Feynman Paths and Weak Values

Abstract

There has been a recent revival of interest in the notion of a 'trajectory' of a quantum particle. In this paper, we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are able to conclude that a Bohm 'trajectory' is the average of an ensemble of actual individual stochastic Feynman paths. This implies that they can be interpreted as the mean momentum flow of a set of individual quantum processes and not the path of an individual particle. This enables us to give a clearer account of the experimental two-slit results of Kocsis et al.

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entropy
Article
Feynman Paths and Weak Values
Robert Flack and Basil J. Hiley *
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK;
r.flack@ucl.ac.uk
*Correspondence: b.hiley@bbk.ac.uk
Received: 16 April 2018; Accepted: 9 May 2018; Published: 14 May 2018


Abstract:
There has been a recent revival of interest in the notion of a ‘trajectory’ of a quantum
particle. In this paper, we detail the relationship between Dirac’s ideas, Feynman paths and the
Bohm approach. The key to the relationship is the weak value of the momentum which Feynman
calls a transition probability amplitude. With this identification we are able to conclude that a Bohm
‘trajectory’ is the average of an ensemble of actual individual stochastic Feynman paths. This implies
that they can be interpreted as the mean momentum flow of a set of individual quantum processes
and not the path of an individual particle. This enables us to give a clearer account of the experimental
two-slit results of Kocsis et al.
Keywords: Feynman paths; weak values; Bohm theory
1. Introduction
One of the basic tenets of quantum mechanics is that the notion of a particle trajectory has
no meaning. The established view has been unambiguously defined by Landau and Lifshitz [
1
]:
“In quantum mechanics there is no such concept as the path of a particle”. This position was not
arrived at without an extensive discussion going back to the early debates of Bohr and Einstein [
2
],
the pioneering work of Heisenberg [3] and many others [4].
Yet Kocsis et al. [
5
] have experimentally determined an ensemble of what they call ‘photon
trajectories’ for individual photons traversing a two-slit interference experiment. The set of trajectories,
or what we will call flow-lines, they construct is very similar in appearance to the ensemble of Bohmian
trajectories calculated by Philippidis et al. [
6
]. Mahler et al. [
7
] have gone further and claimed that
their new experimental results provide evidence in support of Bohmian mechanics. However such a
claim cannot be correct because Bohmian mechanics is based on the Schrödinger equation which holds
only for non-relativistic particles with non-zero rest mass, whereas photons are relativistic, having
zero rest mass.
The flow-lines are calculated from experimentally determined weak values of the momentum
operator, a notion that was introduced originally by Aharonov et al. [
8
] for the spin operator. When
examined closely, the momentum weak value is the Feynman transition probability amplitude (TPA) [
9
].
In fact, Schwinger [
10
] explicitly writes the TPA of the momentum in exactly the same form as the weak
value. Recall that the TPA involving the momentum operator plays a central role in the discussion of
the path integral method, an approach that was inspired by an earlier paper of Dirac [
11
] who was
interested in developing the notion of a ‘quantum trajectory’.
Weak values are in general complex numbers, as are TPAs. The real part of the momentum
weak value is the local momentum, sometimes known as the Bohm momentum. The imaginary part
turns out to be the osmotic momentum introduced by Nelson [
12
] in his stochastic derivation of the
Schrödinger equation. In this paper, we will show how the weak value of momentum, Feynman
paths and the Bohm trajectories are related enabling us to give a different meaning to the flow-lines
constructed in experiments of the type carried out by Kocsis et al. [5] and Mahler et al. [7].
Entropy 2018,20, 367; doi:10.3390/e20050367 www.mdpi.com/journal/entropy

