# Experimental Comparison of Bohm-like Theories with Different Ontologies

Arthur O. T. Pang<EMAIL_ADDRESS>Hugo Ferretti  Noah Lupu-
Gladstein  Weng-Kian Tham  Aharon Brodutch<EMAIL_ADDRESS>Department of Physics and Centre for Quantum Information Quantum Control
University of Toronto, 60 St George St, Toronto, Ontario, M5S 1A7, Canada
Kent Bonsma-Fisher<EMAIL_ADDRESS>Department of Physics
and Centre for Quantum Information Quantum Control University of Toronto, 60
St George St, Toronto, Ontario, M5S 1A7, Canada  National Research Council of
Canada, 100 Sussex Dr, Ottawa, Ontario, K1A 0R6, Canada  J. E. Sipe
<EMAIL_ADDRESS>Department of Physics and Centre for Quantum
Information Quantum Control University of Toronto, 60 St George St, Toronto,
Ontario, M5S 1A7, Canada  Aephraim M. Steinberg<EMAIL_ADDRESS>Department of Physics and Centre for Quantum Information Quantum Control
University of Toronto, 60 St George St, Toronto, Ontario, M5S 1A7, Canada
Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1M1, Canada

###### Abstract

The de Broglie-Bohm theory is a hidden variable interpretation of quantum
mechanics which involves particles moving through space with definite
trajectories. This theory singles out position as the primary ontological
variable. Mathematically, it is possible to construct a similar theory where
particles are moving through momentum space, and momentum is singled out as
the primary ontological variable. In this paper we experimentally show how the
two theories lead to different ontological descriptions. We construct the
putative particle trajectories for a two-slit experiment in both the position
and momentum space theories by simulating particle dynamics with coherent
light. Using a method for constructing trajectories through the primary and
derived (i.e. non-primary) spaces, we compare the ontological pictures offered
by the two theories and show that they do not agree. This contradictory
behaviour brings into question which ontology for Bohmian mechanics is to be
preferred.

##  I Introduction

Bohm’s hidden variable interpretation of quantum mechanics Bohm (1952a, b),
also known as Bohmian mechanics or de Broglie-Bohm theory de Broglie (1927);
Holland (1993a), is an alternative formulation of quantum mechanics with a
clear deterministic ontology, and experimental predictions that match those of
quantum theory. The theory continues to attract attention Philippidis  _et
al._ (1979); Kocsis  _et al._ (2011); Mahler  _et al._ (2016); Zhou  _et al._
(2017); Xiao  _et al._ (2017, 2019), perhaps due to the fact that it allows
physicists to visualize the dynamics of quantum systems. As is the case in
classical physics, Bohmian particles have well defined properties at all
times. In Bohmian theory all properties can be determined from the particle’s
actual position and the guiding wave, giving position special ontological
significance. Wiseman Wiseman (2007) showed that it is possible to
experimentally extract the velocities attributed to Bohmian particles by
taking conditional averages of weak measurements on an ensemble of post-
selected systems. Extending his ideas, some of the present authors and others
were recently able to construct the putative Bohmian trajectories in various
two-slit experiments Kocsis  _et al._ (2011); Mahler  _et al._ (2016); Xiao
_et al._ (2017, 2019). These results have been considered by some as evidence
for the validity of Bohm’s interpretation and the preferred status of position
Goldstein (2017).

The choice of position as the preferred ontological variable introduces an
asymmetry which is foreign to both classical and quantum mechanics. In
classical Hamiltonian mechanics, position and momentum act as the canonical
phase space variables and are both equally important in formulating the
theory, and in orthodox quantum mechanics position and momentum are treated on
equal footing. So the importance placed on position in the Bohmian approach
was one of the main criticisms of Bohm’s work by the pioneers of quantum
theory Pinch (1977). Shortly after Bohm’s paper appeared, Epstein Epstein
(1953a, b) pointed out that there is nothing inherent in the formulation that
requires position to be the preferred variable, and that other possible
choices can lead to other results, i.e. different ontological descriptions,
while still yielding experimental predictions identical to those of quantum
theory.

In this paper, we demonstrate how different choices of the primary variable
can lead to qualitatively different trajectories. Using light to simulate the
mechanics of massive particles For commentary on simulations of massive
particles using photons see R. Flack, B. Hiley (2018), in a double-slit setup
similar to those used earlier Mahler  _et al._ (2016); Kocsis  _et al._ (2011)
we construct the trajectories in both Bohm’s theory (which we refer to as
$x$-Bohm) and an alternative theory in which momentum is the preferred
variable ($p$-Bohm). The differences between the trajectories in the two
theories illustrate why the results of previous experiments Mahler  _et al._
(2016); Kocsis  _et al._ (2011) should be understood as specific instances of
the many possible ontological descriptions of the same system. This multitude
of possible theories, and corresponding ontological pictures, makes it
difficult to decide which theory, if any, should be ascribed to reality,
emphasizing one of the weaknesses in Bohm’s approach.

We begin in Sec. II by describing some of the basic features of the $x$-Bohm
and $p$-Bohm theories, and the method for constructing trajectories through a
sequence of weak and strong measurements. In Sec. III we lay out the details
of our experiment, including the specifics of the lens system and the weak
measurement procedure. The results of the experiments, including plots of the
trajectories and phase space snapshots at the near and far field are presented
in Sec. IV for both the $x$-Bohm and $p$-Bohm theories. The implications of
our results are discussed in Sec. V.

##  II Bohmian theory

Contrary to classical mechanics, which allows for the deterministic prediction
of the motion of particles, quantum mechanics only offers statistical
predictions of the results of measurements. Yet in 1952 David Bohm introduced
Bohm (1952a, b) a deterministic dynamical theory that its advocates argue
provides an underlying description more fundamental than quantum mechanics
Bohm and Hiley (1993); Bacciagaluppi and Valentini (2009). In his
generalization and extension of earlier ideas by de Broglie Bacciagaluppi and
Valentini (2009), the positions of particles play the role of hidden
variables; their motion is characterized by well-defined trajectories, as the
particles are “guided” by the Schrödinger wave. In this approach position
variables, together with the Schrödinger wave, have a special significance as
the primary ontological variables; the momenta of particles simply follow from
their velocities, determined by the gradient of the Schrödinger wave at the
positions of the particles. The symmetry of position and momentum that
characterizes orthodox quantum mechanics is broken, with position variables
more fundamental than momentum variables.

Shortly after Bohm’s work appeared, Epstein Epstein (1953a, b) noted that
different choices of the primary ontological variable can lead to different
theories. In particular, one could work with the momentum representation of
the wave function and build a theory where particles are characterized
fundamentally by their momenta111The possibility of a velocity-based theory
had already been raised by Pauli at the 1927 Solvay conference in response to
de Broglie’s pilot wave theory Bacciagaluppi and Valentini (2009); Bohm
(1952a).. In contrast to Bohm’s original theory, which we refer to as
“$x$-Bohm,” in Epstein’s proposal, which we refer to as a “$p$-Bohm” theory,
it is momentum that has primary ontological status. In his reply to Epstein
Bohm (1953), Bohm pointed out technical difficulties in implementing a
“$p$-Bohm” approach when the Hamiltonian involved the Coulomb potential. But
he also argued that an “$x$-Bohm” approach, where particle position and the
coordinate representation of the wave function are the primary ontological
variables, seemed more favored because “in all fields other than the quantum
theory, space and time have thus far stood out as the natural frame for the
description of the progress of physical phenomena.” Nevertheless, alternate
approaches were developed further a few decades later by Bohm and his
collaborators Brown and Hiley (2000), and a general framework for such
theories was discussed by Holland Holland (1993a, b, 1998) and others Struyve
and Valentini (2008); Struyve (2010).

