ‘photon trajectory’ has no meaning, but the notion of a ‘momentum flow’ does have meaning.

Bliokh et al. [

32] have presented a beautiful illustration showing the results of a two-slit interference experiment.

Figure 2a shows the real part of the momentum flow lines in the electromagnetic field, while the imaginary component (osmotic) momentum flow lines are shown in

Figure 2b. It is then clear that we can regard

${v}_{B}(x,t)={p}_{B}(x,t)/m$ as a

local velocity, while the osmotic velocity

${v}_{O}(x,t)={p}_{O}(x,t)/m$ can be regarded as a

localising velocity as discussed in Bohm and Hiley [

33]. The osmotic velocity behaves in such a way as to maintain the form of the probability distribution.

#### 3.2. Where Is the Quantum Potential?

One of the features that many find ‘mysterious’ [

34] is the appearance of the ‘quantum potential’ in the Bohm approach. Is there any trace of it in the Feynman paper [

9]? To answer this question, we must first refer to de Gosson and Hiley [

18] where it is shown that this energy term is absent in quantum processes when taken only to

$O(\Delta t=\u03f5)$ so we must consider terms to

$O(\Delta t={\u03f5}^{2})$.

Feynman shows that the kinetic energy is of

$O\left({\u03f5}^{2}\right)$ when written in the form

$\mathrm{K}.\mathrm{E}.={[({x}_{k+1}-{x}_{k})/\u03f5]}^{2}$, and diverges as

$\u03f5\to 0$. Feynman points out that this quantity is not an observable functional. However, let us now define the kinetic energy to be

This function is finite to

$O\left(\u03f5\right)$ and therefore is an observable functional. Feynman then shows that if we allow “the mass to change by a small amount to

$m(1+\delta )$ for a short time, say

$\u03f5$ around

${t}_{k}$” we can obtain the relation

the extra term arising from the normalising function

A. Thus, we must add a ‘correction’ term to the K.E. in order for the total energy to be finite to

$O\left({\u03f5}^{2}\right)$.

This is the forerunner of mass renormalisation used in quantum electrodynamics. In that case the charged particle is subjected to electromagnetic vacuum fluctuations. The particle we are considering here is not charged and so the fluctuation must arise from a different source, but however it arises, it changes the TPA by $\delta $.

Later in the same paper, Feynman shows that any random fluctuation in the phase function will produce the same effect. A random fluctuation at the point

${x}_{k}$ implies we must replace

$S({x}_{k+1},{t}_{k+1};{x}_{k},{t}_{k})$ by

${S}_{\delta}({x}_{k+1},{t}_{k+1};{x}_{k},{t}_{k}-\delta )$. Thus, to the first order in

$\delta $ we have

where

${H}_{k}$ is the Hamiltonian functional

Apart from the minus sign, the last term is identical to the last term in Equation (

19). Thus Feynman required extra energy to appear from somewhere. A more detailed discussion of this feature appears in Feynman and Hibbs [

35]. The Bohm approach indicates that some ‘extra’ energy appears in the form of the quantum potential energy at the expense of the kinetic energy. Could it be that the source of the energy is the same?

To explore this possibility, let us use the method explained in

Section 2.3 to obtain a more general result for the K.E. The real part of the weak value of the momentum operator squared is obtained from

$\left(\langle \psi \left(t\right)|{\widehat{p}}^{2}|x\rangle +\langle x|{\widehat{p}}^{2}|\psi \left(t\right)\rangle \right)/2$. Under polar decomposition of the wave function, we find the real part of the weak value of the kinetic energy is

With the identification

$\nabla S\leftrightarrow m({x}_{k+1}-{x}_{k})/\u03f5$, we see that the quantum potential is playing a similar role as the mass/energy fluctuation in Feynman’s approach. In fact, de Broglie’s original suggestion was that the quantum potential could be associated with a change of the rest mass [

36].

Notice that the quantum potential appears essentially as a derivative of the osmotic velocity, which in turn is obtained from the imaginary part of

${S}^{\prime}(x,{x}^{\prime})$. Any fluctuating term added to the real part of

${S}_{\u03f5}(x,{x}^{\prime})$ should also be added to the imaginary part. This would also introduce some change in the energy relation shown in Equation (

20). This interplay between the real components of the complex

${S}_{\u03f5}(x,{x}^{\prime})$ is clearly presented as an average over fluctuations arising from some background. Here we can recall Bohr insisting that quantum phenomena must include a description of the whole experimental arrangement. More details will be found in Smolin [

37] and in Hiley [

38].