Entropy 2018,20, 367 2 of 11
Feynman [
9
] also shows that in his approach the usual expression for the kinetic energy becomes
infinite unless one introduces a small fluctuation in the mass of the particle. We will show that this is
equivalent to introducing the quantum potential, a new quality of energy that appears in the real part
of the Schrödinger equation under polar decomposition of the wave function [13].
2. Dirac’s Notion of a Quantum Trajectory
2.1. Dirac Trajectories
To make the context of our discussion clear, we will begin by drawing attention to an early paper
by Dirac [
11
] who attempted to generalise the Heisenberg algebraic approach through his unique
bra-ket notation, not as elements in a Hilbert space, but as elements of a non-commutative algebra.
In this approach the operators of the algebra are functions of time. Dirac argued that to get round
the difficulties presented by a non-commutative quantum algebra, strict attention must be paid to the
time-order of the appearance of elements in a sequence of operators.
In the non-relativistic limit, operators at different times always commute. (In this paper, we will,
for simplicity, only consider the non-relativistic domain. Dirac himself shows how the ideas can be
extended to the relativistic domain.) This means that a time ordered sequence of position operators
can be written in the form,
hxt|xt0i=Z···Zhxt|xtjidxjhxtj|xtj−1i. . . hxt2|xt1idx1hxt1|xt0i. (1)
This breaks the TPA,
hxt|xt0i
, into a sequence of adjacent points, each pair connected by an
infinitesimal TPA. Dirac writes “. . . one can regard this as a trajectory .. . and thus makes quantum
mechanics more closely resemble classical mechanics”.
In order to analyse the sequence in Equation (1) further, Dirac assumed that for a small time
interval ∆t=e, we can write
hx|x0ie=exp[iSe(x,x0)/¯h](2)
where we will take Se(x,x0)to be a real function in the first instance. Then Dirac [14] shows that
p0
e(x,x0) = hx|ˆ
P0|x0ie=i¯h∇x0hx|x0ie=−∇x0Se(x,x0)hx|x0ie(3)
and
pe(x,x0) = hx|ˆ
P|x0ie=−i¯h∇xhx|x0ie=∇xSe(x,x0)hx|x0ie. (4)
Here
ˆ
P
is the momentum operator. The remarkable similarity of these objects to the canonical
momentum appearing in the classical Hamilton-Jacobi theory should be noted, a fact of which Dirac
was well aware. They are also the canonical momenta appearing in the real part of the Schrödinger
equation under polar decomposition of the wave function exploited by Bohm [
13
] who identified the
momentum with the gradient of the phase of the wave function.
In an earlier paper, Dirac [
11
] did not specify how
Se(x
,
x0)
could be determined. It was
Feynman [
9
] who later identified its relation to the classical Lagrangian
L(˙
x
,
x
,
t)
through the relation
Stt0(x,x0) = Min Zt
t0L(˙
x,x,t)dt. (5)
However, this Lagrangian determines the classical path, so using the exponent of the classical
action seems puzzling. Is there a mathematical explanation for such a choice? The answer is ‘yes’
and is discussed in Guillemin and Sternberg [
15
]. The essential reason for this lies in the relation
between the symmetry group, in this case the symplectic group, and its covering group. Exploiting
this structure, de Gosson and Hiley [
16
] have shown in detail how it is possible to mathematically