In the rest of this section we sketch both $x$-Bohm and $p$-Bohm theory,
discuss the trajectories that follow from each, and show how – under the
assumption that one of the theories is correct – its associated trajectories
can be revealed by weak measurements. We begin with trajectories of the
primary ontological variable of the particles – position for $x$-Bohm and
momentum for $p$-Bohm – and then turn to the trajectories that can be
associated with non-primary variables. This allows us to compare the two
theories by contrasting their predictions for trajectories in the same space.
We focus on the one-dimensional motion of a single particle, where the
classical Hamiltonian as a function of position and momentum is $H(x,p)$, and
denote the coordinate wave function by $\psi(x,t)$ and the momentum wave
function by $\tilde{\psi}(p,t)$. The Schrödinger equations for these two wave
functions are

$\displaystyle i\hbar\frac{\partial}{\partial t}\psi(x,t)=$ $\displaystyle
H\left(x,-i\hbar\frac{\partial}{\partial x}\right)\psi(x,t)$ (1)
$\displaystyle i\hbar\frac{\partial}{\partial t}\tilde{\psi}(p,t)=$
$\displaystyle H\left(i\hbar\frac{\partial}{\partial
p},p\right)\tilde{\psi}(p,t).$ (2)

###  II.1 Position Ontological Bohmian Theory ($x$-Bohm)

In Bohm’s original theory Bohm (1952a, b), the particle’s position and the
wave function $\psi(x,t)$ constitute the objectively real elements from which
all other properties can be derived222For an $N$-particle system the wave
function is a function over the $3N$-dimensional configuration space of the
system, and since the wave function is granted ontological significance that
configuration space must be taken as the underlying arena of reality; the wave
function and a point in this configuration space, identifying the positions of
all $N$ particles, are best taken to identify the ontology of the theory.. In
describing an ensemble of experimental runs, at some initial time $(t=0)$ each
particle is assumed to have a definite position according to a probability
distribution function $|\psi(x,0)|^{2}$, and each particle is guided through
space by the wave function. Writing
$\psi\left(x,t\right)=R_{x}\left(x,t\right)\exp\left[iS_{x}\left(x,t\right)/\hbar\right]$,
where $R_{x}\left(x,t\right)$ and $S_{x}\left(x,t\right)$ are real functions
of position and time, for a Hamiltonian of the form $H(x,p)=p^{2}/2m+V(x)$ the
guidance equation is

$v_{x}\left(x,t\right)=\frac{1}{2m}\frac{\partial S_{x}(x,t)}{\partial x},$
(3)

and the trajectory for each particle is given by

$\frac{dx\left(t\right)}{dt}=v_{x}\left(x\left(t\right),t\right).$ (4)

Since the expression (3) for the velocity $v_{x}(x,t)$ can also be written as
Holland (1993a)

$\displaystyle
v_{x}\left(x,t\right)=\frac{j_{x}\left(x,t\right)}{\left|\psi\left(x,t\right)\right|^{2}},$
(5)

where $j_{x}(x,t)$ is the usual probability current density of orthodox
quantum mechanics,

$\displaystyle
j_{x}\left(x,t\right)=\frac{\hbar}{2mi}\left(\psi^{*}\left(x,t\right)\frac{\partial\psi\left(x,t\right)}{\partial
x}-\psi\left(x,t\right)\frac{\partial\psi^{*}\left(x,t\right)}{\partial
x}\right),$ (6)

it follows that as the particles in the ensemble move, and as $\psi(x,t)$
evolves according to Schrödinger’s equation (1), the evolution of the
distribution function characterizing the positions of the particles follows
the evolution of $\left|\psi(x,t)\right|^{2}$.

Although the Bohmian trajectories had been studied theoretically and discussed
in the literature since 1952 (see, e.g., Philippidis et al. Philippidis  _et
al._ (1979)), it seems it was not until Wiseman’s work in 2007 Wiseman (2007)
that a strategy for identifying them experimentally was investigated. Wiseman
pointed out that the expression (3) for the velocity of a particle at $x$,
which can be written as Holland (1993a)

$v_{x}\left(x,t\right)=\frac{1}{m}\text{Re}\left[\frac{\left<x\right|\hat{p}\left|\psi\left(t\right)\right>}{\left<x|\psi\left(t\right)\right>}\right],$
(7)

where $\hat{p}$ is the momentum operator ($\left\langle
x|\hat{p}|x^{\prime}\right\rangle=-i\hbar\partial\delta(x-x^{\prime})/\partial
x$), can be connected with the theory of weak measurements introduced by
Aharonov, Albert, and Vaidman (AAV) Aharonov  _et al._ (1988). Weak
measurements are those with small back action and consequently high
uncertainty, and Wiseman noted that the expression (7) corresponds to the
operational prescription of a weak momentum measurement followed immediately
by a strong (projective) position measurement. The apparent simultaneous
measurement of two conjugate variables respects the uncertainty relations
since the momentum measurement is weak, and consequently the measurement
scenario must be repeated many times with the averaging done separately for
every final value of position. This fits neatly into the Bohmian perspective
in general: Since all variables in the theory are uniquely determined by the
primary ontological variable, it could be argued that ensemble averaging can
be justified as long as post-selection onto the primary ontological value for
each experimental run is sufficiently accurate and the measurement back action
for the weak measurement is sufficiently small.

Of course, the identification of the right-hand-side of (7) with a weak
momentum measurement followed by a strong position measurement can be made
operationally, independent of any proposed explanation of quantum mechanics in
terms of a deeper theory. Nonetheless, the trajectories that are predicted by
$x$-Bohm theory can be formally constructed from the results of weak
measurements; this has been done by Kocsis et al. Kocsis  _et al._ (2011) for
a single particle in a double-slit interferometer, and by Mahler et al. Mahler
_et al._ (2016) for entangled light. Advocates of $x$-Bohm theory then
identify these constructed trajectories with trajectories that are held to
really exist.

###  II.2 Momentum Ontological Bohmian Theory ($p$-Bohm)

In $p$-Bohm theory one adopts momentum as the primary ontological variable,
and the fundamental dynamics take place in momentum space. Here, one relies on
the momentum representation of the wave function $\tilde{\psi}(p,t)$, and for
Hamiltonians of the form $H(x,p)=p^{2}/2m+V(x)$ there is no general expression
for the time derivative $v_{p}(p,t)$ of the momentum of a Bohmian particle,

$\displaystyle\frac{dp\left(t\right)}{dt}=v_{p}\left(p\left(t\right),t\right),$
(8)

which would be analogous to the corresponding expression (3) for the time
derivative $v_{x}(x,t)$ of the position of a Bohmian particle in $x$-Bohm
theory. This can be traced to the fact that all such Hamiltonians exhibit the
same dependence on $p$ but, depending on the potential, can have very
different dependences on $x$; thus, while Schrödinger’s equation in coordinate
space (1) involves only second derivatives with respect to $x$, Schrödinger’s
equation in momentum space (2) can involve any number of derivatives with
respect to $p$. Nonetheless, one can look for an expression for $v_{p}(p,t)$
analogous to the expression (5) for $v_{x}(x,t)$, writing

$\displaystyle
v_{p}(p,t)=\frac{j_{p}(p,t)}{\left|\tilde{\psi}(p,t)\right|^{2}},$ (9)

where $j_{p}(p,t)$ is a current density in momentum space. In a one-
dimensional problem it must satisfy