Entropy 2018,20, 367 3 of 11
‘lift’ classical trajectories onto this covering space. It is from this structure that the wave properties
emerge. The lift is achieved by exponentiating the classical action, namely using
exp[iSe(x
,
x0)]
. It is
the existence of this structure that the close relation between the Dirac quantum ‘trajectories’ and the
de Broglie-Bohm ‘trajectories’ first calculated by Philippidis et al. [
6
] emerges. We will bring out this
relationship in the rest of this paper.
2.2. The Feynman Propagator
Equation (5) allows us to write the propagator in the well known form
K(x,x0) = Zx
x0eiS(x,x0)Dx0
where the integral is taken over all paths connecting
x0
to
x
. We have written
Dx0
for
dx0
A
,
. . .
,
dxj−1
A
where
(x0
,
x1
,
. . .
,
xj−1)
are points on the path and
A
is the normalising factor introduced by Feynman.
Clearly here S(x,x0)is real.
For a free particle with mass m, we have L=m˙
x2/2 and one can show that
Ktt0(x,x0) = 1
Aexp im(x−x0)2
2¯h(t−t0)(6)
where
A=2πi(t−t0)
m1/2
. With this propagator, Feynman was able to derive the Schrödinger equation
by assuming the underlying paths were continuous and differentiable.
However if we examine the terms
hx|x0ie
for
e→
0, we find the curves, although continuous,
are non-differentiable. To show this let us introduce the TPA of a function F(x,t)defined by
hφt|F|ψt0iS=Lime→0Z···Zφ∗(x,t)F(x0,x1, . . . , xj)
×exp "i
¯h
j−1
∑
k=0
S(xk+1,xk)#ψ(x0,t0)Dx(t).
Here Dis now written as Dx(t) = dx0
A. . . dxj−1
Adxj.
These TPAs can be evaluated by using functional derivatives. In fact, the average of the functional
derivative of a function F(x,t)is given by
δF
δx(s)S
=−i
¯hFδS
δx(s)S
(7)
at the point x(s)on the path x(t). In the case of the specific integral
Z∂F
∂xk
exp[(i/¯h)S(x(t))]Dx(t),
Equation (7) can be written in the form
∂F
∂xkS
=−i
¯hF∂S
∂xkS
.
Feynman notes that the quantities in this expression need not be observables, nevertheless the
equivalence is true [17].

Entropy 2018,20, 367 4 of 11
Let us now consider three adjacent points
xk−1
,
xk
,
xk+1
, each separated by a small time
difference e, we have
−¯h
i∂F
∂xkS
=F∂S(xk+1,xk)
∂xk
+∂S(xk,xk−1)
∂xkS
.
This equation is correct to zero and first order in
e
. If we choose the action for a particle moving
in a potential V, we have
S(x,x0) = m(x−x0)2
2e−eV(x,x0).
Then at the point xkthis gives us
−¯h
i∂F
∂xkS
=F−mxk+1−xk
e−xk−xk−1
e−e∂V
∂xk
(xk).
If Fis unity and we divide by ewe get
0=1
e−mxk+1−xk
e−xk−xk−1
e−∂V
∂xk
(xk). (8)
If we follow Feynman and call
(xk+1−xk)/e
a ‘velocity’, then this equation gives the ‘average’
over an ensemble of individual velocities. It is the quantum equivalent of Newton’s second law of motion;
the potential
V
at
xk
gives rise to a force which changes the incoming momentum
m(xk−xk−1)/e
to the outgoing momentum
m(xk+1−xk)/e
. Notice to order
e
, no extra term corresponding to the
quantum potential appears. de Gosson and Hiley [
18
] have shown in a detailed analysis that this is to
be expected.
These paths are reminiscent of Brownian motion, a characteristic feature of which is the appearance
of two ‘derivatives’ at
xk
, a ‘forward’ and a ‘backward’ derivative, illustrating the non-differentiable
nature of the path. In this paper, we need not discuss the precise nature of these paths to arrive at
our conclusion. It is sufficient for us to note that the substructure of a quantum process is certainly
not classical. In passing we should also note that the ‘velocities’, being of order
(¯h/me)1/2
, diverge as
e→0 and therefore, in Feynman’s terms, are not observables.
2.3. TPAs Involving the Momentum
In 1974 Hirschfelder [
19
,
20
] introduced a quantity
ψ(x
,
t)−1ˆ
pψ(x
,
t)
, which he called a
‘sub-observable’ as he could see no way of measuring it directly, although integrating it over the
whole of configuration space gave the measurable expectation value. Using the polar form of the wave
function,
ψ(x
,
t) = R(x
,
t)exp[iS(x
,
t)/¯h]
, this ‘sub-observable’ is the weak value of the momentum
operator which can be written in the form
ψ(x,t)−1ˆ
pψ(x,t) = hx|ˆ
p|ψ(t)i
hx|ψ(t)i=m[vB(x,t)−ivO(x,t)], (9)
where explicitly
vB(x
,
t) = ∇S(x
,
t)/m
is the local Bohm velocity and
vO(x
,
t) = ∇R(x
,
t)/mR(x
,
t)
is
the localising osmotic velocity, originally introduced by Nelson [
12
] in a stochastic theory. The meaning
of these velocities is discussed in more detail in Bohm and Hiley [
21
]. Much later Hiley [
22
] showed
exactly how these expressions emerged directly from the weak value of the momentum operator.
It should be noted that weak values are essentially TPAs of the type considered by Feynman [
9
] and
Schwinger [23].
In the spirit of Schwinger [
10
], where he argues that “the quantum dynamical laws will find
their proper expression in terms of the transformation functions” that is TPAs, we can introduce two

Entropy 2018,20, 367 5 of 11
momentum TPAs,
hx|−→
P|ψ(t)i
and
hψ(t)|←−
P|xi
where
−→
P=−i¯h−→
∇
and
←−
P=i¯h←−
∇
. Notice by placing
the arrows over the momentum operators, we are emphasising the distinction between left and right
multiplication and it is this distinction that is equivalent to the forward and backward derivatives.
In fact we may identify
hX|−→
P|ψ(t0)i=hX|−→
P|x0iψ(x0,t0) = −ilim
(x0→X)
ψ(X)−ψ(x0)
(X−x0)
with the forward derivative at X, a point that lies between x0and x.
hψ(t)|←−
P|Xi=ψ∗(x,t)hx|←−
P|Xi=ilim
(X→x)
ψ∗(x)−ψ∗(X)
(x−X)
corresponds to the backward derivative. Note that the words ‘forward’ and ‘backward’ here have
nothing to do with time order.
If we again evaluate these TPAs using ψ=Rexp(iS/¯h), we find
1
2
hx|−→
ˆ
P|ψ(t)i
hx|ψ(t)i+hψ(t)|←−
ˆ
P|xi
hψ(t)|xi
=∇S(x,t) = PB(x,t), (10)
and
1
2i
hx|−→
ˆ
P|ψ(t)i
hx|ψ(t)i−hψ(t)|←−
ˆ
P|xi
hψ(t)|xi
=∇R(x,t)
R(x,t)=PO(x,t). (11)
Notice how the sums and differences of the left/right operators produce real values.
We can immediately connect these results with those of Dirac [
11
] if, in Equations (3) and (4),
we replace the real value of
Se(x
,
x0)
by a complex value which we will write as
S0
e(x
,
x0) = Se(x
,
x0)−
iln Re(x,x0). In this case, we find
p0
e(x,x0) = −∇x0Se(x,x0)−i∇x0Re(x,x0)
Re(x,x0)(12)
and
pe(x,x0) = ∇xSe(x,x0)−i∇xRe(x,x0)
Re(x,x0). (13)
Notice also the connection with the classical relations obtained in Equations (3) and (4).
2.4. The Relation between Weak Values and TPAs
In the previous two sections, we have shown how TPAs of the form
hφt|ˆ
F|ψt0i
arise from some
underlying non-differentiable process. The original assumption was that these quantities could not be
investigated experimentally. However starting from a different perspective, the notion of a weak value,
introduced by Aharonov, Albert and Vaidman [
8
], allows us to experimentally measure these quantities.
A weak value of an operator ˆ
Fis defined by
hˆ
Fiw=hφt|ˆ
F|ψt0i
hφt|ψt0i.
Clearly these weak values are Feynman TPAs. Using the suggestions of Leavens [
24
] and
Wiseman [
25
], Kocsis et al. [
5
] have actually measured the weak value of the transverse momentum in
an optical two-slit experiment and as a result have constructed what they called photon ‘trajectories’.
We refer to their paper to explain the details of how this is done.