$\displaystyle\frac{\partial j_{p}(p,t)}{\partial p}=-\frac{\partial}{\partial
t}\left(\left|\tilde{\psi}(p,t)\right|^{2}\right),$ (10)

and since the right-hand-side is determined by the Schrödinger equation in
momentum space (2), a unique $j_{p}(p,t)$ can be identified,

$\displaystyle
j_{p}(p,t)=\frac{2}{\hbar}\int_{-\infty}^{p}\text{Im}\left(\tilde{\psi}^{*}(p^{\prime},t)\left(V(i\hbar\frac{\partial}{\partial
p^{\prime}})\tilde{\psi}(p^{\prime},t)\right)\right)dp^{\prime},$ (11)

under the physically reasonable assumption that $j_{p}(p,t)\rightarrow 0$ as
$\left|p\right|\rightarrow\infty$ Struyve and Valentini (2008). The situation
is more complicated in higher dimensions; in three dimensions, for example,
the continuity equation for a momentum current density $\bm{j}_{p}(\bm{p},t)$
only restricts the divergence of $\bm{j}_{p}(\bm{p},t)$ and not its curl, and
it is not immediately clear how it should be assigned. The range of possible
choices for current densities in general de Broglie-Bohm theories, and the
criteria one might want to apply in making a choice, have been investigated by
Struyve and Valentini Struyve and Valentini (2008). Our focus in this paper
will be on free particles ($V(x)=0)$, where the distribution
$\left|\tilde{\psi}(p,t)\right|^{2}$ of Bohmian particles in momentum space is
time independent, and from (9,11), and in agreement with physical intuition,
we have $v_{p}(p,t)=0$.

###  II.3 Trajectories in a non-primary space

Consider then a $p$-Bohm theory for free particles. In an ensemble of
experimental runs there would be a distribution of particles characterized by
$\left|\tilde{\psi}(p,t)\right|^{2}=\left|\tilde{\psi}(p,0)\right|^{2}$ in
momentum space, and each Bohmian particle would maintain its momentum. Is the
question of what each particle is doing in real space even meaningful? After
all, momentum is here the primary ontological variable, and so the arena of
reality is momentum space. Each particle has a momentum and it is constant;
there seems nothing else that can be said. The theory does not concern itself
with trajectories in spacetime, despite the fact that, at least as argued by
Bohm, that is “the natural frame for the description of the progress of
physical phenomena.”

For a Bohm-like theory with a given primary ontological variable, Holland
Holland (1993a, b) suggested a strategy for identifying the values of
variables other than the primary ontological variable. For one-dimensional
systems and in our notation, if we consider a “$\xi$-Bohm theory,” where here
$\xi$ is an eigenvalue of a Hermitian operator $\hat{\xi}$ that we take to
have continuous eigenvalues, the value $\omega$ of a continuous variable
associated with a Hermitian operator $\hat{\omega}$ is taken to be

$\displaystyle\omega_{\xi}(\xi,t)=\text{Re}\left[\frac{\left<\xi\right|\hat{\omega}\left|\psi\left(t\right)\right>}{\left<\xi|\psi\left(t\right)\right>}\right]$
(12)

at time $t$, if the ket is $\left|\psi(t)\right\rangle$ and the primary
ontological variable has value $\xi$. Holland did not take this suggestion to
be at the level of a new postulate, and even considered other approaches for
some physical systems. Nonetheless, the proposal has the physically comforting
feature that the average of the values granted to a variable $\omega$ over an
ensemble of Bohmian particles described by a $\xi$-Bohm theory does agree with
the expectation value of the operator associated with that variable in the ket
describing the ensemble,

$\displaystyle\left\langle\psi(t)|\hat{\omega}|\psi(t)\right\rangle=\int\omega_{\xi}(\xi,t)\left|\left\langle\xi|\psi(t)\right\rangle\right|^{2}d\xi.$
(13)

As an example, consider momentum in an $x$-Bohm theory. For a particle at
position $x$ at time $t$, from (12) we see that the value of momentum that
would be assigned is

$\displaystyle
p_{x}(x,t)=\text{Re}\left[\frac{\left<x\right|\hat{p}\left|\psi\left(t\right)\right>}{\left<x|\psi\left(t\right)\right>}\right].$
(14)

Comparing with the $x$-Bohm expression (7) for $v_{x}(x,t)$, we find

$\displaystyle p_{x}(x,t)=mv_{x}(x,t),$ (15)

as would be physically expected. Yet we can now also assign evolving position
variables to particles in a $p$-Bohm theory, for the prescription (12) gives

$\displaystyle
x_{p}(p,t)=\text{Re}\left[\frac{\left<p\right|\hat{x}\left|\psi\left(t\right)\right>}{\left<p|\psi\left(t\right)\right>}\right],$
(16)

and following $x_{p}(p,t)$ as $t$ advances allows us to assign a trajectory in
real space to a Bohmian particle in $p$-Bohm theory with momentum $p$.

Furthermore – and somewhat remarkably! – Holland’s prescription (12) is
precisely that which operationally characterizes a weak $\hat{\omega}$
measurement followed by a strong $\hat{\xi}$ measurement. Thus, just as
velocities of particles in an $x$-Bohm theory can be constructed by weak
momentum measurements followed by strong position measurements, so the
positions of particles in a $p$-Bohm theory can be constructed by weak
position measurements followed by strong momentum measurements. And so we have
a route to identifying trajectories of Bohmian particles in “non-primary”
spaces, by which we mean spaces associated with variables other than the
primary ontological variable. This is done by first constructing the
trajectories of the ontological variable, leading to an equation for $\xi(t)$
and then using Eq. (12) to construct the trajectory given by
$\omega(\xi(t),t)$. We can experimentally construct trajectories in momentum
space for particles in an $x$-Bohm theory, as indeed has already implicitly
been done Kocsis  _et al._ (2011); Mahler  _et al._ (2016) relying on (15),
and we can also experimentally construct trajectories in position space for
particles in a $p$-Bohm theory. We turn to this in the following sections.

##  III Experimental scheme

In our experiment, we simulate the evolution of a massive particle under
$x$-Bohm and $p$-Bohm theories with light from a laser diode, using the fact
that light propagating in the paraxial regime can be modelled with the
Schrödinger equation. The propagation of monochromatic light can be modelled
with the Helmholtz equation Saleh and Teich (2007)

$\nabla^{2}A+|\textbf{k}|^{2}A=0,$ (17)

where $A$ is the vector potential and $\textbf{k}=(k_{x},k_{y},k_{z})$ is the
wave vector. In the paraxial regime where $|\textbf{k}|\approx k_{z}$, this
equation can be reduced to

$i\frac{\partial}{\partial z}u=-\frac{\partial^{2}}{\partial
x^{2}}\frac{u}{2|\textbf{k}|},$ (18)

where $A=u\cdot\exp\left(ik_{z}z\right)$, $u$ is the envelope function of the
propagating light, $z$ is the longitudinal position, and the $y$ coordinate is
factored out through a separation of variables. Equation (18) has the form of
the one-dimensional Schrödinger equation (1) for a free particle. Defining an
effective mass through $|\textbf{k}|=mc/\hbar$, the correspondence of
variables between optical and massive particle regimes is summarized in Table
1. Note that we use the transverse angle $\theta=k_{x}/|\textbf{k}|$,
equivalent to a normalized momentum, when plotting results.