Entropy 2018,20, 367 6 of 11
Unfortunately photons cannot be treated as particles that satisfy the Schrödinger equation.
They have zero rest mass and are excitations of the electromagnetic field. Nevertheless this does
not invalidate the notion of a momentum flow line; the question remains “How are we to understand
these flow lines?” Flack and Hiley [
26
] have shown that if we generalise the Bohm approach to include
the electromagnetic field [27], each flow line emerges as the locus of a weak Poynting vector.
To connect with the non-relativistic approach we are discussing in this paper, we need to use atoms.
Indeed experiments are being developed at UCL to measure weak values of spin and momentum,
hˆ
piw
, for helium atoms [
28
] and argon atoms [
29
] respectively. The experimental details can be found
in these references. In this paper, we will clarify further the relation between the Feynman paths and
weak values.
3. Weak Values Are Weighted TPAs
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,
∆V0(x0)
surrounding the point
A=x0
and
∆V(x)
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
∆V0(x0)
, each with a different momentum.
Over time we will have a spray of possible momenta emerging from the volume
∆V0(x0)
, the nature
of this spray depending on the size of
∆V0(x0)
. Similarly there will be a spray of momenta over time
arriving at the small volume ∆V(x)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
∆V(X)
as shown in Figure 1. To see how the local
momenta behave at the midpoint X, we will use the real part of S0
e(x,x0)defined by
Se(x,x0) = m
2
(x−x0)2
e. (14)
X
(x',t')
(x,t)
Figure 1. Behaviour of the momenta sprays at the midpoint of hx,t|x0,t0ie.
Let us define a quantity
PX(x,x0) = ∂Se(x,x0)
∂X=∂Se(X,x0)
∂X+∂Se(x,X)
∂X, (15)
then using the action (Equation (14)), we find
PX(x,x0) = m(X−x0)
e−(x−X)
e=p0
X(x,x0) + pX(x,x0). (16)
Not surprisingly, this is exactly what Feynman [
9
] is averaging over at the point
X
, agreeing with
the term between the brace [. . . ].

Entropy 2018,20, 367 7 of 11
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 1which we regard as an image of an ensemble of actual individual quantum
processes. We are interested in finding the average behaviour of the momentum,
PX
, at the
point X
.
However, we have two contributions to consider, one coming from the point
x0
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
ψ(X)
at
X
. Feynman suggests [
9
] that we can think of
ψ(X)
as ‘information coming from the past’ and
ψ∗(X)
as ‘potential information appearing in the future’.
This suggests that we can write
lim
x0→Xψ(x0) = Zφ(p0)eip0Xdp0and lim
X→xψ∗(x) = Zφ∗(p)e−ip X dp.
The
φ(p0)
contains information regarding the probability distribution of the incoming momentum
spray, while
φ∗(p)
contains information about the probability distribution in the outgoing momentum
spray. These wave functions must be such that in the limit
e→
0 they are consistent with the wave
function ψ(X).
Thus, we can define the mean momentum, P(X), at the point Xas
ρ(X)P(X) = Z Z Pφ∗(p)e−ipX φ(p0)eip0Xδ(P−(p0+p)/2)dPdpdp0(17)
where
ρ(X)
is the probability density at
X
. We have added the restriction
δ(P−(p0+p)/
2
)
since
momentum is conserved at X. We can rewrite Equation (17) and form
ρ(X)P(X) = 1
2πZ Z Pφ∗(p+θ/2)e−iXθφ(p−θ/2)dθdP
or equivalently taking Fourier transforms
ρ(X)P(X) = 1
2πZ Z Pψ∗(X−σ/2)e−iPσψ(X+σ/2)dσdP
which means that
P(X)
is the conditional expectation value of the momentum weighted by the Wigner
function. Equation (17) can be put in the form
ρ(X)P(X) = 1
2i[(∂x1−∂x2)ψ(x1)ψ(x2)]x1=x2=X(18)
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
ψ(x) = R(x)exp[iS(x)]
, we find
P(X) = ∇S(X)
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