Paraxial Light | Particle in 1D
---|---
[plotted units] | normalized units
Transverse position: $x$ [mm] | Position: $x$
Longitudinal position: $z$ [m] | Time: $tc$
Transverse angle: $\theta=\frac{k_{x}}{\left|\textbf{k}\right|}$ [rad] | Momentum: $\frac{p}{mc}$
Table 1: Variable correspondence between paraxial light and a particle in 1D.
Note that in the derivations we use $k_{x}$ and $|\textbf{k}|$ while the
normalized momentum $\theta$ is used in the plots. Figure 1: Illustration of
experimental setup in three parts. The slit setup is used to initialize the
double-slit experiment. A diagonally-polarized beam is split into two co-
propagating beams using a displaced Sagnac interferometer, where the beam
separation (set to 2 mm throughout the experiment) can be tuned by translating
one of the mirrors. The system of lenses is used to simulate propagation of
the light along $z$ over a large range (see Fig. 7 and Fig. 8 for further
details). The two plots above the lens system indicate the intensity profiles
of the beam before and after the lens transformation. A thin piece of calcite
is used to weakly couple momentum to polarization (the weak measurement). The
calcite is positioned before the lenses for the weak momentum measurement in
the $x$-Bohm experiment and between the lenses for weak position measurement
in the $p$-Bohm experiment. The imaging setup, consisting of a polarizing beam
displacer and a CCD camera, is used to obtain two interference patterns, one
for each polarization.

Drawing the analogy between Eq. (18) and the Schrödinger equation we simulate
the trajectories of a particle double-slit experiment by sending 915 nm laser
light through a double-slit apparatus, employing the experimental setup
outlined in Figures 1 and 2. In the rest of this section we describe the
details of this setup, beginning with the gadget used for the weak momentum
measurement and the procedure employed in the experimental construction of
position trajectories in $x$-Bohm, which closely follow those outlined earlier
Kocsis  _et al._ (2011); Mahler  _et al._ (2016). We then describe how the
same procedure is used to construct momentum trajectories in $x$-Bohm theory,
and how the setup is modified for constructing position trajectories in
$p$-Bohm theory.

###  III.1 Weak momentum measurements

A weak measurement is performed by coupling the desired observable to a
pointer variable, often a different degree of freedom of the same physical
system, followed by a strong pointer variable measurement Ritchie  _et al._
(1991); Dressel  _et al._ (2014). Here we use polarization as the pointer. Our
observable of interest is momentum, which maps to $k_{x}$ for the light beam.
We use $\hat{k}_{x}$ to denote the operator form of $k_{x}$. Specifically, in
the position representation $\left\langle
x\right|\hat{k}_{x}\left|x^{\prime}\right\rangle=-i\partial\delta(x-x^{\prime})/\partial
x$. As described below, the shift in polarization will be proportional to the
weak value which can then be extracted through a standard polarization
measurement. We use the notation $\left|{H}\right\rangle$,
$\left|{V}\right\rangle$ for horizontal and vertical polarization
respectively,
$\left|{D}\right\rangle=(\left|{H}\right\rangle+\left|{V}\right\rangle)/\sqrt{2}$,
$\left|{A}\right\rangle=(\left|{H}\right\rangle-\left|{V}\right\rangle)/\sqrt{2}$
for diagonal and anti-diagonal respectively, and
$\left|{R}\right\rangle=(\left|{H}\right\rangle+i\left|{V}\right\rangle)/\sqrt{2}$,
$\left|{L}\right\rangle=(\left|{H}\right\rangle-i\left|{V}\right\rangle)/\sqrt{2}$
for right- and left-circular polarizations, respectively. The pointer is
initially set to the diagonal polarization, $\left|{D}\right\rangle$.
Polarization is coupled to the transverse momentum of the light using a thin
calcite crystal. The interaction can be described by the Hamiltonian

$H_{I}=\hbar g\frac{\hat{k}_{x}}{|\textbf{k}|}\hat{\sigma}_{z}$ (19)

where

$\hat{\sigma}_{z}=\frac{1}{2}\left(\left|{H}\right\rangle\left\langle{H}\right|-\left|{V}\right\rangle\left\langle{V}\right|\right)=\frac{1}{2}\left(\left|{D}\right\rangle\left\langle{A}\right|+\left|{A}\right\rangle\left\langle{D}\right|\right),$
(20)

and $g$ is the coupling strength. If the joint state of the transverse
position and polarization before the calcite is
$\left|{\Psi}\right\rangle=\left|{\psi}\right\rangle\otimes\left|{D}\right\rangle$,
then with a sufficiently weak interaction, i.e., sufficiently small $\zeta
k_{x}=gtk_{x}\ll 1$ over a range of interest for $k_{x}$, the joint state
after the calcite is

$\left|{\Psi^{\prime}}\right\rangle\approx\left|{\psi}\right\rangle\otimes\left|{D}\right\rangle-i\frac{\zeta}{2}\frac{\hat{k}_{x}}{|\bf{k}|}\left|{\psi}\right\rangle\otimes\left|{A}\right\rangle.$
(21)

The interaction is followed by a projective $x$ measurement, post-selected on
the result, $x_{f}$. The state of the pointer following this post-selection is

$\left\langle{x_{f}}|{\Psi^{\prime}}\right\rangle\approx\psi(x_{f})\left(e^{-i\frac{\zeta}{2|\bf{k}|}\left\langle\hat{k}_{x}\right\rangle_{w}}\left|{H}\right\rangle+e^{i\frac{\zeta}{2|\bf{k}|}\left\langle\hat{k}_{x}\right\rangle_{w}}\left|{V}\right\rangle\right),$
(22)

where

$\left\langle\hat{k}_{x}\right\rangle_{w}=\frac{\left\langle{x_{f}}\right|\hat{k}_{x}\left|{\psi}\right\rangle}{\left\langle{x_{f}}|{\psi}\right\rangle}$
(23)

is the weak value, in general a complex number. The real part of the weak
value shows up as a phase shift between $H$ and $V$ polarizations which can be
extracted by the projective measurement
$\hat{\sigma}_{y}=\frac{1}{2}\left(\left|{R}\right\rangle\left\langle{R}\right|-\left|{L}\right\rangle\left\langle{L}\right|\right)$.
This corresponds to making measurements of the right- and left-circular
intensities, resulting in

$\left\langle\hat{\sigma}_{y}\right\rangle=\sin\left(\frac{\zeta}{|\bf
k|}\text{Re}\left(\left\langle\hat{k}_{x}\right\rangle_{w}\right)\right).$
(24)

Figure 2:  Illustration of the system of lenses configured to make
measurements for $x$-Bohm (top) and $p$-Bohm (bottom) theories. The focal
lengths of the lenses are $f_{1}=$ 15 cm and $f_{2}=f_{3}=$ 10 cm from left to
right, and the total length, from Lens 1 to the imaging setup, is 55 cm. Lens
1 focuses the beam and remaps the position variable of planes from 0 m to
infinity onto the position variable of planes from 0 cm to 15 cm after the
lens, with a scaling factor. The grey axis indicates the correspondence
between the location of the focus of Lens 2 and the effective propagation
distance being imaged by the lens setup, and a detailed plot of the
propagation distance vs the displacement $d$ of Lens 2 is plotted in Figure 7.
Top: Lens 2 and Lens 3 map the position variable at the dotted line to the
imaging setup (solid line); Bottom: Lens 2 and Lens 3 are set to be 20 cm away
from each other, forming a one-to-one telescope.