Entropy 2018,20, 367 8 of 11
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
vB(x
,
t) = pB(x
,
t)/m
as a local velocity, while the
osmotic velocity
vO(x
,
t) = pO(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.
Figure 2. (a) Local field momentum; (b) Localising field momentum.
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(∆t=e)so we must consider terms to O(∆t=e2).
Feynman shows that the kinetic energy is of
O(e2)
when written in the form
K.E. = [(xk+1−xk)/e]2
, and diverges as
e→
0. Feynman points out that this quantity is not an
observable functional. However, let us now define the kinetic energy to be
K.E.0=m
2xk+1−xk
exk−xk−1
e.
This function is finite to
O(e)
and therefore is an observable functional. Feynman then shows that
if we allow “the mass to change by a small amount to
m(
1
+δ)
for a short time, say
e
around
tk
” we
can obtain the relation
m
2xk+1−xk
exk−xk−1
e=m
2xk+1−xk
e2
+¯h
2ie, (19)
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(e2).
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 δ.

Entropy 2018,20, 367 9 of 11
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
xk
implies we must replace
S(xk+1,tk+1;xk,tk)by Sδ(xk+1,tk+1;xk,tk−δ). Thus, to the first order in δwe have
hξ|1|ψiS−hξ|1|ψiSδ=iδ
¯hhξ|Hk|ψiS
where Hkis the Hamiltonian functional
Hk=−∂S(xk+1,tk+1;xk,tk)
∂tk+1
+¯h
2i(tk+1−tk). (20)
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
hψ(t)|ˆ
p2|xi+hx|ˆ
p2|ψ(t)i/
2. Under polar decomposition of the wave function, we find the real part
of the weak value of the kinetic energy is
1
2mhˆ
p2iw=1
2m(∇S)2−∇2R
R. (21)
With the identification
∇S↔m(xk+1−xk)/e
, 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
S0(x
,
x0)
. Any fluctuating term added to the real
part of
Se(x
,
x0)
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
Se(x
,
x0)
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].
4. Conclusions
Our explorations of the weak values of the momentum operator [
22
] have led us to reconsider the
basis on which Feynman [
9
] developed his path integral approach. We have shown that there is an
unexpected close connection between the Feynman propagator, the weak values of the momentum
and the original Bohm approach [13].
Feynman had already noticed that to prevent the kinetic energy tending to infinity as the time
interval between steps tends to zero, it was necessary to introduce a ‘fluctuation’ in the mass/energy
of the particle. This extra energy can be thought of as arising in a way similar to the way the quantum
potential energy appears as an effect of some background field. Indeed, as we have remarked above,
de Broglie [
36
] had already proposed that the quantum potential could be included in the mass term
M=√[m2+ (¯h2/c2)R/R]
,
R
being the amplitude of the wave function. Hiley [
38
] has shown a
similar conclusion arises for the Dirac equation.
The approach outlined in this paper shows that the basic assumption made in Bohmian mechanics,
namely, that each particle follows one of the ensemble of ‘trajectories’ calculated by
Philippidis et al. [6]
from
PB(x
,
t)
cannot be maintained. Rather the trajectories should be interpreted as a statistical average
of the momentum flow of a basic underlying stochastic process.

Entropy 2018,20, 367 10 of 11
It is now possible to experimentally explore weak values, perhaps clarifying the nature of this
stochastic process. In the case of the electromagnetic field this has already been done by
Kocsis et al. [5]
,
but as we have seen the notion of a ‘photon trajectory’ has no meaning. However, the average
momentum flow does have meaning [
26
]. As mentioned above, new experiments using argon and
helium atoms are now being carried out at UCL by Morley et al. [
29
] and by Monachello, Flack, and
Hiley [
28
]. It is hoped that these future experiments will throw more light on the nature of individual
quantum processes.
Author Contributions: Both authors contributed equally to this manuscript.
Funding: This research was funded in part by the Fetzer Franklin Memorial Trust.
Acknowledgments:
Special thanks to Bob Callaghan, Glen Dennis and Lindon Neil for their helpful discussions.
Thanks also to the Franklin Fetzer Foundation for their financial support.
Conflicts of Interest:
The founding sponsors had no role in the design of the study; in the collection, analyses,
or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.
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c
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