###  III.2 Position trajectories in $x$-Bohm theory

A double-slit pattern is generated by separating a Gaussian beam ($1/e^{2}$
diameter of 0.55 mm) into two, using a horizontally-displaced Sagnac
interferometer (slit setup in Figure 1) that gives an effective slit
separation of 2 mm. The light is then diagonally polarized and sent through a
thin calcite crystal (0.2 mm, cut at 45 degrees) to weakly couple the
transverse momentum of the light to polarization via a birefringent phase
shift (see Sec. III.1 above). Importantly, the interaction Hamiltonian (19)
commutes with the Hamiltonian for free propagation. This implies that the
calcite crystal can remain fixed at a single $z$ position before the lens
system independent of the plane of interest.

Next, the co-propagating beams traverse a system of three lenses (Fig. 1,
middle pane), labelled Lens 1, 2, 3 with respective focal lengths 15 cm, 10
cm, 10 cm (see Figure 2). By translating Lens 2 along the $z$-axis, we
simulate different propagation distances for the light, resulting in effective
distances ranging from $0.66$ m to $3.5$ m. In other words, the three-lens
system maps what would have been the transverse position of the light beam
propagating in free space onto the transverse position at the end of the lens
system333With our experimental parameters, $\lambda=915$ nm, slit separation
$s=2$ mm, and slit width $w=0.55$ mm, we expect the near-to-far field
transition to occur at $\frac{s}{2}/(\lambda/(\pi w/2))=0.77$ m.. The
calibration of the lens system is discussed in Appendix B.

Finally, the co-propagating beams enter the imaging setup (Fig. 1, right
pane), where the resulting intensity patterns at the end of the lens system
were measured on a CCD camera. In addition to the intensity of the
interference pattern, the polarization is measured by a quarter wave plate and
a polarization beam displacer that effectively separates the left- and right-
circularly polarized light in the vertical direction. Since the interference
occurs along the horizontal transverse axis444This is the axis that is
horizontal and perpendicular to the axis of propagation., the interference
pattern of the left- and right-circular polarizations can be measured
independently. The intensity patterns of the two polarizations ($|u|^{2}$ in
Eq. (18)), given by $I_{R}$ and $I_{L}$, differ by an amount directly related
to the real part of the weak value of transverse momentum, which in the limit
of an infinitely weak measurement can be extracted as

$\frac{\text{Re}\left(\left\langle\hat{k_{x}}\right\rangle_{w}\right)}{\left|\textbf{k}\right|}=\frac{1}{\zeta}\left[\sin^{-1}\left(\frac{I_{R}-I_{L}}{I_{R}+I_{L}}\right)-\phi_{0}\right],$
(25)

where the $\sin^{-1}$ term comes from Eq. (24) and $\phi_{0}$ is a momentum-
independent phase shift acquired in the calcite crystal, set by tilting the
calcite; $\zeta=134.49\pm 0.13$ is the coupling strength555The quantity
$\zeta$ corresponds to rotation imparted to the polarization of light per
transverse angle of the light, and is hence dimensionless. which depends on
the length of the calcite. The calibration of $\zeta$ is discussed in Appendix
B. Examples of measured intensity patterns for near- and far-field propagation
distances are shown in the top row of Figure 3, while measured values of the
momentum, in the same two planes, are shown in the third row of Figure 3. By
performing this measurement for each $z$-plane, we extracted ensemble-average
values of the transverse momentum as a function of position, from which we
construct particle trajectories. Experimentally constructed $x$-Bohm position
trajectories are shown in the top row of Figure 4, with theoretically
calculated trajectories shown in the bottom row. We will discuss all
experimental results in greater detail in Section IV.

###  III.3 Momentum trajectories in $x$-Bohm theory

To construct the momentum trajectories in $x$-Bohm we follow the procedure
outlined in Sec. II.3, where again we use Eq. (14). In our case the momentum
is proportional to the velocity (see Eq. (15)), which implies that the
measurement for the $x$ trajectories in Section III.2 suffices for
constructing the momentum trajectories. The resulting trajectories are shown
in the first row of Figure 5.

Note that proportionality between velocity and momentum is only valid when the
potential term in the Hamiltonian is independent of $p$. In cases where the
potential has $p$ and/or $p^{2}$ terms, Eq. (7) is no longer valid, while Eq.
(14) remains valid generically.

###  III.4 Momentum trajectories in $p$-Bohm theory

As described in Sec. II.2, the conservation of momentum for a free particle
implies that the $p$-Bohm momentum trajectories follow lines of constant $p$.
There is no need to construct these trajectories experimentally; however, the
relative probabilities (or density of trajectories) can be measured by making
a strong $p$ measurement. In practice this is accomplished by strong $x$
measurements in the far field, using the fact that momentum maps to position
at infinity (see Figure 8 in Appendix B). The same setup is used to calibrate
the coupling strength of the weak measurement (again, see Appendix B). The
resulting trajectories are shown in the second row of Figure 5.

We emphasize that $p$-Bohm theory in three dimensions is not unique and that
different theories lead to different velocities, ${\bf{v}}_{p}({\bf{p}},t)$
Struyve and Valentini (2008). However, in one dimension the continuity
constraint (10) essentially identifies (11) as the current density in momentum
space, which in the limit of a free particle leads via Eq. (9) to the
conservation of the $p$-Bohm momentum. The results in Fig. 5 are therefore
free of any ambiguity that would affect $p$-Bohm theories in higher
dimensions.

###  III.5 Position trajectories in $p$-Bohm theory

Position is a derived variable in $p$-Bohm theory, and so position
trajectories can be constructed following the procedure in Sec. II.3 using Eq.
(16). Operationally, this amounts to making a weak position measurement
followed by a post-selection on momentum. This is achieved by using lens
transformations to map between position and momentum since a lens performs a
position-momentum Fourier transform at its focus. Practically, Lens 2 and Lens
3 are kept at a fixed distance from one another, making a one-to-one
telescope, and are translated together (Figure 2). In this way, the transverse
momentum between Lens 2 and Lens 3 corresponds to transverse position in a
fixed propagation plane. As such, the calcite crystal is placed in between
Lens 2 and Lens 3 to perform a weak position measurement. Additionally, the
one-to-one telescope relates the light one focal length before Lens 2 to the
light one focal length after Lens 3 by an identity transformation. This
effectively places the far field of the interfering beams onto the imaging
setup, causing it to perform a strong momentum measurement. To read out the
weak measurement, the quarter wave plate and polarization beam displacer, once
again, are used to separate the left- and right-circularly polarized component
of the beam in the vertical direction and, with procedures similar to those in
Section III.2, we can extract the weak position value post-selected on
momentum. The fourth row of Figure 3 shows results of the corresponding weak
measurements in near- and far-field planes. Position trajectories, with
momentum as the ontological variable, are constructed in the same manner as
before. Constructed trajectories are shown in the second row of Figure 4.

##  IV Comparison of $x$-Bohm and $p$-Bohm trajectories

We now consider the $x$-Bohm and $p$-Bohm particle trajectories in detail. The
trajectories are constructed by interpolating data points taken at discrete
$z$-planes ranging from an effective distance of $0.66$ m to $3.5$ m after the
slits. The results presented and discussed below show the qualitatively
different behavior of the trajectories in the two theories, especially in the
near-field. The experimental results are in good agreement with the
theoretical predictions, and illustrate the dependence of the ontological
description on the choice of the preferred ontological variable.

Figure 3:  Near field and far field snapshots. Near field (left column) data
were taken at an effective $0.70$ m from the slits, and far field (right
column) taken at $3.5$ m. The transition from near- to far-field occurs around
0.77 m, given by the ratio of the slit separation to the full divergence angle
of the beams. The intensity profiles displayed in the first row, i.e., the
position distributions, are the sum of left- and right-circular intensities
and show typical two-slit interference patterns. The asymmetry in the
intensity distributions results from a slight difference in intensities
between the two slits. Note that the far-field intensity profile differs
slightly from the momentum distribution, which has a higher interference
visibility. Numerical simulations of the Bohmian position-momentum profiles
for $x$-Bohm (red) and $p$-Bohm (blue) theories in the second row show
qualitative differences between the predictions of the two theories, in
particular in the near field. The results of the two theories are expected to
converge at infinity. Third and fourth rows show the measured position-
momentum profiles for $x$-Bohm and $p$-Bohm theories respectively. A slight
difference in the slope of the $p$-Bohm experimental near field weak value
comparing to that of the theoretical weak value can be found. This is due to
systematic uncertainties not reflected in the error bars while calibrating the
Gaussian beam width and effective propagation distance. A qualitative
difference between the predictions of the two theories can be seen, notably
the presence of strong peaks in the near-field data for $x$-Bohm and not for
$p$-Bohm. Error bars correspond to the standard deviation between weak values
given by different calcite tilt angles. Error bars are larger in areas where
the overall intensity of the interference is small (see Section IV.1). Note
that in the $p$-Bohm experiment post-selection is always at infinity so that
the minima do not correspond to those in the intensity profiles in the first
row.

###  IV.1 Single-time position-momentum snapshots

For a given $z$-plane, which corresponds to an instant in time, data were
taken by fixing the lenses and post-selecting on the preferred ontological
variable producing a complete description of the functions $p\left(x,t\right)$
and $x\left(p,t\right)$ for the $x$-Bohm and $p$-Bohm theories respectively.
Results for two of these instants of time, one in the near-field and one at
far-field, are presented in Figure 3 and compared with theoretical
predictions. To illustrate the difference between the two theories we begin
with a numerically simulated plot (Figure 3, second row), where we overlay two
ontological momentum-position snapshots, based on Eq. (12). Experimental
results are shown in the third and fourth rows of Figure 3.

In $x$-Bohm, peaks in the momentum $p\left(x,t\right)$ appear when a particle
approaches a minima in the double-slit interference pattern (i.e. the minima
in the top row of Figure 3). These peaks get progressively narrower, with
width approaching zero, as the measurement is taken further into the far
field. The asymptotic large $x$ behavior of the function $p\left(x\right)$
corresponds to

$p/m=\frac{x-\text{sgn}(x)w/2}{t},$ (26)

with $t$ being propagation time and $w=2$ mm being the slit separation. This
can be roughly interpreted as the consequence of the guiding wave $\psi(x,t)$
at the near field having two distinguishable parts with a small overlap so
that particles away from the overlap are effectively guided by one or the
other, leading to behaviour similar to what one would observe if only one slit
were open.

In $p$-Bohm theory, we expect a linear relation $x\left(p\right)=p\cdot t/m$.
It is important to note that, experimentally, data with momentum post-
selection always projects the far field onto the imaging setup. Similarly, the
guiding wave $\tilde{\psi}(p,t)$ has the form of the far-field interference
pattern.

Due to the nature of our measurement, the weak value of the variable of
interest is very sensitive to background noise when the post-selection
probability is small. Background light and other systematic errors dominate
the measured signal, and as a consequence the probability of registering a
measurement in the left- and right-circular polarization basis becomes roughly
equal. As a consequence, and by referring to Eq. (25), one can see that the
weak value tends to the incorrect result of $-\zeta^{-1}\phi_{0}$ near the
minima of the interference patterns. As mentioned in Section III.2, the value
of $\phi_{0}$ in our experiment was controlled by the horizontal tilt angle of
the calcite crystal. In an idealized noiseless measurement, the value of
$\phi_{0}$ exists purely as a calibration parameter of the weak measurement
(see Appendix B) and does not affect the measured weak value. However, with
some amount of noise present in the measurements this is not the case. To
account for this imperfection, we measure weak values using various calcite
tilts. The final weak value at each time is tabulated by averaging measurement
results with different calcite tilts. Error bars in the third and fourth row
of Figure 3 correspond to the standard deviation of the measurement given a
set of values for $\phi_{0}$. The effects of the calcite tilt are particularly
pronounced in $p$-Bohm experiments, where the measurements at the minima go
very close to zero and the standard deviation between measurement results
increases significantly.

Figure 4: Constructed position trajectories based on $x$-Bohm (red) and
$p$-Bohm (blue) theories. Top (middle) plot corresponds to $x$-Bohm ($p$-Bohm)
position trajectories constructed experimentally. Bottom plot corresponds to
numerical simulation of both $x$-Bohm (red solid line) and $p$-Bohm (blue
dotted line) position trajectories overlaid. $x$-Bohm trajectories originate
from the location of the two slits, while $p$-Bohm trajectories originate from
mid-point between the two slits. In the far field, trajectories from $x$-Bohm
and $p$-Bohm theory converge to the same values as expected. The $x$-Bohm
trajectories are also plotted in Figure 6 (top) with one position trajectory
highlighted.

Figure 5: Constructed momentum trajectories based on $x$-Bohm (red) and
$p$-Bohm (blue) theories. Top(middle) plot corresponds to $x$-Bohm($p$-Bohm)
momentum trajectories constructed experimentally. Bottom plot corresponds to
numerical simulation of both $x$-Bohm (red solid line) and $p$-Bohm (blue
solid line) momentum trajectories overlaid. At far fields both sets of
trajectories bunch in the manner of a far field interference pattern. The
momentum trajectories for $x$-Bohm are equal to the first derivative of the
position trajectories under $x$-Bohm, whereas the momentum trajectories for
$p$-Bohm are flat lines due to the conservation of momentum. Peaks in the
$x$-Bohm trajectories corresponds to the crossing of the particle over an
interference minimum, and their time of occurrence is highly sensitive to the
initial conditions of the ontological variable, causing inconsistencies
between the numerically simulated and experimentally constructed trajectories.
The alignment procedure for our system of lenses and the calibration of the
weak measurement strength $\zeta$ rely on the conservation of momentum.
Similarly, the flat lines given by $p$-Bohm are derived from the conservation
of momentum and a single measurement at the far field which gives the momentum
distribution, as reflected in the distribution of lines. The $x$-Bohm
trajectories are also plotted in Figure 6 (bottom) with one momentum
trajectory highlighted.  Figure 6: $x$-Bohm experimental trajectories for
position (top) and momentum (bottom) with a single initial setting highlighted
for clarity. Note that the highlighted, non-primary variable, trajectory of
momentum (bottom) corresponds to the highlighted ontological trajectory of
position (top). Peaks in the momentum trajectories correspond to the
ontological position trajectories crossing a minimum in the two-slit
interference pattern.

###  IV.2 Constructing Trajectories

We construct a set of trajectories for both position and momentum in both
$x$-Bohm and $p$-Bohm theory, chosen so that the density of the selected
trajectories in the primary ontological space corresponds to particle
distribution probabilities. This is possible due to the fact that the
velocities are defined through the probability current (see Sec. II). However,
there is no a priori reason to expect this feature to be preserved in the
derived space, and indeed we will see that it is not. Similarly, the
trajectories in the primary space cannot cross, since given a wave function,
the velocity is uniquely defined by the value of the primary ontological
variable. As we will see, the $p(x)$ trajectories in $x$-Bohm do cross. The
trajectories for $x$-Bohm and $p$-Bohm experiments are shown in Figures 4 and
5 respectively along with theoretical trajectories derived from a numerical
simulation.

####  IV.2.1 Position Trajectories

The $x$-Bohm position trajectories shown in the top row of Figure 4 are very
similar in nature to those obtained earlier Kocsis  _et al._ (2011). These
trajectories originate from one of the two slits and, while providing the
signatures of interference for the probability density, they generally diverge
away from $x=0$ while displaying a rapid ‘acceleration’ through each region of
destructive interference, where the density becomes low and the ratio of flux
to density correspondingly large. As required by the Bohmian formalism, the
trajectories of the primary ontological variable do not cross.

For $p$-Bohm, the position trajectories (Figure 4 middle) originate from a
single point in between the two slits and spread out in a manner that
preserves momentum, resulting in straight lines given by $x=pm/t$. A crossing
(or in this case convergence to a point at $t=0$) is possible since these are
not the trajectories of the primary ontological variable. Due to imperfect
translation of Lens 2 and Lens 3, causing some transverse displacement, a
systematic error is introduced to the weak value measurement and the
trajectories are displaced by a different amount at each plane. This causes
the trajectories to shift in the y-axis of Figure 4, resulting in the
experimental weak values of position deviating from simple straight lines.

Apart from the obvious discrepancy between the position trajectories in
$x$-Bohm and $p$-Bohm, the $p$-Bohm position trajectories exhibit the
potentially surprising phenomenon of originating at $x=0$ rather than in
either or both of the slits. That is, the initial position for all the
particles according to p-Bohm theory is a position which has vanishingly low
probability according to x-Bohm; moreover, a detector placed at that position
would never be expected to register a photon. Placing a detector at this point
and not registering a detection is, however, consistent with the $p$-Bohm
description of quantum mechanics.

####  IV.2.2 Momentum Trajectories

Next, we construct particle momentum trajectories, tracking the change of
momentum over time (see Figure 5). As the conservation of momentum of light in
free space is an assumption used in the alignment, the $p$-Bohm momentum
trajectories are constructed from theory as flat lines with a distribution
derived from the strong momentum measurement (position at the far field).

The $x$-Bohm momentum trajectory functions are proportional to the time
derivative of the position trajectory functions. The peaks observed in
$x$-Bohm momentum trajectories correspond to time intervals when the position
trajectories are crossing the minima of the interference pattern (as
emphasised in Fig. 6). The time instances at which these peaks appear are
highly sensitive to the initial conditions of the $x$-Bohm position
trajectories, and as such, they do not align with the peak positions in the
numerical simulation.

To further explain the trajectory behaviour at these peaks, we highlight a
single trajectory line in Figure 6, where the top and bottom plots correspond
to a position and momentum trajectory in $x$-Bohm for the same initial
conditions. A peak in the momentum trajectory directly corresponds to the
portion of the position trajectory where the particle crosses a minimum in the
double-slit interference pattern.

##  V Discussion

> When everyone is somebody, then no one’s anybody.
> W.S. Gilbert, The Gondoliers
>

The lack of an ontological interpretation has been criticised as a serious
drawback of quantum theory since its early days Bacciagaluppi and Valentini
(2009), for without such an interpretation the visualization of quantum
dynamics is not possible. Apart from any philosophical considerations, such
visualizations are arguably essential for developing the intuition necessary
for scientific development. At the same time, incorrect visualizations (such
as those involving the aether in electrodynamics) can lead us astray. Bohm’s
interpretation, with its deterministic particle trajectories, presents an
attractive visual picture of quantum dynamics at the cost of some non-trivial
assumptions. Among these is an assumption of the role of position and the
coordinate representation of the wave function as the fundamental variables
that determine the dynamics. Indeed, in contrast to classical physics, where
both the initial position and momentum are necessary for predicting the
dynamics that follows, in an $x$-Bohm theory the initial conditions are just
the initial position of the particle, together with the initial wave function.
While the theory ensures that in current experiments the real position remains
hidden and that the dynamics appears non-deterministic, the significance of
this hidden variable for the real dynamics hints at a fundamental asymmetry in
nature. It is therefore tempting to view the identification of this asymmetry
as a profound discovery, suggesting that position is indeed more important
than other variables. One could even hope that the realization that position
plays a special role would lead to new experimental predictions. However, as
emphasized in this work, the specific choice of position is not unique,
leaving us with an infinite number of possible ontological variables – each
allegedly more fundamental than all the others – or, as Gilbert’s line above
implies, with none.

Our main results show that a variation of Bohm’s theory ($p$-Bohm) can be used
to construct a very different ontological picture, one in which the dynamics
are based on the momentum variable. In this theory the fundamental
trajectories are paths through momentum space, and the equations of motion
take a form which is closer to that of Newton’s laws, with a first time
derivative for momentum. If the underlying ontological pictures of both
theories were the same, one might say that the symmetry between position and
momentum had been restored, removing a non-trivial assumption from Bohm’s
theory. The pictures are, however, very different (see Figures 3, 4 and 5) and
so, the asymmetry in a Bohm-like theory is confirmed but ambiguous. One is
left to wonder which of the two theories, or indeed of the infinite
intermediate theories with other ontology, is correct, and possibly more
importantly what is the preferred variable.

A striking example of this conundrum arises when we consider a harmonic
oscillator, where the Hamiltonian operator $\hat{H}$ can be written in terms
of the usual raising and lowering operators $\hat{a}^{\dagger}$ and $\hat{a}$,

$\displaystyle\hat{H}=\hbar\omega\left(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right).$
(27)

We can also write

$\displaystyle\hat{H}=\frac{1}{2}\hat{p}_{\theta}^{2}+\frac{1}{2}\omega^{2}\hat{x}_{\theta}^{2},$
(28)

for any real $\theta$, where the operators $\hat{x}_{\theta}$ and
$\hat{p}_{\theta}$ are defined as

$\displaystyle\hat{x}_{\theta}=\sqrt{\frac{\hbar}{2\omega}}\left(\hat{a}^{\dagger}e^{i\theta}+\hat{a}e^{-i\theta}\right),$
(29)
$\displaystyle\hat{p}_{\theta}=i\sqrt{\frac{\hbar\omega}{2}}\left(\hat{a}^{\dagger}e^{i\theta}-\hat{a}e^{-i\theta}\right).$
(30)

Since $\left[\hat{x}_{\theta},\hat{p}_{\theta}\right]=i\hbar$, we can
construct what might be called a $\theta-$Bohm theory by taking
$\hat{x}_{\theta}$ to be the “position operator” for the particular $\theta$
chosen; in this representation the Schrödinger equation is

$\displaystyle i\hbar\frac{\partial}{\partial
t}\psi(x_{\theta},t)=-\frac{\hbar^{2}}{2}\frac{\partial^{2}\psi(x_{\theta},t)}{\partial
x_{\theta}^{2}}+\frac{1}{2}\omega^{2}x_{\theta}^{2}\psi(x_{\theta},t),$ (31)

and following the usual Bohmian procedure and taking
$\psi(x_{\theta},t)=R_{\theta}(x_{\theta},t)\exp(iS_{\theta}(x_{\theta},t)/\hbar)$,
with $R_{\theta}(x_{\theta},t)$ and $S_{\theta}(x_{\theta},t)$ both real, the
guidance equation is

$\displaystyle v_{\theta}(x_{\theta},t)=\frac{1}{2}\frac{\partial
S_{\theta}(x_{\theta},t)}{\partial x_{\theta}}.$ (32)

At least following Holland’s suggestion Holland (1993a), this would be taken
as the value of the non-primary variable associated with $\hat{p}_{\theta}$
(compare Eq. (12)). Here for each $\theta$ a different physical picture
emerges, and weak $p_{\theta}$ measurements followed by strong $x_{\theta}$
measurements would allow the construction of trajectories for each
$\theta$-Bohm theory, yielding entirely different visualizations.

The significance of this is apparent if one considers the Hamiltonian (27) to
describe a mode of the radiation field associated with a standing wave. Then
in a standard treatment Gerry and Knight (2004) the operators $\hat{x}_{0}$
and $\hat{p}_{0}$ are (within factors) associated with the electric and
magnetic fields respectively. Thus in the $0$-Bohm theory the ground state (or
indeed any energy eigenstate) would be associated with an ensemble of
different values of the electric field but, following Holland’s suggestion,
the magnetic field would vanish in each member of the ensemble. On the other
hand, in the $\pi/2$-Bohm theory it would be $\hat{p}_{\pi/2}$ that would
correspond to the electric field and $\hat{x}_{\pi/2}$ with the magnetic
field, and so a description of the ground state (or indeed any energy
eigenstate) in the $\pi/2$-Bohm theory would be associated with an ensemble of
different values of the magnetic field, but with the electric field vanishing
in each member of the ensemble. Yet other descriptions would arise for the
ground state for other values of $\theta$. Considering more general quantum
states of the radiation field, weak measurements associated with one field
quadrature followed by strong measurements associated with the complementary
quadrature would allow for the formal construction of very different sets of
trajectories.

What would be the physical motivation for granting reality to one set of
trajectories or the other? For massive particles, Wiseman has suggested that
the question can be settled in favour of the usual position variable if one
tries to construct the trajectories of particles in a potential with an $x$
dependence more than quadratic Gambetta and Wiseman (2004). Unfortunately the
measurement of trajectories in such theories remains experimentally
challenging even with the simplification of a photonic simulation. We expect
that continued work in this direction, ideally experiments involving massive
particles, would lead to results that shed further light on the question. For
states of the radiation field, weak and strong measurements of field
quadratures would extend the discussion of the kind of issues raised here to
Bohmian descriptions of field theories.

###### Acknowledgements.

This work was supported by NSERC and the Fetzer Franklin Fund of the John E.
Fetzer Memorial Trust. A.M.S. is a fellow of CIFAR. We thank David Schmid for
useful discussions.

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##  Appendix A ABCD Matrix Transformation

To calculate the effective imaging plane, we employ the ray transfer matrix
analysis on our lens system. The propagation and thin lens matrix is given by

$\displaystyle M_{\rm{prop}}\left(d\right)=\left(\begin{matrix}1&d\\\
0&1\end{matrix}\right),$ (33a) $\displaystyle
M_{\rm{lens}}\left(f\right)=\left(\begin{matrix}1&0\\\
-\frac{1}{f}&1\end{matrix}\right).$ (33b)

To see the effective imaging plane at some distance $y$ from the slits after
the Lens 1, we analyze the ray matrix of light being back propagated for a
distance of $y$ and then forward propagated through a lens and some distance
$f_{1}-d$. This results in a ray matrix of

$\left(\begin{array}[]{cc}1-\frac{f_{1}-d}{f_{1}}&-y\left(1-\frac{f_{1}-d}{f_{1}}\right)-d+f_{1}\\\
-\frac{1}{f_{1}}&\frac{y}{f_{1}}+1\\\ \end{array}\right).$ (34)

For the position distribution of the plane after the lens to be equivalent to
the effective imaging plane, we require

$d=\frac{f_{1}}{1+y/f_{1}}.$ (35)

This results in the transformation matrix

$\left(\begin{array}[]{cc}\frac{f_{1}}{f_{1}+y}&0\\\
-\frac{1}{f_{1}}&\frac{f_{1}+y}{f_{1}}\\\ \end{array}\right),$ (36)

which corresponds to a transformation that relates the position of the beam as
a function of the position of the beam before the transformation only and does
not depend on the momentum of the beam before the transformation. The
resulting transformation also results in a scaling of $f_{1}/(f_{1}+y)$ in the
position variable from the effective image plane to the plane at distance
$f_{1}-d$ after Lens 1. Placing the focus of Lens 2 at a distance of $f_{1}-d$
after Lens 1 results in a transformation matrix of

$\left(\begin{array}[]{cc}\frac{f_{1}(f_{2}-d_{2})}{f_{2}(f_{1}+y)}-\frac{f_{2}}{f_{1}}&\frac{f_{2}(f_{1}+y)}{f_{1}}\\\
-\frac{f_{1}}{f_{1}f_{2}+f_{2}y}&0\\\ \end{array}\right)$ (37)

where $d_{2}$ is the distance after the second lens. As one can conclude from
inspecting this matrix, the momentum of the light after the second lens is
independent of the momentum at the effective imaging plane and proportional to
its position distribution. Placing a calcite to perform a weak momentum
measurement after Lens 2 thus results in the weak position measurement at the
effective image plane.

Lens 3 controls whether position or momentum of the effective imaging plane is
projected onto the imaging setup. When placed one focal length away from the
imaging system it transforms momentum after Lens 3, which reflects the
position of the effective imaging plane, onto the position distribution at the
imaging setup. This is the configuration for $x$-Bohm measurement where post-
selection was performed on position. Alternatively, Lens 3 could be placed at
$f_{2}+f_{3}$ away from Lens 2. The resulting transformation with all three
lenses and back propagation combined is

$\left(\begin{array}[]{cc}0&-f_{1}\\\
\frac{1}{f_{1}}&-\frac{f_{1}+y}{f_{1}}\\\ \end{array}\right).$ (38)

Note the absence of $f_{2}$ in the equation. This is due to the fact that Lens
2 and Lens 3 are identical and they were configured such that a one-to-one
telescope is formed. The transformation effectively places the focus of Lens 1
onto the imaging setup, performing a momentum post-selection for measurements
in $p$-Bohm.

##  Appendix B Calibration

###  B.1 Lens system

The system of lenses ( Figure 2) must be calibrated in order to infer the
effective propagation distance. This is done by noting the magnification of
the setup given the position of lenses. By measuring the waist of the beam
from an individual slit, as well as the distance between the centroid of
individual beams from the two slits, the effective propagation distance and
magnification can be calculated. The results of the lens calibration is shown
in Figure 7.

Figure 7: Calibration of the position of the second lens from the left and the
corresponding propagation distance of the effective image plane. The origin
position of the second lens is the place where far field is projected onto the
camera. The blue vertical bars indicate the calibrated distance with
uncertainty, where the uncertainty mostly originates from the beam profile of
the individual slit not being perfectly Gaussian. The orange dotted line is
the fitted curve of the calibration data.

###  B.2 Weak measurement calibration

Calibration of the strength of the weak measurement $\zeta$ was determined by
performing weak measurement and post-selection on the same variable, where
$\omega$ and $\xi$ in Eq. (12) were both set to be either position or momentum
when calibrating for $x$-Bohm or $p$-Bohm, respectively. The corresponding
setup and calibration results can be found in Figure 8.

Figure 8: Illustration of calcite location for calibration of weak measurement
strength $\zeta$. Top (middle) figure corresponds to the setup where we
calibrate the strength of weak position (momentum) measurement, where position
(momentum) of the effective plane was measured both weakly and strongly. While
the strength of weak momentum measurement is constant ($\zeta=134.49\pm
0.13$), the weak position measurement strength is a function of effective
propagation distance and is plotted in the bottom figure. Error bars, mostly
due to the precision of the translation stage, are too small to show.